AN INVESTIGATION OF CERTAIN COMBINATORIAL PROPERTIES OF PARTIALLY BALANCED INCOMPLETE BLOCK EXPERIMENTAL DESIGNS AND ASSOCIATION SCHEMES, WITH A DETAILED STUDY OF DESIGNS OF LATIN SQUARE AND RELATED TYPES Thesis for the Degree aI Ph. D. MICHIGAN STATE UNIVERSITY DaIe Marsh Mesner 1956 I“: I. . I . ., H I .I‘ ~ ‘ v w .n.- ..c‘rpn...’ f.~‘Ir:t- ‘ -‘L. . LIBRARY Michigan State University This is to certify that the thesis entitled An Investigation of Certain Conbinatorial Properties of Partially Balanced Incomplete Block Designs for Statistical Experiments, with a Detailed Study of Designs of Latin Square and Helatecpggpfiéfl b9 Dale Marsh Mesner has been accepted towards fulfillment of the requirements for _Ph_.__D__._ degree in Mains /\ r4, -’ 44 _... r Major proM Date May 221 1956 0-169 "bVIESIHJ "—n RETURNING MATERIALS: Place in book drop to LIBRARIES remove this checkout from ‘31—. your record. FINES will be charged if book is returned after the date stamped below. M 7 ntlufizgosg All INVESTIGATION OF CERTAIN WIBINATORIAL PMPHITIES 0F PARTIALLY BALANCED IRWIN-ETD BLOCK EXPERIIDITAL DESIGNS AND ASSOCIATION 80m, IITH A DETAILED STUDY OF DESIQTS 0F LATIN SQUARE AND RELATED MES W Dole larch leaner ATHEIB submitted to the School of Advanced Graduate Studies of lichigan State University in partial fulfill-ant of the requirements for the degree of [OCTOB OF PHILOSOPHY Doped-ant of 1956 Statistics Dale Marsh Mesner candidate for the degree of Doctor of Philosophy Final examination, May 22, 1956, 1:00 P. M., Physics-Mathematics Conference Room Dissertation: An Investigation of Certain Conbinatorial Properties of‘Partially Balanced Incomplete Block Experimental Designs and Association Schemes, with a Detailed Study of Designs of Latin Square and Related types Outline of Studies Major subject: Mathematical statistics “130? subjects: Algebra, analysis, ancestry Biographical Items Born, April 13, 1925, Central City, Hebraska Undergraduate Studies, Nebraska Central College, 1941-45, Uni- versity of’Nebraska, 1946-48 Graduate Studies, Nbrthwestern University, 1948-49, Michigan State College and Michigan State University, 1950-56 Experience: Instructor, Candler College, Marianas, Cuba, 1949- 50, Graduate Teaching Assistant, Michigan State Col- lege, 1951-52, Temporary Instructor, Michigan State College, 1952-54, Instructor, Purdue University, 1954-56 Member of Society of the Sign 11, Pi Mu Epsilon; member of Insti- tute of Mathematical Statistics, American Matheaatical Society, Mathematical Association of America ACKNOWLEDGMENTS The author is sincerely grateful to Prof. Leo Kata for the guidance and encouragement received from him during the study presented here, and for his patience while the writing was being completed. Thanks are also due to Dr. W. 8. Connor, Jr., for suggesting one of the problems taken up in Chapter IV. iii AN INVESTIGATION OF CERTAIN COMBINATORIAL PROPERTIES OF PARTIALLY BALANCED INCOMPLETE BLOCK EXPERIMENTAL DESIGNS AND ASSOCIATION SCHEMES, WITH A DETAILED STUDY OF DFSIQIS OF LATIN SQUARE AND RELATED TYPES By Dale Marsh Mesner AN ABSTRACT But-titted to the School of Advanced Graduate Studies of Miehiganratatsflnmrsity irr partial fulfillment of the requirements for the degree of MOTOR OF PHILOSOPHY Department of Statistics Year 1956 A partially balanced incomplete block (PBIB) design is an arrange- ment of a set of experimental treatments into smaller subsets, or blocks, in accordance eith a certain definition. Except for an introductory section in which the role of P913 designs in the statistical analysis of experiments is discussed, this thesis is concerned with the combinator- ial problems that arise in the construction of the designs. The defini- tion states requirements for a relation of association between any two treatments, and the term "association scheme" is used for any method by which a relation of the kind specified can be set up. A considerable portion of the thesis is devoted to the study of association schemes rather than actual designs. Incidence matrices are used throughout the thesis to study the properties of designs and association schemes by al- gebraic methods. A method of enumerating combinatorially possible PBIB designs with two classes of associates is outlined, based on both new and old methods. While tables of known designs have been published, no exhaustive tables of all possible PBIB designs hare appeared heretofore. An extensive table of the possible parameter values of association schemes is compiled, along eith tables of possible parameter values of the designs themselves in the cases of special interest in this study. Known PBIB designs eith tee classes of associates have been class- ified according to the nature of their association schemes, and designs of Latin square type eith g constraints, in ehich the number of treat- ments is a square I? and the association relation maybe defined by a set of g mutually orthogonal n x n squares, are singled out for spe- cial etndyherea - _ tnlatechlass of new designst introduced and given V the name .'negative Latin square“. While association schemes for the new designs cannot be canstructed from Latin squares, a method based on finite fields is developed and used to construct some schemes of both types, including four in the new series. A fifth is constructed by other methods. Several new designs are constructed from the new associa- tion schemes. Some examples are given to show the possibility of association schemes which have exactly the same parameter values as those of Latin square type with g constraints but in which the association relation cannot be defined by a set of g orthogonal squares. It is then proved that for a fixed value of g, this can be the case only for a less than a certain value, which is expressed as a function of g, and that for larger values of n the Latin square type association scheme is unique. The proof is based on a series of theorems on the structure of incidence matrices, some pertaining only to association schemes and others applying more generally. Some other applications of the methods are suggested. vi TABLE 01" CONTENTS CHAPTER I GENERAL PROPERTIES OF PARTIALLY BALANCED DESIGNS ANDASSOCIATIONSCHDIES............... 1 lel IntrOdUCtian e e e e e e e e e e e e e e e e e e e e 1 1.2 Association Schemes and Incidence Matrices . . . . . 10 1.5 Applications and Algebraic Properties of the Matrices ‘1 e e e e e e e e e e e e e e e e e e e 16 CHAPTER II ENUIERATION 0F POSSIBLE DESIGNS AND ASSOCIATION SCHEMESWITHTVDASSOCIATECLASSES. . . . . . . . . . 25 2.1 The Class of PBIB Designs with Two Associate 0188838 e e e e e e e e e e e e e e e e e e e e e e 25 2.2 Enumeration Of ABHOCiCtion SChGEOS e e e e e e e e e 59 2.5 Enumeration of Possible Designs for Particular ABBOCiItiOn SChBIGS e e e e e e e e e e e e e e e e 68 CfiAPTER III NEGATIVE LATIN SQUARE TYPE ASSOCIATION SCHEMES . . . . 84 5.1 Relationships between Latin Square and.Negative Latin Square Association Schemes . . . . . . . . . 84 5.2 Construction of Negative Latin Square Type Association Schemes by a.Iethod Based on Filit. Fields e e e e e e e e e e e e e e e e e e 96 5.5 Construction of a,Negetive Latin Square Type Scheme with 100 Treatments by Enumeration . . . 151 CHAPTER IV THE STRUCTURE OF LATIN SQUARE TYPE ASSOCIATION SCHEMES 150 4.1 Preliminary Discussion of Uniqueness, with Counter-Examples e e e e e e e e e e e e e e e e e 150 4.2. On the Uniqueness of L2 Association Schemes . . . 158 4.5 On the Uniqueness of L8 Association Schemes, 8;Z_5 e e e e e e e e e e e e e e e e e e e e e e 176 CHAPva WMOOOOOOOOOOOO00.0.0.0... 225 APPMIXOOOOOOOOOOOOOOOOOOOOOOOOOOOO. 243 A.l Tables of Parameter Values of Association Schemes 245 A.2 Tables of Parameter Values of Possible Designs for Particular A880618tion SChOMGB e e e e e e e e 260 A.5 Construction of Two Particular Designs; Impossi- bility Proofs of’Particular Designs . . . . . . . 279 vii Il.‘llll|ll|l!l Ii: .l‘llllll III in... . TABLE 013‘ CONTENTS (Continued) A.4 List of Negative Latin Square Association Schemes A.5 Fifteen Squares with Special Orthogonality Properties.................... LIST OF WMCES O O O O O O O O O O O O O O O O O O O O O O O C Figure 1. Figure 2. HmmS. Figure 4. Figure 5. Figure 6. TABLE Ia TABLE Ib TABLE II TABLE III TABLE IV LIST OF FIGURES ExampleofPBIBDesign................. Example of Group Divisible Association Scheme . . . . . . Example of Latin Square Association Scheme . . . . . . . AnotherPBIBDesign................... Example of L3 Association Scheme for 16 Treatments mu Elma P0831b19 Values 0‘ piz “d [3&2 e e e e e LIST OF TABLES Preliminary Computation of the Parameter Values of Association Schemes by Means of Theorem 2.0 . . . . . . Preliminary Computation of the Parameter Values of Association Schemes.bw leans of Theorem 2.1 . . . . . . Parameter Values of Association Schemes not of GroupDiViaibleTypo................. Preliminary Computation of Parameter Values.of'Possible Designs, Illustrated for Several Association Schemes Pumter Values of Poafiible “signs a e e e e e e e e 0 viii 285 288 289 11 52 46 244 257 262 268 PREFATORI NOTE Chapter I is introductory in nature. The next three chapters, talc- up special aspects of this study, are long and somewhat complex. For this reason a detailed sumary or synopsis has been included as Chapter V. The reader may find it useful to appraise the scope and methods of Chap~ ters II, III and IV by a preliminary reading of the summary. I. GENERAL PROPERTIES OF PARTIALLY BALANCED DESIGNS AND ASSOCIATION SCHEMES 1.1 Introduction Statistical analysis of many types of experimental data may be facilitated hy proper planning of the experiment. Partially balanced incomplete block (PBIB) designs are a.particuler class of arrangements for this purpose. A definition of PBIB designs will be preceded by a simple example which illustrates the concepts involved. Ag Illustrative Example gith,gistorical Remarks. The average.yields of seven new varieties of hybrid corn are to be compared in a field experiment. A possible plan is to divide the available land into seven plots and to plant one variety in each plot, as indicated in the follow- ing figure. (Throughout this example, varieties will be indicated hy numbers from 1 ts 7.) f 1 2 5 4 5 6 7 i I n L Under conditions of strict control ef soil fertility, eater supply and drainage, and other extraneous factors, this might furnish the desired information on the varietal differences, but in experiments in the biological and social sciences such control is not usually possible. It will be impossible with this arrangement to know whether an observed difference between two plots can be attributed to differences in the two varieties or whether it is due to differences between the plots of ground. If the effects of extraneous factors cannot be controlled, the next best thing is to estimate their importance. This can be done by planting several plots to each variety and observing the variation among them. It is intuitively reasonable and proves to simplify analysis of the data to plant the same number of plots to each variety so that in effect we have e.number of repetitions, or replications, of the original experiment. Three replications will be used in this example. Cemparison of varieties .grown under similar conditions will be easier if the 21 plots are grouped into blockSof seven plots, each block to contain a complete replication. Soil conditions are likely to be more homogeneous within a block than over the entire experimental area and will have a correspondingly smaller effeCt on comparisons made withtn a block. The blocks may er'may not be contiguous in the field. This design is indicated in the following die. gram, with blocks inclesed by heavy lines. 1 g 2 5 4 5 ’ 6 7 q [M2 5 4 515 q 1 2 5 4 5 i A defect of this plan is that the same arrangement of varieties is used a: _I_mu-+ r~l_.v in each block, so that effects of location within blocks may be impossible to distinguish from differences between varieties. For instance, an ob- served difference betueen varieties l and 7 could have been caused hy a gradient in soil fertility from left to right. Other extraneous sources ef variation which are less obvious may introduce a similar bias in favor of certain varieties. To insure that no variety or group of varieties will be systematically favored in all replications of the experiment, a device known as randomization may be used. In our example this would mean assigning the numbers from I to 7 to each block in such a way that each of me 7! possible arrangements is equally likely to result. In addition, the three blocks might be assigned to the three positions in the field in a random manner. The effect is that in each replication, each variety has an equal chance of being tested under favorable condi- tions. While the results of any particular randomisation may favor cer- tain treatments, this happens only to an extent that can be allowed for in the analysis and interpretation of the data. The plan that results is called a randomized .p_l_g_¢_:_k_ m. It might appear as follows. l ‘* I l6}2[s 5§4§7 1J { : .7 1 l 3 7 4 1'76 5 I2 5 t .LA 7 3 t 2 i l i? ls 6J5 7 1&4 R. A. Fisher was the first to realize the importance ef randomisation as a scientific technique and to introduce it into designs for experiments. It is discussed in his book I"The Design of Experiments" [7 27 with some illuminating examples . It frequently happens that, within a block which includes an entire replication of an experiment, there is too much variability of conditions to allow useful measurements te be made. This may make it necessary to Q arrange the experimental plots in blocks of smaller size, with direct comparisons to be made only between varieties in the same block. In our example we shall suppose that it is necessary te cut down the block size to three plots. There is some loss of information here, as suggested by the fact that the number of possible direct comparisons is reduced from 5(;) : 65 to 7(2) 3 21, but the gain in precision ef compari- sons may more than offset this. If some of the comparisons are less important than others, it may be possible to arrange the blocks so that the unimportant information is lost and the important information is mostly retained. However, in many situations all comparisons may be con- sidered equally important; it will be assumd in this example that infor- mation is desired en the comparative yields of each pair of varieties. the term W £935 1953 covers any experimental desigi in which the blocks are of size mailer than the number of treatments, while the term balanced ,incoggete M (BIB) 929.32% is used for the important special case in which an equal amount of information is retained on each pair ef treatments. A BIB design may be defined as an arrangement of v varieties or treatments into b blocks each containing 1: distinct varieties, each variety being used the same number of times r, and each pair of distinct varieties occurring together in the same block the same number A of times. It is easily verified that the following arrange- ment of our example satisfied these requirements, with v 8 b = 7, r 8 k = 5, A- 1. (Blocks are enclosed by heavy lines.) TWTW‘F‘: ... U! q ‘fis “fl N N L..._... 0| b 03 hula.L ‘ .3 J . I Randomization would be applied to this design by assigning the numbers i “l 1 l, 2, ... , 7 to the varieties at random, assigning the three numbers in each block to the three plots in a random way, and assigning the blocks to the seven positions in the field by a third random procedure. Balanced incomplete block designs were introduced by Iates in 1936 [36], The construction of a BIB design for a given set of values of v, b, r, k, A, is a combinatorial problem which may be considered apart from the analysis of experimental data. It is clear that the five para- meters are not all independent. Considering the total number of plots we have (1-1) vr 3 ha , and by counting pairs of varieties two ways we obtain Am ' at) These two results may be combined to give a more useful form of the latter. (1.2) A: rfii— Other necessary conditions for the existence of these denials have been obtained, along .with some methods for constructing large classes of them. In 1938, Fisher and Yates [E g] published all the BIB designs then known, with a list of the possible parameters of other designs of prac- tical interest. (A desigi is of practical interest if it does not re- quire more experimental material than the experimenter can afford: for a given number of treatments, this means “for r sufficiently small.') The construction of many of these designs was made possible by methods introduced by a. C. Bose in 1959 [4]. The set of constructiblo BIB designs was soon found to be inadequate for the needs of experimenters. A simple case in which no convenient balanced design is available is obtained from the first example by con- sidering eight varieties of hybrid corn instead of seven, again to be planted in blocks of three plots. With v 3 8 .d k =- 3, the smallest value of r which can be. used in (1.1) and (1.2) to give integral values of b and A is found to be 21, and the blocks of the design are all the combinations of the eight varieties three at a time. It was to pro- vide useful designs for such values of v and k that arrangements like the following were introduced. 1 2 3| 1 4 a] i 1 via N 1F rmm‘t ‘1 N 0| 0 I L" l OI . 0‘ 1, [sis s [51. . Figure 1. Example of PBIB design. This is not a balanced design because the pairs of distinct varieties do not all occur equally often. Every pair occurs once with the excep- tions (1,6), (2,6), (5,7), (4,8), which do not appear at all in the same block. The remaining requirements for a balanced desip are satisfied. This is an example of a My we; new“ £133 -(PBIB) deal... Partially balanced incomplete block designs were introduced by a. 0. Boss and x. a. an: in 1939 [a]. They are a generalization of balanced incomplete block designs and include them as a special case, along with certain other incomplete block designs which were already known. Their analysis is somewhat more difficult than that of balanced designs, though conditions are specified (paragraph iii, 0, of the definition which follows) which simplify it greatly. They have not been studied as extensively as balanced designs. Some of the literature on the subject will be discussed in later chapters. Combinatorial properties of partially balanced incomplete block designs will be the principal subject of this thesis. The problems of analysis and interpretation of experimental data will not be taken up. For our purposes from now on, a PBIB design is an arrangement of objects know: as varieties or treatments into blocks according to certain rules. A definition of P313 designs will now be given. An incomplete block design is said to be partially balanced if it satisfies the following conditions: (i) The treatments or varieties being tested are grouped into b blocks, each consisting of 8 distinct treatments. (ii) There are v treatments, each of which occurs in r blocks. (iii). There can be established a relation of association between any two treatments satisfying the following requirements: (a) Two treatments are either 1st, 2nd, . . . , or mth associates. (b) Each treatment has exactly n1, 1'“ associates. (c) Given any two treatments which are ith‘associates, the number of treatments common to the 3th associates of the first and the a“ associates of the second is 931; and is independent of the pair of treatments with which we start. Also pa = 9%. (iv) Two treatments which are 1th associates occur together in exactly A 1 blocks. It should be noted from (iii) that the association relation is symmetric but not necessarily transitive. It was proved by Bose and Hair [8] that the following relations hold among the parameters. (1.5) bk = vr , (1.4) n1+n2+- ... +n-3 v-l, (1e5) HA1+32A2+0 0 0+3-A. 3 r(k’1) , m (1.6) F19}; : ‘33 (if 1 i J) , “1-“ (if i = j) , (1'7) “19%;: z “191: = "1:91;: ' For fixed 1 the parameters pi are conveniently displayed in an m x m matrix with J and k as row and column indices, denoted by Pi' By the final remark of (iii)(c), these matrices are synetric. It is easily verified that the example given in Figure l is a PBIB design with two associate classes and the parameters y:b:3, r:k=5s “1:1, ”2:6: A1:os Azzle oo 01 P1'[ ]oP2'[ ] 06 14 It may also be verified that these parameters satisfy relations (1.5) to (1.7). 10 1.2 Association Schemes 9313 Incidence Iatricesi The definition of the previous section is not in the same form as originally given by Bose and Nair, but follows closely several papers around 1952, notably Bose and Shimamoto, «Classification of P318 designs with two associate classes" [I 0]. The definitions are logically very similar} but as pointed out by Bose and Shimamoto, the association relations among the treatments do not depend on how they are distributed in blocks. In this form of the definition the association relations are completely specified in paragraph (iii). They may be taken up without considering the parameters b, r, 1:, A1. — in association scheme is a convenient device for describing the association relations of a design. It is a table or other arrangement listing for each treatment its 1”, 23d, . . . , nth associates. The treatments may be assigned the numbers from 1 to v in any convenient order for such a table. Bose and Shimamoto found it possible to classify the association schemes of all known designs with m = 2 association schemes into five types, some of which can be set down very concisely. Perhaps the simplest type of scheme is the group divisible (GD), in which v - In and the treatments aredivided into m groups of n each, treat» ments in the same group are first associates and treatments in different groups are second associates. A compact form for the association scheme is an m by n rectangle, with the n treatments in a row constituting a group. The example given in Figure l is a GD design with the following association scheme . l. The original definition contained the specification that the 1 be distinct. m: was dropped in a 1942 paper by stir and Rao [is ,generali- singhthe class of P318 designs somewhat. Their definition is equivalent to a one given here. .0010.“ 5 6 7 8 Figure 2. Example of Group Divisible association scheme. The first associate of treatment 1 is treatment 5, etc. It is natural to attempt to generalize this by taking two treatments as first associates iftheyappearinthesameroworthesamecolumn, butifmfln itis easy to see that condition (iii)(c) of the definition is violated. For example, the number 9&1 of treatments common to the first associates of treatments 1 and 5 would be 0; the number for treatments 1 and 2 would be 2. If m = n so that v = n2, this generalisation leads to an association scheme described by Bose and Shimamoto as of Latin square type. The following array is given as an illustration. 1 2 3 4 5 6 7 8 9 Figure 5. Example of Latin square association scheme. 1 This array defines a GD association scheme in which treatmentfihas as its first associates treatments 2 and 3; it also defines a Latin square type scheme in which the first associates of treatment 1 are 2, 5, 4, 7. 12 Tu various types of association schemes will be discussed further in Section 2e1e It may be noted that for a group divisible scheme the relation for first associates is transitive as well as symmetric; that is, the first associates of a treatment are pairwise first associates. This is a sufficient condition for the scheme to be GD, for it implies that the treatments may be divided into groups such that two treatments in the same group are first associates and two treatments not in the same group are second associates, while the condition that each treatmentihave n1 first associates requires that.the groups be of equal size. The Latin square scheme described above illustrates that in general two treatments which are first associates of the same treatment may not be first associates or each other. There may be several designs for any one association scheme. The following is another design using the GD association scheme of Figure 2. vsm rza ksg b=m A1=m A2=L dis 1 5 L1 5 I} 6] [2 6 re 7 1E L1L1°“ ‘4‘ TBI #000000 Figure 4. Another PBIB desip. _’o 15 A number of possible designs for the Latin square association scheme of Figure 3 will be enumerated in Table IV of the Appendix. The portion of an association scheme corresponding to 1th associates, i-i,...,m,meyberepresentedbya vxv matrix at ... (a, v) where a1 v has the value 1 or 0 according as treatments ,1. and 1/ ,o are or are not it!1 associates. A1 will be called the incidence atr__i_x_ for it"h associates, or simply the ith association matrix. It follows from paragraph (iii) of the definition that it is a symmetric matrix with exactly n1 1's in each row and column. Before further properties of the A1 are derived, a connection will be pointed out with another incidence matrix pertaining to the design. The incidence matrix for treatments and blocks of a PBIB design is a .. ... (3,”) = 1 if treatment Pi occurs in block 1/ , vxb matrix where (1.8) an '3 0 otherwise. That is, positions of 1's in row ’4. of the matrix indicate the blocks of the design which contain treatment v . We shall consider the product of N on the right by its transpose N'. This product NN' will be a symetric v x v matrix. Let (1.9) Ni" = (b/W). 14 The diagonal element b PP of NW is equal to the number of 1's in row [4 of II, or the umber of blocks of the design which contain treat- ment In . For ,4 it v , b,” is equal to the inner product of rows ,u and v taken as vectors, or the number of blocks which contain both of treatments P- and v . For a PBIB design we have by paragraphs (ii) and (iv) of the definition, (1.10) bft’s = r , bray =A 1 when pqév andtreatments ,0. andv are ith associates. That is, m (1.11) mm : ”Iv + 12:1 A1A1,where Iv is the vxv identity matrix. The matrices N and NN' have been used extensively since about 1950 in various studies of balanced and partially balanced designs} The matrices A1 do not seem to have received much attention. There follow as examples matrices N and NN' for the design given in Figure 4, preceded by the matrix A1 for the association scheme of this desim, given in Figure 2. 1. The following papers referred to in this dissertation-eke makesubstantial use of N, RN' and related matrices: [7].. figs 1.75 71-127 [327 15 z um: 16 1.5 Applications and Algebraic grogrties 9_i_‘_ _t_h_§_ Matrices A1 . Consider the product of two of the association incidence matrices, not necessarily distinct. (1.12) AjAk = (ch) , where v (1.15) cf: 3 : aJ a.k In (1.18), each term of the sum has the value 0 or 1 and is equal to 1 only if treatment 0’ is a 1th associate of treatment [A and a km associate of treatment 7/ . Thus cikv is equal to the number of treat- ments which have this property. From (iii)(c) of the definition, page 8, we have (1.14) cliky = p‘h‘ when ,u 1: v and ’u and 1/ are it‘1 associates. A diagonal element (:3: of the product is equal to the number of treat- ments which are simultaneously 1th associates and km associates of treatment ,u . Therefore (1.15) c/j‘kf‘ 3 51]: n), where 5 j]: is the Kronecker function defined as 1 if j = k and 0 if .1 f k. Statements (1.12 to 1.15) lead to the following theorem. Theorem _l_._,_]_... If ‘1 denotes the incidence matrix for 1th associates in a PBIB design with m associate classes, then I - .. Z i i 3 l llllinjl. TI.’ 17 where S it is the Kronecker delta function and Iv is the v x v idea“ t, matrix e Proof: Statements (1.12) to (1.15) show that AjAk has the indicated form. The statement, in (iii)(c) of the definition, that p3}: =- p?” then implies that the product is commutative. The statement that products of the 11 are commutative is equivalent to the statement that the products are symetric, for if A and B are symetric matrices, then BA = B'A' =- (AB)‘ and (AB)' is equal to AB if and only if AB is symetric. Formula (1.16) for forming products is almost a sufficient as well as a necessary condition that the matrices A1 satisfy the conditions of partial balance. The sufficient conditions are stated in the following theorem. Theoremlég. If A1 , i=1, 2, . . . ,m,areasetofsymmetric v x v incidence matrices whose sum is the matrix with 0's on the main diagonal and 1's elsewhere, and if there exist mom-negative integers m1 and p}k such that (1.16) holds for 1, k = 1 , 2 , . . . , m , then the ‘i are the association matrices of an association scheme satisfying the conditions of partial balance. Proof: It must be verified that parts (iii) (a) , (b) and (c) of the defi- nition on page 8 are satisfied. The statement that the sum of the 18 incidence matrices is a matrix with 1's in all off-diagonal positions shows that every pair of distinct treatments are ith associates for some i , which is equivalent to (iii)(a). The number of jth associates of treatment I“ is equal to the number of 1's in row lu. of A , which is in turn equal to the element in the f‘ , ,4. position of the product matrix Ajbj' . By symmetry of Aj , this is identical with if and may be computed by (1.16), which shows that all diagonal elements of A12 are equal to nj . Therefore each treatment has n j Jth associates and (iii)(b) is satisfied. The set of 1th associates of treatment la. is determined by the positions of the 1's of row IU- of A: , and the set of kth associates of 7/ is determined in the same way by row 2/ of Ak . The number of treatments common to these sets is equal to the inner product of these two rows taken as vectors and appears as the element in the ,M , 7/ position of the product Ajak' , which by symmetry of Ak is identical with Ajhk and has the form of (1.16). m The only term of the sum 2 pg]: A1 which contributes to the element 1 s l in the [u , 7/ position is the term with i such that /4. and 'U are 1”“ associates. Therefore the number is equal to pit When fl- and are any pair of ith associates, proving most of (iii)(c) . The final statement follows from the fact that AJ. A]: 3 AKAJ , and the proof is complete. The stipulation that the A are incidence matrices is necessary in 1 Theorem 1.2. It is possible to construct matrices having elements other than 0'8 and 1's which satisfy all the other hypotheses of the theorem, 19 but are of course not association matrices. An example with m = 2 is memnwmg 0000 111 0001. 010 A130010 1-11 . 0100 010 L1000 111 looooo—q . 3’ u bHHHoi 'OHHHOI A typical product is 122 = 3 I + “1+ A2 . Next consider matrices which are linear combinations of the identity Iv and the association matrices Ai , say (1.1?) onvt- >‘1A1+ ...+ AnAm, where the )\1 are scalars. A product of two such matrices will be a linear combination of tonne of the form Iv , A and AiAj , and by 1 Theorem 1.1 will reduce to the form.(l.17). Thus the set of matrices of this form is closed.under multiplication. Some consequences of this are mentioned below. Among the products of matrices which are readily com- puted by application of (1.16) are integral powers of the A1 matrices and of the matrix NN' . The square and cube of the matrix A1 for first associates in a design with m = 2 associate classes will now be given as illustrations. 2 .. 1 2 (1°13) A1 ’ ”11¢ p11“1 "' p11‘s ' s 1 2 2 ‘1 " n1“1"'1’11*1 + l’119‘1“2 20 (1'19) ‘13 : n1911“ + (”1 + p112+ p1:83:95 + (911911 +9119? A:2 ‘ It was pointed out.that the set of matrices of the form (1.17) is closed under multiplication. It is obviously also closed under addition, and if negative coefficients are allowed, under subtraction. It follows from these remarks and from general properties of matrices that the set forms a commutative ring of matrices with a unit element. This has a number of interesting consequences, of which one may be mentioned. The matrices Iv , A1 , . . . , An are easily seen to be linearly inde- pendent and form a basis of m +-l elements for the ring. For any matrix c in the ring, the set 1‘, , c , 02 , . . . , 0“1 contains m +-2 elements which must be linearly dependent. Therefore 0 satis— fies an equation with scalar coefficients of degree at most m +-1. This means that the minimum equation of C has degree at most m + 1 , or that any matrix of the form (1.17) has at most m +-l distinct characteristic roots. In particular, this applies to NN'. It is possible to use methods based on the commutative ring to find the char- acteristic roots and their multiplicities. The same results on the number of distinct characteristic roots of NN' , together with a com- putation of the values and multiplicities of the roots, appear in a paper of Connor and Clatworthy D Z7 which does not use the A1 matrices. This paper was published before the present work on the association matrices was completed. Several theorems of [717 will be used in Chapters 11 and III. Credit is also due to R. C. Bose for some work on the association matrices A1 , including the equivalent of Theorem 1.1, which has not 21 been published but was presented at a meeting of the Institute of Mathe- matical Statistics at Ann Arbor, Michigan on September 2, 1955. The portions of this research making use of the association matrices had already been completed at that time. Another possible interpretation of association schemes is by means of linear graphs. (A linear graph may be defined for our purposes as a finite set of points, certain pairs of which-are joined by non-directed lines.) The association scheme for it‘h associates in a PBIB desim m be identified with a linear graph on v points by identifying points with treatments and joining points which are it“ associates. Since each treatment has 111 itb associates, each point of the graph will lie on n1 lines. Since any two treatments which are i"h associates have as common 1th associates ph other treatments, each line of the graph will lie on ,pi’i triangles. Ilore generally, if an arbitrary line of the 1th graph joins points A and B then there are just p3]: other points which are joined to A by a line of the 3th graph and to B by a point of the kth graph. In the case of P318 designs with two associate classes the graphs may be described more simply. Each of the two graphs is the complement of the other and it is sufficient to describe the one for first associates. In this graph there are nl lines on each point, each line lies on p11 triangles, and each pair of points not joined by a line is joined by pil chains of two lines. The incidence matrix A1 of 1th associates may also be interpreted as the incidence matrix of the it'h graph, 5. 1 in the ft ,v position of 22 the matrix indicating that points F. and V of the graph are joined by a line. Incidence natrices are useful in analysis of the structure of graphs. The terminology of linear graphs will be used in parts of Chapter IVfor the investigation of the structure of association schemes. II. ENUIERATION 0F POSSIBLE DESIGNS AND ASSOCIATION SCHEMES WITH no ASSOCIATE CLASSES 2.1, The Class of PBIB Desigs with 130 Associate Classes In this chapter, attention will be confined to partially balanced incomplete block designs with two associate classes. Bose and Shimamoto discussed these desigls thoroughly in 1952 [397 and introduced a classi- fication of them into five types according to the form of the association scheme. An extensive set of tables of these desigls was compiled by Bose, Clatworthy and Shrikhande and published in 1954 [a], following the classification of Bose and Shimoto. Over 570 designs are listed, about three-fourths of them of group divisible type. The authors state that the compilation includes all designs that were known at that tine, but do not claim that additional desims cannot be constructed. The classification of association schemes seems also to be a summary of known types, and is not represented as a listing of all possible schemes. Some new association schemes to be constructed in Chapter III fall outside the classification, showing that it is not exhaustive. The classification is described later in this section. A computing procedure developed from some known necessary conditions on association schemes is introduced in Section 2.2 and used in Tables I and II of the Appendix to list the parameters of all possible association schemes with two associate classes and not of group divisible type, for all numbers of treatments v 5 100. Several new necessary conditions are 24 also proved in Section 2.2. Necessary and sufficient conditions for the existence of an association scheme are not known. Of 101 sets of parameters tabulated in the Appendix, 56 correspond to schemes which are already known or constructed in this dissertation, and 4 are proved impossible. These schemes are identified in Table II. There remain 41 sets of parameters for which the existence or non-existence of an association scheme is unknown. Such a list was frequently promised by the early writers on PBIB designs, but appears never to have been compiled and is offered here as an original contribution, along with the computing scheme and the necessary conditions of Section 2.2. The next logical step is to list all combinatorially possible designs for each association scheme, identifying those known to exist or to be imp possible. The counterpart of this list for BIB designs was mentioned on page 6; it was published by Fisher’snd Iates in 1958 and revised in 1945 and 194? [i J. It includes 16 sets of parameters about which nothing was known in 1938 but which were subsequently attacked.so assiduously by various writers that by now all but two (at most) have either been con- structed or shown impossible. Such a list for PBIB designs would be much longer, even for the association schemes so far constructed, and its cam- pilation has perhaps been deterred by the fact that enough PBIB designs are already available to satisfy most of the needs of experimenters. The parameters of all possible designs with r 5 10, k 5 10 will be listed for the schemes under special study in this dissertation, mostly of Latin Square type. The list appears in Tables III and IV of the Appendix and the method by which it is constructed is.developed.in.Section 2.3, 25 using known theorems for the most part. The parameters of schemes which have been constructed or proved impossible are identified in Table IV. Some new designs and impossibility proofs are included for reference in SectionA3 of the Appendix. before. It is believed that this list has not appeared The parameters for desips with m = 2 associate classes are v, b, k, r, Arlyn '- H 1 2 vi a a ’ n2, 2 2 P P -12 22-4 These parameters satisfy relations (1.5) to (1.7), which are restated here for this special case. (2.1) (2.2) (2.5) (2.4) (2.5) n+n - v—l n1>\l+n2x2= r(k’l). 2 1+Pi1+Pizz Pn+P§2= 1+p1 : 2 2 pm a l+fi2+p 22 1 z . nl p12 n2p§l l : 2 nl p22 n2 p12 n 1 n2 26 The classification by Bose and Shimamoto of association schemes for P818 designs with two associate classes will now be taken up. This classification was introduced by Shimamoto in a master's thesis written under the direction of Bose, was first published in a joint paper in 1962 [is], and has been used, with minor changes, in later papers by the same authors and others. The five types of designs are Group Divisible (GD) , Triangular, Simple, Cyclic and Latin Square (Lg) and will be described separately . £1332 Divisible designs are defined when the number of treatments v may be expressed as a product an. The treatments are divided into n _ groups of n treatments each, treatments in the same group beingtaken as first associates while those in different groups are second associates. GD designs have been mentioned with some examples in Chapter I. They form the most important class of partially balanced designs and the largest known class, and have been studied more extensively than any other partially balanced design. In 1952 Bose and Connor [7] published several results on these designs, one of which will be generalized to Latin square type designs in Chapter IV. One feature of [7] was the division of the designs into three subclasses, essentially on the basis of the rank of the matrix NN' , though the connection with the character- istic roots of mm was not brought out clearly until 1954 in a paper by Connor and Clatworttnr [71.7. A similar classification of Latin square type designs will be mentioned in Section 2.5. Some other important publi- cations on CD designs are [TS], [7 9]. D L7. 27 Trium. designs are defined when v is equal to a triangular number n n- . The association scheme is an array of n rows and n colunnswith the following properties: (i) The positions in the main diagonal are left blank. (ii) The n 3" . positions above the sain diagonal are filled . with the nunbers“.l, 2, . . . , 9.1-3:le- eorresponding to the treatments. (iii) The positions below the nain diagonal are filled so that the array is syIIIetric. * (iv) For any treatment 0 the first associates are those treat- nentswhichlieinthesanerow(orinthesanecolunn)as 0. gap}; designs include designs with various values of v. In the 1959 paper [a] in which PBIB designs were first introduced, Bose and Hair gave some examples of designs obtained by dualising BIB design, that is, interchanging the roles of treatments and blocks. The treatments in one particular block of the dual design then correspond to the blocks of the original design which contain a particular treatment. The duels of some BIB designs fail to satisfy the conditions of partial balance, but in any case the dual desigl will have the property that any two blocks will have the same number of treatments in com. This led to the designation "linked block“ for such designs [327. The duals of several classes of BIB designs were shown by Shrikhsnde [3;] to be partially balanced with two associate classes. Sons of these designs fall within the triangular class discussed above; the others have the property that A1 = 1, A2 = o. In the classification by Bose and Shimamoto in their 1952 paper [T Q7, these were listed as a separate "linked block" type of designs. In the tables published by Bose, Clatworthy and Shrikhande in 1954 [a] this classification is enlarged somewhat to include soae designs which are not obtained by dualization and do not have the linked block property, but which do have the property that Al 3‘ 0, A2 = 0. They are referred to as simple desims. The classification was enlarged a little too much in the 1954 tables, as the three designs listed for v 3 19 are not partially balanced. Table II and Theorem 2.2 will each show the impossibility of a PBIB design with two associate classes and v = 19. In Cyclic designs the first associates of treatment 0 are the treatments 0 -I- d1, 9 + d2, . . . , O +dn1, reduced modulo v, for a suitably chosen set of integers d1 . has the special property that each row is a cyclic permutation of the The association matrix A1 thus first row. In every know design of this type v is a prime of the form 4t + l and the set of d's may be taken either as the set of quadratic residues of v or as the set of quadratic non-residues. {£3.15 m type designs are defined when v is equal to a square n2. The association scheme consists of an n x n array of the numbers 1, 2, . . . , n2, possibly with an orthogonal set of one or more n x n Latin squares superimposed. Two treatments are taken as first associates if they 'occur in the same row or colunn’Iif they coincide with the same letter in any of the Latin squares. A scheme of this type using 3-2 Latin 29 squares is said to be of Latin square type with g constraints and is denoted briefly by the symbol I.g . This type of designs will be treated in some detail in this dissertation. Parameters of possible L design Will be enumerated in Section 2.5 and tabulated in Tables III algld IV of the Appendix, and a number of properties of the association schemes will be investigated in Chapters III and IV. An n x n Latin square is an arrangement of n letters into the cells of an n x n array in such a way that every row and every column of the array contains every letter exactly once. A Latin square may be con- structed for every n, for example by taking each row as a cyclic per- mutation of the first, as in this example. UGO)? mucus UPUO onus-u Two Latin squares are said to be orthogonal if, when one is superimposed on the other, every ordered pair of letters occurs exactly once in the resulting square. Thus the following 3 x 5 Latin squares are orthogonal. ABC ABC BCA CAB CAB BCA On the other hand, there exists no 4 x 4 Latin square orthogonal to the example above. It has been shown that at most n—l mutually orthogonal n x n Latin squares can be constructed, and that the construction of such a set, called a complete orthogonal set, can actually be accomplished if n is a prime or a power of a prime [fig], [3.7.],[5]. The following three rm- ... I‘ll]. III 1.! a squares form a complete orthogonal set for the case n a 4. ABCD ABDC ABC D BADC CDAB DC BA CDAB DCBA BADC DCBA BADC C DAB Knowledge about sets of orthogonal Latin squares when n is not a prime power is rather sketchy. In no such case has a complete set been con- structed, though methods are known for constructing a smaller number in certain cases (e.g. two orthogonal 12 x 12 squares). For n satisfying certain conditions it is known that the maximum number of squares in an orthogonal set is less than n-l [72], while in the case n = 6, enumera- tion methods have been used to show that no orthogonal pair exists. This case was mentioned by mile:- 53] but not finally settled until 1900 [5.17 (see also [a 9]). The existence of a complete orthogonal set of n x n Latin squares is equivalent to the existence of a finite projective plane geometry in which each line contains n +1 points. Either of these systems can be constructed from a finite field of. order n , so that a sufficient condition for their existence is that n be a prime power, but this condition is not known to be necessary. An open question at present is whether any set of two or more 10 x 10 orthogonal Latin squares exists. Orthogonal squares which are not Latin squares can be 'useful in the construction of association schemes. The two following squares are ob- viously not Latin squares but they are orthogonal; that is, when they are superimposed every ordered pair of letters occurs exactly once. AAAA ABCD ,ssss sacs R: cccc 0: sacs DDDD ABCD An association scheme obtained by superimposing them on an array of the numbers 1, 2, . . . , 16 and taking numbers as first associates if they occur with the same letter in either square will be identical with the scheme L2 in which associates are defined by rows‘and columns of the array. Ioreover, any a x 4 Latin square is orthogonal to each of them, and any 4 x a square which is orthogonal to both must be a Latin square. The analogous statement for n x n squares is clearly true. Therefore a set of g—2 orthogonal Latin squares is equivalent to a set of g orthogo- nal squares of which two are R and C.1 If the n2 cells of each of g orthogonal squares are subjected to the same permutation, the resulting squares will still be orthogonal, though not necessarily Latin. Given any set of orthogonal squares, simultaneous permutation, of the cells can be used to place any two of the squares in the form of R and C, still preserving orthogonality. The association scheme L g may now be redefined by a set of g n x n orthogonal squares superimposed on an array of the numbers 1, 2, . . . , n2, taking numbers as first associates if they occur with the same letter in any of the squares. Permutation of the num- bers of the array together with the cells of the superimposed array will . preserve all association relations, so that any such association scheme is equivalent to one in which two of the squares are R and C and any -h‘ 4 __—_A ... fl __..~_———— '— 1. The notion of non-Latin orthogonal squares is not new. In at least one recent publication [F 1!] there is a description of the squares R and C and their relation to Latin squares. 52 remaining squares are necessarily Latin. In particular this shows that for a given n, all pairs of n x n orthogonal squares lead to L2 schemes which are equivalent except for numbering of treatments. It say be noted that an L1 schene is a special case of a group divisible scheme. It is convenient to use this definition of the L8 scheme to derive expressions for the parameters n1 and pgk and to slaw that they satisfy the requirements for a PBIB design. This derivation will be illustrated with an example of an L scheme for 16 treatments based on 5 the squares R, C, and a Latin square of the orthogonal set. These squares are listed below for easy reference, along with the array of ma- bers with which they are to be superimposed. The orthogonal squares are numbered from 1 to s for identification in the discussion. 1254 AAAA ABCD ABCD 5 6 7 8 B B B B A B C D B A D 0 9101112 0000 ABCD CDAB 15141516 DDDD ABCD DCBA Array Squarel Squarez Squares Figure 5. Example of L5 association scheme for 16 treatments. It follows from the orthogonality of the squares that two cells occupied by the same letter in one square must be occupied by different letters in each other square. Thus the n-l associates of a particular treatment in one square will be distinct fron its associates in each other square, and the treatment will have as the nunber of its first associates (2-6) n1 7- g(n-1) . 55 In the example, treatment 10 has as its first associates 9, ll, 12 in square l; 2, 6, 14 in square 2; and 4, 7, 15 in square 5, for a total of 9 first associates. Let two treatments 0 and ‘p occur with letters at and oh respectively in the 1:“ square, k = l, . . . , g'. ak and bk will be distinct for all values of k if the two treatments are second associates, and equal for Just one value of k if they are first associ- ates. If 0 and [6 occur with distinct letters in the nth and l:tn squares, so that ‘h # hh and at f bk , and if these two squares are superimposed, the pair of letters ah , bk will occur in just one cell. The treatment in this position will be a common first associate of 0 and fl . The total number of such treatments will be equal to the number of ordered pairs h, k such that 1113 it ch and at i bk . The number of pairs may be expressed u(u—l), where u is the number of squares in which 0 and fl occur with distinct letters. If 0 and [d are second associates they occur with distinct letters in all g squares, u = g , and the num- ber pfl of first associates the treatments have in common is (2.?) oil = c . In the example, let 9 = 5 and £6 3 10 . Then a1 = B, b1 3 C; 32 . A, b2 = B; as = 13, b5 = D; and since ak # bit for all 1:, treatments 5 and 10 are second associates. The pairs of letters ah, bk obtained when squares are superimposed and the corresponding common first associates of the two treatments are as follows. Pair of superb Ordered pair Cell in which the pair imposed squares of letters of letters occurs h, k ab, bk 1, 2 B, B 6 2, l A, C 9 l, 5 B, D 7 5, 1 B, C 12 2, 5 A, D 15 5,2 3,3 2 The six cells singled out represent the six common first associates of treatments 5 and 10. If 0 and fl are first associates, occurring with the same letter in one square, say the first, then u 3 g—l . In addition to the (g—1)(g-2) common first associates found by superimposing pairs of dis- tinct squares, 0 and fl will have as common first associates the n-2 other treatments occurring with the same letter in the first square. No additional first associates are found by superimposing the first square with any of the others, for the pair a , b in this case is a , b , h k l k which is identical with bl’ bk, and this pair of letters occurs in the position of ¢ itself rather than any distinct treatment. The number 1 . Pll of-common first associates of the two treatments is therefore 35 (2.8) PL 3 (g-l)(g-2)+n-2 3 g2- 33+n. In the example, let 0 3 9 and i = 10 . Then a1 3 C, b1 3 C; a5 = 0, b5 3 D; and since a1 3 b1 , treatments 9 and 10 are first associates. They occur with distinct letters in squares 2 and 5 . The pairs of letters an, bk obtained when these squares are superimposed and the corresponding common first associates of the two treatments are as follows. Pair of super- Ordered pair Cell in which the pair imposed squares of letters of letters occurs h, k ah,bk 2, 5 A, D 15 5, 2 C, B 14 Treatments 9 and 10 occur with the same letter (3 in square 1, and the other n-2 3 2 cells of the Square which also contain the letter C are 11 and 12 . The four cells singled out represent the four common first associates of treatments 9 and 10 . The remaining parameters are quickly computed from n1, pil and 9&1 to give the following set. 36 (2.9) v 3 n2 , ( ) 82 - 53 r n (s-l)(n-g +1) n 3 g n-l , P 3 1 1 (g-1)(n-g+1) . ' s(s~1) 8(n-s) P2 3 2 . son-g) (n-s) + 3-2 The nondnegative nature of n2 implies the inequality (2.10) g _<_ n +1 , proving the statement previously made that the maximum number of Latin squares in an orthogonal set is n-l . If a complete set of :1 +1 orthogonal squares is constructed and if g of them are used to define an L8 association scheme, then second associates are precisely those treatments which occur with the same letter in one of the n-g + 1 re- maining squares. It will be convenient later if the letter 1' is intro- duced to represent this number: (2.11) f =n-g+l. It is clear that if the designation of first and second associates is interchanged in the L8 association scheme, the result will be the Lf scheme based on the f remaining squares. A scheme with the parameter values and properties of I.f can be obtained in this way from any Lg scheme, whether or not the f orthogonal squares are actually constructed. Some examples will be given in Chapter IV of schemes of this kind for which the orthogonal squares can be shown not to exist. Since the schemes L 8 and Lf are equivalent for any value of g , each Lg scheme is 57 equivalent to one in which g g 2...}; , and in particular the Ln scheme is equivalent to Ll , which is simply a group divisible scheme. The dual roles of g and f are most clearly seen from the following expressions for the parameters. (2.12) n 3 (g «Pf-l) , . 2 (g—l) + r-2 an) v 3 n2 , P1 = f(g-l) r(£-1) n1 3 8(331) 9 .— n2 = f(n-l) , _ 3(3-1) s(f-l) 92 - 2 . 8(f-l) (f-I) + 3-2 These expressions give the values of parameters of L8 schemes if g and f are positive; if group divisible schemes are to be excluded, g and 1‘ must be taken as 2 2. . It may be verified that certain negative values for g and f (and hence n) lead to values for the above parameters which are non-negative and different from those obtained with positive g and f . 'Conditions (2.2), (2.4) and (2.5) are algebraic identities in g and f and are satisfied in either case, so the new values represent the parameters of a possible new series of association schemes. Some connections of the new schemes with the ordinary Lg series will be discussed in Chapter III, and several ofthem will be constructed. They are found to fall outside the five known classes of association schemes. The name "negative Latin square" will be used for the series .of schemes whose parameter values are 1! 8 g negative, will be used as an abbreviation. Parameter values of schemes given by (2.12) with g, f and n negative, and the symbol L , with in the L; series are identified in Table II of the Appendix, and possible desips for the new schemes will be listed in Table IV. It should be mentioned that the ordinary Latin square, or L8 , series was defined to include only schemes in which first associates can be defined by means of a set of g orthogonal n x n squares. The term "scheme with L g parameter values” will include the L8 schemes and any other schemes whose parameter values are given by (2.12) with positive values of g, fand n. 59 2.2 Motion 9‘; uggciation Sch-sea. The parameters by which an association scheme is specified are v, n1, pg], . A procedure will be developed in this section for the con- struction of a table of all possible sets of values of these parameters for P318 designs with two associate classes. For a given value of v , a unique group divisible (GD) scheme exists for each pair of integers m 2 2, n 2 2, such that mm = v; that is, there is one GD‘association scheme for each proper divisor of v.. The construction of these associ- ation schemes is trivial and it is not considered necessary to list their parameters. The enmeration’ of possibledsssociation schemes of other types will be carried out for all values of v s 100.1. Theorem 2.0, due to Connor and Clatworthy 52], defines parameter values of one series of possible association scheees in terms of a para- meter t . Theorem»2.1 uses two parameters s and t in the deriva- tion of expressions for the parameters of all possible schemes not given by Theorem 2.0. Several necessary conditions on .the parameters are also derived. Table Is of the Appendix lists» the sets of parameters given by Theorem 2.0. Table Ib also makes use of Theorem 2.5. Tables Ia and lb give 101 possible sets of parameters for '$ 100. These “are listed in Table II. The necessary conditions. applied in. constructing these tables are by no means sufficient for the existence of n association scheme and v w 1.< An easy computation shows that there are 285 GD schemes for values of v 100. . 40 it is easy to derive additional necessary conditions applying to certain classes of the parameters. Theorems 2.2 to 2.8, which are of this nature, show the impossibility of four of the schemes in Table II and place restrictions on several others, as well as giving some general information on the structure of association schemes. The schemes which are proved impossible are indicated by the letters x in Table II, followed by a reference by number to the applicable theorem. Parameters of known schemes are indicated by the letter C , followed by a reference. The remaining 41 schemes have neither been constructed nor proved impossible. Further explanation of the tables precedes them in the Appendix. Several necessary conditions satisfied by the parameters of a PBIB design are derived by Connor and Clatmorthy [T Z] by using the matrix NN' . They show that this matrix has only three distinct characteristic roots, obtaining eXpressions for the roots and their multiplicities. The same results may be obtained rather easily by methods mentioned in Section 1.5. In their notation, rk is a root with multiplicity l, r - 21 is a root with multiplicity 0(1 , r - zz is a root with multiplicity0<2 . The 0(1 depend only on the parameters of the association scheme, and are this the same for all designs having a given association scheme. The mi depend in addition on A l and A2 and will not be needed in this section. Equations (5.9) and (5.10) of [[2] give 41 (2.15) 0Q : (V—l)(—Y+— W + 1) “an; 1 2 \FE (2.14) 0(2 2 (v~l)(Y-: TA— +- l) -2n2 , W 2 FE" where I (a) . Y 3 piz " p12 s (2.15) (b) F: p12 + pf, . (c) A = Y 2 + 25 + l . Useful necessary conditions on the parameters may be obtained from the fact that the multiplicities 0< 1 and 0(2 must be non-negative integers. They are of course not independent; 0(1 + o<2 = v—l . Connor and Clatworthy in their theorems 5.3 to 5.5 investigate the nature of A . One of their results will be stated as a theorem. THERE}! 2.0. (Connor and Clatworthy) 517. If A is not a square, it is necessary that (a) pig 3 pig. 3 t! (2.16) (b) h1 = h2 = 0(1 = 042 = 2t, (c) v = A = 4t+1 , where t is a non-negative integer defined by (2.16) (c) . This series of possible association schemes is easily enumerated. The possible parameters are listed in Table In. In every other case 42' A is an integral square. This will be used to develop a mettnd of systematic enumeration of other possible association schemes. From (2015) (c): 2. (2.17) p = A-i 211 - Solving (2.15) (a) and (b) for the pig , then using (2.17), (2.18) pl 3 LL: MA-YZA-2Y-1 : A-;Y+1)2 12 2 4 4 (l2: -VY' - l) {1Y7§-+'V’1';J.se 2 2 (2.19) 2 : fir. .-. A-Y2+gr-1:é_‘x ~le 4 4 912 . 2 (W (ILL?! L1). . Statement (2.17) shows that the integers A and Y must be of opposite parity. Therefore W 1' Y must be odd integers, V3- 1 Y 3"— 1 must be even integers for all choices of signs, and s and t defined as follows will be integers. s 3 :12:;_52\{’ ’4;. o (2.20) - tzfltY‘l- Equations (2.18) and (2.19) may now be rewritten as follows. (2.21) pig == s(t + 1) , (2.22) p; = (s + 1)t . 45 Also, (2e23) E = 8 + t + 1 e A preliminary enumeration of possible pairs of values pig , piz now reduces to the listing of pairs of integers s, t and application of (2.21) and (2.22). Bose and Connor [7] show that a PBIB design with two associate classes is of GD type if and only if piz '3 0 for i = 1 or 2. This case will be excluded by requiring a and t to be asitive integers. For each pair 9&2 , pig , it is next desired to enumerate possible sets of the remaining parameters, particularly n1 and n2 . It will be convenient to do this by finding values of 13:2 and pi . lultiplying equations (2.. 5), we obtain 2 n1 ll‘2 Pi2 1"12 n1 (2'24) Pia p:122 922 932.1 ° Pairs of possible values of 922 and pfl may thus be obtained by ex- pressing the product piz 93.2.2 in every possible way as the product of two positive integers 922 and pfl . Relations (2.4) then give values of the remaining parameters, including 2; 1 e = 2 2 . “2 p12 + 922 ’ 111 p12 + p11 To avoid duplication, we make the restriction 1 2 . 1 : 2 (2.25) p12 5 p12 , if 1312 912 . then n2 _<_ 9, . 1 2 -1 2:: 2 A design for which p12 > P12 01' 912 912 and ”a. > a! may be 44 reduced to one for which (2.25) holds by changing the designations of first and second associates. The enumeration will be carried out only for values of v s 100. Simv=n+n+1-pf2+p§1+p{2+912+1. 1 2 2 thisaeans pig + pi]. +pi2 +1922 S 99, implying apt, + pf, 4- pg, + p21) < 25. Since the geometric scan of any set of positive numbers is 5 their aritlnetic mean, (1 92 Pl pa“ <25- 1)12 12 22 11 Using (2.24) we obtain (2.26) (p1 92% < 25 . 12 12 Only finitely any values of p12 and piz satisfy (2.21), (2.22) and (2.26). A convenient fern for listing them is a table of we. of the function a- (YK- 0' ), where 0' is an integer. The following portion of the table will illustrate its form. 0" 0‘ O- 0! N H O O O O O O O O ...o O I 3' 45 The table is most easily constmcted by noting that the diagonal entries are 0's and that the entries in column a- forn an arithaetic progres- sion with difference a" . In the row of the table corresponding to a fixed value of VT = s + t +1, the consecutive entries for 0' 3 s and o- - s +1 are precisely s(t +1) = p}, and (s+l)t = pfz . For example, for TX 2 4, the possible pairs of values of pie , p§2 are 0, a, s, 4; 4, s; 3, o. 1111 values or the pig satisfying 0 (pig S Pig 3 (Pig P3242)i < 25 are given in the portion of the Thus the sets of parameters to be listed in Table 1b include only . 51 possible pairs of values of pig and 9&2 . For a given pair, the number of values for the remaining parameters depends in part on the nuaber of divisors of the product p12 pig . The relation n2 = p52 + p§2 + l and the non.negative nature of p22 lead to 2 _ - 1 1 . I’12 5 "2 1 " l’12""P22 ' 1' 1 2 1 P22 3 I’12 " p12"‘1' 1 (2.27) p22 2 Y + 1 . This restriction on the value of p?22 will be used to shorten the °°nputation sonewhat. A similar restriction on pil turns out to be Vacuous in view of (2.25). It will be shown in Theorem 2.5 that at 13881; one of n1 and né must be an even number. If both are‘odd in a Figure 6. V23 1 2 5 4 5 5 2 2 4 5 4 5 4 6 6 6 5 8 9 7 6 10 12 12 s 7 12 15 16 9 8 14 18 20 2o 10 9 16 21 24 25 11 10 18 24 12 11 20 27 15 12 22 14 15 24 15 14 26' 16 15 28 17 16 so 18 17 52 19 18 54 Array giving possible values of p12 and pie . l 46 4‘7 line of Table Ib, their values are omitted and the rest of the line is left blank, further shortening the computation. The full strength of the positive integral condition on 041 and 0K2 has not yet been imposed. Using v-l = n1+ n2 , expression (2.15) may be written (2.28) o<1 = “ZWZ’Y + 1)+nl(V—- Y '1) =52;('+1)+9fi 2V7)— . V7: The value of this quotient is readily computed for each set of values of piz , pig , 111 , 112 in Table Ib. If it is not an integer, no association scheme exists and the letter f is entered in column “I of the table. If 0( 1 is an integer, it is entered and followed by the value of v = n1+ n2 4- l. , The results that have been obtained for the construction of Table Ib are collected for convenience and stated as the following theorem. THEOREM 2.1. An association scheme for a PBIB design with two associate classes and not of group divisible type must have parameter Values given either by the expressions stated in Theorem 2.0 or by the f allowing conditions. V7: is a positive integer, s and t are positive integers satisfying (2.25): s. + t + l 3 TX (2.21) 11:2 ‘ e(t +1) , (2.22) p122 - (s +l)t , 48 l 1 2 . p22 is a divisor of p12 p12 satisfying (2.27). P222Y+1=P§2’Pi2*1: ”)2 9&2 (obtained from (2.24) ), 9&2 pi = 2 2 n1 p11 + 912 ' (2.4) g 1 n2 922 + ’12 ’ (2.28) (' + 1) n2 + ' nl = d must be an integer; m“ l moreover, if the requirements 1 2 , 1 : 2 (2-25) 912 S 912 2 if 912 912 s then a; S n, vSlOO are imposed, then 9&2 , p12.2 must be a pair=of consecutive entries in 1‘01! A of the array in Figure 6. The proof of Theorem 2.1 has already been completed. One additional necessary condition. used in Table Ib will appear as Theorem 2.5. This condition is a special case of (2.26) but seems to be of sufficient 1Interest to be stated separately. It may be remarked that taking negative inhgral values for s and t (which is equivalent to using the negative ‘q‘uu‘e root of A ) leads to parameter values which are positive but no different from those alreachy obtained. 49 The enumeration in Tables Ia and lb gives 101 sets of possible parameter values for association schemes. These are reproduced in Table II, in order of increasing values of v, and numbered serially. These serial numbers are given for reference in Tables Ia and lb. Table II gives values of v, n1, 9:1: , 0(1 and m . The parameter TE will be found convenient in locating particular sets of parameters in Table Ib, which islarranged in order of increasing values of V7: . Table II is standardized by listing only association schemes for which al 5 n2 . In some cases this requires that desigiations of first and second associates be interchanged in the corresponding scheme of Table lb. The same parameter values occur, but with the indices 1 and 2 inter- changed wherever they appear. Inspection of Table II suggests a number of remarks about the possible association schemes. Their abundance when v is a square is somewhat striking; so is their scarcity when v is a prime. 111 and n2 have a factor in'comon in every scheme in the list; 0( and O< 2 l have a common factor in many cases but not all; there seems to be a high proportion of cases in which at least one of the d 1 has a factor in common with v. The following theorems, some of which were suggested by this sort of observation, show that at least part of the apparent r'zi-fldlrity is a result of general properties of association schemes. More of impossibility of several association schemes are obtained as Particular results of some of the theorems. THEOREII 2.2. In a PBIB design with two associate classes, if the number of treatments v is a prime, then 17 must have the form 4t + l and the parameters of the association scheme must satisfy (2.16). PROOF: Except in the case specified in (2.16), the values of the association scheme parameters are given by Theorem 2.1. The following makes use of (2.2), (2.4), (2.21), (2.22) and (2.24). = = l 2 2 v n1+n2+l p22+p%2 +912+pn+l p12‘2+s(t+l)+(s+l)t+1+£tLa+:)£:t+l) (pé2)2+ (2st + s + t +1) 9:2 +st(s + 1)(t + 1) P122 [Iéz + 8?] [912.2 + (s+ 1)(t -+- 1)] 922 (2e29) V The greatest common divisor of two integers f and g will be denoted by the usual notation (f, g). Define c and d by O N (9&2 s 3t) 9 Cd 3 p%2 e 9&2 + at is then divisible by c. Since (1 is a divisor of pi, , it 1 2 ' P mstbeprimeto st. But-Lag??- = “8+1 ‘34, isaninteger Bil P22 - 51 so d must be a divisor of (s +1)(t +1) and hence of pig-+- (s + 1)(t + 1). We may therefore write (2.30) v : [p%2+ st] [pig-)- (8+ 1)(t-#1)], c d where both factors in the right member are integers. If s and t are both positive, (2.29) shows that both factors are greater than 1; v is then composite. If s or t is equal to O , it has been shown that if the design exists, it must be of group divisible type, which is defined only for composite values of v . This completes the proof of Theorem 2.2. THERE]! 2.5. If a PBIB design with two associate classes is not of group divisible type, then the number of treatments v cannot be of the form p+1 foranyprime p. PROOF: This theorem is a particular result of some general relations connecting the parameters 111 , n2 , p12 and 9&2 , which will now be developed. Using (2.4), ‘ 2 2 = .911‘H’12 n1 V 2 2 .. n2 pu+n2 912 - n1 ‘2 e Applying (2.5), 1 2 '- The following form of (2.51) was found useful as a check during the con- Strucuon of Table Ib. 52 (1.) 2 (2.52) £3 + 31—2- 1 . n2 “1 Next introduce the greatest common divisor (nl , n2) of nl and n2 anddefine 8,fll,n2 by (“l n '12) = a 9 (2.33) Then (m1 , m2) = l . (2.31) may now be written 1 2 .. 2 am1p12+am2p12 - a m1m2, (2.34) m1 ph-f-mz 932.2 = a m1 m2 . Equation (2.54) in integers, with 'l and m relatively prime, implies 2 that pk is divisible by 1:12 and pfz is divisible by m1 . Say (2.55) piz a um 2 9 2 - I’12 " " “'1 ° Substituting (2.35) in (2.54) and simplifying, (2.56) u+w 8 a. If the desigi is not to reduce to a balanced design, both n1 and n2 4..- l- This admits a geometric interpretation if p12 and pig are taken as rectengular coordinates of a point in a plane. Then the point (pi2 , p§2 ) mat lie on the straight line with intercepts n2 and n1 . 55 must be non-zero, so that m1 and m are non-zero (tacitly assumed in 2 some of the preceding statements). If the design is not of GD type, piz and pfz must also be positive, so that each of u and w is _>_ 1. Then (2.56) shows that a _>_ 2, so that ml and n2 have a proper divisor a in common. Their sum nl + n2 If v - l is also divisible by e , completing the proof of Theorem 2.5. Relations (2.55), (2.55) and (2.56) can be used as the basis for an enumeration of possible sets of values :11 , n2 , 5&2 , 9&2 . It appears to be considerably less efficient than the method based on Theorem 2.1. THEOREM 2.4. In a PBIB design with two associate classes, if 1&1 2 1, then ph _<_ .1; —l,where i and j areequalto 1 and 2 in some order. PROOF: The proof will be carried out for i = l and j = 2 . The other case is similar. Let 0 and fl be two treatments which are first associates. Since ph _>_ 1, there is at least one treatment which is a first associate of both. Denote one such treatment by 1T. Of the 111 first associates of TT , ph are first associates of O and pil are first associates of ¢ . At most 2p11-1 of them are firstassociates 01' 9 or ¢ or both. At least n1 - th are first associates of neither, and thus are second associates of both. But 9 and fl are first associates and the number of treatments which are second associates of both is precisely péz . This proves the inequality _ 1 1 ”1 2p11 3 p22 ' 54 Using the relations p1 3 n1 - 912 - 1 and 9&2 - 32 "' Pig 1 from (2.4), l l n1-2n1+2p12+2 S n2"1312’ spi-zfi n1+n2-2 = v-5, 7 (2.57) pi, 5 3-1 . The following sets of parameters from Table II violate Theorem 2.4 and are thus impossible. 1 1 # v p 11 p 12 56 so 4 16 40 56 5 1e The following theorems make use of the association matrices A1 . The details will be carried through for A1 , the incidence matrix of first associates. Let the numbering of the treatments be chosen so that the treatment 1 has treatments 2, 5, . . . , 111 + l as its first afiscaciates. Treatment 0 corresponds to row and column 9 of A1 . The matrix may then be partitioned as follows. 55 F— l ' i (2.58) 0.1 . . . 1'0 . . . o L..._.L..._._...-__.;_. ______ ---..-_._ 1} . e. I ’ 'I R : S n1 rows '1 I .. ll 1 ____________ a -b-r ------ .-- O. I eI | e: S. | T n2 rows 0| I _O| ' .. R is a symmetric 11 mm matrix; T is a symmetric n2 1: n matrix; 1 l 2 both have 0's on the main diagonal. S' is the transpose of S. The rows and columns of Al Will be taken as vectors. The inner product of row 1 with row .9 is equal to the number of common first associates of treatments 1 and 0 , and is equal to pi]. or pfl according as treatments 1 and O are first or second associates, respectively. This shows that each row of block R contains I’ll 1's and each row of block 8' contains pil 1's . Each row of A1 contains :11 1'5, and by subtraction the number of 1's in each row of S is equal to - . 1 : 1 . n1 1 p11 p12 , the number in each row of T is equal to -2 g 2 "1 p11"1:2° If the matrix A is partitioned in an analogous way, blocks R and 2 T are n2 3: ha and n1 x ml matrices respectively, and the row totals °f R , S , S' and T are p22 , pig , p.122 and 912 respectively. THEOREM 2.5. In any PBIB desigl with two associate classes, the f0llewing statements are equivalent and true. 56 (a) The products 111 pi]. , nl pl2 , n2 p22 , and n pig are all even. 1 2 (b) n and 11 cannot both be odd numbers. 1 2 1 PROOF: Each of the :11 rows of suhnatrix R of A1 contains pll 1's , and R therefore contains :11' pi]. l's. Since R is symmetric with 0's on the main diagonal, it contains 1's only in symmetrically 1 1 pll shows that 112 9&2 is even. The argument may be repeated for the matrix located pairs and n must be even. Similar reasoning applied to T A2 to show that :11 912 and n2 p22 are even. (An equivalent argument using A1 is based on the remark that :11 pig and n2 p22 are equal to the numbers of off-diagonal 0's in R and T respectively.) This completes the proof of (a). Since both terms in the left member of (2.51) are even, must be even, proving (b). It remains to show that (b) ml 32 implies (a). Let (b) be true. If both ml and n are even, (a) is 2 true trivially. If one is odd, say n2 , than n is even, and l n pl = n p2 is even implying 2 is even Therefore 1 22 2 12 ’ p12 ' 2 - .. 2 z: 2 2 n2 1 pl2 p22 is even and the products n2 pl2 and n2 p 22 both even, as well as n1 ph and n1 pig . A similar argument is used are When n1 is odd, completing the proof that (b) implies (a). Statement (b) is used in the construction of Table Ib. It can be Bhown that it is weaker than (2.28), a condition which is also used, but, it is used because it shortens the computation. Additional information is now needed about the partitioned matrix A1 . squaring according to the rule for products of partitioned matrices, ([L]: Do 24), 57 (2.59) I n “ RS+ST .74»- ‘.- . " ... _ —— —.—- -‘ .— .J’zo H T— l l .4 i I l l l I I l l .J s'a + rs: s's + T2 — — .- .-- .- _ — _ — ~ _J where u is a matrix all of whose elements are 1's. By (1.18), 112 has diagonal entries n1 , entries pi]. in the positions of 1's of A1 , and entries pil in the positions of off-diagonal 0's of A1 . This proves LENA 2.1: If R, S, S' and T are the submatrices of A1 depicted in (2.58), (a) 112 - 1 , entries 1 1 1 p11 " in any positions occupied by Its in matrix R, and entries p121 - 1 elsewhere; + 58' has diagonal entries 11 (b) S'S + T2 has diagonal entries :11 , entries pi]. in any positions occupied by 1's in T , and entries 912.1 elsewhere. 58 mm: 2.6. A necessary condition for the existence of an associ- ation scheme for a PBIB design with two associate classes and p11 = O, is the existence of a BIB desigi with parameters v 3 n1 , r = :11 - l , 1:: p111, bznJ , A =pii-1,wbere i and j areequalto 1 and 2 in some order. loreover, given any block of the BIB desigi, there exist at least pj other blocks which have no treatments in common with 1.1 the given block. PROOF: The proof will be carried out for the case i = l , j -"-' 2 , using the matrix Ll . A similar proof using A2 applies in the other case. When 9111 - o , the submatrix it contains no 1's and 32 isa zero matrix. According to statement (a) of Lemma 2.1, 88' then has entries :11 - l on the main diagonal and entries pil - l elsewhere. 3 1s thusan nlxn2 matrix with uniformrow totals pig-sal-l, “niform column totals equal to the row totals 9&1 of S' , and uniform ”0‘ inner products 912.1 - l , identifying it as the incidence matrix of the 318 design described in the Theorem. The number of treatments which a 314nm block of the design has in common with another block is equal to the 1Jitter product of the two corresponding columns of S , which in turn is Name to an off-diagonal entry in the given row of 8's , and is _<_ the entry in the same position of S'S + T2 . The number of 1's in a row or T is equal to pi? and by statement (b) of Lemma 2.1, an equal number of entries in the same row of 3's 4- 1'2 are equal to p1 = o. 1.1 Thus the given row of 8's must have at least pfz entries equal to O , Proving the final statement of the theorem. 59 There are also sets of parameters in Table II with p11 8 O , five of which belong to constructed association schemes. The others are schemes #15, 54, 59, 50. None of the balanced designs specified by the theorem are known to be impossible: the first three are known designs and the other has not been studied. Existence of the BIB design does not imply the existence of the association scheme, though it may give useful information about the structure of the scheme if it does exist. Whether or not the BIB design exists, the condition on blocks may be impossible, as fact which will now be used to show the impossibility of schemes 15 and 50. Let 'ft 7 denote the number of treatments which blocks [4 and v of an incomplete block design have in cannon. Where N is the incidence Matrix of the design, spy will be the value of the element in the P, 1/ position of the b xb matrix N'N. Let f(n) denote the number 01‘ blocks of the design which have precisely :1 treatments in common with a Chosen block, say the first. f(n) m be interpreted as the number of 1“slices ‘V for which s“, = n, or as the number of occurrences of the entry :1 in the first row of BM! (disregarding the entry in the 1,1 Position). In the case of a BIB design, Bussain [Effproves the follow- ing identity in the integers x and y. k (2.40) xy(b-l)-k(x + y - 1)(r-l) + k(k-l)(/\-1) '3 Z (x-n)(y-n)f(n). n = o s‘tting x=y=0 weobtain k (2.41) k(r - 1) + k(k - 1)(1- 1) = Z n2 :(n). n=O Setting x80, y=1 eeobtain k “hum-1) :- Z (n2~n)r(n). n'O ' loadingto k (2.42) k(r-1) = Z nf(n) . n20 Statements (2.41) and (2.42) give expressions for the sum and the sum of aquares of the b - 1 off—diagonal entries of a row of NW . These results, valid for all BIB designs, will be applied to the Particular designs introduced in Theorem 2.6. For these designs, at least 932.2 of the b - 1 entries are equal to O. The remaining b - l - pie entries are integers n satisfying 0 _<_ n f k, with cum and sum of 8Queues still given by the left members of (2.41) and (2.42). In some cases it may be impossible to find such a set of integers. This will be demonstrated in the cases of schemes #15 and 50 by computing the variances or the proposed sets of integers. The pertinent parameter values are these. 3 :2 g .. 2- 2 # b n2 1: p11 r 11:L 1 A-pn 1 p12 50 42 10 20 9 11 61 Using (2.41) and (2.42) these lead to the following values. I} Number of Sun Sun of ~ Variance integers squares 15 12 28 52 .112 . 52 : €82 3 --160 144 144 50 50 190‘ 910 50 ~ 910 - 1902 = ~880 900 900 Since negative values of variance are impossible, no such sets of integers can exist. This proves the impossibility of schemes 15 and 50. THEORm 2.7. A necessary condition for the existence of an associ— ation scheme for a PBIB design with two associate classes and pi}. = l is the existence of a PBIB design of GD type with parameters v = “i , 1‘ 8n1-2,k=pii, bSnJ, )(1=0,)(2Cpii-1, basedonan 7) o ' aBsociation scheme with parameters n; = l, n; = 111 ~ 2, P1 3 s - q _0 11:1 - 2-1 0 1 Pa: ,Ihere i,jareequa1toland21nsomeorder ; _1 find starred quantities refer to the GD design. Ioreover, given any block of the GD design, there exist at least p311 blocks which have at most one treatment in common with the given block. PROOF: The proof will be carried out for i 3 l and j = 2, using 62 the matrix 11 . A similar proof using A2 applies in the other case. If ph 3 1, then in (2.88) R has a single 1 in each row and column and is a synetric permutation matrix. 32 is equal to I , the idontity matrix. It is easily verified that B has the necessary pro- parties for the incidence matrix of first associates in the GD scheme specified in the theorem. By Lens 2.1, 83' has diagonal entries equal to :11 - 2, entries equal to pil - l = O in the off—diagonal positions that are occupied by 1's in block it , and entries equal to pi1 - 1 elsewhere. All the requirements are now satisfied for S to be the incidence matrix of the GD design specified in the theorem. The number 01' treatments finish a given block of the design has in common with another block is equal to the inner product of the two corresponding colums of S, which in turn is equal to an off-diagonal entry in the given row of 8'8, and is $ the entry in the same position of 8'8 + T2. The number of 1' I in a row of T is equal to pig and byustatement (b) of Lemma 2.1, ‘31 equal number of entries in the same row of svs + T2 are equal to PL 3 1. Thus the given row of 3's must have at least pi2 entries S. l, proving the final statement of the theorem. Seven schemes of Table II have p1 = 1, including schemes #41, 45 ll and 90 which are unknom. The GD design to which these lead do not seem to have been investigated and will not be taken up here. It is therefore tlot clear whether Theorem 2.7 can be used to prove the impossibility of any of these schemes. A remark which will be used in the proof of the next theorem will now be stated as a lemma. 63 LENA 2.2. Sufficient conditions for a v x v matrix A1 of 0's and 1's to be the incidence matrix for first associates in a PBIB design with two associate classes are (a) A1 is symetric, (b) the diagonal elements of A1 are 0's, (c) 112 = al I + p11 A1 -)- Film-I’LL) , where I is the identity matrix, 0 is a matrix all of whose. elements are 1's , and n1 , p11 , pil are non—negative integers. PROOF: Define A2:U-I-kl' Then ‘1 and A2 incidence matrices whose am is the matrix with 0's on the main diagonal are symmetric and 1's elsewhere. By Theorem 1.2, they are the association matrices 01‘ a PBIB denim with two associate classes if the products ‘12. , A112 , ‘2‘]. , A: have the form of (1.16) , where the constant coefficients “1 and pi;k are non-negative integers. By hypothesis -this is true for A? . It is easy to compute the remaining products, but not necessary for this proof. The equality of the diagonal elements of A: implies that Al has equal row and column totals n1 , implying .. _, 2 _ 110 - 0A1 - n10 . Also U - v0 . _Each of the promicts Ail: reduces to a linear combination of I , A1 , A2 , and u = 1 + 11+ 12 , with constant coefficients. Since the elements in any product of incidence matrices must be non-negative integers, the c=<>efficients are of this form and the proof is complete. The values of the coefficients are easily computed by (2.2) and (2.4) . 64 THERE 2.8. The existence of an association scheme with two associ- ate classes and parameters v, n1 , pg} satisfying the condition given in (a) or (b) below is equivalent to the existence of the BIB desiy with 7 treatments described in (a) or (b) respectively. ~i and J are equal to l and 2 in some order. s i " (a) Condition. p11 - p11 . ‘ BIB design: v 3 b; r = k a B1; A‘ Fifi = Pin); “10 incidence matrix N is symetric with 0's on the main diagonal. (b) Condition: pig-)2 = p1 . «)1 BIBdesign= v b; r k 3 n1+1; A: Ph'fN-‘Piih the incidence matrix I is symmetric with 1's on the main diagonal. PROOF: The proof will be carried out for the case i 3 1 and j = 2. The other case is similar. Case (a) The treatments in block 0 of the design will be taken as the first associates of treatment 0 in the association scheme. Then 1“ =51, sndby(1.18) m' ' a“; = 112 ' anvilwvil‘w DQi'ining r and A asin(a), 1111' = rI+ Mala-12) = rI-rAw—I) . Thus I! is a v x v incidence matrix with uniform row and column totals r and uniform row inner products A , identifying it as the incidence matrix or the BIB design described in (a). Conversely, lets N be the incidence 65 matrix of such a design. Defining A1 =- N , the conditions (a) and (b) of Lemma 2.2 are satisfied imediately and condition (c) follows from the expression for Ni" which holds for all BIB designs. M" = rI+A(U-I) = r1+AN+ 1(0-1-1!) . Case (b). The treatments in block 0 of the desigi are taken as treatment 0 and its first associates. Then I! 8 A1+ I and me = (11+1)(11+ I)‘ = (11+ n2 = A12+2A1+I ‘ (n1+1)I+(pL+2)A1+p§112 Defining r and A asin(b) , mm = r1+?\(11+a2) = rI+A(U-I). Thus N is the incidence matrix of the design described in (b). Con- “reely, let I be the incidence matrix of such a design and define A1 a a - 1, . Again conditions (a) and (b) or Lena 2.2 are satisfied and we have 112 = (s-1)2sw2-2w+1 sum- 2w+I-r1+l(u-1) -2s+1 - (n-1)1+(A-2)(n-1)+A(o-n) . Therefore by Lemma 2.2, A1 so defined leads to the required association Scheme. This completes the proof of Theorem 2.8. Parts (a) and (b) of Theorem 2.8 are not independent. If either of them applies to the matrix A1 of an association scheme the. other applies 66 to la . The two BIB design will be complementary, a given block of one containing exactly the treatments not occurring in the corresponding block or the other. The conditions ’11 a pfl and p22 + 2 2 pie are easily shown to be equivalent, either by direct application of (2.4) or by the device of applying one part of Theorem 2.8, taking the complement of the resulting BIB desig, then applying the converse of the other part of the theorem. The conditions of Theorem 2.8 are satisfied by 18 of the sets of para- Deters listed in Table II, of which 11 belong to known association schemes. There remain schemes #22, 39, 84, 85, 92, 100, 101. These lead to 5 dis- tinct BIB design, all of which have r >10 and have not been studied so far. Scheme #22 is equivalent to a design with v = b =- 36, r II k 8 15, A: 8, N sy-netric with 1's on the main diagonal. A desig with these Parameter values is constructible from the known scheme #28, but with an incidence matrix I having 0's on the main diagonal. These designs all 1‘all within the class of 'eymmetric" BIB desigs which have the property tlint v = b; symmetric BIB design have been investigated more thoroughly the any others. However, none of the known necessary conditions exclude any of the design in question. In particular, some deep conditions due to Shrikhende [597 are satisfied automatically whenever r - A is a perfect Square, which it is for all of these design.1 Therefore Theorem 2.8 does not furnish conclusive information about any unknown association schemes. . ‘ w— 1. _Weremarh without (proof that the value) of r - A is a perfect square 1‘ or all the BIB design specified by Theorem 2.8. This is a fairly direct I‘esult of the conditions of the theorem and the expressions given in Theorem 2.1. 6? Theorems 2.2 to 2.8 prove the impossibility of four association schemes of Table II and may provide the .basis for other such proofs. They do not represent an exhaustive list of theorems on the structure of association schemes, but they shes that such theorems may be proved rather easily, and illustrate some methods of proof. lost of them make use of algebraic properties of the expressions for parameter values of the schemes, or of properties of the association incidence matrices. It is not illustrated here but deserves to be mentioned that empirical attempts to construct an association scheme may lead quickly to a con- structed scheme or to a proof that the scheme is impossible. This method requires too much enmeration to be practicable for most schemes with more than 20 treatments, but there are exceptions. Some empirical proofs of inpossibility of design will be mentioned in Section 2.5 , and two association schemes are constructed in Section 5.5 by methods which are largely empirical. 68 2. a Inneratim 2; Possible Design £21; Particular Association Schemes. If a balanced or partially balanced desig is to be used in an ex- periment, the first parameters to be specified by the experimenter are likely to be v and k , which are determined by the number of treat- ments and the variability of the experimental material. From the design available for the particular v and k , he will try to choose one for which the number of replications r is large enough to provide the pre- cision desired but not too large to be economically feasible. This will determine the value of b and will leave little-or no choice in the Values of the other parameters. a somewhat different procedure is used for our purpose of enumerating Possible design. It has been convenient to classify design first by useciation scheme, so that the first parameters specified are v , n1 , Park , leaving the parameters b , r , k , A1 and A2- . Since these f in parameters must satisfy relations (2.1) and (2.5) , at most three of them may be chosen independently. The requirement that all be non-negative 1Integers is also a considerable restriction. The existence of any design 1|nplies the existence of an infinite class of other design obtained by uBing each block r1 times , r1 = 2 , 5 , ... The parameters v and It will be unchanged for design obtained in this way, while the para- Ietara r , b , A1 and A2 will be multiplied by r1 . Only a finite num- ber of these will be useful to experimenters, since there are practical limits to the amount of experimental material that can be used. Fisher and Yates 15 L7 enumerated only design for which r g 10 , and other 69 writers have followed their example. Extremely large block si see are likely to defeat the purpose of having homogeneous experimental conditions within blocks, and some limitation on k is also desirable. It is a property of balanced design that k _<_ r , , so -thatho- qualification was necessary for Fisher and Yates. k is somewhat larger-then r in some PBIB design, and Bose, Clatworthy and Shrikhande [6J‘enumerate only design for which k S 10 also. The same restrictions will be adopted here, admitting only a finite number of design for a given association scheme. A fairly efficient enumeration of the possible design for an asso- ciation scheme may- be begun by choosing a pair of values- for A1 and A2 , then computing the quantity n1 Al-r- n2 2.2-...By (2.5) ,, this is equal to r(k-l) and by factoring it in every possible way as the Product of two integers, possible pairs of values for r and k may be Obtained. By (2.1) , the fraction “7k is equal to b and must be integer valued. This, along with upper bounds on r and k , will eliminate some sets of values. Some additional restrictions depend on the characteristic roots of the matrix M" , which have been mentioned in Sections 1.5 and 2.2. In the notation of Connor and Clatmrthy [77], rk is a root with multiplicity l , 1 r - 22 is a root with multiplicity 0&2 , ~ r - z is a root with multiplicity 0(1 , Where the o<1 may be obtained from the parameters of the association Scheme and the 21 depend in addition on A1 and A2 . Since ml is the 70 product of a real matrix by its transpose, it is positive semi-definite, meaning that each of its roots must be non-negative. This gives the re sults r-zizo , or (2.45) r2z1, i=1, 2.‘ If both of the multiple roots are positive, the v x v matrix NN' is non-singular and has rank v , meaning that the v x b matrix N has rank at least v , which is impossible if b < v . Therefore in this case b 2 v . This is identical with Fisrer's inequality for balanced de sign and is equivalent to the following statement. (2.“) If r>zi, 1:1 and 2, than rgk. If one of the multiple roots r—si is equal to O , the rank of NN' is v-O(1 , meaning that R has rank at least v-o<1 , which is im- possible if b < v--O(:l . This leads to the following statement. (2.45) If r=s, i=1 or 2, then b2v-O( . 1 1 The situation that both 21 and 22 are equal to r does not arise, alace it can be shown that any desig for which zl = 22? will be a balanced desig. Attention will now be restricted to association schemes of the L8 and 1.; series. The following expressions for parameter values, which “39 the notation of (2.12), apply to both series. For the 1.8 schemes, ‘ e f and n are all positive integers; for the L; , they are all “gative integers. 71 n g+f-1, (2.46) :11 = 201-1) . n2 3 f(n - 1) e For the L3 schemes, in the same notation, 8(3'1) s “2 . f(fl-l) p p II (2.47) z .3 (l-f)Al+f32 , $2 = gAl'i’u- g)A2 . The expressions of (2.46) apply to schemes of both series, but for reasons Which will be stated in Section 5.1, the desigation of the multiple roots is reversed for L; schemes, giving the following expressions instead of (2.47) . d1 = f(n " 1) , 0‘2 : 8(n ‘ 1) e (2.48) 21 = gxl+(l-3)A2, 22 8 (l-f)A1+fA2e It will be noticed that for either series, the multiplicities 0&1 or the characteristic roots of him are equal in some order to the nunborn m1 of treatments in the associate classes. This relation holds. . Onlq for certain classes of association schemes and will be discussed in SQCtion 5.1. 72 The work of enumerating desig parameters is shortened by some pre- liminary restrictions placed on k 1 and A 2 , which will be described first for the L8 case. Taking r $10 and k_<_10 implies r(k—l) _<_90. Using (2.45) and (2.47), (l .- f)A1+f>\2 $10 , 8A1+(1’8))\2 $.10 9 and from (2.5), nlAl-i- n2>\2 $90 . Solution for A 1 leads to the following inequalities, which define the quantities m , I and I' for L8 schemes. f 10 : (“9’ A12 7372 ' Tr: “‘ ' 10 - c- (2050) A1 h -g- +374), " I , <2-51) Alsw = I' - “1 In the L; case, g and f are negative, leading to the following inequalities and different definitions for m and I . Inequality (2.51) and the definition of 10 hold without change. (2.52) Algl§+aidgz = m, f - (2.5s) Alsf-1)2 -39... - a. For a particular set of association scheme parameters 8 s f 9 “l and 75 n2 , the lower bound m and the upper bounds I and I' for A l are quickly listed for each non-negative value of A 2 . The enumeration which has been outlined in this section is carried out in Table III of the appendix. The section of the table for each association scheme is preceded by a list of the values of g , f , 211 and :12 and the expressions for m , I , I' , ”l , 22 , and r(k-l). In the table, values of A 2 are listed, followed by the value of m if it. is positive and the value of the smaller of I and' I' . The possible Values of A 1 are then listed. The value A l = A2 is omitted, since ' it leads only to balanced design. Also if n1 = n2 , values A 1) A2 are omitted since they lead to design which can be obtained from design with A l< )(2 by interchanging the designation of first and second associates. For each pair Al , A2 , the quantities zl , 22 and P(k—l) are entered in 'the next columns of the table for use in computing values of r and k . Only values r 5 lo and 1: $10 consistent with (2.45) and (2.44) are listed. The value of b is then computed and entered in. the table if it is integral. Finally, in case r = 21 , is applied, eliminating a few more sets of parameters. Table III is in- (2.45) tended as an illustration of the computations and is presented only for a representative sample of the association schemes. Table IV is a list of those parameter values which satisfy all the conditions applied in Table III. The designs for each association scheme are listed together, preceded by a list of the scheme parameters for reference. Design parameters are identified by the numbers given to the 74 scheme in Table II, and by a serial numbering of the designs for each scheme. All known schemes of the Lg and Lg* series are included, and values of v , r , k , b ’Al ,A2 , zl and z are listed. Designs 2 which are known to have been constructed or have been proved impossible are marked by the letter C or X respectively, followed by an explana- tory remark or reference. Several methods which are frequently of use in constructing designs will now be listed in the form of theorems. These are presented here for easy reference and no claim is made that they are new, although the author is not aware of any publications which include theorems 2.12 to 2.14. THEOREM 2.9. A PBIB design with k = 2 treatments per block may be formed from any association scheme by taking as the blocks all pairs of 1th associates. The parameter values will be v , r = n k = 2 , i 1 b=ivni, A1=l, 31:0. i and 3 represent 1 and 2 in Ekmme order. PROOF: Since each pair of 1th associates occurs together in a 1flock exactly once and since no treatments which are not 1th associates Occur together in any block, the design satisfied the requirements Specified. THEOREM 2.10. In a Latin square type asSociation scheme with v = n2 treatments and g constraints, a PBIB design with parameter values . v=n , r=g, k=n, b=ng, Al=l, A230 may be formed by taking as blocks the sets of n treatments occurring in 75 the rows of the g orthogonal squares. If there exists a set of f '-'-' n — g + 1 additional squares which may be adjoined to form a complete orthogonal set, a PBIB design with parameters vznz, r=f, k=n, b=nf, Al‘zo, A2=l may be formed by taking as blocks the sets of n treatments occurring in the rows of the 1‘ additional orthogonal squares. PROOF: These are m lattice designs, whose properties are well known. They are discussed, for example, in Chapter 10 of [71-]. By the orthogonality property of the n x n squares, no pair of treatments occurs together more than once in a row of any of the squares. By definition Of the I.g association scheme, the treatments occurring together are precisely those treatments which are first associates in the case of the first design described, or second associates in the second. All desigs of either of these types will be identified in Table IV of the appendix by the word "Lattice". THmREI 2.11. Let two PBIB desigs based on the same association a(theme have the same number k of treatments per block, so that their Parameter values may be represented by Vs”: ks 13*, Al‘.’ A2* v, r“, k, b“, A1“, A2“ I‘etspectively. Then a desig with parameter values 76 v , r = r*+r**, k , b = b*+b**, Al =A13-XI‘,A2 3A3)? may be formed by taking each block of the two original designs as a block of the new desig. PROOF: It is obvious that the set of blocks obtained in this way leads to the values specified for b and r and that the total number of occurrences within blocks of a given pair of treatments is equal to the sum of the numbers of occurrences in the two original designs. Since the two designs have the same association scheme, the number of occurrences of a pair of treatments is liti- X i“ when they are 1th associates, 1 = l or 2 . It is an immediate extension of the theorem that three or more com- Ponent desigs with the same association scheme and the same value of k new be combined in the same way. The designs need not all be distinct. In Table IV, a desig which may be formed in this say from other designs for the same association scheme will be identified by the letter R , fonoved by the serial numbers of the other designs. THmRDl 2.12. Given any association scheme with 2 associate classes, a PBIB desig with the parameter values V'be r'k3nia Al-‘Piio A2zpii’ “here i and j are equal to l and 2 in some order, may be formed by taking block 0 as the set of 1th associates of treatment 0 . PROOF: If the desig is formed in this way, its incidence matrix w35-11 be identical with the association matrix A1 of 1th associates, 77 giving the result : : 2 :. NN' Alai' A n11 + p1 This shows that each treatment occurs in n1 blocks and each pair of treatments occurs together in p'h or pii blocks, according as the two treatments are first or second associates. Each block contains ni treatments, and all the requirements for a PBIB design are therefore satisfied. In Table IV, a design which may be formed by applying this theorem is identified by the statement N=Ai,(i =lor 2). THEDBEI 2.15. Given any association scheme with 2 associate classes, a PBIB design with the parameter values -.- = = :1 v b, r k ni+1, Al pii+2' A2=pii, Where i and J are equal to l and 2 in some order, may be formed by taking block 9 as the set of treatments consisting of treatment 9 and its 1th associates. PROOF: If the design is formed in this say, its incidence matrix N “111 have the form A1+ I , giving the result 0 = + o : 2 : 2 NN (A1 1)(Ai+ I) (5.1+ I) 81 + 2A1 ‘1' I g 1 (ni+ 1): + (p11 + 2):;i + piiaj . 78 This shows that each treatment occurs in n1 4- 1 blocks and each pair of treatments occurs together in p11 two treatments are first or second associates. Each block contains 4- 2 or p11 blocks, according as the 111+ 1 treatments, and all the requirements for a PBIB design are there- fore satisfied. In Table IV, a design which may be formed by applying this theorem is identified by the statement a = 31-4-1 ,(izl or 2). THERE] 2.14. In a Latin square type association scheme with v n n2 treatments and g constraints if a balanced incomplete block design with parametervalues ”‘39 1": k*s bi.» A*s is constructed on each of the sets of n treatments in the rows of the g Orthogonal squares, the .result is a PBIB design with parameter values v=nv*=n2,r=gr*,k=k*,b=g1b*, Al=A*,A2=O. If there exists a set of f =- n - g+l additional squares which may be fidjoined to form a complete orthogonal set, and the same BIBD is constructed 0!! each of the nf rows, the result is a PBIB design with parameter values v=n2,r=fr',k=k*,b=fnb*,A1=O,A2=A* . PROOF: The proof will be stated for the first case. The necessary Changes in wording for the second case are inserted in parentheses. Since 15" blocks are constructed from the treatments of each of the n rows of each of the g (or f ) squares, the total number of blocks will be 9113* ( or an." ) and each block will contain 1: treatments. Since each treatment occurs in just one row of each square, it will occur r* times in the desig) formed from each of the g (or f ) squares, leading to the stated value for r . By definition of the association scheme, each pair of first (or second) associates occurs together in just one row of one of the g (or f ) squares, so that thenumber of occurrences within blocks of the PBIB desig; of the pair of treatments is equal to the number A“ of blocks of the BIB design in which two treatments occur together. Two treatments which are second (or first) associates do not occur together in any rows of the squares used and will not occur together in the PBIB design, which means that A 2 (or A1 ) is equal to 0. THEOREM 2.15. In a Latin square type association scheme with v- = n2 treatments and g constraints, if the rows of each of the g orthogonal Squares are identified with the treatments of a BIB design with the para- the ter values 1Eben a PBIB design may be constructed with b* blocks formed from each of the n x n squares by replacing the treatments in each block of the balanced desig: by the sets of n treatments in the corresponding rows 01‘ the square. The parameter values of the partially balanwd desig: ‘111 be v=nv*=n2,r=gr*,k=nk*,b=gb*, A1 8 r*+(s-1)A* . A2 = 23* - If there exists a set of f = n—g+l additional squares thich may be adjoined to form a complete orthogonal set, and the rose of the f squares are used in the same way with the same BIB design, a PBIB design is ob- tained with the parameter values v=n2, r=fr*, k=nk*, b=fb*, PFDOF: The proof will be stated for the first case. The necessary changes in wording for the second case are inserted in parentheses. Since b" blocks are formed from each of the n x n squares, the total number of blocks is gb" (or rb‘i) . Since each treatment of a block of the balanced design is replaced by n treatments of the partially balanced design, the block size is nk* . Each row of an n x n square occurs in r" blocks and each pair of rows occurs together in A" blocks. Since each treatment occurs in Just one row of an n x n square, it occurs in r* of the blocks formed from each square, for a total of gr* (or fr*) occurrences. If two treatments are in the same row of a square, they will occur together in r* of the blocks formed from that square; if they occur in different rows of a square, they will occur together in A * of the blocks formed fmm that square. First associates occur in the same row of one square and in different rows of the remaining g-l squares, while second associates occur in different rows of all g squares. (In the case of f squares, first associates occur in different rows of all 81 f squares, while second associates occur in the same row of just one square.) The total numbers of occurrences of pairs of treatments are therefore equal to the values given for A 1 and A 2 . The method of construction outlined in Theorem 2.15 is a rather direct extension of a construction given by Bose and Connor [7] for group divisible designs, and of a generalization by Zelen 5397 . There are other general methods of generating PBIB designs, but the ones Just given furnish constructions for most of the known designs of Table IV, which is sufficient for the purpose of this section. Of the remaining hioun designs, some are tabulated by Bose, Clatuorthy and Shrikhande and are identified in Table Iv by a reference to [6] . Others that have been constructed by miscellaneous methods are listed in Section LS of the Appendix. It is known for em incomplete block design and is probably true for many of those listed in Table IV that two or more solutions exist Which are distinct under permutation of treatments or hlocks. This is certainly the case for those designs which can be constructed from either or two inequivalent association schemes. The question of uniqueness of desigis based on the same association scheme will not be- taken up in this dissertation. Broofs of impossibility of designs, which are given for several parti- cular designs in Section A.3 of the Appendix, may involve the question of Uniqueness of association schemes. Design {7.5 furnishes a useful example. The design is in the L2 series with v = 16 , and when the association 82 scheme is based on a pair of orthogonal 4 x 4 squares, is easily shown to be impossible. However, the design can be constructed by using a different association scheme with the same parameter values, which will be used as an example in Section 4.1. This shows that different associ- ation schemes with the same parameter values may have different properties and that any proof of impossibility of a PBIB. design must cover all asso- ciation schemes with the appropriate parameter values. It will be shown in Section 4.2 that for L2 denials with n f 4- , a the association scheme defined by n x n squares is unique, so that the. squares may be assumed in any discussion of these desips. This is a necessary step in the proof of impossibility-of designs such as #20-2, #50-2, ande#95~l. 0n the other hand, desig) #12-2 in the L series may be shown impossible with an 5 association scheme based on three 5 x 5 squares, but another example in Section 4.1 will show that the scheme is not unique and the existence of the desig remains in‘doubt. A singular incomplete block design is one for which the matrix NW is singular, and for PBIB designs with two associate classes, this means a design for which one of the values :1 and $2 is equal to, r . It 13 easy to verify that Lattice designs, designs constructed by the method of Theorem 2.15, and designs formed by replicating a design of either of these types, are singular. These desigis all have the property that the blocks may be partitioned into subsets of n treatments which are the 8ets occurring in the rows of the orthogonal squares. It is conjectured by the author that every singular design based on an association scheme of the Lg series has this property and may be formed in one of the ways 83 described. This mould be an extension of results proved by Bose and Connor [7] on the structure of singular group divisible design. If this conjecture were proved, a necessary condition for the existence of a Latin square type design, with the parameter values stated in Theorem 2.15 would be the existence of the BIB design described in the theorem. This would prove the impossibility of designs #7-20 and (12-8 of Table IV, Since ttm BIB denials involved would have fractional values for some of the parameters r , k and b and are obviously impossible. Table II gives parameter values of 20 schemes in the Latin square series, of which 18 are known and are listed in Table IV. 167 sets of design parameters are listed for these schemes, designs are constructed or indicated for 125, and three are proved impossible by enumeration methods. There remain 59 unknown designs. Table II gives parameter values of 10 schemes in the negative Latin Square series, aside from schemes which are also in‘the 1.8 series. Five of these schemes will be constructed in Chapter III and are included in Table IV. 22 sets of desig parameters are listed, and designs are con- Structed for nine. The remaining 15 desips are unknown. The constructed schemes and designs of the L; " series are believed to be new. In all, Table IV gives parameter values of 189 designs, of which 154 are constructed, three are sham to be impossible, and 52 are unknom. III. NEGATIVE LATIN SWARE TYPE ASSOCIATION SCHEMES 5 .1 Relation s between Latin guare and negative Latin square association schemes. It was pointed out in Section 2.1 that formulas (2.12), developed 1‘ or Latin square type (Lg) association schemes, give parameter values of a. possible new series of association schemes when the arguments n , g , 1‘ were given negative integral values. This new series of "negative Latin square' type (Lg) scheme will be the principal topic of this Chapter. Five of the schemes will be constructed in Sections 8.2 and 5.8, and have already been included in the tables discussed in Chapter II. In the present section it will be shown that the family resemblance in the parameter values is not the only thing the new series has in common with 1'8 schemes. A property related to the characteristic roots of NH' , Where It is the incidence matrix of a design, is shown to be shared by both series of association schemes and to come close to characterizing them, holding for only one other class of schemes. Formulas (2.15) and (2.14), due to Connor and Clatworthy [:2], for the multiplicities ok 1 of the characteristic roots of HIV are easily used to find general expressions for» the 0&1 for any family of designs for which general expressions for the other parameters are available. lhen the formulas are applied to Latin square designs it is found that the ex— pressions for 0k 1 and at 2, are identical with those for the parameters 85 Di and n2 , given in (2.9). This is not true for group divisible or triangular designs except in special cases, showing that it does not hold in general. an the other hand, reference to Table II of the Appendix shows that about half of the non-group-divisible schemes with v 5 100 , including all L and L; schemes, have one of the two following 8 properties. 3 0< = 0 Property A 1 n1 Property 8: (X1 = 112 . Since dl+ok2=n1+n2= v-l , property A or 8 implies that 0K2 is equal to n or m respectively. In this section we determine the 2 1 Class of designs which have either property A or property 8 . First it will be shown that the two properties are practically iden- tical, and that basically the difference between them is one of notation. ml and n2 second associates respectively of a treatment in the design. The two denote the numbers of other treatments which are first and Classes of associates play dual roles in many respects and nothing more than a choice of notation is involved in designating one class as the first. Once the choice is made for a particular desig, the values of :11 , n2 , A1 , A2 , and the pig-k are uniquely determined. The designation of OK 1 and 0&2 , however, depends in addition on the designation of the two characteristic roots r - 21 and r - “2 of RN; . These are obtained as the two roots of a quadratic equation whose coefficients are functions of r , X1 , A 2 , p12 and pie . Solution of the equation leads to 86 (5.1) z =- Q1 +123} (‘Ag jinx/i WK) 2 where z: 2 .. 1 = 2 - 1 2 2 Y p12 p11>. and A (912 912) + “912+ 912) + 1 ° This result for the two 2.1 is given in [7 Z7 . The orpreseions for zl and z differ only in the sign of the terms involving A , which is a 2 symmetric mnction of ph and pi? and is thus independent of the desig- nation of associate classes. Connor and Clatworthy denote by 21 the root obtained by taking the + sign, giving the expressions (5.2) . Mlu-v-vzwilzch-WE). (3.5) 22 — Min -Y+\fZ)+ flan-FY 475) . This anounts to desigiating the ith characteristic root r - '21 as the one in which the coefficient of A 1 is positive. It needs to be empha- sized that this convention is arbitrary and does not identify 21 with the it'h associate class. An expression which involves a positive multiple of X1 and a negative multiple of A2 is not thereby more closely related to one than to the other. While it is convenient to be able to refer to r - 21 and r - 22 without ambiguity, this does not reveal any intrinsic connection between the designation' of these two char- ficteristic roots and the designation of the two associate classes, and none Should be inferred. One choice for the designation of z1 and 22 seems to be as good as another, and it is sensible to stick to the choice already 'made by Connor and Clatworthy. The values of 0( l and (X 2 are then 87 uniquely determined. This is the notation used in Tables I to IV of the Appendix. If the other choice of notation were made for any scheme of Table II which has property B , it would have property A instead. It is now possible to clear up a discrepancy in the notation which has been used in this dissertation for" association schemes of the negative Latin square 0.8") series. It was stated at the beginning of this section that schemes of the ordinary Latin square (Lg) series have property A . It is stated in section 2.1 that the expressions for the parameter values of the 1.3 and L; schemes are identical, which would imply that the L; schemes also have property A . However, the schemes of this series listed in Tables II, III, and IV have property B . The parameter values of the L * schemes are given by the eXpressions (2.12) 8 used for the L‘ schemes provided the parameters 11 , g , f of those 8 expressions are taken as negative integers. For both classes, A = n2 and m— = n ; taking n as a negative integer means using the negative square root of A in the expressions for 2.1 and 22 . If this is done, these schemes have preperty A . But this is the opposite of the sign convention agreed on in the previous paragraph and used in the tables, explaining wry they appear there with property B . This concludes the discussion of the nature of properties A and B and we now return to the problem of finding the class of designs which have property A or property 8 . For group divisible designs the values of n and 041 , which are 1 given, for example, in [l- Z7 , are as follows. 88 n1 n-l , n2 n(m-1) , (X1 m-l ,0(2 m(n-l) , where m and n are positive integers. It is easily verified that these designs have property B only if m = l or n = l , in which cases the design reduces to a balanced design. They have preperty A only if m = n . A group divisible design with m = n is the simplest case L l of a Latin square type design. All partially balanced designs with two associate classes and not of group divisible type have association schemes whose parameter values may be determined by the conditions of one of Theorems 2.0 and 2.1. For all schemes of the class defined by Theorem 2.0, ml = n:2 =0(l =- 0‘2 , so that both of properties A and B hold. These schemes are defined only for v of the form v 8 4t 4- l 5 a scheme of the class may be constructed for each such value of v which is a prime or prime power, for example by the method to be described in Section 5.2. No schemes of this class are known at present for other values of v . We now turn to the schemes specified by Theorem 2.1, in which ex- pressions for the parameter values are given tenns of positive integers a and t . Some of these expressions are now repeated for reference. (2.21) pi- : s(t + 1) , 2 (2.22) piz = (s + l)t , (2.2a) 0(1 -.— (s + ”“2 + 3'11 ° W D 89 using (2.25) , O2. nlz‘8*(‘*+1)=8*(n*’l) 9 pig = -f*(-s*+ 1) =f"(a* -1) . These and the other expressions in n* , g* , f* are of the form of (2.12), identifying the present series of schemes as the negative Latin souare series. Therefore every scheme specified by Theorem 2.1 which has property B must be in the Lg" series. In some work with Li schemes it is convenient to have expressions for the parameters as functions of positive integers. The letters 11 , g , f will still be used, but with the following relation to the para- meters of Theorem 2.1. t, f=s, (5.16) s+t+l. I3 H In this notation the expressions for the parameter values are the following. n = g+ f + l , 2 . (a +1) - f+ 2 f(g+1) V =3 n2 P a ' l (5.17) f(g + 1) f(f + 1) n1: g(n + l) , h2 = f(n+ 1) , [a(a+ 1) g(f+ 1) 92 = 2 a(f+1) (£+1) -g +2 The classes of association schemes which have been characterized by properties A and B are not disjoint. When n is odd, say n = 2a+ l , the design with parameter values 95 v=n2=4a2+4a+l, a2+a-l a2+a a2+a a2+a n sn=2a2+2a 2 ’ (5.18) 1 a2+a a2+a p :- a2+a a2+ a-l is in the class defined by Theorem 2.0, with t = nz-r a . It is also an Lg scheme with gs f = a+l , and an Lgfl scheme with g= f = a . There are no other duplications. There are clearly no other Lg or Lg* schemes which have the property 111 = oz of Theorem 2.0 , and for a scheme with v = n:2 treannents to be simultaneously an Lg and an L? scheme, it is necessary that n be simultaneously a multiple of n-1 and 1 n-+ 1 . The only possible value less than n2 -1 is é(n2-l) , with n odd. The results that have been proved in this section will now be stated as a theorem. THEDREII 5.1. Let N be the incidence matrix of a partially balanced incomplete block design with two associate classes. (In order for the multiplicities O( l and OK 2 of the multiple characteristic roots of RM to be equal in some order to the numbers ml and n2 of treatments in the associate classes, it is necessary and sufficient that the desig: be in one of the following classes. (1) The class specified by Theorem 2.0 ; (ii) The L3 series, Latin square type designs with g constraints, 3 21 , or other schemes with the same parameter values:' 94 (iii) The LgIt series, negative hatin square type designs, intro- duced in Section 2.1, with parameter values given by (5.17) . For v an odd square there is one possible association scheme with n1 2 n2 which falls in all three of these classes; otherwise they have no schemes in common. The specification of Lg’ schemes in terms of the negative integers of , g' , f” , is not very helpful in suggesting possible ways of con- structing the schemes. The parameters g and n in the Lg series are related to a set of g orthogonal n x n squares, and there seems to be no analog to this for negative integers g* and n* . Expressions (3.17) in terms of positive arguments are a little more promising, at least in any case in which a complete set of orthogonal squares exists. This is more easily described in terms of the finite Euclidean plane geometry which may be constructed from such a set of squares. ~This geometry has n2 points, any two of which determine a line; there are n points on each line and n +~l lines on each point. For an association scheme the v = n2 treatments are identified with the points of the geometry. If the scheme is of Lg type, each treatment has n = g(n-l) first asso- l ciates, which for a given point may be taken as the n-l remaining points on each of g suitably chosen lines through the points This is discussed in further detail in the following section. If the scheme is of Lg, type, the value g(n +-l) for the number n1 of first associates suggests that the first associates of a particular point might be g suitably chosen points on each of the n e-l lines through the given point. It appears that it would be a difficult combinatorial problem to select these points 95 in a way that would satisfy all the requirements of partial balance, although two schemes constructed by another method in the following section have precisely this geometrical interpretation.1 It should be remarked that not all .Lg schemes are associated with finite geometries. They may be constructed from sets of g orthogonal squares which cannot be extended to a complete set of n +-l , and there are examples of association schemes which have the parameter values of; the Lg series but which correspond to no set of g orthogonal squares. Some examples of this kind will be given in Section 4.1, while an example appeared in Section 2.1 of a 4 x 4 Latin square not belonging to a come plete orthogonal set, which is equivalent to a set of g = 8 orthogonal squares which cannot be extended to a set of Vn+-l . Such squares are known for many values of n and presumably exist for all values of n >~3 . By analogy with this, there is no reason to expect all schemes of the Lg* series to be related to complete sets of orthogonal squares or to finite geometries. On the other hand, there is at least the possibility of such a relation for each of the five schemes of the series which are known at present. The four which are constructed in the next section are all based on finite fields of order n2 , and in every case where such a field exists, the geometry and set of squares also exists. The one constructed in Section 3.3 is for v 8 100 treatments, and while no field of this order exists, it has never been proved that the geometry and orthogonal squares do not exist. 1. These are schemes #6 and'Sl, for 16 and 64 tregpments respectively. 96 5.2 Construction 22 Negative Lgtin Square 22 2 Association Schemes “flaw-.4- A.method is developed in this section for the construction of an infinite class of association schemes with two or more associate classes. The method is applied.to the construction of four schemes of the negative Latin square, or Lg* , series. The method is applicable when the number of treatments is equal to a power of a prime, v = pq , so that there exists a finite field with v elements, denoted by the standard notation GF(pq) . The treatments will be identified with the field elements or marks in any conrenieat order. It is well known that the multiplicative group of non—zero marks of the field is cyclic; denote a.generator of this group by z . The marks of the field may be represented by c , 1 , z , 32 , , qu‘Z , where 399“} = l . Each non-zero mark. x may be represented uniquely in the form x = 2k , 0 _<_, k _<_, pq‘z . The integer I: so defined is usually called the index of x relative to the base 2 and will be denoted by the symbol ind x , but the term 'exponent' will be used in discussion, in order to reserve the term "index" for a different use. Express the order of the multiplicative group as the product of two integers c and d , pq - l = cd , and define a field mark e by e - 2° . Then e is the generator of a subgroup of order d , with elements 97 e0 = l , e , e2 , ...., ed‘l . Since the group is cyclic, the subgroup of order d is unique. This sub- group and its coasts provide a partition of the non-zero field marks into c sets, each containing d marks. The 1th set contains the marks 31 , ezj , 0223 , ... , ed'lsj , and 3 may have the values 0 , l , ... , c-l . It will be necessary to impose the condition that a coset contain with each element its addi- tive inverse. It is easily seen that an equivalent condition is that the subgroup have this property, and that this reduces to the requirement that this (multiplicative) subgroup contain the additive inverse of the element 1 , denoted by -l . If the prime p is equal to 2 , l is self- inverse (as is every other mark) and the condition is satisfied for every subgroup. If p is.odd, l and -l are the two solutions of the equation xz-l = O . The corresponding exponents are the solutions of 21ndx§0modp3-1. The solution corresponding to -l is ind(-l) =12§=L-= c 7'2" "’1 = Zed/2 e n. , leaning A.necessary and sufficient condition that this be in the subgroup is that it is an integral power of the generator e 3 2° , or equivalently, that d is an even integer. Accordingly it will be required that the chosen subgroup be of even order if the order of the field is odd. 98 We now define an association relation by saying that a mark 9 is 2 a 3th associate of 01 11‘ their difference 02 - 01 is a mark of the 1th coeet. In this case, 91 - 92 is in the same coset by the condition just imposed, so that the association relation is symmetric. In order to shoe that it satisfies the definition of the association relation in a PBIB design it remains to show that condition iii(c) of the definition is satisfied. Let G and 0» be ith associates, so that for some to , l 2 0 ~91 = etoz1 . The J?“ associates of 01 are of the form, 01'+-et1z3 , 2 8 0 , l , ... , dpl . The kfih associates of 02 are of the form = O , l , ... , 691 . It is necessary to show that t1 Oz'r'etzsk , t2 the number p of marks in the intersection of these two sets is inde— i it pendent of the particular pair of 1th associates chosen as 91 and 06 . This number is equal to the number of pairs t1 , t2 for which t1 : tZK 01+e 33 02+e z . This reduces to 92 ~ 01 ’- etlzk - otzzj , (5.19) ems1 3 stick - .t2,j , 21 ’ etl‘t°zj - etibtozk . Now as t1 (or t2) , rwns over all the values 0 , l , ... , d-l modulo d , tl-to (or t2- tb) runs over the same values, so the number of solutions pix is the same as the number of solutions of the equation 99 This number pit is independent of the pair of 1th associates chosen as 91 and 02 . loreover, et'2 and -et2 run over the same set of values, so we may replace the previous equation by (3.20) z1 8 et'1 zj .+ at2 zk . 1:1. 11:ka This is the last condition necessary for the association relation for a Since this equation is symmetric in j and k , we have p PBIB desig: with c associate classes. The classes are of equal size, 313d,1=0,1,oao, 3-1. The standard relations (1.?) hold and reduce in this case to (5.21) pik3pik=p§1, 1, j, kzo, 1,..., c-l. lultiplying (3.20) .by z gives the following equivalent equation, which must have the same number of solutions t1 , t2 , 21*13et1zj*1 etzzk+1. This proves the relations 1 - 1 + l - 5.22 " 1 k "’ 0 1 see C"1 e ( ) pjk pj + 1’k + 1 , , J , , ’ , In apphing (5.22), the indices may be reduced modulo c if necessary. The method used for the construction of these schemes has led to a notation in which the c associate classes are numbered from ‘0 to c-l , ratherthsnin‘thsusualwayfrom l to c . The matrices P willbe i 100 numbered in the same way, and in particular their rows and column will be indexed from O to c-l . A change of notation would be easy enough, but would necessitate another definition and more symbols; instead, the reader is asked to bear with minor inconveniences such as referring to the 0th row of a matrix. These association schemes will be used only in the present section, and only for the purpose of constructing schemes with two associate classes, for which the usual notation will be resumed. The addition table of the field will serve as a convenient form for the association scheme if the first row is arranged with 0 as the leading mark and the marks of each coast in adjacent positions. The associates of any mark 0 are read from the row of the table containing 0 in the first column. The 1th associates are the marks appearing in the columns corres- ponding to the 1th coast. If 0 and fl are any two it'h associates, than pk is by defi- nition equal to the number of treatments which are 3“ associates of 0 and kth associates of ¢ . In determining the value of pg-k there is no loss of generality in taking {1 = 0 and 0 any mark of the. ith coast; the kt". associates of ii are then the marks of the kth coset. p?k is than equal to the number of marks of the kth coast in the set obtained by adding 0 to each of the marks in the a“ west. with a fixed value of O , using only one row of the addition table, and assigning all of thevalues 0, l,‘..., c-l to j and k, alltheelementsofthe ... matrix P1 may be determined for a particular 1 . The elements of P0 may be determined by using only the row of the table corresponding to 101 9 8 so 8 l . When the values pgk are obtained, the remaining pg-k values are easily obtained from (5.22) without further use of the addition table. Equations (3.21) and the symmetry relation p5,} = pi, may be used to shorten the work and check the values. J The derivation of this association scheme has made use of a parti- cular primitive mark a and the subgroup of order d generated by the mark e 3 3°. . Since this subgroup is unique, any other primitive mark y will lead to the same subgroup and hence to the same a coasts. The use of y leads to a notation in which the 1th coset is the one con- taining .,’J , rather than 23 . The numbering of the cosets, other than the 0th , which is the subgroup itself, may thus be different. This means that the c classes of associates may be numbered differently for different choices of the primitive mark, but are otherwise identical. That is, the association scheme is unique except for numbering of the associate classes. The results that have been obtained in this section will now be stated as a theorem. . THEOREM 5.2: For any number v of treatments of the form v 3 pg for p a prime and q a positive integer, identify the treatments with the marks of the finite field of order pq . Let a divisor d of p‘1 - l be chosen subject to the requirement that d be‘even if p is odd, and define c = g1? . Let a subgroup of order d of the multiplicative group of the field, and the coasts of the subgroup, be used to partition the p‘l - 1 group marks into c disjoint sets, each containing d marks. 102 For any primitive field mark 2 refer to the coast containing :1 as the 1th coset, i = o , 1 , , c-l . Let the set obtained by adding the field mark 0 to each of the marks of the 1th. coset be de- fined aspthe set of 1th associates of O . This defines an association scheme with c associate classes, with n = d , i =0 , l , ... , c-l , i and satisfying all the conditions of partial balance. The value pix is equal to the number of marks of the kth coast in the set obtained by adding a fixed mark of the it“ coast to each of the (1 marks of the ' 3th coset. The following special relations hold for the pit , i , j , 1:30, 1,..., c-l. 1 z (5021) Pix 3 9:1 9 5.22 1 = i + 1 . ( ) 9.11: p: + 1, k + 1 For particular values of p(1 and d , the scheme is unique except for numbering of the associate classes . An example will now be given to illustrate the procedure Just des- cribed. An.association scheme which will be useful later is based on the fieldoforder v=p<1=24=1e , with d=5 , c=5 . The field marks will be represented by the integers O , l , ... , 15 . Rules for forming sums and products in this field are easily stated but it will suffice here to give the addition and multiplication tables. ms mmumenu 9878545210 R u at. 0 1. n8 9:7452501 9 m n 8n054761052 890 45670 23 in i 9 14 15 12 8‘” M non R52107654 8 11 10 15 12 15 14 8 9 9 10‘11 12 15 14 eggn u u‘u 410525476 H R H 12 15 14 1 u H u n .m 15 12 15 1512 9 m n 8 u m B 9 9 8 4 4 5 6 7 O 1 2 5 12 15 14 8 u M 8 9 15 12 11 13 225016745mn89 R urn 11052547698n 00125‘5678910. 12 12 15 14 15 14 14 15 12 15 1O 11 15 15 14 u u n n m #0125456789m ”MUHMHWTMU 4.4135 2605‘ 798 1 u 11 u n o .5 55526 5.52 n 11 1 11 1793 88 ‘1155u28054279 l 11 l 9 1 l 7798n4155526054m 44279814155526 5 ll 1 01 1 1 1 4R798nu155fl26 .0 1 05‘2798 4155 ll 1 n1 6 NU‘n‘nlganulH 95u41555260u4u7v 42798nu1555260. T55R26mu4n798nu ThmflfiflkfihnhfleuinawumfumMMdmtbpdmun In the notation of this section, element 5 as a generator. 104 2:5, inc-55:15, and the subgroup of order d 8 5 consists of the marks 1 , 15 , 10 , 12 , 8 . Note: Since the order of the field is even, it is not necessary that d be even. The subgroup and its cosets are displayed in the horizon- tal rows of the following array, which is formed by filling in one column at a time with the entries of the first row of the multiplication table in owl's Subgroup (o-coset) 1 15 10 12 8 l-scoset . 5. 2 ' 15 7 11 z-coset 5 6 4 9 14 We can now define an association scheme with three associate classes by saying that two field marks are 1‘11 associates, i = 0 ,‘ 1 , 2 , if their difference is a mark of the i-coset. The following is the addition table rearranged to serve as a table of the association scheme. O-coset l-eoset 2-coset 0 11510129 5215711 564914 1 0141115 9 2 512 610 4 7 5 615 15 14 o 5 5 7 12 15 2 s 4 10 9 11 6 1 1o 11 5 0 6 2 9 e 7 15 1 15 12 14 5 4 12 15 5 e 0 4 15 14 1 11 7 9 10 s 5 2 9 9 7 2 4 0 11 10 5 15 5 15 14 12 1 6 5 2 12 9 15 11 0 1 14 4 e 6 5 7 1o 15 2 5 15 s 14 10 1 0 15 5 9 7 4 6 11 12 15 12 2 7 1 5 14 15 0 10 6 e 11 9 4 5 7 6 e 15 11 15 W4 5 10 0 12 2 1 5 14 11 10 4 1 7 5 s 9 6 12 o 14 15 15 2 5 5 41015915 676214 0511211 6 79121014 5411115. 50215 9, 4 51114912 76951511201510 ‘9 e 6 5 5 1 10 1 4 14 2 12 15 15 o 7 14 15- 1 ., 4 2 6 15 12 5 9 5 11 e 10 7 o 105 for use in obtaining the values pgk the onset designation of each of the 1th associates of treatment 1 will be noted, first for 1 = o . 0th associatesof 1: 0 14 11 15 9, coset designation: - 2 1 l 2 . The frequencies of marks of the 0th , first and second coasts are 0 , 2 and 2 respectively, giving the values p30 - 0 , his remaining pg: are obtained similarly. 0; 0 p01 2’ I’02'2’ 1'tassociatesof 1: 2 5 12 6 10 , coset domination: 1 l O 2 O , 0 shine 910'29 921'29 92231- 2ndassociatesof 1: 4 7 5 8 15 , coeet desipstion: 2 1 2 0 0 , ~ 0 = o : o = . . giving p20 2 , pa‘ 1 , p22 2 (5 22) gives such results as 0 3 a 2 p01 p11.2 I’20 and the following set or matrices 9.1 2 (pk) is obtained. 0 2 2 2 2 1 2 1 2 90:221,rl=202,92=122. 2 1 2 1 2 2 2 2 O The set of field marks consisting of O and the multiplicative sub- group 1 , e , e2 , ... , ed‘l has some of the properties of an addi- tive group. Addition is simply field addition and is commitative and 106 ‘ associative, and the condition that the element -1 is included insures that the set contains with each element its additive inverse. The set is not necessarily closed under addition. This is the only remaining require- ment for the set to be an additive abelian group, and will be satisfied if and only if the set is afinite field. It follows from general pro- parties of finite fields that this met be a subfield of the original fields of -p‘1 e1emente,-containing p3 elements, where s is a divisor of q . This case has been studied rather 5xtensive1y in connection with a variety of combinatorial problems; a recent application to incomplete block designs will be mentioned at the end of this section. In 1958 Singer [527 showed that it may be used to generate finite projective geometries, in particular projective planes. In the present setting we make use instead of the finite Mplane which may be obtained from the projective-plene by' designating one line as the line at infinity and deleting it withethe points-en it. The number of- remaining points is a square, say n2 , where n is the number of points on a line and in every known case is a prime power, say u = p’ . Two particular associ- ation schemes will now be discussed for v 2 n2 = p23 treatments. The first scheme will be shown to be equivalent to this geometry and leads to association schemes of the I.“ series. The second scheme is used in con- structing the new schemes of the 1.8" series. In-the finite field with pa 8 n2 ‘marks, the order of the multiplicative group is n2... 1 = (n-l) (n + 1) . The first association scheme, leading to the finite geometry, uses (1 = n-1 and c I n + 1 3 in the other scheme the same values are used in the opposite order. 107 in arbitrary treatment 0 in either schsme will new be considered. in arbitrary pair of distinct it‘h associates of 0 may be represented by . O -f-etls1 , 0 +et221 , where on and et'2 are distinct . we investigate uhether these two treatments are i"h associates. This will be the case if and only if the difference between the field marks is a mark of the 1th coset. O -!-et']-s1 - (O + etzzi) = 21(et1 - eta) . This expression represents a mark of the ith coset if and only if etl - at? is an element of the multiplicative subgroup generated by e . Since et1 and et2 are distinct, their diffbrence is not 0 . The additive inverse of et2 is an element of the subgroup, say eta , shins etl - etg = etl +et5 . This sum will be an element of the subgroup for all choices of 31¢ et2 “.9“ only if the set consisting of 0 and the supr is closed under addition, or equivalently if and only if the set is.a field. Therefore the 1th associates of an arbitrary treatment 0 are pairwise 1th- associates if and only if the subgroup used in defining the association scheme is the mltiplicative group of a subfield. This remark is used in the discussion of both association schemes for m2 treatments. lhile properties of the Euclidean geometry are well known and closely similar constructions of it have been piblished [SJ , [a] , enough of 108 the derivation of it will be presented here to be used in describing the association scheme. This will be useful for comparison with the second scheme. The marks of the finite field will be identified with the treat- ments of the desig as for all of the association schemes treated in this section; for the present scheme they will also be identified with the points of the geometrical system. The set- consisting of a treatment and its i"h associates will be identified with the nee—points-on. a line. In order for this line to be well-defined it is necessary to show that-the treatments in such a set are pairwise 1th- associates- The subgroup of order d I n-l may be taken as the multiplicative group of the subfield of order n 8 p' . Therefore, by the previous paragraph, the treatments are pairwise ith associates. Since each of them has theesame number n-l of 1th associates, each of them determines the same set of it marks, and the line is well-defined. Every mark of the field must lie in such a set of n marks related as it” associates and defined as a line. This implies that the n2 points of the system m _be divided into n disjoint sets, each containing the n points of a line determined by the i"h association relation. Since the lines have no points in common, they will be called parallel lines and will be described for convenience as lines in the ith direction. Corresponding to the n + l associate classes there are n +1 systems of parallel lines in as many different directions, each system containing ‘21 lines of. n points each and eidiausting the set of n? points. Since any two distinct treatments are 1“ associates for some i , any two distinct points of the geometrical system determine a unique line. 109 his implies that the number of points canon to two distinct lines can- not be as large as 2 , and must be either -0 or 1 . The n points of a line in direction 1. --must be distributed over the n- lines of the set of lines in direction 3 in such a way that not more than one point falls on each line. Since the number of points is equal to the number of lines, this means that a line of direction i intersects each line of direction 1 in Just one point , i 7 J . This completes the proof of the relevant properties of the geometrical system, showing that it is indeed a finite niclidean geometry, and furnishing a convenient way. of computing the parameter values of the association scheme. The parameter u1 has the geometrical interpretation of the amber of additional points on the line through an initial point in the 1“ direction, and it is clear from the geometry or general properties of the association schemes under discussion that n1 8 n - 1 for all i . The parameter p:k may be defined by means of any two points 0 and fl Joined by a line of the i9 direction. pg} is equal to the amber of points other than possibly 0 and fl themselves, common to the line through 0 in direction 1 and the line through d in direction I: . It is clear from geometrical reasoning that Pil'nvzs (5.2s) pi" 3’113911'0 when ifij, phi-l when 1,1, k arealldistinct. In view of the known relations among the pit , particularly for the class of association schemes of the present section, the three preceding fl..- 110 statements are far from independent. Straightforward use of (1.6) and (3.21) shows that each of the first two implies the other, while the third implies both of the first two. . Finally, the first two new be shown to im imply the geometric structure and hence the third statement. In the second scheme a subgroup of order d = n +~l = p‘ +'1 of the' field multiplicative group is used. This subgroup and its cosets deter- mine c = n-l associate classes. The zero element of the field and the subgroupformasetof n+2=p‘+2 markswhichwillnotbeasub— field, and the set of treatments consisting of an element and its 1th associates will accordingly not be pairwise 1th associates. ~It is therefore~not possible to use association relations in this case to define lines in a plane geometry, and there is no obvious way to compute the p§k: values. However, direct computation gives the values fairly easihy in a particular case. The example already given for n2 = 16 is an association scheme of this class and illustrates-the computation involved. The p§k values fer several other cases will be given later.“ The association schemes constructed by Theorem 5.2 have in general more than two associate classes. In most cases where schemes with two classes are derived, it will be by the device of combining classes, that is, by forming a set C1 of one or more of the associate classes of the original scheme, and defining two treatments to be first associates in. the new scheme if and only if they are associates of one of the classes of the set C . An association scheme formed in this way does not 1 necessarily satisfy the conditions of partial balance. Conditions that'- 1.11 it will do so, in a: more general setting, are derived in the next theo- rem. It will then be easy to show that the schemes related to the Euclidean plane lead to a wide class of schemes of the 1.8 series. The second family of schemes is more difficult to deal with but will be used to construct several schemes of the 1.: series. THERE] 5.5. Let an association scheme with m classes of associ- ates be formed from a scheme with a larger number of classes by parti- tioning the classes of the original scheme into m disjoint sets Cl , ... , cm , with two treatments defined as OK th associates in the new scheme if in the original scheme they are associates of one of the classes of set C“ . The notation Cu will be used interchangeably for- the set of associate classes and for the set of indices by which they are identi- fied. Parameter values will be denoted by hi , pix in the original scheme and by 30‘ , pry in the new scheme. Then no, and a‘ necessary and sufficient condition that the new scheme satisfy the conditions of partial balance is (5.25) p; 3 ; Zp§k forallOK,P ,Y, and ‘ Cp keC‘y uniformly for each i E C 0g . PROOF: The association matrices of the original scheme will be denoted by B1 . It may be recalled that the Bi are symetric matrices of 0's and 1's whose sum is the matrix with 0's on the main dilzonal and 1's elsewhere, and that they satisfy .relation (1.16) , 112 BJBK'S ajkjnI'l'Zkafii (szBj). . Forfixed 1. thematrixwith pik inthe 1,1; positionwillbe denoted as usual by Pi . The definition of the new scheme implies that its association'matricee, denoted by A, , 0‘2 l , ... , a must have the form (5.26) A“: Z 81 . “C. It follows from Théorems 1.1 and 1.2 that in order for the new scheme to satisfy the conditions of partial balance, it is necessary and sufficient that the association matrices A, satisfy (1.16), that is, that there exist constants no. and p2; such that m - - W (5.27) A‘B A1r - A? A, - 5,, n'I +022. p" A.‘ . Since not has the interpretation of the uniform row total of Ag. and n:l is the uniform row total of Bi , is satisfied. Using (5.26), the product A; AY may be written (5.28) is =(Z B)(ZBk)=Z:BJkB JG C, KaCY J; CF [(5ch Since BkBj == 53: , it is clear at this stage that AP), A = dyAfi , and only ‘9 Av will be discussed. Using (1.16) , (5.28) may be written (5.29) AflAY =ZZ(§HnI+Zp§ki.B) JeC, secy Since the sets 0. are disjoint, J and k can be equal in this sum- it is clear from (5.26) that (8.24) nation only if F = Y ,, in which case the first term in the parentheses leads to n31 . By (5. 24) , this is equal to nPI and (5. 29) may be written C, (5.30) AFAY = éflnal + ZZZp}kBi . '5‘ Cs KGCy i This expression for the product A flAY will now be compared with the following, obtained from (5.27) when A“ is written in the form given m ct (5.51) ApAY - spynpl + 2 p" 2: Bi . “=1 sec.) Since the B1 and the identity matrix I are linearly independent, (3.50) reduces to the form of (3.51) if and only if the coefficients of 131 in the two expressions are equal for all i . The coefficient of B1 in (5.31) reduces to the single term p2} , and the necessary and sufficient condition is identical with (5.25) , completing the proof of Theorem 5.5. Thus in order for the matrices A“ defined by (5.26) to multiply in accordance with (1.16), it is necessary and sufficient that each ptv value be equal to the sum of the elements in the submatrix of Pi deter- mined by the row indices 1 belonging to the set of and the column indices k belonging to the set CY , where i is a member of set Cog . The crucial condition is that the same total be obtained from every P1 matrix for which i 6004 . Because of the relations (1.6) and (1.7) satis- fied by the parameter values of every PBIB design, it is not necessary to verify (5.25) for all of the ng . In the special case of a scheme with two associate classes, the values plil and pil , uith 111 and n2 , are sufficient to determine the remaining 1 values, and the following corollary results. 114 COROLLARY 5.5. If m = 2 in Theorem 5.5, condition (3.25) may be replaced by the simpler condition (5.52) pr]. =2 Z pi'i'k forok =1 or 2 and uniformly jec'l REC) for each iECog. If the new scheme has m = 2 classes, set C is the complement of set 2 C and the association relation is most simply defined by saying that 1 D two treatments are first associates in the new scheme if in the original scheme they are associates of one of the classes of set 01 ,- and the two treatments are second associates otherwise. In the application of the corollary, the same symmetric submatrix, determined by the rows and columns whose indices are in set Cl , is used in each of the original P 1 matrices. The necessary and sufficient condition that the new scheme satisfy the conditions of partial balance is that the sum of all the ele- ments of the sutnatrix be the same for all P1 value being taken as pil for the new scheme, and the same for all P1 with i i (31 , the common value being taken as pil for the new scheme. with 1€01 , the common Equations (5.22) show that for the schemes with n +- 1 classes, obtained from the melidean plane, each P1 matrix may be obtained by a cyclic permutation of rows and columns of the matrix Po , .which is an (n + l) x (n + 1) matrix with the following form. 115 n-Z O 0 . . . 0 0' '0 ll 0 1 l . . e O 1 L011...10 d In the matrix P1 , the diagonal element 9:1 is equal to n-2 , the remaining entries in row i and column i are 0's , and the remaining diagonal entries are 0's . Application of Corollary 5.5 to find the values of pi and pil requires finding the sum of the elements of the symmetric sutmatrix of each P determined by the rows and columns whose i indices are in class C Suppose that g classes of the original scheme 1 . are to be combined into set cl 3 than the symmetric submatrix will be of order g x g . If the diagonal element n-2 is not in the submatrix, the g(g-l) off-diagonal elements will all be 1's 3 if the diagonal element n-2 is in the submatrix, then (g-1)(g1-2) off-diagonal elements will be 1's and other elements will be 0's . This means that the sum is n-2 + (g-l)(g-2) = g2-5g+ n whenever the index i of the matrix P1 is in the class 61 Ihich determines the sumtrix, and the sum is g(g—l) for all P1 with 1 ¢ (31 . But this is precisely the requirement of Corollary 5.5, proving that the association scheme defined by taking two treatments as first associates if they are associates of one-of the g classes of set C of the original, Euclidean geometry, scheme, satisfies 1 the conditions of partial balance with two associate classes. There was 116 no restriction on the value of g . The expressions obtained for n1 , oh and pfl are identical with those derived in Section 2.1 for Latin square type schemes with g constraints. Some of the special features of the association schemes constructed by Theorem 5.2 may be used to simplify the application of Theorem 5.5 and Corollary 5.5. This will be discussed in the case of Corollary 5.5 for the present purpose of constructing schemes with two associate classes. In the schemes of Theorem 5.2, the c associate classes all have equal numbers of treatments n = d , so that if first associates are defined in i the new scheme by a set CI of g of the original associate classes, the number of treatments in the first associate class of the new scheme is ex = gd , regardless of the particular set of g classes chosen for set 1 . Suppose that a set 01 is known to satisfy conditions (5.52) of Corollary 5.5; define a new set C C 1' by'adding 1 to each index in the set Cl 3 that is, i+l€cl' if and only if 1601 i +1 is reduced modulo c if necessary. The following equalities are I obtained by successive use of ( 5.22), of the definition of set C 1 , and of a change of notation in the indices of sit-ation. ijk ‘2 Z Pie-1.1”]. Z :9} j+l,k+l ”Z Zap): ° JEZC, REC. 16C 3+!e'c,' (met. J.‘=C Since the first sum is equal to tin same value p11 uniformly for 1 £01 and equal to pfl uniformly for i¢c 1‘, the last sum will be equal to _ ! ph uniformly for 1601' and equalto pil uniformly for iFCl . 117 Thus the set 01' satisfies conditions (5.52) of Corollary 5.5, giving the same values for the parameters n1 , pil and 9:1 , and hence for the remaining association scheme parameters as well, as are given by the set C1 . The operation of increasing by unity the index of each associ- ate class in C1 may evidently be repeated as often as desired, giving in each instance a scheme equivalent to that obtained with C1 . A suffi- cient number of repetitions will result in a set which.contains the 0th class of associates. Therefore, in application of Corollary 5.5 to scheme constructed by Theorem 5.2, it may be assumed without loss of generality that the set C1 of classes which are combined to form the class of first associates in the new scheme contains the 0th class of associates in the original scheme. The submatrix determined by set C will then contain the leading diagonal 1 element of each P1 matrix. This fact may be used to reduce the amount of empirical search necessary to find a suitable set C although it was 1, not needed in discussion of the schemes related to the finite geometry. The search may also be simplified if the parameter values of the possible new scheme are known. In the second family of schemes discussed for v = n2 treatments, each associate class has. d = n +-l treatments, and the only schemes which can be formed by combining classes are those in which n1 is a multiple of n +-l . .Inspection of Table II of the Appen- dix shows that most of the schemes with appropriate values of v and 111 are in the L" series, and that in this case the number of classes to be 8 combined in set C1 , which is equal to the order of the symmetric sub- matrices of the Pi matrices specified in Corollary 5.5, is given by the 118 numerical value of the subscript g . For a particular possible scheme, the values of ph and pfl may also be obtained from Table II. The sumof the elements of the submatrix of P mustbeequalto pi, e 0 condition which is easily checked for any animatrix and may eliminate many of the possible submatrices. Corollary 5.5 and the remarks which have just been made will now be used to attempt to construct schemes with two associate classes from the second family of schemes constructed by Theorem 5.2 for v = n2 = p2s treatments. For v = 9 , the scheme of this family has only two classes of associates and there is no need to combine classes. The schem is listed in Table II as #2, and is also in the Lg series. For v 8 16 , the scheme has three associate classes of 5 treatments each and has been given as an example. The P1 matrices are repeated here for reference. 022 221 212 P =221 ,P =202 ,p =122 . ° 212 1 122 ”2 220 The only scheme of Table II in which n1 is a-multiple of 5 is scheme #6, in which n = 5 . If this scheme is to be formed by combining l associate classes, the set C must consist of a single associate class 1 th and will be taken as the 0 class. The submatrix in this case is the single element p30 , and is equal to O for i = O , and 2 for i = l or 2 . These are the required values for pil and pi]. in scheme {6, proving that the scheme can be constructed by this method. n9 Table II includes no association schemes for v 8 25 treatments which cannot be constructed by other methods. Scheme #22 is an IL; scheme but the number of treatments is 56 , which is not a prime power, and the present method is not applicable. The scheme constructed by Theorem 5.2 with v = 49 treatments has six associate classes, each containing 8 treatments. The schemes of Table II with v = 49 and n1 a multiple of 8 are schemes #51 and #55. The latter is a known 1.:5 scheme. Scheme #31 is an 1:2 scheme with I’ll = 5 . The set C must therefore determine a symmetric 2 x 2 l submatrix which may be assumed to contain the leading diagonal element of each P1 matrix, and the sum of the four elements of the submatrix of P0 must be 5 . latrix PO , computed by the methods already illus- trated in the example with v = 16 , is as follows. —I MHNNOO NONMNO . Po NHOHMN owmomm NMNPOH Lgmommm If the elements of a symmetric matrix are integers and their sum is odd, it is clear that a diagonal element must be odd. The only odd diagonal element of Po is p22 = 1 , which must b; 12m the 2 x 2 submatrix. This determines a submatrix with the form 2 l , which has sum 5 instead of the required 5 . It is therefore impossible to choose a set CI of associate classes which satisfies all the conditions of Corollary 5.5, and scheme #51 cannot be constructed by the method of this section. 120 The next scheme constructed by Theorem 5.2 has v = 64 treatments, with seven associate classes, each containing 9 treatments. The schemes of Table II with v = 64 end nl ‘a multiple of 9 are schemes #48 and #51. at l g 2 scheme with p11 2 . The set C]. therefore must determine a synetric 2 x 2 subatrix containing the leading dia- Scheme #48 is an I. gonel element of each 14’1 matrix, and the sun of the four elements of the submatrix of P0 must be 2. The matrices P1 are listed below and it is easily verified that no submatrix of P0 satisfies these require- ments. Therefore scheme #48 cannot be constructed by the method of this section . Scheme #51 is an IL; scheme with pi = 10 and pfl = 12 . The set c1 must contain three associate classes whose indices determine a 5 x 5 submatrix of each of the seven P1 matrices. The sum of the elements of the submatrix must be 10 or 12 , according as the index 1 of the matrix is or is not the index of a class in the set 01 . The Seven matrices will now be listed. lfl P I. ’ 1 J. . _ _ I». 2212200 2222010 2021202 1022220 2200221 0202220 2201022 2120220 0212022 0022122 0222002 2200221 2022021 1202202 2220102 .2200202 0022122 1022220 0220212 2010222 2122002 r _ _ _ _ _ : = : 1 5 5 P P P e. 3 2 0 _2202120_ .010222&4 .12200224 110202224 2122002 2220102 0212022 2021202 0222201 2002212 2022200 1220022 2010222 1202202 2120220 2222010 0221220 2220020 2002212 0102222 0220212 2022021 2201022 2212200 2002022 0221220 0222201 2202120 _1 b _ L r L _ L = = : s 0 2 4 6 P P P P 1cmummgduus0, The association 5 satisfies the requirements and leads to a construction of scheme 0 The seven 5 x 3 submatrices are the following. 1 #51. Beheme is given in section A.4 of the Appendix. It is not difficult to verify that the set 1 122 202 021 i = 0 z 0 2 l 3 i 3 1 z 2 2 0 ; 210 102 022_ 202 i=2: 202; 1:5: 002 , . 220 222 020 210 i=4: 222; i=5: 102; 022 022 222 i=6: 220 ' 200 The next scheme constructed by Theorem 3.2. has v = 81 treatments, with eight associate classes, each containing 10 treatments. The schemes of Table II with v = 81 and n1 a multiple of 10 are schemes #68 and #70 in the 1.; series, #75, a know scheme of the Lg series which is also in the L; series, and #72, in neither series. Scheme #68 is an L3; scheme with pil = 1 and pi: = s . The set 0 must contain two of the eight associate classes and determine a 2 x 2 1 submatrix of each of the eight P matrices; the sum of the elements of i the submatrix must be equal to 1 if i is the index of either class in set 0 and equal to 6 if i is any of the six other indices. The 1 I usual assumption that 01 contains the 0th associate class means that the submatrix of each P1 matrix includes the leading diagonal element P30 e submatrices have the required totals, showing that the construction of andtheset 013(0,4)isquicklydetermined. The 2x2 125 the scheme is possible. The eight 8 x 8 P matrices appear below, 1 With the 2 x.2 submatrices. The association scheme is given in Section A.4 of the Appendix. Scheme #70 is an L;; scheme with pil 1 9 and pfl = 12 . The set 01 must contain three of the eight associate classes and determine a 5 x 5 submatrix of each of the eight P matrices; the sum of the i elements of the submstrix.must be equal to 9 if i is the index of any of the three classes in set 01 , and equal to 12 if i is any of the five other indices. The set 01 3 ( 0 , 1 , 6 ) is feund to be satisfactory. The 5 x 5 sub-atrices appear below and the associatidn scheme appears in Section A.4 of the Appendix. No construction has been found for scheme #72. The P1 matrices for the scheme with 81 treatments and eight associate classes are the following. 124 O, O, O, . _220121034 .2120212Wm 101210222. .020022214 02222200 22200022 02202020 20220202 20212021 21022201 00222102 01210222 20002222 20202202 22020220 22220002 02220121 22210200 21022201 21202120 20220220 02022022 22000223: 00222220 21020022 02220121 20212021. 22012102 02202202 00022222 22222000 22022020 _ a PI 1 _ up _ P x a = a 1 5 5 7 P h P P P O, O, O, O, _220220201d .0022222Wm 122220002. .202202024 20121022 12021202 12102220 20022210 22222000 22000222 22020220 02202022 02120212 10222012 02221020 12102220 00022222 02022022 20202202 22200022 22201210 22102002 10222012 12021202 02202202 20220220 20002222 02222200 10200222 22201210 02120212 20121022 _ a .1 _ _[ _ r L : : . = : pm i? 24 he The 2 x 2 submatrices used in the construction of scheme #68 nememuwmm 125 . The 5 x 5 submatricaaased in the construction of scheme #70 are the following. 0, 3 2 -i=5: 0 2 O 2 i=2: 0 2 5: 2 L1 0 2 126 e Theorem 5.2 is not/applicable to any schemes with v = 100 , since there is no finite field of order 100 . The schemes for which a construction by the method of this section has been discussed thus far have all been in the L8 and 1.: series. It has been shown that no more schemes of these series with v_<_ 100 can be constructed in this way. lost of the author's attempts to construct new schemes forwhich v is 'a prime power but which are not in these series have been inconclusive. When the number of treatments' v ‘6 pq is of the form 4t + l , Theorem 5.2 m be used directly with d 2 2t and c = 2 to form an association' scheme with two classes. When q 8 1 this is the known scheme of the cyclic series in which the first associates of' 0 are the quadratic residues of p ; when q is even, the scheme is in the I.“ series. For odd q 2 5 the scheme is not of cyclic or' Latin square type, but the first examnle is for v = 125. This example illus- trates that the methods of Theorems 5.2 and 5.5 are not limited to the cyclic, Latin square and negative Latin square association schemes, but it does not seem likely that they will provide solutions to any more schemes within the range of Table II. It was pointed out in Section 5.1 that if the Euclidean plane geo- metry with n2 points and the I." association scheme with n2 treat- ments both exist, the number of first associates of a treatment of n 1 the scheme is equal to me + 1) , where (n +1) is equal to the number of lines through a point of the geometry and the numerical value of the negative integer g is used. It could therefore be conjectured that for 127 some correspondence between the treatments of the scheme and the points of the geometry, the first associates of a given treatment would corres- pond to lg‘ points on each of the .(n‘+-l) lines through the given point. The treatments of the four Lé’ schemes which have been constructed in this section are already identified with the elements of a finite field, which in turn are identified with the points of the Euclidean plane geo- metry, giving a very natural one-to-one correspondence between treatments and points. It will be shown that with this correspondence, schemes #6 and #51 have the geometric property described. It will be convenient to discuss both the scheme and the geometry in terms of the finite field. For any element 0 , the first associates of e in the L: scheme are obtained by adding a to each of the first associates of the additive identity 0 . The points of the line through‘ 0 in direction 1 are obtained by adding 0 to each of the points of the line in direction 1 through. 0 . There is therefore a one-to-one correspondence between the first associates of O which lie on the line through 0 in direction 1 , and the first associates of 0 which lie on the line through 0 in direction 1 . The distribution of the first associates of any treatment 0 in the Lg? scheme over the ‘n +-1 lines through 9 in the plane geometry is therefore the same for any element 0 as it is for the element 0 and it is sufficient to consider the element 0 . The first associates of 0 and the remaining points on any line through 0 all correspond to non-zero field elements and will be replaced for the rest of the discussion by their indices or exponents 'With respect to a primitive element of the field. The remaining points on the line through 0 in direction 1 have exponents which are congruent to i modulo n 1‘ l . The first associates of 0 in the L; scheme have the '4“ + 1) exponents obtained by com- bining the sets of (n + l) exponents which are congruent to j modulo n—l , for Ig| suitably chosen values of j . A typical set of this kind Will be considered and may be written J+u(n-l) , u'-’O,*' l , ... , n. Suppose that two exponents in this set are congruent modulo n + l . j + u1(n-l) = j + u2(n-l) mod (n + l) , u1(n-l) E u2(n-l) mod(n + l) . Only the case in which n is even will be considered. In this case, n-l is prime to n + l and may be cancelled; since n1 and u2 are both between 0 and n this gives the result “1 = u2 , showing that no two of the exponents of the set fall into the same residue class modulo n + l , and that the n + 1 points corresponding to the set lie one each on the n + 1 lines through 0 . Since the same is true for each of the lgl sets of first associates of 0 , exactly Ig‘ of the first associates of 0 must lie on each of the n+ 1 lines through 0 . Finally, this shows that if an 11.; association scheme with n2 treatments is obtained by the method of this section, and if n is even (meaning that n is a power of 2), then the Igtn + 1) first associates of any treatment 0 correspond to lg] points on each of the n + 1 lines through the point corresponding to O in the finite Euclidean plane geometry with n2 points. 129 In the case of scheme #6, n = 4 and g = -l , and the 5 first associatis of any treatment 0 lie one each on the five lines through the point 9 . In the case of scheme #51, n = 8 and g = -5 , and the 27 first associates of any treatment 9 lie three each on the nine lines throng! 0e If n is odd, it is still true that the distribution of the first associates of a treatment 9 over the n-+ 1 lines through the corres- ponding point is the same for all choices of 9 , but the distribution is not necessarily uniform. It proves not to be uniform for the schemes constructed for 81 treatments. The results of Theorem 5.2 have also been found by D. A. Sprott and were published in two papers [3175327. The second of these, which is the only one dealing with partially balanced designs, appeared in 1955, after the present work.had been completed. The first article appeared in 1954 but did not come to this author's attention until after the second had been published. Both articles are on the construction of incomplete block designs from finite fields and make use of sets of field elements equivalent to the subgroup'used in Theorem 5.2. The designs described in sections 4 and 5 of the second paper have association schemes which are identical with those constructed in Theorem 5.2. Theorem 3.2 was moti- vated hy the desire for a class of association schemes for v = n2 treat- ments in which the numbers of treatments in the associate classes are multiples of n +-l , and the method of proof was originally suggested by some work published by Bose in 1942 [5']. The author 13 indebted to 130 Sprott's papers, however, for the realisation that the final statement of Theorem 5.2 was necessary in the proof that schemes such as #51 and.#48 cannot be constructed by the present methods. Sprott's work is different from that appearing here in many details. The presentrdiscuseion is limited to association schemes, while Sprott constructs actual designs, including some classes of them whose association schemes are not related to Theorem 5.2. He treats a field as an instance of a module and bases his construction on a general theorem of Bose and Hair [8] on the con-4 struction of partially balanced designs from a module. The proof of Theoren.5.2, dealing directly with properties of the finite field, is self-contained, may be simpler in some respects, and is certeinly different in its arrangement. Sprett does not consider combining associate classes to form designs with.fewer classes, and the only designs of the L8 or L; seriesobtainedarethosewith v=p9=4t+1, =2t and c=2 . In particular, the new schemes #6, #51, #68 and #70 are not obtained. 131 3.5 Construction of _a_ Negative Latin Sguare ms. Scheme with _igg Treatments fl Emotion. It is possible to solve some combinatorial emblems by making syste— matic trials of possible solutions until a solution is found or all possibilities are shown to fail. This method entails too much compu- tation to be usable in the. construction of most incomplete block designs or association schemes, but it will be used in this section to construct the L; scheme which appears in Table II as #94. The parameter values of the scheme are v=100, o 21 6 16 11222, P: , Pa . 1 1 21 so 2 16 so n2=77, Since no Galois field of order 100 exists, the method of Section 3.2 cannot be applied here. The scheme would seem to have some special interest because of its possible connection with the unsolved question of the existence of orthogonal 10 x 10 Latin squares. The reason this problem is amenable to empirical study is the parameter value pi]. = O , which permits use of Theorem 2.6. In this section the symbol He will be used to denote a c x d 3 d matrix all of whose elements are 1's . The subscripts will sometimes be omitted when the order is clear from the context. The orders of matrices and matrix products occurring in certain equations will be indicated in parenthetic statements which appear to the right of the equations. 152 If scheme #94 exists, let an arbitrary treatment be designated as treatment 0 , and its 22 first associates numbered from 1 to 22 . Then the matrix A1 of first associates may be partitioned in the form of (2.3g),‘with submatrix R a zero matrix. ' I 0'1 0 e e 1.0 e e e 0 war-—-—~—-r--—--——1I———-—-— 1 l e. I . i 0 l S 22 rows , I ° l (353) A = 1' ----- I -------- -- --I-I'-—- 1 0' I .' I ~ ' 3' ' T 77 rows . el ' . i no! ' .JL Theorem 2.6 may be applied to show that submatrix S is the incidence matrix of a BIB design with parameter values (5.34) v=22, r=2l, kzs, b=77, A25. Moreover, each row of the 77 x 77 matrix S'S must have at least p§2 3 16 off-diagonal elements equal to 0 . By (2.42) and (2.41) the 60 remaining off-diagonal elements of each row of S'S must have sum k(r-l) = 6(20) = 120 and sum of squares k(r-l) + k(k-—l)(7\-l) 3: 6(20)~+ 6(5)(4) = 240 . The variance of this set of 60 numbers is then Egg...(l%g,2 = 0 , showing thatrthey must all be equal to their mean value 2 . Thus each row of S'S contains a 6 on the main diagonal, 16 0'3 and 60 2'8 . This means that each block in the balanced design has.no treatments in common with 16 of the other blocks, and exactly 2 133 treatments in common withheach of the remaining 60 blocks. The existence of such a BIB design is therefore a necessary condition for the existence of scheme #94.1' As pointed out in the discussion accompanying (2.58), each row of submatrix '1' contains 1's in pig = 16 offodiagonal positions. By statement (b) of Lemma 2.1, s's + T2 must have entries ph = o in these positions and entries 0:1 = 6 in the other 60 off—diagonal positions of each row. Since T2 has non-negative elements, the 60 2's in each row of svs must fall in these same 60 positions. By difference, the element of 12 in each of these positions must be a 4 . This determines the following structure for T2 . 1‘2 = 16 I + cor + 4(U-I-T) , (77 x 77 matrices) . Lemma 2.2 may now be applied to show that T is the matrix A1 of first associates in an association scheme with two classes of associates and the parameter values :60, 1 “o, 2 =4. v 8 77 , p11 — pll 111 = 16 , n2 This is scheme #64 of Table II . Thus the existence of scheme #64 is another necessary condition for the existence of scheme #94.1 Either of submatrices S or T would presumably be easier to inves— tigate than the 100 x 100 matrix, and it will be shown below that the lé In other cases in Table II where Theorem 2.6 applies, either matrix T is not determined or *T is not an association matrix of wry scheme with two classes. 154 balanced design corresponding to S can actually be constructed. It is therefore important to show that the existence of S is sufficient as well as necessary for the existence of scheme #94. This will now be done. Let S be the incidence matrix of a BIB design with parameter values (3.34). This implies that 88' has the form (5.55) as. a 16 I + 50 , (22 x 22 matrices) . Also, let 8 have the property that each column has inner product 0 with each of 16 other columns. This was shown to imply that it has inner 2 with each of the 60 remaining columns. Then S'S has the form (5.56) 8'8 8 6 I‘+ 0°B +'ZB 1 2 , (77 x 7? matrices) , where B2 is a symmetric matrix with 0's on the main diagonal, .60 1's in each row, and 0's elsewhere, and B1 is defined by (5.37) 751 = u - I - 132 , (77 x 77 matrices) . The following useful equations are easily derived from the fact that S has uniform row totals 21 and uniform column totals 6 . 3.38 so = 21 u . ( ) 77,d 22,d (5.39) Uc,228 - 6 Uc,77 . (s-S)2 will now be computed in two ways. From (5.56) , (5.40) (39302 2 as I + 2432 + 41322 . 155 The next chain of equalities uses (5.55), (5.56), (5.59) and (5.57) in the order stated. (S'S)2 = s'(ss')s = S'(16 I + SU)S = 163's + 5905 316(61+2B2)+1800 3961+ 5232+ lacunal +32) , (3.41) (S'S)2 2 27s I + 180131 + 212132 . Solution of (5.40) and (5.41) for 322 gives 2 32 =601+458 +47B2 . 1 Application of Lemma 2.2, with the designations of first and second associates interchanged, shows that B2 is the incidence matrix of second associates in scheme #64. in incidental result is that S' is the inci- dence matrtx of a.PBIB design with this association scheme. This is the dual of the BIB design represented by S and is obtained by inter- changing the roles of treatments and blocks. So far, it has been shown that the existence of S with the given properties implies the existence of a matrix with the preperties required for T , defined by T - B1 . It remains to show that if 8 and T so defined are used as the submatrices in (5.55), then Al will meet the requirements for the association matrix of scheme #94. Using A1 in this form, and squaring according to the rule for partitioned matrices, 136 _. l _. 22:0. 0 e0|6e e e 6 --_-’~ T“? ““““ r--------- 0I | ', U "' 53' l 331 22 rows , ° I .I | I 2 O, I ______ (5042) A1 =L-6-' —————— .1. —————— p—e— .' : .' 0 g 2 . 0| 313 | S S +B1 q 77 rows l 6' I a "l _J The forms of the suhnatric‘es of A12 will be computed separately. (5.45) U+SS'=U+l6I+SU=22I+6(U-I), (22x22). The value of 8B1 may be obtained by solving (5.56) and (5.57) for Bl . then multiplying by S . 898=BI+2(U-I-Bl) . B1821+U-§S'8, (77x77matrices) . 8B1 = 28 + SD - $8.98 (22 x 7? matrices.) =28+210-§(161+5U)8 =28+210-88-150 (5.44) sol = 6(0 - s) . The value of 312 is easily obtained from (1.16), recalling that B1 is the incidence matrix of first associates in scheme #64. Then s's +3 2 + = +6 1 (GI-+282) (161+4B2) 221 82, (5.45) s's + 312 22 I + 6(0 - 1 - B1) , (77 x 77 matrices) . 15? Using equations (5.45), (5.44), (5.45), equation (5.42) may be written 52=2zx+6(0-1-A1)=22I+6i2, (100x100), and by Lemma 2.2, A1 and A2 are the association matrices of scheme #94. This completes the proof that the existence of the BIB design implies the existence of scheme #94. It will also furnish an easy way of constructing the scheme from the design. The existence of scheme #94 is therefore equivalent to the existence of the specified BIB design. This does not mean that the number of dis- tinct design is equal to the number ofschemes. For any matrix A1 in scheme #94, there are 100 possible choices of the treatment to be desig- nated as treatment 0 , each leading to a different set of rows and colums in the sub-atrix 8 . These determine BIB designs which have the same parameter values, but which are not necessarily all equivalent under permutation of treatments and blocks. However, any given matrix 8 determines the rest of matrix ‘1 uniquely, showing that the number of solutions of A1 is at most equal to the number of solutions of S . It will appear below that there are at most 4 solutions for S , so that if the solution for scheme I94 is not unique, there are certainly no more than 4 solutions distinct under permutations of treatments. The question of uniqueness might be of interest if a connection is found between scheme #94 and sets of orthogonal 10 x 10 Latin squares. The structure of S will now be taken up. S is the incidence hatrix of a BIB desig: with 77 blocks, each containing 6 of treat— llemts 1 , 2 , ... , 22 . Each treatment occurs 21 times and each 158 pair of treatments occurs together in the same block 5 times. The ‘ design satisfies the additional requirement that for any choice of an initial block, there are 16 blocks which have no treatments in common with the initial block; this was shown to imply that each of the 60 remainingblocks has exactly 2 treatments in cannon with the initial blocks There is no loss of generality in assigning numbers I , 2 , 5 , 4 , 5 , 6 to the treatments in the initial block, then considering separately the set of 60 blocks each of which contains 2 of the treat- ments 1-6 and the set of 16 blocks which contain only treatments 7-22. Denote these by Set I and Set II respectively. In what follows the number of special cases will be reduced greatly by showing that certain cases are equivalent under changes of’notation, that is, under permutation of treatments and/or permutation of blocks. The reader may verify that the only treatments or blocks involved in any of these permutations are those which play syn-atrial roles in the part of the design which has been previously specified. At the present stage this permits any permutation of treatments 1-6 among themselves and any permutation of treatments 7-22 , but no interchange of treatments not in the same set. Similarly, blocks may be renumbered within Set I or Set II but the sets of blocks Will be left intact. Repeated use will be made of the fact that the number of treatments common to any two blocks of this design must be either 2 or 0 . Each of the 15 pairs of treatments 1-6 occurs once in the initial block,notatallinthe 1s blocksofSetII,andA-‘-’5 timesinall, 139 so that it must occur in exactly four of the 60 blocks of Set I. Since none of the 60 blocks can contain more than 8 of treatments 1-6 . the blocks must fall late 15 sets of 4 , blocks in different sets con- taining different pairs of treatments 1-6 , and the .4 blocks of each set containing the same pair. No two blocks in .the same set of 4 can have more than 2 treatments in cannon, meaning that no two of them can contain the same treatment of set 7-22 . Therefore .each of. the 16 treatments in this set must occur just once in each of the 15 sets of 4 blocks. It will be convenient to identify the sets of 4 blocks by the pair of numbers of set 1-6 which they have in common. The symbol [341/ will be used to denote the set of 4 blocks containing the treat- ments 1 and j of set 1-6. Since each treatment must occur 21 times in the BIB design and 15 occurrences of each of treatments 7-22 have been accounted for, each must occur just 6 times in the 16 blocks of Set II. Let I denote the sutuatrix of 8 whose rows are determined by treatments 7-22 and whose columns correspond to the blocks of Set II. II is a 16 x 16 incidence matrix whose row totals are all 6 by the final sentence of the preceding paragraph, whose column totals are all equal to 6 , the number of treatments in a block, and whose column inner products are at most equal to 2 . This means that the symmetric matrix I'll has diagonal entries equal to 6 and 15 off-diagonal entries in each row which are at most equal to 2 . It is not difficult to show that (2.42) holds for any incidence matrix with equal row totals 15’ and equal column totals k , whether it is the matrix of a BIB design 140 or’not. This shows in the present example that the sum of the 15 off-diagonal entries of each row of -I'l is 30 , proving that each of these elements is equal to 2 . This proves that l' is the matrix of a BIBdesignwithparameters v=b=16, rzkss, A=2. Since this is a symmetric design, a well~known result originally obtained by Fisher shows that the column inner products of I' are also all equal to 2 , the same as the row inner products. This is the same as saying that I is the matrix of an equivalent BIB design. This is useful in the con- struction of Set II-of 16 blocks. Also, since sash pair of treatments 7>22 occurs 2 times in Set II and must occur— 5 tines in all, each pair must occur exactly 5 times in the 60 blocks of Set I, a fact which is helpful in the construction of Set I. The fact is not essential, and rather than digress to prove that (2.42) can be applied, the construc- tion of Set I will be based on the fact that no two blocks of the set can have more than 2 treatments in common. The fact that Set II determines a BIB design then follows without any appeal to (2.42). The numbering of treatments 7-22 will be chosen so that [1.2/ has the for- 2 7 8 9 10 2 11 12 15 14 2 15 16 17 18 2 19 20 21 22 PHHH llext consider the sets [5,4/ , [5,5/ , [4,5/ . Each contains the treat- ments 7-22 once each, and the rows of each must have either 2 treat. nRents or no treatments in common with the blocks of [1,2/ . This means 141 that the last 4 numbers of any row of M must occur in pairs in two of the rows of each of [_5‘5/ , M , M . Rearrange rows of each of these sets if necessary so that the first blocks of each set contain treatments 7 , 8 , 9 , 10 , with 7 in the first row. The blocks 547______, 5 5 7____, 4 57.___alreadvhavethe maximum number of treatments in common, so they must contain the treat-— nents 8 , 9 , 10 in some order. Renumber these treatments if necessary so that they occur in the order given. This determines the following port“"‘“t 5°t54£§nél awééhél eWZSm§f ‘ 0003010! :5th I 0'4 ...: l O I I I I III! “0'00! U'IUUUIUI .QQ ...: O I I I I I I I I ##hh (names I 1 l I l l I I Renumber treat-ants 11422 if necessary so that the remaining treatments inblock 5 4 7 8 _ __ are 11 and 12 . Then a repetition of the reasoning used for treatments 7 , 8 , 9 , 10 shows that the pairs of treatments 11 15 and 11 14 must occur in sets [54;] and [21.5] 3 renumber treahnents 15 and 14 if necessary so that 11 15 occurs in M . Since no block can have more than 2 treatments in common with the block 5 4 7 a 11 12 , no‘other block can have an 11 or 12 along with a 5 or 4 and a 7 or 8 . This determines the following. 54781112_5579__45710__ 54910 sssio__4sse__ 54____551115__451114__ s4 551214__451215__ 142 There is new a choice of placing the pair l5 14 in the second block of set 45A] or in another block, say the third. The latter case will be investigated first. Renumbering treatments of the sets 15 , 16 , l7 , . 16 and 19 , 20 , 21 , 22 if necessary and remembering that not more than two treatments of either of these sets can occur in the same block of [353/ , the following is obtained. ' 5 4 7 8 ll 12 5 4 9 10 15 16 5 4 15 14 19 20 5 4 17 18 21 22 Since neither 15 nor 16 can new fall in a block of M which con- tainsa9or 10,theset 15,16,17,18 mustfallinthelast two blocks of [35/ , in the following arrangement after renumbering them if necessary. 7 9 8 10 11 15 1'5 1? 12 14 16 13 01010!“ (”0100“ The pair 15 18 must now occur in some block of L445], but it is easy to verify that any such block would then have 5 treatments in common with some block of M or [_5_._§/ , showing that this case is impossible, and that treatments 15 14 must occur in the second block of M . Treatments 7-14 , which occur in blocks 1 2 7 8 9 10 and l 2 ll 12 15 14 0f [Lg/ , have now been assigned to the blocks of M . Notation was chosen so that treatments 7 , 8 , 11 and 12 all occurred in the same block. This was found to imply that treatments 9 , 10 , 13 and 14 all 145 occur in the same block. By symmetry, this shows that if treatments from any two blocks of set 043/ occur in the same block of any of sets M , Q‘s/1 , M] , than alLeight treatments (other than 1 and 2) of those two blocks mist occur in the same two blocks of the set. This fact is useful in completing the blocks of M , M and - [3.1.57 . Treatments 15—22 may be renumbered if necessary so that number 15 is given‘ to one of the remaining treatments in block 5 5 7 9 _ _ . Then after possible further changes of notation, the following is quickly obtained. 54781112 55791517 45 7101922 or 2021 54 9101514 558101618 45892021 or 1922 5415161920 5511151921 4511141518 or 1617 5417182122 5512142022 4512151617 or 1518 The blocks of M m be completed in any of four ways. There seems to be no iuediate way of reducing this number of cases by choice of notation, and from this point on only the first case will be considered (rez- each block, the first pair of the two possible pairs listed). It may be veri- fied that the other three cases give similar results. . The next blocks to be constructed are those in sets [w , M , M . The block containing treatments 5 6 7 must contain another treatmentof set 7,8,9, 10; compsrisonwithblocks 54781112 and 5 5 7 9 15 17 shows that. the treatment mist be 10 . Further com- parison with the blocks already constructed shows that the block met also contain treatments 20 and 21 . This sort of arms“ quickly deter- mines that the three sets of blocks are as follows. 144 56 7102021 46 7E91618 56781514 56891922 46 8101517 56 9101112 5611141617 4611152022 5615162122 5612151518 ~4612141921 5617181920 hwu°¢°fuu M.M.M.M.LW.M. Lag/.159] alreadyhas-treatnentl or. 2.in canon with each block of set M , unmet. contain just one treat-exit of the set 7 , a , 9,10,onetreatnentofset 11,12,15,14-and-soon. Canaries- witheitherblockss'lalllzors6781514showsthat treatments 15 and 14 , in none order, Inst occur in blocks 157__-and257___.Treatnents1andzoccupy synetrical positions in the part of the design which has been specified so far, and they say be interchanged if necessary to give the blocks 157l5__and25714___. Further conparison with blocks containing treatnsnts 5 , 7 and 15 determines the following. 157151622 257141819 15814 25815 __ 159__18___ 259_1e___ 1510_____19 251o_____22 Comparison with blocks alreazb' containing the pairs 5 l4 , 5 18 or 5 19 then determines the remaining treatments. Entirely similar con- siderations determine the blocks of sets (1.3] , M ,. (1,6/ ”[2,5/ , [2,5/ , Aw , completing the construction of all the blocks of Set 1. Each pair of treaments must occur together in exactly A = 5 blocks. Enuneration shows that each of the pairs of treatments 7 to 22 145 has occurred together. 5 times in the blocks of Set-I, leaning that each pair nust occur twice in the 16 blocks of Set 11. It has already been , noted that each of treatments 7 to 22 occurs 6 tines in the set. This verifies that the blocks of Set 11 for: a BIB denim with parameter values w: :16, r=k=6, 1:2. Consider the two blocks of Set II which contain treatnents 7 and 8 , Comparison with blocks of Set 1 "Mon contains treatnents 7 and a shows that the renaining eight treatments in these two blocks nust be treatments 15 to 22 in sons order. Comparison with block 4 5 11 14 15 18 shows that the pair 15 18 least occur in the sane one of these two blocks. Comparison with block 5 4 15 16 19 20 shows that block 7 8 15 18 _ _ lust contain either 19 or 20 ; comparison with block 2 5 7 14 18 19 shows that it cannot contain 19 3 comparison withblock s 6 7 '10 so 21, forinstance , thenshows thatthere- naining treatment in the block must be 22 , determining the structure 7815182022 7816171921. A similar procedure determines the renaining blocks of Set 11. This con- pletes the construction of the 77 blocks of the 818 design desired. The blocks are listed on the following page: 146 Tufdhuuamtabhasfltubuwudemhmbuadwun Jutwuumumwuhmmnuruhu v=%, r=a4 R86, b-77,A=5. Imuuhma: kth HHHH NNNN 04 ..4 CG :5 K p 07 ...o 03 H 0 Stan HH H memo ~HHH~ ammo HHHH HHHH HHHH HHHH aaaa mama eeee mama &tnz mm (g--1)2 treatments which form a complete configuration cannot have more than one treatment in common with any complete n-point unless all h treatments are contained in the n-polnt e 162 THEOREH 4.1. If an association scheme has the parameter values of the L8 series, if each pair of treatments is contained in a set of n treatments which are pairwise first associates, and if n ) (g-l) 2, then there exists a set of g orthogonal n x n squares which may be used to define first associates in the scheme. PROOF: The language of linear graphs will be used. Treatments will be referred to as points, pairs of first associates as lines, and a set of k treatments which are pairwise first associates will be termed a “complete kbpoint' , or briefly a 'k—point' . By hypothesis, each line on an arbitrary initial point of the graph is contained in a complete n-point. Each such n-point contains exactly n—l lines through the initial point. By Lemma 4.2, since n )(g-l)2 , no line through the initial point can be in more than one appoint, and by Corollary 4.1, no complete configuration can contain more than n-l lines through the initial point. Therefore the set of n1 3 g(n-l) first associates of the initial point must fall into disjoint sets of n-l , each forming a complete n-point with the initial point. Therefore there are just g n-points containing the initial point; it was an arbitrary 2 points. From this point and the same remark applies to each of the n or from the remark that there is just one n-point on each line, it follows that there are exactly ng n-points in the entire graph. Denote an arbitrary initial n-point by A . Each point of A lies on g-l additional n-points, for a total of n(g~l) n-points inter- secting A . If any of these were counted twice, for different points of A , it would have more than one point in common with A , which is 165 impossible by Lemma 4.2. Therefore A intersects exactly n(g-l) of the remaining ng-l n-points, leaving n-l of them with which it has no points in comon. Take any n-point B disjoint from A . By Lemma 4.1 , a particular point of B is joined by lines to g-l points of A . Each of these lines lies on an n-point. If two of them lie on the same n-point, then that n-point would have more than one point in common with A , which is impossible, so the g-l lines determine the remaining g-l n-points through the point of B , for a total of n(g-l) n-pointe intersecting A and B and distinct from both. If any of these were counted twice, for different points of B , it would have two points in common with B , which is impossible. Therefore the n(g-l) n-pointsare all distinct, and must be the entire set of n-points which intersect B . Therefore the n—l n-points disjoint from B must be A and the other n-2 in the set of n-points disjoint from A . B was any one of the n-l n-points disjoint from A , so each of these n-points is disjoint from all the others, and the whole set of n are mutually disjoint, exhausting the n2 points. The argument carried out for A applies to any of the g n-points through an arbitrary point of the graph, showing that there are g systems of n "parallel“ n-pointa. n-points in the same system are dis- joint; any two in different systems have just one point in common. Let the n2 points be identified in a l—to-l manner with the cells of an n x n square array. Identify the n “parallel“ n-points of one of the 164 g systems with n distinct letters, and form a square array of these n letters by assigning each letter to those cells of the array corres- pending to the points of the n-point. The g squares which.may be formed in this way satisfy the requirements of a set of g orthogonal n x n squares, and may be used to define first associates in the associ- ation scheme. Some terminology to be used in the next lemma and in some of the theorems of this chapter will now be introduced. In any association scheme with two associate classes, choose notation so that treatments 1 and 2 are first associates and number the remaining treatments so that the treatments in each of the following sets have consecutive numbers and the four sets are numbered in the order listed. Set 1: p11 common first associates of treatments 1 and 2 , Set 2: pig treatments which are first associates of treatment 1 and second associates of treatment 2 , Set 5: p%2 treatments which are first associates of treatment 2 and second associates of treatment 1 , Set 4: 9&2 common second associates of treatments 1 and 2 . In the incidence matrix A1 of first associates, the treatments of each set correspond to a set of consecutive rows and columns, and the four sets of treatments determine sixteen submatrices, which are indicated in the diagram belowu Orders of the submatrices are shown in the margins of the diagram. The notation Ar‘V will be used for the submatrix with rows corresponding to Set/Il and columns corresponding to Set 1/ . 165 F0 1; 1 e o 1:]. e 0 1:0 e e 0' 0 e o 0‘”, _i__o_. _1_,_.,1_i_o_._._o_ ._1_._._1_; 9_._._9_ ,_ _ __, _ _ 1 1: : : s . . ' I A I ; A11 . A12 ; 13 AM ii" 11.3% -i__i.=.-_---i _____ ......... 1.1--. ' 1 0: : T : 4e : e e' ‘ A ' t 2 ( 1) 51 ' A21 : A22 : 23 ‘ A24 Si ’ e e. : . : p12 rows. 1 0' I t ’6'I;”“" ’f” ‘ ”:’"""g"' "” """“ . . i ' A t 5 . : ‘51 I 52 : as : 54 ,3“ O. : : ' 12 0 -3-%,:----«.- ------ :r ---------------- l . . A . t 4 2 A41 : 42 . ‘43 i ‘44 :3 ,3“ e e. I ' J 22 e 0 0' n The number of 1's in a row of if”, will be denoted by tfly; the row totals of a submatrix are not necessarily equal but the symbol will be used only in statements which are true for all rows. The symbols Tluy and Z [w A/“V and the total number of 0's in A/uy which are not on the main will be used, respectively, for the total number of 1's in - diagonal of A1 . LEMMA 4.3. If the incidence matrix A1 of first associates in an association scheme with two associate classes is partitioned in the form of (4.1) , then the number 211 of off-diagonal 0's in submatrix All satisfies the following inequality. (4.2) 211$ Pizu’il " 1) o ’ x 9:1 are 0's of the main diagonal of A1 , PROOF: Since ‘11 is a p11 matrix whose diagonal elements 166 - 1 1. Considering inner products of row 2 with rows of Set 1 , t t = 1 - 1 11'+ 15 p11 ' Summing over the pi rows of Set 1 , (4.4) T + '1‘ - 1 1 11 13 ’ p11(911 ' 1) ’ From (4.5) and (4.4) , 5 = . (4. ) z11 T13 Considering inner products of row 1 with rows of Set 3 , t t - 2 1 ‘ zl't 52 ‘ p11 ’ ' Summing over the pig rows of Set 3 , . 1 2 4.6 T T - - o ( ) :51+ 52 "12(p 11 1) By symetry of A1 , .7 3 . (4 ) T15 T31 Combining (4.5) , (4.6) and (4.7) , = : 1 2 .. .. 211 T51 p12(911 1) T52 ' 1 2 < " o This completes the proof of the lemma. 167 THERE}! 4.2. If an association scheme has two associate classes and v = n2 treatments, n f 4 , then necessary and sufficient conditions that it be a.Latin square type scheme with 2 constraints are (4.8) u1 = 20! - l) . (4.9) p11 - n - 2 . If n 2 4 , the condition is necessary but not sufficient. PROOF: Necessity is proved by the general expressions (2.9) for the parameter values of Latin square type association schemes with 3 con- straints, which reduce in the case of g = 2 to n = 2(n..1) , Vn-z n-l j 1 P = , n2 = (11-1)? , 1 Ln-l (n-l)(n-2) 2 2n-4 ‘ p = V - 2 L2n-4 (n-2)2 Also, the parameters specified in (4.8) and (4.9) determine the remaining values, so that any of them may be assumed in the sufficiency proof. The sufficiency proof will make use of the incidence matrix ‘1 , partitioned in the form of (4.1) . An important step will be to show that any pair of first associates and their pil 3 n-2 common first associates form a set of n treatments which are pairwise first associates. In the notation of (4.1), this amounts to showing that submatrix All has 1's in all off-diagonal positions or equivalently, that Zll = O . Lemma 4.5 provides an upper bound for 211 which reduces in the present case to . .<: - 1 . (4 10) Z11 _ n 168 Suppose that 21170 , meaning that among the n-2 treatments of Set 1 there is at least one pair of second associates. For convenience, number treatments so that numbers 3 and 4 are second associates. Then the entries in the 5,4 and 4,3 positions of Al Will be 0's. Since treatments 3 and 4 are second associates, the inner product of rows 5 and 4 of A1 must be equal to pil = 2 , meaning that the submatrix consisting of these two rows must contain exactly 2 columns with 1's in both positions. But columns 1 and 2 are of this form, meaning that each of columns 5 , 6,..., n2 of this sub-atria: must contain at least one 0. Since n-4 of these columns are in submatrix All , it must contain at least n—4 0's in rows 5 and 4 in addition to the two 0's originally assumed. By symmetry of the matrix there are also at least :14 additional 0's in columns 3 and 4 , for a total of at least 2n-6 off-diagonal 0's in A Therefore, ll. (4.11) if zll>o, then 211221145. The latter inequality contradicts (4.10) for n 2 6 , proving that for n26, 211:0 . For n=3, All isa lxl matrixwhich trivially has no off-diagonal 0's . For n = 5 , it will be proved below that Z11 3 O . Therefore for all n i 4 , the n x n suhnatrix of Al whose rows and columns are determined by a pair of first associates and their n—2 common first associates has all off-diagonal elements equal to 0 , which means that the n treatments of this set are pairwise first associ- 81:88. 169 This completes a proof that for n f 4 , every pair of first associates is in a set of n treatments which are pairwise first associ- ates. By Theorem 4.1 , there exists a set of g = 2 orthogonal squares which may be used to define first associates in the scheme, which means precisely that it is a Latin square type scheme with two constraints. In the special case n 3 4 , Counter-example l of Section 4.1 shows the existence of a scheme whose parameter values satisfy conditions (4.8) and (4.9) but in which it is not possible to define first associ- ates by a set of two orthogonal squares. Therefore the conditions are not sufficient in this case. it remains to prove that 211 = 0 when n = 5 . (4.10) and (4.11) show that if lej> O , than 211 3 4 . Assume that for some choice of two first associates as treatments 1 and 2 , 211 3 4 . That is, the 1 p11 associates of treatments 1 and 2 has 4 offodiagonal elements equal 3 x 5. submatrix All determined by the set of = 3 common first to 0 . After assigning the numbers 3 , 4 , 5 in a suitable order to these three treatments, the leading 5 x 5 principal minor of Al will have the form 0 1 1 1 l 1 0 1 1 1 1 1 O 1 O . 1 1 l 0 0 1 1 0 0 0 Since treatments 1 and 3 are first associates, they must have p11 3 5 first associates in.common, of which two are treatments 2 and 4 . 170 Number the remaining one as treatment 6 and adjoin row and column 6 to the suhnatrix, remembering that no further treatments can be common first associates of treatments 1 and 2 . HHHHHO OHHHOP HOHOHH HOOHHH uoooww OMHHOH Treatments 2 and 6 are second associates and the inner product of rows 2 and 6 cannot exceed pil = 2 . Therefore the letters x and y must represent 0': . Treatments 1 and 4 are first associates and must have three first associates in common, of which two are treat- ments 2 and 3 . Number the remaining one as treatment 7 and adjoin row and column 7 to the submatrix, remembering that no further treat— ments can be common first associates of treatments 1 and 3 . 0 11 ll 11 l 01 ll 00 l 10 10 10 ll 10 00 1 ll 00 00 z 10 10 00 w l 00 12 wO Treatment 7 is a second associate of treatments 2 and 5 and the inner product of row' 7 with either of rows 2 and 5 cannot exceed 2 . Therefore the letters 2 and w must represent 0's . Treatment 1 has nl = 8 first associates, of which the remaining two may be numbered 8 and 9 . Then the next two elements of the row 1 of A1 will be 1's and the remaining elements will be 0's . Treatments 5 , 6 and 7 are 171 first associates of l and rows 5 , 6 and 7 must have inner product 5 with row' 1 , meaning that each of these rows must have 1's in the next two positions. Then the inner product of any two of rows 5 , 6 and 7 will be 6 , which is impossible since these treatments are pairwise second associates. This contradiction disproves the assumption that le:> O for some pair of first associates and proves that every pair of first associates is contained in a set of n = 5 treatments which are pairwise first associates. This completes the proof of Theorem 4.2. The principal object of the remainder of this chapter is to prove as much as possible of the statement that if an association scheme has the parameter values of a Latin square type scheme with g constraints, 32 5 , there must exist a set of g mutually orthogonal squares which may be used to define first associates in the scheme. The counter-examples of Section 4.1 show that this statement is not true without exception, but it will be shown in Section 4.5 that for any g , the statement is true except for a finite number of values of n . When it is attempted to prove this by the methods used in the proof of Theorem 4.2, difficulties are encountered which will be described in the case g = 5 . For g 2 4 , the difficulties are of the same kind but more severe. The proof of Theorem 4.2 hinged on showing that an arbitrary pair of first associates, corresponding to an arbitrary line of the graph, was contahned in a complete n-point. This was equivalent to showing that All , an (n-2) x (n-2) submatrix, contained no off-diagonal 0's , and was accomplished by showing that if any such 0's were present, the 172 restrictions on the inner product of rows corresponding to second associ- ates implied the existence of enough additional 0': in All to violate inequality (4.2) . In the case of g = 3 constraints, All is an n x n submatrix, and rather than prove that it has no off-diagonal 0's , it is desired to show that it has an (n-2) x (n-2) principal minor which is of this form. The symbol 3 to be used in the next section, will 1 : be borrowed for the sake of brevity. In this section, sl will denote the maximum order for a principal minor of All which has no off-diagonal 0's . If any contradiction to inequality (4.2) is to be obtained, it must be on the assumption that slf§_n-5 . Inequality (4.2) is weaker in the case g - 5 , reducing to o < "’ e (4 12) 2:ll __ lOn 20 This is consistent with forms of All such as the following, in which the leading (n-S) x (n-S) principal minor has 1's in all off-diagonal positions and the other submatrices have 0's in all positions. 011...1:ooooo lOl...1.00000 110...1aooooo eee eteeeee n-5I'OVI8 _1__11 o;oooo-g g A = o6'0".".”.""o’!'0‘o"0'6o‘ ““ 11 ooo...o.ooooo ooo...o'ooooo 5rows ooo...otooooo ooo....o:ooooo ooo...o;ooooo The number of off—diagonal 0's is lOn-SO, satisfying (4.12), and 173 el = n-S . Moreover, it appears that s1 can be still smaller without giving:any easy proof that z is large enough to contradict (4.12). 11 The principal tool used in showing that Z is large is the restriction 11 on the inner product of rows of All Corresponding to second associates, and this restriction is also weaker than in the case of Theorem 4.2. With two constraints, the inner product of such rows was necessarily O 5 with three constraints, the maximum value for the inner product is pfl-Z = 4 . It may be possible with the methods used in Theorem 4.2 to prove the existence of a complete krpoint on each line of order sl-+ 2 = n-5 but no better result than this can be hoped for. This falls short of the hypothesis of Theorem 4.1. If it is not possible to prove the existence of a complete n-point on every line of a graph, it may still be possible to prove the existence of one n-point, somewhere in the graph. It will now be shown that this weaker result would actually have been sufficient in the case of Theorem 4.2 for a proof of the remainder of the Latin square structure. In other words, an association scheme with L parameter values either has a com- 2 plots n—point on each line, implying the existence of the orthogonal squares, or it has no n—points at all. It will be possible in Section 4.5 to extend this part of a uniqueness proof to some L3 schemes not covered by the main theorem of that section. The present result, while vacuous for L2 schemes with most values of n , does show that for n = 4 , the one case in which a non-Latin square scheme can have L2 parameter values, the graph of such a scheme cannot contain any complete 4-points. It was verified for the first counter-example of Section 4.1 174 that a particular line was not contained in any complete 4-point; the following theorem shows that the same is true for each of the 48 lines of the graph. THEORDI 4.3. If an association scheme with two associate classes 2 ’ n1 : 2(3.1) , pil = n-2 and there exists has parameter values v 3 n a set of n treatments which are pairwise first associates, then every pair of first associates is in such a set. PROOF: Denote the set of n treatments in an n-point by A and an arbitrary treatment of A by 0 . Of the 2(n-l) first associates of 0 , n-l are in A ; denote the set of the remaining n-l first associ- ates of 0 by B and an arbitrary treatmsnt of B by fl . It follows from Lemma 4.1 that fl has no first associates in set A except 0 , so the n92 first associates which fl has in common with 0 must be the remaining n-2 treatments of set E . Since ¢ was an arbitrary treat— ment of set B , vit follows that each treatment of the set must be a first associate of each of the others, meaning that the set consisting of 9 and its first associates not in A form a complete n-point. O ‘was an arbitrary treatment of set A and the same argument applies to each of the n treatments of A , proving the existence of n additional appoints, each having one treatment in common with A . Since any treat- ment in two of the additional n-points would be a first associate of two treatments of A , Lemma 4.1 shows that these n—points are disjoint, exhausting the n2 treatments of the scheme. They may be called parallel n-points. The same reasoning applied to the original n—point A may 175 now be applied to any one of the new ones to show the existence of another set of n parallel n-points, of which one is A . This shows that each treatment of the scheme is in two complete n-points, the 2(n-l) other treatments of the two n-points being first associates of the treatment. The first associates of all treatments are accounted for by the two sets of n—points, showing that every pair of first associates is contained in an n-point. This completes the proof of the theorem. Theorem 4.1 may then be applied to show that the scheme must be of L2 type. 176 4.5 933.15 Uniqueness 91: Lg _A_ssociation Schemes, g > . In this section methods will be develOped by which Theorem 4.2 can be extended to an infinite class of Latin square type association schemes with 5 or more constrabnts. Theorems 4.4 to 4.6 are devoted to ob- taining a lower bound analogous to (4.11) for the number 211 of off- diagonal 0's in the submatrix All defined by (4.1) . The bounds ob- tained apply to a wider class of incidence matrices and are in a form which gives direct information on the value of k. for which a complete kepoint is known to exist. Theorem 4.7 applies the results to association schemes. Theorem 4.8 and Lemma 4.5, also concerned with association schemes and valid for all schemes with two associate classes, introduce a different line of reasoning concerning the existence of complete kppoints in association schemes and are somewhat similar to Lemma 4.2. Finally in Theorem 4.9 the case of association schemes with parameter values of the Latin square series is taken up and it is shown that for a fixed number g of constraints and sufficiently large n , the Latin square type scheme is unique in the sense used in this chapter; that is, the scheme can only be constructed by the use of some set of g mutually orthogonal n xzn squares to define first associates. In Corollary 4.9 the suffi- ciently large values of a are stated explicitly. Theorem and Corollary 4.9 are the main results of the section and the chapter. The chapter is concluded by a discussion of some extensions and possible extensions. The most important of these, Theorem 4.10, is analogous to Theorem 4.5. Some ideas to be used in Theorem 4.4 to 4.6 will now be given in two definitions. 177 DEFINITION 4.1. An incidence matrix A will be said to satisfy this definition if it is a t x t symmetric matrix with 0's on the main diagonal and if it satisfies the requirement that if any two rows contain a pair of 0's which are symmetrically located with respect to the main diagonal, then the inner product of those two rows must not exceed D . Z will denote the number of off-diagonal 0's in A . If A is the incidence matrix of a linear graph, the rows being identified with points, then two rows containing a pair of symmetrically located 0's represent two points not joined by a line, and the require- ment on inner products has the interpretation that two-such points are joined by at most D 2—chains. If A is a principal minor submatrix of an association matrix of a PBIB design, the rows being identified with treatments, then two rows containing a pair of symmetrically located 0's represent treatments which are not associates, while the inner product of two rows is equal to the number of treatments (of the set corresponding to the submatrix) which are common associates of the two treatments. DEFINITION 4.2. This paragraph constitutes the definition of a set of integers s , s , ... , er and a set of submatrices Qi.1 of a 1 2 symmetric incidence matrix A with 0's on the main diagonal. 31 will denote the maximum order for a principal minor submatrix of A which has 1's in all off-diagonal positions. If there are no 1's , s1 3 l . Denote an 31 x sl minor of this form.by this form. Q11 . The value of 81 is uniquely determined but possibly not the set of rows and columns in 011 . It is not essential but will simplify the later discussion of 178 rows and columns of A are permuted (simultaneously) so that Q11 is the leading principal minor. In the remaining principal minor submatrix, determined by the remaining t~s1 rows and t~s1 columns, let s2 denote the maximum order for a.principal minor submatrix with 1's in all off-diagonal positions, denote such a submatrix by , and permute Q22 these rows and columns so that Q22 is in the next diagonal position. Clearly s1 2 s2 . Different choices of Q11 may lead to different values of 32 ; for a particular choice of Q11 , the value of a2 is determined but possibly not the 52 x 32 submatrix Q22 . Repeat for the remaining diagonal submatrix, and so on until A has diagonal blocks of order s .> s l " 2 the maximal property described for s1 and s2 , elements of each block are 1's . The ith diagonal block will be denoted .> ... where s s ‘+-... +~s = e ch h ...-.852 23f, 1+ 2 f t, a 81 38 and all off-diagonal by Q11 ; the submatrix determined by the rows of Q11 and the columns of _ij ‘will be denoted by Qij . This partition of A is illustrated in the fellowing diagram. Q11 5 Q12 ’ . . . : Qlf s1 rows , __ -_._.— _ ...-..- ..-— — -- 4 ....... -921. -1- .2? - - 3.1:- T. - -8. ."_°‘I’.’ (4.15) A: O C 0:. O O 0:. O O O 0:. O O O O C - _ - -.- _ - 7 - -..- -.- -..- ....... Qfl 1 Qrz . ' ' ° ' fo 3r r°ws ’ It is desired to investigate the possible values of Z in the matrix A specified in Definition 4.1. A matrix with Z = t(t-l) , containing 0‘s everywhere or a.matrix with Z 3 0 , with 1's in all offediagonal Positions, satisfies the requirement, but there may be intermediate values 179 of Z which are impossible. If the matrix is partitioned according to Definition 4.2, the fact that the diagonal blocks Qii contain no off- diagonal 0's will provide an upper bound for Z . 0n the other hand, it is evident that the off-diagonal blocks Q1 must contain some 0's, 3 or it would be possible to form some larger diagonal blocks full of 1's , violating the maximal property of the s1 . This type of reasoning is put into a definite form in the following theorem. THEOREI 4.4. If A is an incidence matrix satisfying definition 4.1 and s1 , s2 , ... , s are determined according to Definition 4.2 , f then the total number 2 of off-diagonal 0's of A satisfies the following inequality. F P (4.14) :1: sJ Iax( s1 + sj-D , 281.2D , 2 ) "' =i+l ‘F _<_ z s t(t-l) - 28551-1) . .g' PROOF: The partition of A depicted in (4.13) will be used. Take any subatrix Q1 with i<5 s an s x 31 matrix, and con- J i sider any m x k submatrix of Q1.1 which contains 1's everywhere. This submatrix, the symetrically located portion of jS , an m x m submatrix of Q11 and a k x k submatrix of Q.“ can be combined to form a symmetric (m + k) x (m +k) matrix with no off-diagonal 0's . By the maximal property of 8i , it is necessary that m + k3 s1 . Next consider the set of si rows of Q and for each column define ij ’ a subset consisting of all the rows which contain 0's in that column. There are 8.1 subsets in all. Take any I: of these subsets, corresponding 180 to a k-columned submatrix of 011 . By the inequality just proved for submatrices containing 1's everywhere, at most si-k rows of this submatrix contain l's everywhere, meaning that at least 1: rows contain 0's . This means that any I: of the subsets of rows of QM contain between them at least k distinct rows. This is true for k = l , 2 , . . , s.1 . By the theorem of P. Hall [521,[2—7] on represen- tatives of subsets, there exists a system of distinct representatives of the s subsets. That is, there are 81 distinct rows of Q which J 1.1 may be ordered so that the luu‘ row contains a 0 in the fit“ column of Qij . So far we have shown that Q1.1 contains at least s 1 0's no two of which are in the same row or column. This seems to be about the best possible result using nothing but the condition that the s are maximal, i but the condition on inner products of A still has not been applied. This will be done next. We will use the suhnatrix consisting of the blocks Qii , Q13 , Q11 , Q” ,. still with i < j . This is a symmetric (s1 + s1) x (81+ s3) matrix. It has been shown that Q“ contains a set of 83 0's , no two of which are in the same row or column. Con- sider one of these 0's and its symmetrically placed 0 in q.11 . The two rows containing these rows must have inner product fl D . Therefore at most D 1's may occur in the remaining cells of these two rows of Qij and jS , meaning that there are at least a + s ~D-2 additional 1 J 0's in these two rows. This can be repeated for each of the initial 0's , 181 r_o' . g . _ '. :Qii: :Qn 11.i51.1:...5... ee | 0 e e | e -----:_--91-_-.°-_-_ . IO . jQJiI “,1an ....0000:1.101.1 I | I'. O ‘ C 0 L . J and since they were in distinct rows and columns, the additional 0's all fall in different rows and are therefore distinct. This proves the existence of at least 8 s s - D - 2 J(i+j ) 0's in blocks Qij and 051 , in addition to the 2sj already discussed. If si-+ sJ $_D +-2 , this is vacuous, but in any case the existence of the 2sj 0's has been proved. Therefore a lower bound for the number of 0's in blocks Qi and J jS is ( ) ax( 8j(8i J ) , sj) Again considering the initial 0 in Qij , note that the number of additional 0's in the row of Q which contains the symmetrically ji placed initial 0 is by symmetry equal to the number of additional 0's in the column of Qi containing the initial 0 . There are therefore 3 at least si-+'sj-D—2 additional 0's in the row and column of Qij 182 J bined, some of the additional 0's may be counted more than once, but containing any of the sJ initial 0's. If these s totals are com- since no two of the initial 0's were in the same row or column, none are counted more than twice, and the number of dupliCations is at most equal to the number of cells of the submatrix which are in the same row as one of the initial 0's and in the same column as another. The num— ber of such cells is sj(sj-l) . A lower bound for the number of addi- tional 0's in Qij is obtained by subtracting this from the combined total, sj(si+ s -D-2) - s j j(sJ-l) = sj(si-D—l) . If s _<_ D + 1 this is vacuous, but in any case the existence of the SJ 1 initial 0's in has been proved. The number of 0's in jS is Q“ equal to the number in (:11 giving as a lower bound for the number of j ’ 0's in both blocks (4.16) lax< 31(231-2D) , Zsj) . This may be combined with (4.15) to show that blocks Qij and Q51 contain at least sJ llax( Bi + sj-D , 231‘2D , 2) 0'8. Summation over all the off-diagonal suhnatrices Q1 gives the 1 lower bound stated in the theorem. The upper bound in the theorem is the total number of cells in the submatrices Qij , since all the off-diagonal 0's are in these submatrices. This completes the proof of Theorem 4.4. 185 The set of values of Z satisfying (4.14) includes the set of possible values of Z for all matrices A satisfying Definition 4.1 and for which the procedure described in Definition 4.2 leads to a particular. partition of t into positive summands s1 . The union of the sets ob- tained from (4.14) for all admissible partitions of t them includes all possible values of Z for a given order t x t of matrixv A and a given value of D . The class of admissible partitions of t may be restricted. For example, if A is the incidence matrix of a graph which.is known to contain complete 3-points but no complete configurations with as many as t—l points, then 5 _<_ s1 3 t-2 . 0n the other hand, if restrictions on the value of Z are known, this theorem may restrict the possible parti- tions of t and in particular the value of s1 . It may be possible to obtain a slightly better lower bound than that given by the theorem“ One of the lower bounds for the number of 0's in submatrices Q11 and Qi which is twice the number of 0's in Q J ’ 13 and is therefore an even number, is 81(81 + sjeD) . If this product is odd in any term of the sum, it may therefore be replaced by the next larger even number. The following_numerical examples illustrate the theorem and the remark just made. NUMERICAL EXAMPLES Each term of the double sum in the lower bound given by (4.14) is a function of D and of two of the s values, and is independent of t . i A preliminary table of values of 8 Max 3 + s.—D , 28 -2D , 2 , J i J 1 184 computed for a fixed value of D and a suitable range of values 85. 2 s. , J is convenient for use in evaluating the sum and can be used for any value of t . A table of this kind follows, computed for the case D 3 4 . sj 1 2 5 4 5 51 l 2 - - - - 2 2 4 — - .. 5 2 4 6 - - 4 2 4 l_0_ lD - The underlined entries in this 5 2 6 12 20 50 table replace the computed values 6 4 8 y; 24 £6 9 , 15 and 55 , For a particular value t and partition 31 + 82+ ... + ef the summation over pairs s1 , s.1 , i < j is quickly carried out. For t = 7 and the partition 5 + 5 + 1 , the sum includes three terms. The term resulting from the pair 81,3J = 5,5 is 6 ; the pair 5,1 occurs twice, each time contributing the term 2 ; and the total is 6 + 2(2) = 10 . There- fore if the process described in Definition 4.2 leads to diagonal blocks of the following form in a 7 x 7 matrix, the number Z of off—diagonal 0's in the matrix must be at least 10 . The upper bound given by (4.14) is so , the total number of cells in the off- 185 diagonal blocks. Similar bounds for all of the partitions of 7 are listed as further examples. Partition Lower bound Upper bound of t on Z on Z 7 0 _ 0 6 1 4 l2 5 2 6 20 5 l 1 2(2) + 2 -‘-'- 6 22 4 5 10 24 2 l 4.1 2-+ 2 i s 28 4111 5(2)+3(2)1l2 50 5 5 l 6 + 2(2) 3 10 50 522 2(4)+4=12 52 3211 4+2(2)+2(2)+2=14 34 31111 4(2)+6(2)=20 56 2 2 2 l 5(4)-+ 5(2) = 18 56 22111 4+6(2)+5(2)=22 58 211111 5(‘2)+19(2)=30 40 lllllll 21(2)=42 42 0f the conclusions which can be drawn from these results, the following are typical. (a) The value 2 = 2 is impossible. (b) If the graph contains no 4-points, meaning s1 _<_ 3 , then Z 2 10 . Note that the proof of this requires consideration of the lower bounds for all partitions with s13 5 , and does not follow from the particular result obtained for the partition 5 3 1 . 186 (c) If Z is known to satisfy Z<8 , then e125 , proving that the graph contains a complete 5-point. The restriction on inner products is essential, as shown by the following example, in which Z = 6 and s = 4 but inner products such as that of rows 1 and .5' l are not €344 . "6111011‘ 1011101 1101110 1110111 0111011 1011101 _1101119, The lower bound on 2 given by Theorem 4.4 is rather complicated, 1 , 32 , eee ’ 8f in a puti‘ tion of t , and it suffers from the disadvantage that it applies only depending as it does on all the terms 8 to a particular partition. In order to get a lower bound which depends only on t and D it is necessary to minimize over a class of parti- tions of t which may be very large. Four lower bounds will now be de- rived which involve ‘1 but none of the other 81 . It is simple to apply these formulas and take the maximum of the values obtained as a lower bound on 2 , Valid for all partitions in which the largest term is 51 . A lower bound which depends only on t and D may then be obtained by minimizing over admissible values of sl . Lower bounds thich are inde— pendent of 82 , ... , 88 are not only simpler but more useful. In the present application of Theorem 4.4 and the following theorems, the object will be to prove that the linear graph formed from the incidence matrix A has a complete subgraph whose order exceeds a certain minimum value. 187 The value 51 is important here, since it may be interpreted as the 0 maximum order of a complete subgraph. The values s2 , ... , sf will be of less interest and their interpretation is not so simple. The first simplified lower bound on 2 follows directly from Theorem 4.4. COROLLARY 4.4. If an incidence matrix A satisfies Definition 4.1 and if 81 is defined by Definition 4.2, then the total number 2 of off~diagonal 0's in A satisfies the inequality (4.17) 222(t-sl)(sl-D) . PROOF: This inequality is obtained from (4.14) by taking only the terms of double sum corresponding to i 3 l and taking the second of the three expressions in parentheses. The sum then reduces to f‘ :5 (2s -2D) = 2(s -D) 28, . . J 1 1 - J 4:2 m. Since sl-+ 32-+ ... +-sr = t , the sum in the right member reduces to t-s and the result is proved. 1 The 0's enumerated in this corollary are those in the first 31 rows of A , that is, in submatrices Q12 , Q15 , ... , Qlf , and the symmetrically located 0's in the first 61 columns of A . If s1 is nearly as large as t , these rows and columns will contain most of the off-diagonal 0's , and inequality (4.17) may be nearly'as strong as 4.14) . For small values of 81 , it becomes much weaker, collapsing mmthfw fiSD- 188 The two lower bounds for 2 given in Theorem 4.5 do not follow from the statement of Theorem 4.4 but use some of its proof. THEOREM 4.5. When A is an incidence matrix satisfybng Definition 4.1 and s1 , 82 , ... , sf are determined according to Definition 4.2, then the total number 2 of off-diagonal 0's of A satisfies both of the follOWing inequalities. (4.18) z 2 §(t-D)(t-sl) ; (4.19) 22%“. - sl)(t + 31 - 2n) . PROOF: The symbol Ii will be used to denote the total.number of off-diagonal 0's in the 81 x t submatrix of A consisting of blocks Q11 , Q12 , ... , Qif . In this notation the proof of Corollary 4.4 implies the statement (4.20) 112 (t - sl)(sl - D) . If two rows of A contain a pair of 0's which are located symp metrically with respect to the main diagonal, then by the restriction on inner products of rows, there can be at most D columns of A which contain 1's in both of these rows, and the two rows together must con- tain at least t—D off-diagonal 0's , including the original pair. In the proof of Theorem 4.4 it was shown that for i<(.J , submatrix Q15 contained a set of 31 0'8 , no two of which were in the same row or column. These 0's and the symmetrically located 0's in jS there- fore lie in 23j distinct rows, forming 8 pairs of rows, each pair 1 Satisfying the inner product condition and containing at least t-D 189 off—diagonal 0's . The 28 rows are all contained.in the (si-p 8)) x.t .1 submatrix whose rows are determined by submatrices Q15 and Q giving 11’ the result (4.21) 11+ niacin-n) , i "’ "‘ e (4 25) 22 If _, (t sl)(t + 81 21)) The term Yf is dropped again and (4.19) is obtained, completing the proof. The two inequalities of this theorem are weaker than (4.1?) for large values of 81 , but give better results when 31 is small. The expres- sion in the right member of (4.19) has its maximum value for s1 3 D , the value for which (4.1?) gives the trivial lower bound 0 . For 81 < D , (4.18) is the strongest of the three inequalities. 190 These three lower bounds on 2 fill the needs of the present section, but may fall far short of the actual minimum value of z for many values of s1 . For example, when 81 = l , the best result obtained from any of the three is Z 2 §(t-l)(t-D) , a very conservative underestimate, since 31 - 1 means that A contains no 1's and the actual value of Z is t(t-l) . A fourth lower bound, which will be given in Theorem 4.6, is stronger for very small values of sl , but makes no use of the restric- tion on inner products and is of little use for large values of ‘1 . It is closely related to a known result in graph theory which will be men- tioned following the proof. THEOREM 4.6. If A is a symmetric incidence matrix with 0's on the main diagonal, and if 31 is determined according to Definition 4.2, then the total number Z of off-diagonal 0's of A satisfies the inequality (4.24) 221‘?- -t . ‘1 PROOF: Inequality (4.14) of Theorem 4.4 is used, taking the third of the three expressions in parentheses. The inequality reduces to F F (4.25) 222: :3 . 1"] Jsigl J This sum is represented graphically by the sum of the areas of the rect- angles in the figure below, which is located with reference to a rectangular coordinate system with origin 0 . 191 1' Vertex Pi has coordinates ( Z sj , i ) 5 J” vertex Qi has coordinates ( t , i-l ) . In particular, the coordinates of P1 and P1. are (s1 , l ) and (t , 1‘), respectively. Rectangle Pic;1 then has altitude l and area equal to the 5 sun Z 33 , and the sum of the areas of all the rectangles is equal Fifi to the double sum in (4.25) . It will be convenient to deal with the area of the polygon OF P . . Pfql , which exceeds the combined area of the l 2 rectangles by exactly it , the combined area of triangles whose altitudes an equal to l and the sun of whose bases is t . 192 r r (4.26) s=Area opp ..PQ-gt. :2: 12 ‘1 Since 3 >8 >...>s the o onalline OPP ..P iscon- 1 ‘- 2 '- " r ’ p I” 1 2 r cave upward and the area of the polygon is not less than the area of triangle ORQl , where R lies on 0Pl extended. 0Ql '- t and by similar triangles Q18 2 I: , giving the result 3 1 A one)1 ‘2 r35 = u- , 281 which is enough to prove the theorem. However, it will be of interest to prove (4.28) below, a slightly stronger result which calls for closer study of the figure. Q18. cannot exceed the integral length f of the sealant Qle 5 if t is written in the form =asl-b, a and b integers, o£b21t-2 , the exact value is given by (4.50) m(t , D = t—2 , 31) = (a-1)(t-b) . It was also remarked that in this case and for fixed t , the function is monotone decreasing in s . not even this is known for most values 1 of D , though it may be conjectured that increasing the order of the largest complete subgraph of a graph will.necessitate an increase in the total number of lines. The lower bounds on 2 derived in Corollary 4.4, Theorem 4.5 and Theorem 4.6 are of course lower bounds on m(t , D , sl) . The following notation will be useful in discussing these bounds. (4.51) Bl(t , D , 31) = 2(t - s1)(s1 - D) , (4.32) 82(t , D , s1) = g(t-slxt + 31 - 20) , t(t - but - a1) . 2.2... 8J. (4.55) 85(t , D , s1) (4.54) s4(t , D . 81) Then the fact that these four expressions are lower bounds on Z may be expressed (4.35) n(t,D,el)231(t,D,al), i=1,2,5,4. The dependence of the four bounds on sl for fixed t and D is shown schematically in the following figures. I96 197 As indicated by the figures, for fixed t the bounds Bl , 32 and 85 become weaker as D is increased. It is probably to be expected that an increase in D will permit an increase in the inner product of some pairs of rows of the incidence matrix, allowing some 0's in such rows to be replaced by 1's and decreasing the value of Z . In the extreme case D Z t-2 , the bound B 4 is the only one needed, as men- tioned above. In the cases shown in the figures, each of the four bounds is stronger than the others for certain values of 31- The figures indi- cate that for s1 = 333:9, the lower bound for Z is “at-mg, and that Z can be less than this value only for s1)? . This observation is essential in the proof of the next lemma. The information needed to show that it is true in general is contained in the following table. was or VALUES ' towns BOUND ON ' VARIATION OF BOUND (mono- or 31 m(t , n , s1) tone within each interval) 1 _<_ 81 g D (1.) ; 85 Decreases from fi(t-D)(t-l) ' to fi(t-D)2 D _<_ “is t ; 2D 82 Decreases from t(t-DY? - to g-(t-d)2 t ‘2: 2D 3 s13 3—3213 X Bl ‘ Increases from g(t-D)2 i to aha-ma t+D 333 , and that for 81 in this range, m(t , D , 81) is bounded below by Bl(t , D , s1) , which is monotone decreasing in el . This implies the following lemma. LEMMA 4.4. If A 18 in incidence matrix satisfying Definition 4.1 and for some a‘ > £L§LD the number Z satisfies the inequality (4.36) zsalu . D . er ) = 2(t-r)(o*-D) . .> . then sl_cr The know lower bounds on m(t , D , sl) admit the possibility that z is less than g(n-n)2 , (but not less than gun-O)2 ) for D <81<-t-'--2~2 . If m(t , D , 81) were known to be monotone decreasing in 81' , Z would be known to be at least equal to t(t-D)2 for all 31 in this interval and the restriction on (7' in the statement of the lam could be weakened to r > 34-32 . The purpose of Theorem 4.4, Corollary 4.4, Theorem 4.5 and Theorem 4.6 has been to provide methods of proof strong enough to extend Theorem 4.2 to Latin square type association schemes with.more than two constraints. It may be recalled that it was desired to prove that the submatrix All , shown in (4.1) , contains a complete (n-2)-point. The proof that this is true for most Lg schemes will be completed in the next 3 theorems, the first of which requires Lemma 4.4. The other theorems and corollary which have been proved in this section will not be used explicitly. Submatrix All was defined by (4.1) but it will be convenient to recall the 199 definition here. Where A1 is the matrix of first associates in an association scheme with two classes, and two initial treatments which are first associates are chosen, All is the submatrix whose rows and columns are determined by the pi]. con-non first associates of the two chosen treatments. It is a ph x ph symmetric matrix with 0's on the main diagonal. THEDREI 4.7. In any association scheme with two associate classes, define .. 1 2 1 2 2 1 2 0" .. .. - . - - (4.37) . t(pn +911 2) +3W11 911+” 2p12(pn 1) and 2 1 + 2 (4.58) a‘ —_‘.’l-.1..3.1.1_.:.3 1 p 3 Then if 0'. is real and 0', > <7; , each pair of first associates in the scheme is contained in a set of k treatments which are pairwise first associates, where k 2 0". 4' 2 . PROOF: Lama 4.4 will be applied to sutmatrix All of the incidence matrix Al of first associates in the scheme. Two rows of All con- taining a pair of symmetrically located off-diagonal 0's correspond to two second associates, and there must be exactly pfl columns of A1 which contain 1's in both of the two rows. Columns 1 and 2 are of this form. Therefore there can be at most PEI - 2 such columns in A 11’ and All satisfies the conditions of Definition 4.1, with t = pil , =p§1-20 200 By Lemma 4.5, the number 2 of off-diagonal 0's in A].1 satis- 11 fies inequality (4.2) , 2 . Z11gpi2(pl1 1) ° (71 Will be defined by setting B1“ , D , 0-1) = p}2(pfl - 1) , where- upon (4.2) assumes the form of (4.56) in the statement of Leanna 4.4. Using the given values for t and D , the definition of 0'1 may be written mph - 0'1)( 0'1 - p§1+ 2) = pimpil — 1) and solved for 0' 1 to give the definition (4.5?) in the statement of this theorem. The other root of the quadratic equation, if real, will be less than L}; and cannot meet the conditions of Lemma 4.4. Definition (4.58) for a- is equivalent to 0- 2 = 3593112 and the 2 hypothesis 0'1 > 0’2 of this theorem is identical with the condition placed on 0“ in Lemma 4.4, where 0’1 here plays the role of 0' in the lemma. The lemma may then be applied to show that All contains a principal minor suhnatrix of order 8 Z a- with Us in all off- 1 l diagonal positions. The s1 corresponding treatments of the association scheme, together with treatments 1 and 2 , form a set of = sl+ 2 Z 0'14- 2 treatments which are pairwise first associates. Since treatments 1 and 2 were taken as an arbitrary pair of first associates in the definition of the submatrix All , this completes the proof of the theorem. 201 The inequality 0.1 > 0-2 can be transformed by straightforward algebra to the form (4.58s) 5 Wail - tail + 2)2 - ZPiJPfl " 1) > Pil " P§1+ 2 Squaring both sides gives the following inequality, which is true only 1 2 if (4.58s) is true, is equivalent to it if p11 - p11 + 220 , and is somewhat simpler to apply. 2 (4.39) 4(1)}1 - pfl + 2) - episzl - 1) > O . It will be shown later that for fixed g , association schemes with parameter values of the Latin square type with g constraints satisfy the conditions of Theorem 4.7 if the number n2 of treatments is suffi- ciently large, proving the existence of a complete k-point on every line of the graph, with each 1: 2 0'1 + 2 . However, for g 2 3 , 0'1 is too small for the existence of a complete n-point to be proved by this theorem alone. The next theorem and lemma bridge the gap in the argument. THmREII 4.8. In any association scheme with two associate classes, let there be a set A of k1 treatments which are pairwise first associ- ates, and a set D of k2 treatments which are pairwise first associates, and let the intersection of the two sets contain u treatments. (1) If u _>_ 2 , then 1 4.40 k k - u< + 2 . ( ) 1‘? 2 __Pll (ii) If there is a treatment in either set which is a second associ- ate of a treatment in the other set, then 202 2 (4o 41) u S- p11 0 (iii) If 5+ k2 > 9h + p§1+ 2 , then either n51 or all treatments in the union of sets A and B are pairwise first associates. PROOF: If u_>_ 2 , then there are at least two treatments which are in both of sets A and B , meaning that each is the first associate of each of the k1 + k2 - u - 2 remaining treatments in the union of A and B . But no two first associates can have more than p; first as- sociates in common. Therefore k1+ k2- u-2fi I’ll , proving statement (1). If there is a treatment 0 in set A and a treatment 95 in set B which are second associates, they can have at most pil in common. But 9 is a first associate of all the remaining treatments first associates in set A , O is a first associate of all the remaining treatments in set D , and the u treatments which are in both sets are common first associates of both 0 and ¢ . Therefore u_<_ pil , proving statement (ii). If the hypotheses of both of statements (i) and (ii) are satisfied, then inequalities (4.40) and (4.41) are both true and may be added to give 1 2 kl-t- lszgpl1 +pu+ 2 . If the contrary is true, then one of the hypotheses of statements (1) and (ii) must be false, meaning either that “S l or that each treatment in each set is a first associate of all treatments in both sets, Which means that all treatments in the union of the two sets are pairwise first associates. This proves statement (iii) . 205 In terms of the linear graph whose incidence matrix is Al ment (iii) of Theorem 4.8 means that if the graph contains two complete , State- configurations, or k-points, of orders kl and k2 , and if k1‘ + k2 > ph + pili- 2 , then either the two configurations have no line in common or the graph contains a complete configuration of which both are subgraphs . LEMMA 4.5. In any association scheme with two classes, if for any treatment 0 there exists an integer k0 satisfying - l 2 k0 > :(pu + 911+ 2) such that every pair of first associates including 9 is contained in a set of k _>_ k0 treatments which are pairwise first associates, then the 111 first associates of 0 fall into disjoint sets, each set together with 9 forming a complete configuration with at least to treatments. PROOF: Each of the :11 first associates of 0 forms with 0 a pair of first associates which by hypothesis are contained in a complete configuration of k 2 k0 treatments. Form one such configuration on 0 and each of its first associates and consider the sets of treatments in the m1. configurations. These are subsets of the set consisting of 9 and its first associates. If any of the sets are identical, drop the duplicates. Since each set contains more than f(Pil +p§l 4’ 2) treat- ments, any two of them satisfy the hypothesis of statement (iii) of Theorem 4.8. 9 is in each of the sets, and if any first associate is in two of the sets, the two sets have u _>_ 2 first associates in common and by Theorem 4.8 their union forms a complete configuration. In this case, 204 drop both sets and use their union instead. This process may be repeated as long as any of the first associates of 9 are in more than one set. After a finite number of repetitions the result will be a set of disjoint sets of first associates, each set together with 0 forming a complete configuration. Each configuration contains at least to treatments, since it is formed by union of sets having at least he treatments. THEOREM 4.9. If an association scheme with two associate classes and v = n2 treatments has the parameter values of a Latin square type scheme with g constraints, and if n exceeds the larger root of each of the equations (4.42) 4n2 — (g-l)(9g2-9g +-7)n +-(g-l)2(9g2-9g +—7) = O , (4.45) .‘Bgn‘a - (gs-2g4 + figs-g2-2g +l)n - '(gG-Sgs + 334 + ng—ng + g+l 1' 0 , then there exists a set of g mutually orthogonal n x.n squares which may be used to define first associates in the scheme, and the scheme is of Latin square type. PROOF: The parameters of a Latin square type scheme with 3 con- straints include the following. P&1 ’ n +'82 ' 53 9 2 p11 pig -(g-1)(n - a +1) . 82 ' 8 9 t 67‘ and 47'2 will be defined as in Theorem 4.7 and 671 has the form 205 \ n L 2 0-1 = 2‘01 +2g‘-4g-2) + 5 V(n-Zg +2) - 2(8-1) (ii-'8 +1)(82-g’l) - Statement (4.42) will be needed in the application of Theorem 4.7 and Lemma 4.5, while (4.43) will be needed in the final part of the proof. As a preliminary step, it will now be shown that the hypotheses imply n2 2g , a fact which will be used to simplify the application of Theorem 4.7. When n = 2g , the expression in equation (4.45) reduces to -5g6 + 7g5—9g4 + 8g3 + 732-534. , which is easily shown to be negative for all g 2 2 , showing that the larger root of (4.43) is greater than 2g for all g _>_ 2 . It is no restriction to take 11 2 2 in the special case g =-' l . It may there- fore be assumed for any g that n 2 2g. Since ph - p§l+2 = n-2g+2 , this implies (4 44) p1 «- p2 + 2 > 0 e 11 n e It was pointed out following the proof of Theorem 4.7 that if (4.44) holds, the inequality 0"1 > 0’2 is equivalent to (4.59). In the present case, (4.39) has the form (4.45) 4(n - 2g+2)2- 9(g - l)(n - g+l)(g2 - g - 1) >0 , reducing to (4.46) 4:12 - (g-l)(9g2-9g+7)n +(g-l)2(9g2-9g+7) >0 . If n exceeds the larger root of (4.42) , this inequality will be satis- fied, implying 0"l > 0‘2 , and by Theorem 4.7 each pair of first 206 associates in the scheme is contained in a set of k pairwise first associates for some kg k0 : 0'1 +- 2 . The relations 0-1 > 0.2 and (4.44) are used in the following inequalities on he . k 0 2 1 2 0.1% > <72”. 2 2112911” = ....491.1+2911+8 : 6 1 2 1 2 1 2 51)11+ 3911+ 6 + {311‘ 91le ? 5911+59A11f§ . 6 6 6 (4.47) Re) $(ph-r- pil-l- 2) . Therefore the conditions of Lemma 4.5 are met for any treatment 9 , proving that the n first associates of any treatment 9 fall into 1 disjoint sets, which will be referred to as special sets, each special set containing at least 0‘1 + 1 treatments and forming with 9 a complete configuration of at least 0”l + 2 treatments. 0" will now be re- 1 quired to satisfy the condition I! -1 (4.43 7+1 1 = n . ) l >g+1 3+1 This reduces to (s + 1) V (rt-2s + 2) 2 - 2(sol) (n-s + 1) (82-3-1) > (3-1) n-2s54- 282-2 and is satisfied if the following inequality, obtained by squaring and simplifying, is satisfied. 5 2gnz- (gs-2g4+ 5g5-g2-2g+l)n - (gs-5g + 5g4+ 2g5~5g2+ g +1) > O . This in turn is satisfied if n exceeds the larger root of (4.45), so that the hypothesis of the theorem implies (4.4.8) . It follows from (4.43) 207 that (g + 1)( 0’1 + 1) ) nl . Since the number of treatments in each special set is at least UPI-t l and the sum of the numbers of treat- ments is n this implies that the number of sets must be less than i 9 g +-l . By corollary 4.1 , 9 does not lie in any complete configu- ration with more than n treatments and none of the special sets can contain more than n-l treatments, and in order for the sum of the num- bers to be n1 3 g(n-l) there must be at least g special sets. There~' fore there must be exactly g special sets, each containing exactly n-l treatments, meaning that 9 , which was an arbitrary treatment, lies in exactly g complete configurations of n treatments, and each pair of first associates lies in such a configuration. Then by Theorem 4.1, there exists a set of g mutually orthogonal n x n squares which may be used to define first associates in the scheme, completing the proof that the scheme is of Latin square type. The requirements placed on n by Theorem 4.9 will now be examined more closely, in a few cases by using the exact solutions of equations (4.42) and (4.45) . When g 3 2 , equation (4.42) becomes 4n2 — 25m +'25 = O and the larger root is n = 5 ; equation (4.43) becomes 4n2 - l7n - 43 = O and the larger root is n = 6.05 3 the theorem applies for n 2 7 . A better result has already been obtained in Theorem 4.2. When g = 5 , equation (4.42) becomes 4n2 - 122n'+-244 = 0 and the larger root is n = 28.5 ; equation (4.45) becomes 6n2 - l48n — 319 = 0 and the larger root is n = 26.6 ; the theorem applies for n 2 29 . 208 When g 3 4 , equation (4.42) becomes 4112 - 54511 + 1055 = O and the larger root is n = 85.2 ; equation (4.45) becomes an2 - ssin - 1957 = o and the larger root is n a 87.9 ; the theorem applies for n 2 88 . When g 8 5 , equation (4.42) becomes 4n2 - 748n + 2992 = 0 and the larger root is n = 182.9 ; equation (4.45) becomes 10h2 - 2216n - 84.31 a o and the larger root is n = 225.5 3 the theorem applies for n? 226 . For use with larger values of g , the general solution of (4.42) is easily obtained, giving the inequality (4.49) For g 2 2 , the expression 9g2 - 9g +7 is positive and dropping the second term in the radicand increases the value of the right member of (4.49), showing that it is sufficient for n to satisfy 5 2 n > (g—l)(9& - 9g+7) ,3 9g - 18g iris; - 7 4 4' Still for g2 2 , an even stronger requirement on n is 9 3 (4.59) n>—§-— . The general solution of (4.45) leads to (4.51) n > g5 - (234-3334 82+Zg-l)+Vglo - (439-1cg8+6g7+1536-23gs-g4.26g5-1032-4g-1) 4“ 4E 209 For g2 2 , the two expressions in parentheses are easily shown to be positive, and a sufficient condition for n to satisfy (4.51) is obtained by dropping them, giving (4.52) n > its“ Therefore, for g2 2 , any n satisfying (4.50) and (4.52) will exceed the larger root of each of the equations (4.42) and (4.45), permitting the theorem to be applied. For g25 , (4.53) is a weaker requirement than (4.52) and may be dropped if (4.52) is used. The results of the last few paragraphs are summarized in the following corollary. COROLLARY 4.9. If an association scheme with two associate classes and v = n2 treatments has the parameter values of a Latin square type scheme with g constraints, then the following conditions are sufficient that the scheme be of Latin square type. If g 5, n229; if g=4, n288; M if g 5, n2226; if g26 , n>§g4 . Theorem 4.9 shows that for any fixed g and for all values of :1 except a finite number of possible exceptions, the Latin square type associ- ation scheme is unique in the sense that it can be constructed only by . means of a set of g orthogonal squares. Corollary 4.9 gives explicit upper bounds below which any exceptional values of n must lie. It may be noted that in the cases g = 2 , 5 , 4 and 5 the bound given by the 210 simplified inequalities (4.53) and (4.52) is considerably larger than the one obtained from the original equations (4.42) and (4.45). The bounds approach each other in an asymptotic sense, as illustrated by two more special cases. H When g a lo , §g4 5000 and the larger root of (4.43) is 4152.9 . 4 When g = 100, 5g 7 5 x 107 and the larger root of (4.45) is 4.90 x 10 . The difference between the bounds is unimportant in any study of designs within the useful range, since no Latin square type designs used in any ordinary statistical experiment at the present time require a value of n larger than 20 . For g;3 5 the question of exactly which values of n admit nonéLatin square type designs with Latin square parameter values is still far from solved. For g 3 5 , the scheme with n 2 4 is easily shown to be unique, it was shown in counter-example l of section 4.1 that the scheme with n = 5 is not unique, and the question of uniqueness has not been answered for the schemes with 6 g 113 28. In each of the three examples in Section 4.1, second associates could be defined by a set of orthogonal squares, suggesting that in any scheme with Lg parameter values, either-first or'secondlassociates can be so de- fined. No proof or disproof of this statement is known, but its implications may be illustrated by a.numerical example. If a scheme with L4 parameter values exists in which first associates cannot be defined by any set of 4 orthogonal squares, and if the statement is true, then second associates can be defined by a set of n-5 orthogonal squares, meaning that a set of n-S orthogonal Latin squares exists, and showing incidentally that 211 L8 schemes exist for all BS n—5 . This is far more than is known for any values of n2 10 which are not prime powers. A number of methods have been employed by the writer in an attempt to find a better result than Theorem 4.9, but without much success. The nature of some of these methods will be mentioned as a guide to possible future work on the problem. The crucial step in the proof is to show that each pair of first associates, together with n-2 of their common first associates, form a set of n treatments which are pairwise first associates. This means for the incidence matrix A1 of first associates that the submetrix All , defined in (4.1), contains an (n-2) x (n-2) suhmatrix with 1's in all off-diagonal positions, or in terms of linear graphs, that in the graph whose incidence matrix is Al , every line is contained in a complete n-point. This step of the proof, which took one paragraph in the case of Theorem 4.2 and L schemes, has occupied most 2 of the present section in the general case, and has been divided into three phases, as follows. Part 1: Lemma 4.5. The number 211 of off-diagonal 0's in All 13 small. Part 2: Theorem 4.7; Lemma 4.4 and Theorems 4.4 to 4.6. If 211 is small, then each line is contained in a complete kfpoint, where k is fairly large. Part 5: Theorém 4.8 and Lemma 4.5; parts of proof of Theorem 4.9. If the k-points are sufficiently large, they can fit into the graph only if the scheme has Latin square structure. 212 The remarks to be made about Parts 2 and 5 are brief and will precede the discussion of Part 1. The principal result used in Part 2 is Lemma 4.4, which makes use of the lower bounds on m(t , D , s1) developed in Theorems 4.4 to 4.6. If a proof along the lines used in Theorem 4.9 is to be improved at this point, it would seem that the thing to look for is a stronger lower bound. As pointed out following the proof of Lemma 4.4, a proof that m(t , D , 81) is monotone decreasing in 31 would immediately permit a stronger state— ment of the lemma. While there is considerable literature on linear graphs and their incidence matrices, the case in which there is a restric- tion on the inner product of rows of the matrix does not seem to have received much attention and it is possible that more information on the number of 0's in A could be obtained. In Part 5, it is shown without difficulty that the first associates of each treatment fall into disjoint sets, each forming with the given treatment a set corresponding to a complete k—point of the graph. Then in the proof of Theorem 4.9 the condition (4.48) is imposed. This insures that each of the disjoint sets of first associates is large enough that the number of sets can be at most g , making it easy to prove that the number of sets must be exactly g . It turns out that (4.48) requires much larger values of n for all g}: 4 than any of the other conditions imposed. If additional information could be obtained about the kppoints in the graph, or about the number of disjoint sets of first associates, it might be possible to replace condition (4.48) by some weaker requirement. 215 Most of the writer's attempts to generalize Theorem 4.9 were concen- trated on Part 1, the derivation of an upper bound for the number 211 of off-diagonal 0's in submatrix A11 , or equivalently, of a lower bound for the number T11 of 1's in A The definition of an 11 ' association scheme for a PBIB design is enough to determine the number of lines in the corresponding graph, the number of triangles on a line, and the number of occurrences of other configurations which involve three points of the graph. The definition does not determine the frequencies with which any subgraphs having four or more points occur. The number of complete 4-points which include treatments 1 and 2 in (4.1) is iden- tical with the number of pairs of symmetrically lecated 1's in submatrix A and is therefore very closely related to the problem. In an effort 1]. i to determine the total number of complete 4-points in the graph, the more 2 general problem of classifying the (2 ) 4 x 4 principal minor submatrices of All , which determine the subgraphs having 4 points, was begun. Apart from permutations of rows and columns, there are 11 distinct symmetric 4 x 4 incidence matrices with. 0's on the main diagonal, corresponding to the ll distinct graphs on 4 points. 0 o .——-o [—4 0—0 V 1———o’ o e e e o o——~ e J—e Several equations were found in the 11 frequencies with which the 4 x 4 submatrices occurred in Al , for example by computing the total number 214 of triangles of the graph in terms of the frequencies of the 4-graphs containing triangles, and equating it to the known total number of tri- 2 angles in the graph. The sum of the determinants of the (2 )submatrices is equal to a known coefficient of the characteristic equation of A1 and led to another equation. The 11 frequencies are expressible in terms of the total numbers T of 1's in the submatrices A lu-V of fv A1 and it was found advantageous to set up all the equations in terms of the T/uv . There are 16 submatrices but symmetry of A1 gives 6 equations of the form T/u'y = T], /,. and reduces the number of independent T/M/ to 10 . Other methods were used to obtain equations in the Ta.” , in particular an enumeration of the 5-chains Joining points 1 and 2 of the graph. The number could be expressed in terms of certain of the Tflv and could be computed directly in terms of products and other operations on the matrix A using methods of Rats [2'5] and Ross and Harary [29]. l s A similar enumeration of chains of 4 or more lines was investigated. In all, over 20 equations were obtained, reducing to a set of 9 indepen- dent linear equations in the 10 totals T,“ y , and a one-parameter family of solutions was obtained. For reasons which will be stated in the following paragraph, it was not to be expected that a tenth independent equation could be determined, but the non-negative nature of the T /“ 1! provides some inequalities and leads to upper and lower bounds on the totals T/uy and on the frequencies of the 11 types of 4 x 4 sub- matrices. No relations loading to further inequalities were found, and the best result obtained for T11 was equivalent to inequality (4.2) of Lemma 4.5. The other information obtained on the T fly appears to con- tribute nothing to the problems of this thesis, and will not be discussed. 215 All of these results apply to any association scheme with two classes. A comparison of inequality (4.2) with the situation in an actual association scheme will show why an improved inequality was hoped for, as well as a possible reason why one was not found. The case of an L5 scheme with 25 treatments will be taken as an example. Suppose that the treatments are represented by the numbers in the following array, and that first associates are defined by rows and columns of the array and the letters of a 5 x 5 Latin square with A B C D E as its first row. 1 2 5 4 5 6 7 8 9 13 ll 12 15 14 15 16 17 18 19 2O 21 22 25 24 25 The pil = 5 common first associates of treatments 1 and 2 are treat- ments 5 , 4 and 5 , the treatment occurring with the letter A in column 2 and the treatment occurring with the letter B in column 1.. Treatments 5 , 4 and 5 are pairwise first associates, accounting for six 1's in the 5 x 5 submatrix All . Neither of the two remaining treatments is a first associate of any of treatments 5 , 4 and 5 . If they are first associates of each other they lead to two more 1's in All and T11 = 8 3 if they are second associates, Tll = 6 . The value of Tll depends on whether or not the array 2 2 occurs in the first' 216 two columns of the Latin square. The folloWing two examples show that either situation is possible. A B C D E A B C D E B A E C D B E A C D C E D A B C A D E B D C B E A D C E B A E D A B C E D B A C Both structures can occur in n x n Latin squares for many values of n , and probably for all n )>5 , and it is easy to show that in the two possible cases the number T11 of 1's in the n x n submatrix All is either n2-5n + 6 or n2-5n-+ 8 . The number 211 of off-diagonal 0's is then 4n-6 or 4n-8 respectively. Thus T11 is not determined uni- quely by the parameter values of the association schemes, and no system of equations can lead to a unique value for it. On the other hand, the assertion of Lemma 4.5 in an L5 scheme is le S lOn-20. This upper bound is considerably larger than either of the possible values. In view of the large discrepancy between this upper bound and either of the possible values, there appear to be good grounds to be dissatisfied with it, at least until a thorough search has been made for a better one. The efforts of the writer to improve Lemma 4.5 have already been described. While the proof of Theorem 4.9 included a demonstration that in the schemes involved, evegy pair of first associates is contained in an 217 n-point, the weaker result that one n-point exists in the scheme would have been sufficient in the case of some schemes. This was shown for L 2 schemes in Theorem 4.5, and is shown for a class of L5 schemes in the following theorem. The parameter values of L3 schemes will now be listed for easy reference. 2 9— “‘1 v 3 n , n 2n-4 P '1' 9 - 1 n1 - 501.1) , L2n-4 (n-2)(n—3)_J n2 .-. (n-l)(n-2) , P = 6 52-9 1 . 2 Lfin-9 n -6n +-1Q_ THEOREM 4.10. If an association scheme with two associate classes has parameter values v = n2 , n1 = 3(n-l) , pil = n , where n 2 l4 , and there exists a set of n treatments which form a complete n-point, then every pair of first associates is in such a set and the scheme is of L3 type. PROOF: Relations (2.2) to (2.5) may be used with the given parameter values to Show that the remaining parameter values are those of an L5 scheme. Number treatments so that the set of treatments in the complete n-point receives numbers 1 to n , with an arbitrary treatment of the set designated as treatment n . Let 0 be a first associate of treat- ment n which is not in the n-point but is otherwise arbitrary. Next 1 l2 ment n and second associates of O . Fewer than n of these can be in consider the p = 2n-4 treatments which are first associates of treat- the n-point, so that (for n 2 4) one such treatment not in the n-point can be chosen. A treatment chosen in this way will be numbered n.+ l . 218 The pair of first associates n , n +-l will now be used in an indirect method of obtaining information about treatment 9 . The first step is a classification of the treatments other than 1 , 2 , ... , n +~l into four mutually exclusive sets. Choose notation so that the treatments in each set have consecutive numbers and the four sets are numbered in the order listed. Set 1: common first associates of treatments n and n +’l , Set 2: the remaining first associates of treatment n (including 0) , Set 3: the remaining first associates of treatment n«+ l , Set 4: common second associates of treatments n and n +.1 . By Lemma 4.1, treatment n +-l has exactly g-l = 2 first associates in the n—point, of which one is treatment n and the other must be one of the common first associate of treatments n and n-+ l . Set 1 consists of the rest of the 9&1 = n common first associates. Therefore Set 1 contains n-l treatments. Treatment n has n1 3 3(n-l) first associates, of which n-l are in the n-point, one is treatment n +-l , and n-1 are in set 1 . By difference, Set 2 contains n-2 treatments. It may be shown that Sets 3 and 4 contain 2n-4 and n2-5n +~6 treatments respectively, but these facts will not be used. The rows and columns corresponding to Sets 1 to 4 determine sub- matrices which will be denoted by D/‘V as indicated in (4.55) below. t/rv and s*py will denote the number of 1's and the number of 219 off-diagonal 0's , respectively, in a row of 9p], ; T/pv and %”Z/ will denote the number of 1's and the number of off-diagonal 0's , respectively, in the entire submatrix Dfiuv . Unless otherwise Specified, statements made for t#z/ (4.53) A1 '61”; l O . . l }-l.2.: 9 l_l_._.;l’ E1 E 2 a . - h'HPL"? '. 3 - *." H L .. He He O_0_Q.90 CID-‘0 H 00 0&0 PLOO O a ‘ a. - - - C I 1 I 1 . . and z/‘II will be true for each row involved. -} Treatments 1 to n-l e on..- ---‘u Set 2. n-l rows: __ - -- J - 912-}??? ..... Set 5. Set 4. By Lemma 4.1, any treatment not in the n-point is the first associate of exactly two treatments of the n-point. one of these two treatments is treatment n , l to 11.1 e exactly one 1 . This implies that every row of E The inner product of two rows of Al 2 3 p11 ates. ment of Sets 1 to 4 may be expressed in terms of the row totals t 6 : If it is in Set 1 or Set 2, and E l 2 is equal to p and one is among treatments contains 1 11-13 or according as the two rows correspond to first or second associ- The inner product of row n with the row corresponding to a treat- fl‘V’ Among the first n-l elements in row‘ n +-l is a single 1 which may 220 contribute to the inner product of this row with other rows, and the inner product cannot be expressed exactly in terms of the tflj/ . How- ever, inequalities can be obtained. The following relations are obtained in this way. Using inner products of row n with rows of Set 2, 4054 t +t 3 "le ( ) 21 22 ’1 Using inner products of row n +~1 with rows of Set 2, tnl+ t S 5 e a 25 Because the row totals t are non-negative, the last statement implies /“V 4.55 t <5. ( ) 21- Each row of D2? has n-2 elements, or vhich n-S are not on the main diagonal of Al , giving (4.56) t -+ z = n — 5 . (‘0 on, (.4 (of. Statements (4.54), (4.55) and (4.56} may be solved simultaneously to give (4.57) Z223 5 , a statement which holds for every row of submatrix Dn2 . 4 Assume that Z2“:> 0 , which is equivalent to saying that submatrix C. D22 metrically located with respect to the diagonal. The rows of Al contains off-diagonal 0's . Consider two such 0's which are sym- 221 containing these 0's correspond to a pair of second associates and have inner product equal to 6 , meaning that exactly 6 columns of Al con- tain 1's in both of these columns. Column n is one such column. Therefore at most 5 columns of D22 contain 1's in both of these rows. D22 contains n-2 columns, and each of the remaining n-7 columns must contain a 0 in at least one of the two rows. This total includes the two 0's first considered. One of the rows must contain at least half of this total, which may be expressed LEE] , where [x] denotes the greatest integer S x . Therefore, under the assumption that 222 > O , there must be _a_t_ least 992 row for which (4.58) 2222 [2%.] . This violates (4.57) for all n 214. Therefore for n214 , D22 con- tains 1's in all off-diagonal positions. The treatments of Set 2 to- gether with treatment n are therefore pairwise first associates and form a complete (n-l)—point. Treatment 0 is therefore contained with treat- ment n in a complete (n-l)-point. But 9 is an arbitrary treatment of the set of first associates of treatment n which are not in the initial n-point. Therefore every first associate of treatment n is contained with treatment n in a complete configuration with at least n-l treat— 2 11 and for such values of n , Lemma 4.5 may be applied to show that the ments. n—l exceeds g(pil+ p + 2) =- 5(n + 6 + 2) for all n 2 ll , Sn—S first associates of treatment n fall into disjoint sets of at least n-2 treatments, each set forming a complete configuration with treatment n . By lemma 4.1, none of the disjoint sets can contain more 222 than n-l treatments. It is easily verified that these conditions can be satisfied only if there are 5 sets, each with exactly n-l first associates of treatment n . Therefore treatment n is in 5 n-points which have no other treatment in common, and is contained with every one of its first associates in a complete n-point. But in the numbering of treatments, treatment n was taken as an arbitrary treatment of the given n-point. Therefore every treatment of the given n-point is con- tained with each of its first associates in a complete n—point. Every one of the n2-n treatments not in the initial set of n is a first associate of two treatments of the set and is therefore in at least one complete n-point. Finally, since every treatment in the scheme is con- tained in an n-point, the argument used here shows that every treatment is contained with each of its first associates in an n-point. Theorem 4.1 then shows that the scheme is of Latin square type and the proof is complete. Theorem 4.10 is vacuous for all values of n2 29 , for which Theorem 4.9 gives the stronger result that any scheme with L5 parameter values must have L3 structure. Theorem 4.10 shows that for 143 né 28 , if the existence of one n-point in an association scheme with L3 para- meter values can be demonstrated, the scheme must have L5 structure. The possibilities for extending Theorem 4.10 to Latin square type schemes with more than three constraints or to smaller values of’ n in the case of three constraints appear to be about as good as the possibilities already discussed for the extension of Theorem 4.9. Example 2 of Section 4.1, which shows that non-L5 schemes with L5 parameter values can exist, is 223 not known to be a counter—example to Theorem 4.10, because it is not known whether the association scheme contains any complete 5-point. It was verified that a particular pair of first associates was not contained in a 5-point, but the scheme contains 150 pairs of first associates, most of which have not been investigated. All of the theorems and lemmas of this section except Theorems 4.9 and 4.10 apply to any association scheme with two associate classes. The same is true of Lemma 4.5 in the previous section. It is possible to use them to investigate the structure of association schemes not in the Latin square series. Theorem 4.7 provides a sufficient condition for the .existence of a complete kppoint, or set of k treatments which are pair- wise first associates, and is easily applied to any association scheme. This was done with the schemes listed in Table II and it was found that in most cases (7' is imaginary and the theorem proves nothing. How- 1 ever, (7.1 is real and satisfies the required inequality for schemes of the Triangular series with 66 or more treatments. It is possible to use Theorem 4.7 as the basis of a proof that for n 2 l2 , the only association scheme with v 3 (3) treatments and the parameter values of the Triangular series is the scheme whose construction was described in Section 2.1. It is not known whether this is a new result. In any case, the proof will not be given here. Speaking rather loosely, the thing which is needed to make Theorem 4.7 work is a large value of pil and a small value of pil . A small value of pil means a small value for the inner product of two rows of the association matrix which correspond to second associates and is closely related to the restriction on such 224 inner products stated in Definition 4.1 and used in several theorems of this section. In most of the schemes of Table II which are not in the 2 1 11 is at least as large as p11 and it can probably not be expected that the methods of this section Triangular or Latin Square series, p will show the existence in these schemes of k—points for any large k . A more significant fact for many association schemes may be the non- existence of k—points. The methods of this chapter were not designed to prove this. V. SWEY CHAPTER I. GMERAL PROPERTIES OF PARTIAILY BALANCED DESIGNS AND ASSOCIATION Ell-Elms Section 1.1. Introduction. This section gives some simple examples of incomplete block designs and partially balanced incomplete block designs in particular, followed by a formal definition of PBIB desigas and a basic list of relations satisfied by the parameters of the designs. Section 1.2. Association Schemes and Incidence Matrices. In this section association schemes are defined with some simple examples, and the incidence matrices of association schemes, denoted by A1, are introduced. The relation of these matrices to the more familiar incidence matrix R of the blocks of the design is discussed briefly. Section 1.3. Applications and Algebraic Properties of the Matrices A1. Theorem 1.1 gives a rule (1.16) for forming products of the ‘ matrices A1. This result is used in several parts of Chapters II and III, and has other applications which are not treated in this disserta- tion. Theorem 1.2 shows that any set of matrices satisfying (1.16) and a few light restrictions may be used to define an association scheme satisfying the conditions of partial balance. This theorem is used in the proof of Theorem 3.3. 226 Association incidence matrices do not seem to have received much study. Nearly all of the results presented or mentioned in this section were obtained as original results by the writer, but some of them have recently been obtained by others, to whom credit is given in the discussion following Theorm 1.2. CHAPTER II. ENUMERATION or POSSIBLE DESIGNS AND ASSOCIATION SCHEMES WITH NO ASSOCIATE CLASSES ’ Section 2.1. The Class of PBIB Designs with Two Associate Classes. The general expressions given in Chapter I for partially bal- anced designs are specialized in (2.1) to (2.5) to the important smial case of designs with two associate classes. It is convenient to classify PBIB designs according to the method of defining the association relation, and a classification due to Bese ani Shimanoto ef the known association schemes with two classes is adopted here. Four of the types, gnup divisible, triangular, simple, and cyclic, are described briefly, and the fifth type, Latin square, is discussed in considerable detail. The association scheme of a design of Latin square type with g constraints, briefly denoted by L8, is ordinarily defined in terms of a set of g—2 mutually orthogonal Latin squares; the present treatment is based instead on a set of g mutually orthogonal squares which do not necessarily have the Latin square property. The symetry of an 1.8 association scheme is emphasized by this point of view, uhich is not new but does not seem to have been discussed much in the available literature. Expressions for the parameter values of Lg schemes are derived, first in expressions (2.9), then in a new 227 notation in (2.12) for use in Chapter III. It is pointed out that for certain negative values of the arguments, these expressions give sets of parameter values which are different from those for any of the schemes classified by Bose and Shimamoto. The possible new schemes are given the name "negative Latin square" and the brief notation Lg, where g is a negative integer, and are studied at some length‘in Chapter III. This section is primarily a collection of knom results, with the addition of some new notation and the definition of negative Latin square designs . Section 2.2. Enumeration of Association Schemes. An enumeration of association schemes may be considered a pre- liminary step in the enumeration of combinatorially possible PBIB designs and is carried out in this section for designs with two associate classes. Group divisible schemes are easily enumerated and are omitted from the present list. The enumeration is arbitrarily limited to schemes with v S, 100, a figure which .was chosen to include most of the schemes within the range useful to experimenters, and to include schemes related to 10 x 10 Latin squares. Soul notation of Connor and Clatworthy is adopted and one of their results is listed as Theorem 2.0. This theorem specifies a one- parameter family of non-group-divisible schemes, whose parameter values are listed in Table Ia of the Appendix. All other non-group-divisible schemes are shown in Theorem 2.1 to be contained in a larger family whose parameter values can be listed systematically. Table Ib of the 228 Appendix is a working table in which this listing is carried out. This list is shortened somewhat by omitting the complement of each scheme listed, that is, the scheme obtained by changing the designations of first and second associates. Table II collects the results of Tables Ia and Ib in an orderly arrangement. The parameter values of known association schemes are identified in this table, along with some which are proved impossible by later theorems of this section. Table II lists 101 sets of parameter values, of which four are shown to be impossible, 50 were already knovm, 6 are constructed for the first time in this dissertation, and the remaining kl are still unknown. Theorems 2.2 and 2.3 show that if the number of treatments in a PBIB design with two associate classes is of the form p+l or p for any prime p, then the only possible association schemes are of group divisible type or the type specified by Theorem 2.0, respectively. Theorems 2.1. to 2.8 state additional necessary conditions for the existence of association schemes with two associate classes. The condition stated in Theorem 2.5 is used to shorten the computation of Table Ib. The other theorems provide the four impossibility proofs mentioned in connection with Table II, and give some information about the structure of any possible. scheme in approximately 12 of the un- known cases. Low 2.2, used in the proof of Theorem 2.8, specializes Theorem 1.2 to the case of two associate classes, giving a simple condition that a given matrix be the incidence matrix of first associates. It is used again in Section 3.3. An exhaustive list of possible partially balanced designs was often 229 promised by the earlier writers in the field but does not seem to have appeared, although Bose , Clatworthy and Shrikhande have published tables which include virtually all designs within the practical range known up to 1953. The present tabulation is believed to be new. It should be of some use in the application of PBIB designs to experiments, and of mrther use in later studies of the structure of designs an! association schemes. Also new in this section are most of the details of Theorem 2.1 and all of Theorems 2.2 to 2.8. Several immediate additions to the tables given here are possible, including an extension to some values of v > 100, and further investigation of the u schemes which are unknown. Another question to be discussed in some aspects for Latin square type designs in Chapter I? but considered only incidentally for other designs, is the question of the number of solutions of a constructible association scheme. Section 2.3. Enumeration of Possible Designs for Particular Association Schemes. Several known facts about PBIB designs are reviewed and used to develop a systematic method of enumerating all possible designs for a given association scheme. The method is outlined in this section and carried out in Tables III and IV of the Appendix. The enumeration is limited to constructed association schemes of the L8 and L8., series, and for each association scheme is limited to designs with r _<_ 10 and k _<_ 10. Many of the desigts in Table IV are easily constructed and a few are easily shout to be impossible; all designs either constructed or known to be impossible are identified in the table. Many of them 250 are easily enumerated by a few methods which are listed for convenience as Theorems 2.9 to 2.15 in this section. Two designs which have been constructed by the author by other methods are listed in Section A.3 of the Appendix. Enumeration proofs of impossibility of three designs appear in the same section. Section 2.3 concludes with a brief men- tion of singular designs. The author is not sure that any of the material in Section 2.3 is new, though no list of possible designs as inclusive as Table IV seems to have appeared and Theorems 2.12 to 2.11. may be new. Tables III and IV could easily be extended to designs with association schemes ofother types, and to designs with r>10 . The latter extension would be of dubious value to experimenters but might give a useful background for further theoretical studies. The large number of unknown designs in any list such as Table 1V suggests a comparably large collection of potential theorems on the construction or impossibility of designs. CHAPTER III. NEGATIVE LATIN SQUARE TYPE ASSW IA'I‘ION SCHEMES Section 3.1. Relationships Between Latin Square and Negative Latin Square Association Schemes. It is pointed out that the Negative Latin square schemes share with the ordinary Latin square schemes the property that the multi- plicities 0‘1 and 0‘2 of the characteristic roots of NW are equal in some order to the mmbers hi and n2 of first and second associates of a treatment. It is shown in Theorem 3.1 that the only other schemes with this prOperty are the one-parameter family specified by Theorem 2.0. There is some discussion of two alternate notations for the parameter values of the negative Latin square series. In one 231 notation, negative integer parameters are used, 11* the negative square root of v: n2 and i, the negative integer which is used as a sub- script in.the symbol Lg*. In this notation, the expressions for the parameter values have the same form as those for the Latin square series. The other notation is based on the positive square not of v and the numerical value of the subscript in the symbol 1.; and does not lead to expressions of the same form but is more convenient for some purposes. The section concludes with some remarks about the relation between negative Latin square schemes and finite Euclidean plane geometries. The existence of the geometry is a sufficient but not a necessary con- dition for the existence of an ordinary Latin square scheme. The existence of a connection either way between the geometry and the nega- t_.j._\_r_e_ Latin square scheme has not been proved or disproved. The computation of the multiplicities O( 1 of the characteristic roots of 1%“ was first carried out for Lg designs by Connor and Clatworthy, using a method which immediately applies to L: designs. The class of negative Latin square designs and association schemes was defined in Section 2.1 and the study of the connection between the 0‘ 1 and the “i is new in this section. Section 3.2. Construction of Negative Latin Square Type Association Schemes by a Method Based on Finite Fields. Theorem 3.2 provides a method of constructing a wide class of association schemes from finite fields. In general the schemes have more than two associate classes. Methods are described for setting down the association scheme and for computing the values of the par- ameters “i and pfik . Following an illustrative example using the 252 field with 16 elements there is a discussion of two families of schemes which can be constructed when the order of the finite field is a perfect square n2 (which requires that it be an even power of some prime). The simpler of these two schemes is shown to be equivalent to the finite Euclidean plane with n points on a line, and the parameter values are computed. The same computation for the later scheme is completed later in the section for several particular values of n , but is not carried out in general. An association relation defined by combining associate classes in a scheme with three or more classes will not in general satisfy the condi- tions of partial balance. Theorem 3.3 states necessary and sufficient conditions for a relation defined in this way to satisfy the definition of an association scheme. In Corollary 3.3 a simplified form of the conditions is stated for the case in which the new scheme has two classes. The proof of the theoremtmakes use of association matrices and applies Theorems 1.1 and 1.2. The method of’Corollary 3.3 is then applied to the schemes constructed for n? treatments by the method of’Theorem 3.2. It is shown that L8 schemes for any g4g,n can be constructed in this way from.the schemes 2 of the first family for each value of ‘v : n which is a prime power. The second family of schemes is related to negative Latin square schemes, four of which are constructed in this section. The method either fails or is not applicable to the remaining Lg* schemes taken up in the pres- ent study. As a result of their common origin from.a finite field, the L3* scheme with n2 treatments constructed here and the finite Euclid- ean plane with n2 points are related in a way which is shown to permit 253 a geometrical interpretation of the scheme. Theorems 3.2 and 3.3 both have applications beyond those developed in this section. Both theorems were derived by the writer but the equiv- alent of Theorem 3.2 was published independently by Sprott in 1955, be- fore the writing of this dissertation was completed. A comparison of the present work with that of Sprott appears in the concluding paragraph of the section. The L; schemes constructed here are believed to be 11“. Section 3.3. Construction of a Negative Latin Square Type Scheme with 100 Treatments by Enumeration. In this section a detailed study is made of a particular association scheme with 100 treatments. The 100 x 100 incidence matrix A1 is studied in detail and because of results proved in earlier chapters of this dissertation and because of some simplifying circumstances for the particular scheme, it is possible to obtain rather complete information about’ properties of certain submatrices of A1 . It is shown in partic- ular that one 22 x 77 submatrix S is the incidence matrix for the blocks of a balanced incomplete block (BIB) design and that if the design is constructed the entire matrix A1 can be constructed from it. The construction of the balanced design is the part of the section which. uses empirical methods, and even though some effective shortcuts are used, the reader has to put up with the individual examination of about half of the 77 blocks of the design, following some of them through several stages of incompletion and false starts. Once the design is constructed, the association matrix A1 can be constructed in short order. The 254 balanced design itself is a bybproduct, and not the only one. The dual of the design, obtained by interchanging the notions of treatment and block, is found to be a PBIB design with a previously unknown association scheme whose matrix of first associates appears as another submatrix.of A1 and which is constructed here for the first time. Other submatrices of A are related to still other designs and to some interesting ar- 1 rangements of h.x.h orthogonal squares. This section applies several methods of'Chapters I and II which may be new, results in two association schemes which are believed to be new, and gives constructions of several other incomplete block designs and other combinatorial arrangements which may be of interest. The scheme with 100 treatments is in the negative Latin square series and cannot be constructed by the method of Section 3.2 because there is no finite field with 100 elements. The scheme may possibly have a connection with the unsolved question of the existence of orthogonal 10.x.lO squares, but the author has no conjecture as to what sort of connection there might be. CHAPTER IV. THE STRUCTURE OF LATIN SQUARE TYPE ASSOCIATION SCHEMES Section h.l Preliminary Discussion of’Uniqueness, and Some Counterbexamples. Given any set of g mutually orthogonal n x.n squares, an L8 scheme can be constructed by using rows of the squares to define first associates. It is not obviously true that all schemes with the 255 parameter values of the Latin square series can be constructed in this way. If it is true for a particular pair of values of n and g, so that the existence of a scheme with the apprOpriate parameter values implies the existence of the set of orthogonal squares, we shall say that the L8 scheme fOr n2 treatments is unique. The term unique 'will be used in this situation whether or not the set of orthogonal squares is unique, and questions of enumeration of Latin squares are not taken up here. in L3 association scheme will be said not to be unique if there exists a scheme having the same parameter values but no set of orthogonal squares exists by which first associates in the scheme can be defined. Three examples are given in.this section of L8 schemes which are not unique. If first associates in a scheme cannot be defined by orthogonal squares, it may be that second associates can; in fact, this is the case in each of the three examples. It may be conjectured that in any scheme with Latin square parameter values, either first or second associates may be defined by a suitable set of Latin squares. No proof or disproof of this conjecture is attempted in.this chapter. Instead it is proved in Sections L.2 and h.3 that for a fixed number g of constraints and sufficiently large n, the L8 .schems for n2 treat— ments is unique in the sense defined above. An alternate statement is that for a fixed number' n2 of treatments and a sufficiently small number g of constraints, the existence of the association scheme is equivalent to the existence of the set of orthOgonal squares. For a comparison of these results with the conjecture just stated, thereader_ is referred to the discussion following Corollary h.9 in Section h.3. 256 Section h.1 contains a statement of some terminology of linear graphs which is used throughout Chapter IV and in this summary. Section h.2. 0n the Uniqueness of L2 Association Schemes. The uniqueness of Latin square type association schemes with two constraints is taken up in this section, though some of the theorems and lemmas apply more generally. The uniqueness of an L8 association scheme for n2 treatments is proved if it can be shown that each treat- ment is contained in g complete n-points which have no treatments in common in addition to the initial one. In Theorem h.l it is proved that for n > (g-l)2 it is sufficient to show that each pair of first associates is contained in one complete n-point. This is pre- ceded by two lemmas. If a scheme with the parameter values of the Latin square series contains n treatments forming a complete con- figuration, then Lama h.l reveals a good deal of uniformity in the association relations of the n treatments with the remaining n2-n treatments. It is an imediate corollary that no caplete configura- tion having more than n points can occur in a scheme with Latin square parameter values. Both lemma l..l and its corollary are repeatedly useful in this chapter. Lemma L.2 deals with the number of treatments which two complete configurations can have in common in an L8 scheme, and is slightly stronger in this case than Theorem 1+.8 and Lenma h.5, which apply to a wider class of schemes. Unlike the other theorems and lemmas of this section, Leanna h.3 is not restricted to schemes with Lg parameter values. It states an upper bound for the number of off-diagonal 0's in a specified submatrix 257 of the association matrix A1 in any scheme with two associate classes. This of course is equivalent to a lower bound on the number of 1's. In Theorem h.2, the principal result of the section, the same submatrix is examined in the case of L2 schemes, and it is shown that with the single exception of the scheme with 16 treatments, the lower bound of Leanna h.3 is inconsistent with the presence of any off-diagonal 0's in the submatrix. The portion of the linear graph correSponding to the submatrix is then a complete configuration and it follows easily that every pair of first associates is contained in a complete n-point. Theorem 4.1 then shows that the L2 scheme is unique. The scheme with 16 treatments had already been shown by one of the examples of Section h.l not to be unique. Additional information in this excep- tional case is given by Theorem h.3. In a passage following the proof of Theorem [“2 it is shown that unless the methods used in this section can be improved, it till not be possible to generalize Theorem L.2 to other L schemes. The 8 new methods and the generalization appear in Section h.3. Section 4.3. On the Uniqueness of Lg Association Schemes, g 2 3. The principal results of this section are Theorem and Corollary n.9, in which are established the uniqueness of an infinite class of Latin square type association schemes. The preparation for this theorem is long and somewhat indirect, involving five theorems and two lemmas in this section, as well as some of the material of Section l..2. Theorems ink to l..6 and Lemma 5.1. are general results on the structure of incidence matrices, all with a bearing on the existence of complete configurations, or equivalently the existence of principal 258 miner’submatrices with 1's in all off-diagonal positions. These theorems are arranged in order of decreasing generality, Lemma h.h stating a particular fact which is used in Theorem h.7. ' A property of association matrices of PBIB designs and of their submatrices is that the inner products of rows or columns taken as vectors are subject to restrictions. In this series of theorems the requirement is imposed that certain inner products must not exceed a fixed value D. ‘While rectangular incidence matrices can be studied from.this point of view, the present investigation is limited to symmetric incidence matrices with 0's on the main diagonal, which will be taken in the applications to be principal minor submatrices of association.matrices. The pairs of rows subject to the restriction on inner products are those which contain a pair of off-diagonal 0's symmetrically located with respect to the main diagonal; in the matrix A1 of first associates in an association scheme, such a pair of rows corresponds to a pair of second associates, and the inner product of the two rows of A1 is equal to pit. The inner product of the same two rows of any submatrix cannot be larger than pil and may be known in some cases to be bounded by some definite smaller value. In this application of’Theorems h.h to h.6 the least upper bound that can be established for the inner product of such rows is taken as the value D. This series of theorems takes up the connection between the number of 1's in a matrix of the form considered and the order of submatrices which have 1's in all off-diagonal positions. When the matrices are interpreted as linear graphs, this becomes a connection.between the number of lines in the graph and the order of complete subgraphs. 259 Numerous theorems of this kind are already known but not for the case in which row inner products are restricted. Theorem l..6 is very closely related to one of these theorems. A number of other approaches to the study of incidence matrices having restrictions on row inner products are possible, and some are discussed in a passage following Corollary h.9. Definition h.2, applying to symmetric incidence matrices with 0's on the main diagonal, describes a certain permtation of rows and columns and a partition of the matrix into blocks in such a way that the blocks lying on the main diagonal are square and contain no other 0's. No- tation is introduced including notation for the orders of the diagonal blocks. For a matrix partitioned in this form, Theorem Li expresses upper and lower bounds for Z, the total number of off-diagonal 0's, as functions of the orders of the diagonal blocks, the order t of the matrix, and the upper bound D on the restricted inner products. Ap- plication of this theorem is complicated by the fact that a particular partition may involve a large number of diagonal blocks and by the fact that to obtain results of any generality it may be necessary to consider a large number of possible partitions. Some numerical examples illus- trate the application of the theorem. The lower bounds on Z are the ones of greatest interest in this study, and more useful lower bounds are obtained in Corollary thh, Theorem 1.. 5 and Theorem L6. In each of these the lower bound is expressed in toms of the order t of the matrix, the bound D on inner products, and the maximum order :1 for a principal minor submatrix with- out off-diagonal 0's. The minim value of Z for given t, D and 240 s1 is denoted by m(t , D , s1) and is primarily considered for fixed t and D , in which case it is a function of s1 . The lower bounds feund for Z may also be regarded as functions of s1 , and are lower bounds for m(t , D , s1) . They are illustrated for two typical cases in two figures. The exact nature of m(t , D , s1) is not known. It may be conjectured that it is monotone decreasing in s1 . If this func- tion or a lower bound for it which is a function of s1 is monotone de- creasing, then certain inequalities on Z are sufficient to imply cer— tain inequalities on s1 . In Lemma h.h, which applies to matrices satisfying certain specified conditions and is used directly in the proof of Theorem.h.7, an implication of this kind is used to establish a lower bound on s1 . This amounts to a lower bound on the order of the maximal complete configuration of the graph. The proof of the lemma includes a demonstration that the lower bounds on Z are monotone decreasing for a certain range of values of sl . Theorem 4.7, the first theorem of this section which applies only to the incidence matrices of association schemes, defines a quantity 61 in terms of the parameters of the association scheme and states sufficient conditions that the scheme contain a complete k-point of order k 2. 01+2 . The proof deals with a submatrix All of the incidence matrix. A1 and makes use of Lemmas h.3 and h.h. oi playsihe role of the lower bound on s in Lemma h.h, and is defined in such a way in (h.37) that l the result of Lemma h.3 provides the inequality on Z needed as a hypothesis. The other hypotheses of Theorem.h.7 guarantee that s1 falls in the range for which Lemma h.h is valid. It can be shown that Theorem.h.7 applies to many association schemes with L.g parameter values and to some schemes of other types. The conclusion of the theorem 241 for these schemes is that every pair of first associates is contained in a complete configuration of order at least equal to a value which is specified. Theorem. h.8 applies to complete configurations in an association scheme. The principal result is that if two complete configurations have at least two treatments in common and if the numbers of.traatments in~. the sets are sufficiently large, then all of the treatments in their union form a complete configuration. This theorem may be described in another way by borrowing a term.used in the sociometric applications of linear graphs and referring to a complete configuration as a clique. In this terminology, Theorem h.8 states that if two sufficiently large cliques have more than one member in common, they must merge. Lemma A.5 states a norther result which in the same language has the following word- ing: if all of an individual's associates are fellow members with him in cliques having more than a specified critical number of members, them none of the associates are members of more than one of the cliques. (Two peOple who meet in a certain clique never meet anywhere else.) The proofs of Theorem. h.8 and Lemma h.5 make use of properties of association schemes and would of course apply to a social group only if they met the rather stringent requirements of partial balance, as defined in Section 1.1. Theorem. n.9, applying several of the preceding results, finally establishes that fOr any fixed number 3 of constraints and for all ex- cept a finite number of possible exceptional numbers n? of treatments, the association scheme of Latin square type with g constraints is unique in the sense that there exists no other type of scheme having the same parameter values. Corollary h.9 uses numerical computations to give 242 explicit lower bounds below which any exceptional values of n must lie. The proof of Theorem 4.9 is summarized and discussed in some detail in a passage following the proof of'Corollary h.9. Theorem.h.10 furnishes some additional information about some of the exceptional cases not covered by Theorem.h.9. Applying to L3 schemes, it is analogous to Theorem h.3. The proof is more difficult than that of Theorem.h.3, illustrating the increasing conplexity of Latin square type association schemes as the number of constraints in- creases. The section concludes with a statement without proof of a unique- ness theorem very similar to those of this chapter, applying to a class of triangular type association schemes. It appears that the methods of this chapter will not apply to the remaining types of association schemes without some modification . Reference has already been made to a passage following Corollary h.9 in which possible further results are discussed. The Opening paragraph of Section h.l contains some re- marks on the significance of the uniqueness proofs of this chapter. The writer believes that most of the theorems and proofs are new. APPENDIX A.l Tables gt; Parameter Values o_f_ Association Schemes. The tables in this section are constructed by methods deveIOped in Section 2.2. Table II gives values of the parameters v , n1 , p§k and OK 1 for all PBIB designs with two associate classes, not of group divisible type, and having v_<__ 100 . The parameter values listed are determined by the association scheme of a design and are independent of the values of r , k , b and A 1 . Tables Ia and lb show the pre- liminary capitation used in constructing Table 11. Each table is pre- ceded by an explanation of the notation used. TABLE Ia. PRELDENARY COMPUTATION OF THE PARAMETER VALUES OF ASSOCIATION SCENES BY MEANS OF THEOREM 2.0. Theorem 2.0, due to Connor and Clatworthy [I7], specifies a class of association schemes whose parameter values may all be expressed in terms of a positive inte- gral parameter t . Values of t from 1 to 2A are listed in the first colum of this table. The values in the next eight columns are obtained from the following equations, stated in Theorem 2.0. 1- 2 .. 1 - 2 - m-m-m-m-t’ v=At+1 . The final colum of the table, headed # , gives the serial number by which the scheme is identified in Table II. 244 TABLE Ia mms BY MS 0? THERE)! 2.0 PREIJNINARX COMPUTATION OF PARAMETER VALUES OF ASSNIATION lzaemummaam m s93nas new uswmnaeeewmseew easemuun mamas men semen 2heemnummmauxao TABIE Ib. PRELIMINARY COMPUTATION OF THE PARAMETER VAIDES OF ASSOCIATION 331181433 BI MEANS OF THEOREM 2.1. Theorem 2.1 specifies all association schemes with two associate classes which are not of group di- visible type and are not given by Theorem 2.0. The additional restric- tion (2.25) . vies»; : 1r ent. e... “as”: is imposed to avoid duplication in the table. The schemes are listed in order of increasing values of \FZS, a parameter which was introduced in [n 7] and is used in this dissertation. For the schemes being tabulated and a fixed value ofVA_ , all possible pairs of values p}; , 9&2 appear as consecutive entries in row {ZS}: the table of Figure 6 in Section 2.2. The cemecutive values of the cola-I index 6‘ in the same table are denoted by s and s +1 and are used in the computation of column 9 and 10 of Table 11:. The values 1&2 and 9:2 are listed in celnms 2 and 3 . The parameter V appearing in colon I. is defined by v: a we . The parameters 922.21 and 9&1 , appearinginoelmmxs 5 and 6 , .‘tnf’ (2.216): 1 2 .. 1 2 papu-oplzplz e met therefore be a divisor ef the product of the entries The value p22 246 in column 2 and 3 . It is also required to satisfy (2.27), pig 2.le . Values of p22 and pil satisfying these requirements but for which 1 2 1 2 - P12+1’12""22“hi-””1”” are also omitted and the mission is indicated by a row of dashes. Fi- 1 - ' 1 2 holly, it p12 .. piz , the restriction n1 3 :12 implies p22 3 p11 , and values such that p352 > pil are omitted in this case. The values of 132 and n1 , appearing in the next eolums, are detemined by (2.1.), If n1 and n2 are both odd, Theorem 2.5 shows that no association scheme con exist. In this case, the word "odd" is entered in each of the two columns and the rest of the row of the table is left blank. The quantities (s +l)n2 and sn1 , listed in the next colossus or the table, are used in the computation of OK 1 , The parameter 04 1 which appears in colum 11 is computed by (2.28), 13(s+1)n2+sn1 US and must be a positive integer for any association scheme. If 0K 1 is o( 247 fractional the letter "f" is listed in column 11 and the rest of the row is left blank. If 0( 1 is an integer, the value v is listed in the next column. It is determined by (2.2), v=n1+n2+1 . The final column of Table Ib, headed # , lists for each complete set of parameter values the serial number by which the set is identified in Table II. PRELIMINARY COMPUTATION 0? PW VALUES 0! ASSNIATION TABIB Ib mm 8! ms OF M 2.1 248 2 1 2 it W: P12 I’12 7 p22 P11 u2 n1("*1)"2 '“1 “1 ' # 3 2 2 o 1 I. 3 6 6 6 h 10 3 2 2 h h 8 It It 9 2 z. 3 I. 1 2 6 5 10 1o 10 5 16 6 3 h 6 8 12 8 5 15 5 1. 3 odd odd 6 2 9 6 18 6 6 l6 7 12 1 odd odd 5 L 6 2 3 8 7 15 1!» ll. f L 6 8 12 16 12 t 6 h 10 10 20 1o 6 21 9 8 3 12 9 21¢ 9 f 12 2 16 8 32 8 8 25 11 21. 1 28 7 56 7 f 5 6 6 o 1 36 7 1.2 21 8:. 21 5o 31 2 18 8 21. 21. 1.8 r 3 12 9 18 27 36 f z. 9 10 15 30 3o 12 26 13 6 6 12 12 36 21. 12 25 12 6 5 8 3 h 10 9 18 18 18 6 28 15 5 8 1o 16 20 16 6 27 11. 8 5 odd odd 10 t. 15 12 3o 12 7 28 16 20 2 25 1o 50 10 1o 36 20 1.0 1 £5 9 90 9 f 6 8 9 1 2 36 10 1.5 30 9o 20 56 39 3 21 odd odd h 18 12 27 36 51+ 15 #0 25 6 12 16 . 21 62 1.2 1h 36 22 8 9 16 18 68 36 1h 35 19 9 8 odd odd 12 6 20 15 60 3o 15 36 23 18 z. 26 13 78 26 r 21. ' 3 32 12 96 2h 20 £15 27 36 2 2.1. 11 132 22 r 72 1 80 10 21.0 20 r 249 TABLE Ib (continued) 2 n 3+1 n on v p22 2 n1( )2 1 041 # 5 12 11 22 22 22 f 6 10 12 20 21. 20 f 10 6 16 16 32 16 r 12 5 18 15 36 15 f ' 15 l. 21 1h 1.2 11.. 8 36 21 20 3 26 13 52 13 f 30 2 36 12 72 12 12 119 30 60 1 66 11 132 11 f 3 60 13 52 39 10!. f 1 30 IL L2 1.2 81. 18 57 kl 5 26 15 36 #5 72 1' 6 20 16 32 1.8 6h 16 A9 31 8 15 18 27 5A 5h 1‘ 10 12 20 21. 60 68 f 12 10 22 22 66 M f 15 8 25 20 75 #0 f 20 6 30 18 ‘90 36 18 A9 32 21. 5 3h 17 102 3h 1' 30 h 1.0 16 120 32 f to 3 50 15 150 30 f 60 2 70 11. 210 28 3h 85 7 5 2 12. 8h 56 252 M 99 90 3 15 60 60 180 f h- 16 £18 6:. 11.1. 1’ 6 18 36 72 108 f 8 20 3O 80 90 f 9 21 28 81. ea 2h 50 35 12 24 2h 96 72 2h A9 33 6 ll. 13 26 26 26 f 7 12 11. 2h 28 21. f 12 7 odd odd ll.‘ 6 21 18 82 18 1' 21 I. 28 16 56 16 9 #5 28 28 3 35 15 70 15 1' le2 2 49 1h 98 1h 11. 6h 1.7 TABLE Ib (continued) ‘_ {5 p12 82 v 82 81 n2 91<°+1>nz 3.1 as 8 12 15 3 h 65 16 6o 28 120 21 5 36 edd edd 6 30 18 65 5!. 9O 18 9 20 odd odd 10 18 22 33 66 66 t 12 15 21. 30 72 6o 1' 15 12 odd odd 18 10 3o 25 9o 50 r 20 9 32 21. 96 1.8 18 57 3o 6 1.2 21 126 1.2 21 66 36 5 1.8 20 w. 10 23 69 65 h odd odd 6o 3 72 18 216 36 r 8 15 16 1 - - - - h 60 19 76 76 228 38 5 1+8 20 6h 80 192 31. 6 no 21 56 81. 168 r 8 3o 23 1.6 92 138 r 10 21. 25 1.0 100 120 r 12 20 27 36 108 108 27 15 16 3o 32 120 96 27 16 15 odd odd - ' 20 12 35 28 11.0 81. 28 21. 1o 39 26 156 78 r 30 8 1.5 21. 180 72 r 60 6 55 22 220 66 I 1.8 5 odd odd 60 A 75 20 300 60 1.5 9 8 u. 6 7 16 15 30 3o 30 r 8 11. 16 28 32 28 r 12. 8 22 22 u. 22 r 16 7 22 21 68 21 r 28 z. 36 18 72 18 10 56 2 61. 16 128 16 16 11818 Ib (continued) 1 2 1 2 E 12L2 p12 p22 911 ng n1(s+l)n2 ml 0(l v # 9 11 18 6 12 20 6o 60 120 20 81 68 7 36 21 56 63 108 19 76 61 9 28 23 a6 69 92 t 12 21 26 39 78 78 r 11 18 28 36 8k 72 r 18 11 32 32 96 61 r 21 12 35 30 105 60 r 28 9 1.2 27 126 56 20 70 58 36 7 50 25 150 50 r 12 6 56 21 168 68 26 81 69 63 z. 77. , 22 231 u. r 9 18 20 8 115 26 115 10!. 135 f 9 50 27 60 108 180 32 88 79 10 36 28 56 112 168 r 12 30 30 50 120 150 30 81 70 15 21; 33 M 132 132 1' 18 20 36 to 111 120 r 20 18 38 38 152 111 r 26 15 62 35 168 105 f 30 12 LB 32 192 96 32 81 71 36 10 56 30 216 90 32 85 77 60 9 58 29 232 87 f 15 8 63 28 252 81 f 9 20 20 - - - - 8 50 28 70 110 280 r 10 to 30 60 150 240 r 16 25 36 1.5 180 180 1.0 82 7h 20 20 1.0 1.0 200 160 1.0 81 73 10 9 16 8 18 17 3h 3h 3h 1‘ 9 16 18 32 36 32 r 12 12 21 28 12 28 7 50 36 16 9 odd add 18 8 27 2!. Sh 26 f 21 6 33 22 66 22 r 36 It 115 20 9O 20 11 66 55 68 3 odd add 72 2 81 18 162 18 18 100 93 TABLE lb (continued) —“ 1 VA— p12 piz p22 p; n2 n1 (s+1)n2 on1 0(1 v I 10 16 21 6 56 22 77 66 15k 22 100 9b 7 138 odd Odd 8 [.2 2‘. 63 72 126 f 12 28 28 159 36 98 f 15 21¢ 30 #5 90 90 18 76 62 16 21 32 [.2 96 8h 18 7 5 60 21 16 Odd Odd 25 1‘6 ’60 35 120 70 19 76 63 28 12 “I 33 132 66 f 1.2 8 58 29 176 58 f 1.8 7 6h 28 192 56 f 56 6 72 27 216 55 27 100 95 10 21 2h - - - - 12 62 33 66 132 198 33 100 98 1‘ 36 35 60 1120 180 32 96 86 18 28 39 52 156 156 f 21 210 162 ‘13 168 1“. 1’ 2‘ 21 Odd Odd 28 18 59 ’02 196 126 f 36 11. 57 38 228 111 1 L2 12 63 36 252 108 36 100 99 10 2h 25 - - - - 20 30 u» 55 220 220 M. 100 100 210 25 198 50 2160 200 M 99 92 25 26 Odd Odd 30 20 5‘ ‘05 270 180 1‘5 100 101 11 10 18 9 20 19 38 38 38 f 10 18 20 36 1.0 36 f 12 15 22 33 u 33 7 56 1+0 15 12 25 30 50 30 f 18 10 28 28 56 28 f 20 9 30 27 60 27 f 30 6 150 2’4» 80 2‘ f 36 5 106 23 92 23 1 L5 13 55 22 110 22 12 78 66 60 3 70 21 160 21 f _TABIE I‘D 46018th) 255 m Pig 9&2 Y Pglgz Pil :12 n1 (3+1)n2 8111 0<1 v # 11 18 21 6 - - - - 9 [.8 27 72 81 w. r 12 36 30 6o 90 120 r 16 27 31. 51 102 102 r 18 2!. 36 1.8 108 96 f 2). 18 62 62 126 81. r 27 16 1.5 60 135 80 1’ 36 12 5!. 36 162 72 r 1.8 9 66 33 198 66 21 100 97 12 11 20 9 10 22 21 1.2 1.2 1.2 7 6:. so 11 20 22 60 u. 60 7 63 as 20 11 odd odd 22 10 33 3o 66 3o 8 6:. 53 M 5 55 25 110 25 f 55 h 66 21. 132 2t. 13 91 81 12 20 27 7 - - - - 15 36 add add 18 30 38 57 m 111. 19 96 87 20 27 60 56 120 108 19 95 83 27 20 odd odd 3o 18 so as 150 90 20 96 88 36 15 56 1.2 168 86 21 99 91 13 12 22 1o 11 21. 23 16 66 1.6 r 12 22 26 u. 1.8 u. r 22 12 3h 36 68 36 f 21. 11 36 33 72 33 f 33 8 as 30 9O 30 f 1.1 6 56 28 112 28 r u. 13 21 11 12 26 25 so so 50 r 13 21 26 68 52 1.8 r 21. 13 odd odd 26 12 39 36 78 36 r 39 8 52 32 101 32 r 52 6 65 30 130 30 t TABLE Ib (continued) 254 2 p12 p12 'Y p22 pm 112 :11 (s+1)n1 8111 0(1 v # 15 16 26 12 13 28 27 5h 56 56 f 1!. 26 28 52 56 52 f 26 11. 60 60 80 1.0 8 81 72 28 13 1:2 39 8h 39 f 52 7 66 33 132 33 11 100 96 16 15 28 13 110 30 29 58 58 58 f 15 28 30 56 60 56 f 20 21 add odd 21 20 36 h8 72 £8 f 28 15 Odd add 30 16 1.5 1.2 90 62 f 35 12 50 I60 100 60 f ‘02 10 57 38 116 38 f 17 16 30 1h 15 32 31 62 62 62 f 16 30 32 60 66 60 t 20 216 36 56 72 56 1' 21 2O ho 50 80 50 f 30 16 1.6 66 92 1.6 r 32 15 1.8 65 96 AS I 160 12 56 62 112 212 f 18 17 32 15 16 Sh 33 66 66 66 f 17 32 36 6h 68 66 f 32 17 Odd odd 3!. 16 51 1.8 102 68 f 19 18 36 16 - - - - 255 TABLE II. PARMKE‘I'ER VARIES OF ASSOCIATION SCHEMES NOT OF GROUP DIVISIBLE TYPE. This table is restricted to schemes in which the number of treatments v does not exceed 100. Schemes are listed in order of increasing values of v, and for fixed v, increasing values of n1. Duplication is avoided in this table by the condition n1 _<_ n2; if Illa-.112, then P12 S 1312‘ Because this differs from condition (2.25) used in Table Ib, it has been necessary to change the designation of first and second associates in about 60 sets of parameter values. The same values occur, but with the indices 1 and 2 interchanged wherever they appear. The entries in most columns of Table II are copied directly rm Tables Ia and Ib. The remaining numerical values are obtained by the relations P11 ' n1 ' P12- 1’ P32 '3 n2 " pfi‘ 1’ “2:? -O(1-1, and by the remark that for the schemes listed in Table Ia , A = v . The parameterv-A- will be found convenient in locating a particular set of parameter values in Tables Ia and Ib, which are arranged in order of increasing values otVA— . Non-integral values of VE occur only in Table Ia. 256 Two columns of Table II are included under the heading "remarks". In the first of these, schemes which are known to have been constructed or tothave been proved impossible are indicated by the letter “C" or "I” respectively. In the second, schemes of triangular, simple and or cyclic types are identified by name, and schemes in the Latin square series are identified by the symbol L8 , where g is the number of constraints. The schemes of these classes which have been constructed are either tabulated by' Bose, Connor’and. Clatworthy, or are easily constructed. Schemes of the negative Latin square series introduced in Section 2.1 are identified by the symbol 18* and, 1t constructed, by a reference to the section in which the construction is described. One constructed scheme, #66, does not fall in any of the categories mentioned and is identified simply by a reference to the section in which it is constructed. The four schemes whose impossibility has been proved by theorems in Section 2.2 are identified by the numbers of the theorems. 257 TABIEII _ 2 V # V n1 n2 P11 P12 P22 P11 P12 1’22 0‘1 “‘2 M Remarks 1 5224011 llOZZfS—CCyclic 29116112222161.3012 3.1036102]. 123 5‘1 CTriangular 11.1366 233 332661300yclich. 511568 lhlt 331095 ltCT')1;i.8ngV-lr2 616 510 01.2 1222212; th-1,Sec.3. 7‘1669123 . 8j1788 31.1 61388VT'17LCCyclic 921101015116 A63616N_iCTriangular 101211010 1.55 551101021 11:25 816 311121269816 50121 12§251212 5666651212 50 ,L_2 13:261015 369 1.681312 SCSnple 11.?2771016 188 5 510206 6CSimple 15;28 918 0 810 1. 51221 6 6XTheorem2.6 1672871215 6 510 1.86720 6CTriangular 17;29u.11 677 7761111¢§00ye11e 18i33l616 788,8871616 3 19 351618 6 9 9 88 92011. 6‘0 Simple 20361025 152028161025 601.2 2173611121 7 615 111010 827 7CT£iangular 22‘361121 1912 68122111. 6 1.2 23 361520 6812 69121520 6013 21371818 899.9981818737‘ccye11e 25101227 2918 181821.15 CSimple 26 1120 20 9 10 10 10 10 9 20 20 1.1 0 Cyclic 271151232 3826 39222026 6:031!!!)10 28 .15 16 28 8 7 21 I. 12 15 9 35 8 AC Triangular 29'15222210111111111022221131 30191236 5630 210251236 7701.2 31691632 31220 610213216 711*? 32191830 71020 612171830 7013* 33 1.921211112121212112121. 7015.1.3 36 50762 0636 16352821 5 357502128 81216 912152526 7OSimP1e2 36 502128 6161212918627101Thsorem.6 37532626121313131312262675300y011e 38 55 18 36 9 8 28 1. IA 21 10 u. 9 0 181111311111- 39 56 10 1+5 0 9 36 2 8 36 35 20 6 1.0 56 22 33 3 18 15 12 10 22 1.8 7 11 X Theorem 2.1. 111571662 11230 110313818 7 TABLE 11 (continued) 258 1' v 1‘1112 211212912 9119129320109 TAX-Rm?” 1.2572132111220915161838'08imple 63572828131616161613282857 1.1. 61 30 30 ll. 15 15 15 15 16 30 30 C Cyclic 1.563221101202011112855712 ' 1.6 63 30 32 13 16 16 15 15 16 35 27 8 0 Simple 1761.11.69 671.2 2123616119 8012 1.8 61. 18 1.5 2 15 30 6 12 32 1.5 18 8 1.4114.. 1.9612112 81230 6152621112 801,2 ' 50 61. 211.2 02022 101130 56 7 12XTeorem2.6' 51 61. 27 36 lo 16 20 12 15 20 36 27 8 c 1*.3800. 3.2 52 61. 2835 121520 1216182835 80 1.. 53 61. 30 33 18 ll 22 10 20 12 8 55 12 51. 65 32 32 15 16 16 16 16 15 32 32 T65 " 55 66 20 1.5 10 9 36 1. 16 28 ll 51. 10 c Triangular 56692018 71236 515322315v1 . 57 69 31. 31. 16 17 l7 l7 l7 16 31. 31. 69 . _ 58702762121128 9182320119 9031111010 59 73 36 36 17 18 18 18 18 17 36 36 73 c cyclic 6o 75 32 1.2 10 21 21 16 16 25 56 18 10 61 76 21 56 2 13 36 7 u 39 S6 19 9 62 76 30 65 821211 1616285718 10 63 76 35 to 18 16 21. 11. 21 18 19 56 lo ‘ ' 61. 77 1660 0151.5 1121.7 5521 80883.3 65 77 38 38 18 19 19 19 19 18 38 38 N77 66 78 22 55 ll 10 1.5 1. 18 36 12 65 ll 0 Triangular 6781166117856 2111691661. 9013"° 68 81 2060 1181.2 6161.5 6020 9c ,2Sec.3.2 698121.56 9111.2 6183721156 9012 " 70 81 30 50 9 20 30 12 18 31 50 30 9 0 1-35». 3.2 718132181318301220273218901,‘ 72 81 1.0 1.0 25 11. 26 11. 26 13 8 72 15 - * 73 8110101920202020191080 901,1... 71. 82 361.5 15 2025 16 202111110 9‘0 Simple 758511170'3106021257311507 76 85 2061.1 3161.8 515185031. 81001....10 77 85 30 56{11 18 36 102033 31150 9 78 85 1.21.2 202121 21212012121133“ 79882760‘620w 918u5532V§<21 80 89 1.1.1.1. 212222 22221.11.“ 40 Cyclic 81 91 21.66 121155 1.20 1.51377 12.0 Triangular L TABLE II (continued) 259 1 .v .. 6 818261 6688061312666 82936616 222323 23232211666135 83956056122727202033751912 81.961976 21660 615605738 8 85 96 20 75 6 15 60 1. 16 58 1.5 50 ‘8 86 96 35 6o 10 21. 36 11. 21 38 63 32 10 87963857102730182036761912 88 96 1.5 50 21. 20 30 18 27 22 20 75 12 ' 899711868 232626 2626236868W00yc1ic 909911.86 11272 2127156111.? 91996256212036152728217712 929968502225252112625511111110 931001881 8972 21665188110013- 91100 2277 02156 616607722 1001,2866.3.3 95 100 2'! 72 10 16 56 6 21 SO 27 72 10 c 96100 3366181652 72639118815 97100 3366 11618118 92161112167511 * 981003366 8211212211166” 10 1.3 99100 36631121121221.383663 10 L- 1001001155182530202130551610 .... 101100 1.551. 2021.30 2025286551. 10 1.5 260 11.2. Tables g: Perameter Values 31; Possible Designs Lo; Particular Association Schemes. The tables in this section are constructed by methods deveIOped in Section 2.3. Table IV gives values of the parameters v, r, k, b, Aiand 21 for all possible designs with v S. 100, r 510 and k S. 10 and having known association schemes in the Latin square or negative Latin square series. Table III illustrates the preliminary computa- tion used in the construction of Table IV. Each table is preceded by an explanation of the notation used. TABLE III. PRELIIUNARY COMPUTATIONS OF THE PARAMETER VALUES 0F POSSIBLE DESIGNS ILLUSTRATED FOR SEVERAL ASSOCIATION SCENES. The method of computation used here requires a separate section of the table for each association scheme, and is presented in this table for schemes #2 and 32 and a portion of #6. For use in the computation, numer- ical values of several parameters of the association scheme are listed at the beginning of the section, along with expressions for the quan- tities m, M, 14', 21, 22 and r(k-l). These expressions are given in (2.117) to (2.53) and (2.3) in Chapter II. Non-negative integral values of k 2 are listed in numerical order in the first mlumn of the table. For a particular value or )2, the lower bound m on 3.1 is listed 1: positive, and the smaller of the upper bounds M and M' is listed. Values of X1 between the bounds are then listed in colunn 5, with the omission of the value _>\ 1 z 12 and of values k1 ) x2 in case 111 : n2. When a value of 7\ 2 is reached for which the bounds admit no integral value of X1, 8 261 row of dashes is entered in the R 1 column. Because the quantities m, H, M' are linear in )‘2, no further values of X2 need to be considered. For each pair of values l1, ’A , the quantities 21, 2 and r(k-l) are listed in the next columns. When the last of z 2 these is expressed in every possible way as the product of two positive integers, the two factors may be taken as values of r and k-l and lead to all possible pairs of values of r 8111 k, thich are then listed in the next oolums. The list is shortened by the restrictions r s. 10 , k _<_ 10 , andbyconditions (2.1.3) and (2.1.1.) , ifr>z i=land2,then 121:. i r Factorizations of r(k-l) which violate any of these conditions are omitted without coment. The last one is illustrated by the row near the end of the computations shown for scheme #6 , with the entries 1 3 -l 25 - -- . For each pair r , k , the value of b is compited from (2.1), b = “7k , and entered in the next column if integral; fractional values of b are in- dicated by the letter 1. Finally, if b is integral, condition (2.1.5) if r=z- , i=lor2, then bzv-O‘ 1 i is imposed in cases where it applies, eliminating a few more sets of parameter values . TABLE III 262 PRELIMINARY COMPUTATIONS 0F PARAMETER VALUES 0F POSSIBLE DESIGKS ILLUSTRATED FOR SEVERAL ASSCXI IATION SCHEMES SCh°m#2,1,2,n:3,v=9’g-2,f:2,n1=h=d1'n28hga2, m a 27k2 - 10, M .1 pug-5, w = 22; — 9&2, 21 = -)\1+2'A2L 22 = 2R1 -22 , t(k-l) = nal-1.12 , )2 m M 14' 7:1- 21 :2 r(k-1) r k b 1 - 55 -- 0 2 -l 1. 2 3 6 >v-Dkl , ox. 1. 2 18 2 -- 6 -- o 1. -2 8 1. 3 12>v- .01. 8 2 36 03 1 3 0 l2 3 5 r h It 9 6 3 18 3 -- 6; -- o 6 -3 12 6 3 18 >v-0<1 , 011. l 5 -1 l6 8 3 21. 2 1. l 20 1. 6 6 >v-0(1 , 011. S 5 9 10 3 3o 1. -- 7 -- o 8 -1. 16 8 3 21. > v-0<1 , 01. 1 7 -2 20 10 3 3o 2 6 0 21. 6 5 r 8 1. 18 3 5 2 28 7 5 r 5 .- 75 —. 0 10 -5 20 10 3 30 7 v-21 , on. 1 9 -3 21. - -- 2 8 -1 28 -- -- 3 7 l 32 8 5 f 1. 6 3 36 6 7 r 9 5 f 6 2 8 .- 2 10 -2 32 .. ..- 3 9 0 36 9 5 r 1. 8 2 1.0 8 6 12 >v-O<1 , 0K. 10 5 18 5 7 h M -- - TABLE III 262 PRELIMINARY COMPUTATIONS 0F PARAMETER VALUES 0F POSSIBLE DESIGNS ILLUSTRATED FOR SEVERAL A3803 IATION SCHEMES Scheme#2, L2,n=3,v=9,g-2, 1:2, n1=1.=0<1, nzcisdz, m =- 27\2 - 10, M 2 9‘2”, 11' = 225 - 32, 31 = -Rl‘.’ 222, $2 3 2A1 -12 , t(k-l) 3 h>\1+kk2 . )2 m H H' 7\1 81 :2 r(k-l) r k b 1 - 5; -- 0 2 -1 1. 2 3 6 >v-03 , ox. l. 2 18 2 -- 6 -- 0 I. -2 8 l. 3 12 > v- 0K. 8 2 36 08‘ ' 1 3 0 12 3 5 r h 11 9 6 3 18 3 -- 6; -- o 6 -3 12 6 3 18 >v-0<1 , OK. 1 5 -1 l6 8 3 21. 2 l. l 20 l. 6 6 >v--(><:L , 0K. 5 5 9 10 3 30 1. -.. 7 -- 0 8 -1. 16 8 3 21. > v-0(1 , 01. 1 7 -2 20 10 3 30 2 6 0 21. 6 5 I 8 1. 18 3 5 2 28 7 5 f 5 .- 75 —- 0 lo -5 20 10 3 30 > v-cxl , 01:. 1 9 -3 211 -- - 2 8 -1 28 .. ... 3 7 1 32 8 5 r 1. 6 3 36 6 7 i' 9 5 t 6 2 8 .- 2 10 .2 32 -- -- 3 9 0 36 9 5 f 1. 8 2 1.0 8 6 12 >v--c><1 , 0K. 10 5 18 5 7 h 116 -- ... TABLE III (continued) 265 7(2 11 M 11' 9(1 31 N 22 r(k-l) r k b 7 1. 83 .- 1. 10 l 1.1. ' -- .. '5 9 3 118 -- -.- 6 8 5 52 .. .;.. 8 6 9 -- 6 10 h 56 -- -- 7 9 6 60 10 7 f 9 8 93 -- 8 lo 7 68 -- -- lo 10 lo .. .. Scheme #6, Lil, n = .1, v = 16, g = .1, r = -2, u1 = 5 :05, n2 = 10 =0‘1. 2 113212-10 ,u=;7\2+3%., m =18-2R2 , 3.2 m 11 14' 7x]; 21 22 r(k-l) r k b _ . o - 3} ..- 1 -1 3 5 5 2 1.0 2 -2 6 10 10 2 80 3 -3 9 15 - -- 1 -- 1. - o 2 -2 10 2 6 r 5 3 f 2 0 1. 2o 1. 6 r 5 5 16 10 3 r 3 -1 7 25 -- - 1. -2 10 3o 10 1. 1.0 > v- o<2 , 0K. 2 .. 1:;- -- o 1. -1. 20 1. 6 r 5 5 16 lo 3 r 1 3 -1 25 .- .. 3 1 5 35 5 8 10 < v- 0‘2 , impossible. 7 6 r 1. 0 8 1.0 8 6 1' 10 5 32 264 TABLE III (continued) SChOIO #32, 13’ n 3 7, v : 1.9, g I 3, f 2 5, n1 3 18 3061, [12 2'. 30 20(2, 2 n : EAZ - 2%, M =3A2+3%, M. - 5 - g12 5 21 = -h>\2+5A2 , 22 = 3A1 " 212 , r(k"’1) : 18A1+30A2 . -‘.--.—--—.-.a-... w~m~ ....- “—— anhw -11.“. ”1311..---.2}--- 22 r(k-l) r k b o - 3%- —- 1 -1. 3 18 3 7 21>v-O(2 , 08. 6 1. t 9 3 11.7 2 -8 6 36 6 7 1.2>v-<><2 , OK. 9 5 r 3 -12 9 51+ 9 '7 63) v— (X2 , 0K. 1 1 -- -- 3- 0 5 -2 3o 5 7 35 >v— 0K. 3 6 6 1.9 0‘1 ’ 10 1. r 2 -3 1. 66 -- .. 3 -7 7 8h - -— 2 2 -- -- 1" 0 10 -lo 60 10 7 70 > v- OK. 3 l 6 -1 78 -- - 0(1 ’ 266 TABLE IV. PARAMETER VALUES 0F POSSIBLE DESIGNS. This is a list of those parameter values which satisfy all the conditions applied in Table III.. The table is limited to designs with known Latin square and negative Latin square association schemes and with v S 100 , r g 10 and k g 10. If n1 = 82 , duplication is avoided by the reetriction 9.10.2 ; designs with 9.912 can then be obtained by changing the designation of first and second associates. The design parameters for each association scheme are listed together, preceded by a list of parameter values of the scheme. Designs are identified by the numbers given to the scheme in Table II, and by a serial numbering of the designs for each scheme. Designs which are known to have been constructed or have been proved impossible are marked by the letter C or X respec- tively, followed by an explanatory remark or reference. The phrase "Pairs of first associates" indicates that all such pairs of treatments are taken as blocks; designs of this kind are described in Theorem 2.9. The word "Lattice" indicates a well-known type of design whose structure is stated in Theorem. 2.10. Some of the designs may be formed by replicating other designs. The procedure is justified in Theorem 2.11 and the designs.are identified by the letter "R" , followed by the serial numbers of the other design or designs used. The statement "N '-"- A1" or "N = A1 + I" indicates a way in mich the inci- dence matrix.of the design may be formed from the association matrix. Further details are given in Theorems 2.12 and 2.13. Some designs are identified as the complements of other designs in the table. Two designs are complements if each block of one contains exactly the treat- ments not contained in the corresponding block of the other. In some 26? cases there is a direct reference to a theorem of Section 2.3 or a section of the Appendix in which the design is constructed. 268 TEE N ’, ahh 9 =: v%% Os.) 322 = 2: “89.. ’ h D 2 Scheme mu-«pflr-el NN' hmms 22 b A1 A2 81 k 1‘ .... L e 00 u 3 rr 8 f . a. 1 a .1 33 t m 8f m 3 mm 6 a e l e t .m% P 3 8 w.“ e333. m. f. .u 8 . e313... C O a .33333 ( If b .08., ,e’e’e, .0t 0 .3333333 sol-unto f Co’s, J30’e’e300de’e’ hf. 1 010033333333 see. a.” e, e’CCe e’e’a’ ’eJ 9e, 1m am. 6 we“ eett es see e.ee . sen-He mes fPRMmRR RRBRRNRHRCR 1412&6061541001212 . . 2&2446385E7636h368 . 121023263§h3252636 “ u . 00019010191212£h26 unseen. measaae m9cen . 223333333L33LLJ566 _ 822h.6688 h 5. I48 1 mmm c c efimuun 23.... 2222 . . . 164 2.22 1 u 2 24 a 2 2-3a 2-7 269 TABIE IV (continued) # r k b A1 A2 z1 22 Remarks 6—1 5 2 1.0 1 O -1 3 C Pairs of first associates. 6-2 10 2 so 2 o ..2 6 c a: 1;1. 6-3 10 2 80 O 1 2 -2 c Rairs of second associates. 6-6 10 I3 I30 0 3 6 -6 --5 -.19--lt -.59” .13- -l- r2..-lo.- -- _ - - -- 6—6 5 5 16 o 2 1. -1. c n = A . 6-7 5 5 16 2 1 0 l. C Appen :Lx A.3. 6-8 10 5 32 0 l. 8 -8 C R: 6;6. 6-9 10 s 32 2 3 l3 0 C R: 6:7. 6219. .1Q --5. - 32-- -4 - 2 - - .0. - -8. - _C. .31 -732. 6-11 6 6 16 O 3 6 -6 6-12 9 6 213 1 h 7 -5 6-13 5 8 10 1 3 5 -3 C Appendix A.3. 6-11. 10 8 20 I. 5 6 2 9:12-49- -3. -20.. -6- .13-- .2--1.0- _ - - - - -..- 6-16 10 10 16 h 7 10 -2 2% ram N hmfimfi) Scheme NN' I rk(r - 21)6(r - 22)9 . mums b )1 AZ 21 32 k O 0.3] se6 mat 31 1%! c o w” sau a m awe. . . . L 5 . rctm u . . ..fl 1.1 .13 O .0 . 7 870 O .0 18.902. 0.2 o o.)- o.’.!.l_2 2 m 3 .7878877+ . ff 0L oooJo’o’o’o’oLo’o’+ . I oolm3meo787778877 m m aeo’ecc.’ oJofio’o’o’oJ090’1 a 2 mmuw3.Wfifi73_77778.n/7Ar .nOAA numeLMtt . .m= PPEmfiTMMRRRRRRERRNM.TN CCCCQCCCCQCCCCQCCCC.CC . . . 242Lfi424hfl6380j5066218 . 232k 52 3&6 1 h9.3 02 90 . ..4_2 . $. 4 ..1. 23 0100020102610221022355 . . 101231102932h1935hh§36 . . . . 2628 w713ama . . . 223333hhhhihhhh.hh78§89 . . 69369§23h6378829076999 _ _ . . . . . . maflazz 6%w1 33mm . 443 77 3A5 7 90 . .rfl .4 .1 2n 3 hum: m: = rk(r — 21)8(r - 22)“ . z2 z1 TABLE IV (continuad) b A1 A2 1‘ Scheme #11, I . . ‘ ‘ .1. I.“ ...,l I" 1.‘J‘La o a O t a .1 C 0 3 3 a t o O o mu . 3. H .1 o 3 35 o 0 f2 0., o o)... o 12 335 335 f coo-9.3.90o)02090+ owe393335§35 n rmmnwmnmnhmnmzmumnAAlm 3 Ram m :-w gnu m 2:. m Pm... R RRRRRRRNN . . CCCCQCCCCCFUCCCC . c . 262hfl6834m30h88 6 4 6 % h a. “Ix-"fl" r.. —“ _____ ’— Mums z 2 21 r k b A1 7&2 # Scheme #12 , L3 , 0000100112 1121023224 . . 55 55 W3W11fihhmm . 555 314145 . 5mm , 6&833099h8 272 TABIE IV (continued) SCheme #20 , L2 , “=6: ! 332,111.10, {25,11 =36 It 5 2 8 P =. P = 1 5 20’ 2 8 16 ' 2 3 25 ’ W! = rk(r - zl)lo(r - 22)25 . # r k b A1 A2 2.1 22 Remarks 20-1 10 2 180 1 0 -l. 2 C Pairs of first associates. 20-2 5 3 60 1 0 -l. 2 X Appendix A.3. 20-3 10 3 120 2 0 -8 h C Theorem 2.11.. 20-1. 10 h 90 3 0 -12 6 30:5- - -5.--§ -3.6.-3- -9- :.._It ........ 20-6 10 5 72 l. 0 --16 8 C Theorem 2.11.. 20-7 2 6 12 1 0 -£. 2 C Lattice. 20-8 1. 6 21. 2 O -8 l. C R: 7;7. 20-9 5 6 30 0 1 5 -1 29-.19_-_6---6 _,3_6-,3__q_-.12 A 6-1.2-1: - 7:37. 20.11 8 6 1.8 1. o -16 8 c a: 7;7;7;7. 20-12 9 6 5h 2 1 -3 3 20-13 10 6 60 O 2 10 -2 20-11. 10 6 60 5 O -20 10 c R: 7575737;7. 320215-_-19--9--u9, --2.-.-3--- -.. - - - - - - .. 20—16 10 10 36 l. 2 -6 6 C N 8 A1. ’W .1913): :r [fir-Trhr' '35:: ‘11" * Schene #2; , L3 , 275 TABLE IV (continued) v=36, 6 8] 6 9 P: P: :12 20, 1 8 12' 2 9 10' NN' = rk(r - zl)ls(r - 22)20 . 6 3 h NW D II II at» :3 1...: “ll ....1 U! :- # r k b A1 A2 21 22 Remarks 23-1 10 3 120 0 1 l. -2 23-2 5 h 1+5 1 0 '3 3 23-3 10 I. 90 2 0 -6 6 23-1. 5 5 36 o 1 1. -2 €3.75 -19--5--72---0---2- 8--.?!1 ......... 23-6 3 6 l8 1 0 -3 3 C Lattice. 23-7 5» 6 2h 0 1 h -2 23.8 6 6 36 2 0 -6 6 C R: 656. 23-9 8 6 1.8 0 2 8 --1; 93:99 - -‘2- -§- -5h_ _ ‘3. - Q - :9 1 - .9- - 9&1:- .6J6,6. 23-11 10 6 60 2 1 -2 11 23-12 10 9 ho I: 1 -8 10 23-13 10 10 36 2 3 6 0 Schene#30,L2,n=7,v'-=h9, 5 6 2 10 g . 2 , HI = 12 , Pl 8 , P2 = , f=6,n2=36, 6 30 10 25 NN' = rk(r -— zl)12(r - 22)30 . # r k b A1 )12 21 22 Remarks 30-1 6 3 98 1 0 -5 2 C Theorem 2.11.. 30-2 h z. 1.9 1 o -5 2 11 Appendix 1.3. 30-3 8 L 98 2 0 ~10 h 0 Theorem 2.11.. 30-h 2 7 1h 1 O -5 2 C Lattice. 20:5- -h_-'l-2_8- _ :2- 51:19- -a --c. B: as»..- 30—6 6 7 L2 3 0 ~15 6 C R: 11316611. 30-? 6 7 L2 0 1 6 -1 c Lattic 30-8 8 7 56 h 0 -20 8 C R: h; :5 h; 6. 30-9 10 7 7o 5 o -25 10 c R: 1.;1;1.;1.;1. 3Qr'10-10_--7-70---2--1--:4.--3---C ”Vi-11395.7.- 30-11 9 9 119 3 1 -9 5 ‘VWW‘. as "IA—A—‘Xr "‘ " 7., EVE-“2'!- TABIE IV (continued) Schene#32,L,n=7,v=h9, 710' 612 3 3:3,n1=18,1=1. ,P2= , f85,n2=30, _10 20 1217 NN' = rk(r - z1)18(r - z )30 . # ' k b 11-3;- 21 Eg-iz'fié- __ 32-1 9 3 M7 1 O ~A 3 C Theorem 2.11.. 32-2 6 6 I19 0 1 5 ~2 . 32-3 3 7 21 1 0 ~l. 3 C Lattice. 3 2~h 5 7 35 O 1 5 ~2 C Lattice. 32:5-__6---7- 112-.2--9-'§--6 10-11: 313.- 32-6 9 7 63 3 0 12 9 C R: 3;3;3. 32-7 10 7 70 0 2 10 ~11 C R: 11:11. Scheme#33,Lh, n87, v=h9, 1112 12 8 . ‘5 ’ n1 8 2h , P1 8 , P2 = , £21.,n2=21., 1212 1211 NN' : rk(r - 21)2h(r ~ 22):“ . _ 14 r k b 11*12 21 22 Remarks 33-1 8 h 93 0 1 h --3 33-2 I; 7 28 O 1 L ~3 C Lattice. 33-3 8 7 56 o 2 8 —6 c a: 2;2 33-1: 9 9 1:9 1 2 5 ~2 SChene#h7,L2,n=8,v=-'6A, 6 7 212 8 = 2 ’ n1 - u , P1 ' , P2 = , f=7,n2=h9, 7 £2 12 2§_ NN' = rk(r ~ zl)u(r ~ 2 )l‘9 . 1f 1' -_ k, 1? >‘1 _>_‘2_,,__ “1 z2 Rw’kf-W .--.-..-...- h7~1 7 7 61. 3 0 ~18 6 117-2 2 8 l6 1 0 ~6 2 C Lattice. 167-3 A. 8 32 2 0 ~12 h C R: 2;2. 47-1. 6 8 1.8 3 0 --18 6 c a: 2;2;2. 117-5 7 8 56 O 1 7 ~1 C Lattice . 17—6 3 8 6:. 1. o -21» 8 C R: 232;st- 117-7 10 8 80 S 0 ~30 10 C R: 2;2;23232. I _‘yagmw' €m’fi' 3‘ u -: "QC-Wo- 275 TABIE IV (continued) Scheme#h9,L3,n=8,v=6h, 812 615 - 833,313219P1' 2P2: 2 £36,n2=h2, 1230 15 26 um: mu-zfiau-zgm. # r k 1) A1 A2 21 :2 Remarks h9-1 7 A 112 1 0 '5 3 1.9-2 7 7 6h 0 1 6 -2 h9-3 7 7 61» 2 0 -1O 6 69-6 3 8 2h 1 0 -5 3 C Lattice. 19:5 _ 6 - .8418- _ -2. _ 9 ~10- -é - 9 .8; 38,. 69-6 6 8 68 o 1 6 2 c Lattice. 1.9-7 9 8 72 3 0 -15 9 C R: 1.3161». Scheme #51 , 1:3 , n = -8 , v = 61. , _ 10 16 12 15 8=’3931=279P1‘ 9P2: a f=-h,n2=36, 1620 1520 mm: = rk(r - 21)36(r - 22)27 . # r k b A1 A2 21 22 Remark: 51-1 9 6 1M 1 0 -3 5 51-2 9 9 6h 0 2 8 -8 51—3 10 10 6h 2 1 -2 6 Scheme#52,LA,n=8,v=6h, 1215 1216 g=h,tu=28 P1' .22: , f = 5 , n2 = 35 , 15 20 16 18 NN' = rk(r - zl)28(r - z )35 _~# :- k b )1 A2 #2; 22 Remarks _ 52-1 1. 8 32 1 0 -h It C Lattice. 52-2 5 8 1.0 0 1 5 -3 C Lattice. 52-3 8 8 6h 2 0 -8 8 C R: 1;l. 52-1 10 8 80 o 2 10 -6 c a: 2;2. ‘1”! .W.. m. ““2 ..2. ‘-' 276 TABLE IV (continued) Schene#67,L2,n=9, 11:81, 7 8 122 ll: 8:2'n1316,P1= ,Pza , f88,n2=6h, 856 11. 1.9 NN' 8 rk(r - zl)16(r - 2:96“ . Au‘ -”‘o“n --_—.¢”mn* W" 0 - fl # r k b M 7&2 21 22 Remarks 67-1 8 3 216 1 0 .7 2 67-2 2 9 18 1 0 -7 2 C Lattice. 67-3 h 9 36 2 0 --1l. 1: C R: 2;2. 67-1. 6 9 55 3 0 -21 6 C R: 2;2;2. .6775. - - .8 - 9 1 -72- -h. -0- :28. _. .8 - C. 81 252.1212. 67-6 8 9 72 O 1 8 -1 C Lattice. 67-7 10 9 90 5 0 -35 0 C R: 2;2;2;2;2. Schene#68,Lf2,n=-9,v=81, 118 61!; 83‘2231=209P1= 9P2: 9 f=-6,n2=60, 1 N1“ = rk(r - zl)6o(r - 2:920 . ————_._—_—_.—.__— -..- . . ”<1...- - -- -uH‘--romc . ., o-u ... ”w“..- -- . .- 1 2-..1 _ .. -.-. .. - 68-1 10 3 270 1 0 -2 7 Scheme#69,L3,n=9, v=81, 9 11. 6 18 8:3, “1:21;,P1- P2: ’ f=7, 1:12:56, L2 18 37 --- # r. _ k‘ b 7x1 X2 21 2 Remarks“ * - 69-1 8 1‘: 162 1 0 -6 3 69-2 8 8 81 O 1 7 --2 69-3 3 9 27 1 0 -6 3 c Lattice. 69-h 6 9 51. 2 o -12 6 c R: 3:3. 69-5 7 9 63 0 1 -2 C Lattice2 ......--..——..-—..o. 7 . -_ 69-6 9 9 81 3 0 :181'9' ‘0“ R: 33333. . -_.__.._ J . 277 TABLE IV (continued) Scheme #70 , Li, , -9 , v = 81 , 9 20 12 18 J '3 2 n1 = 30 2 P1 3 2 P2 = 2 -5 ’ n2 3 50 , 20 30 18 31 NN' = rk(r - zl)so(r - z2)30 . HOQS’ # r k b A1 A2 Z1 22 Remarks ‘1: 70-2 10 5 162 0 1 h -5 Scheme#71,Lh,n=9,v-81, 1318 3:1,,n1=32,p1: , {-6,n2=h8, 18 30_ 70-1 10 3 270 1 0 -3 6 12 20 P : 2 2o 27 ’ NNI : rk(r - zl)32(r - z2)h8 . # r k b )1 kg— 21 22 Remarks ... 71-1 A 9 36 1 0 -5 h C Lattice. 71-2 6 9 5h 0 1 6 -3 C Lattice. 71-3 8 9 72 2 0 -10 8 C R: 131. Scheme#73,L5,n=9,v=81, 1920 20 20 8 = 5 2 n1 = no 2 P1 = 2 P2 = 2 r = s , n2 = no , 20 20 20 121 NN' 1' rk(r - zl)l‘o(r - 22)“) . —.—_——_.-'—— . -,_.-,_.- .-r-..4-«---- —--..—.-_fi ...—.1 ,_ 1‘ _____,...A__. Own-.....-1 ---.”o-—...—-— ... o - « - - —— -« ..._- . - - - o~ .. *-—-—- --.- ....-. ....“ T". 51-3.2131 _‘2- 4 33’9”" 73-1 10 5 0 73-2 5 9 LS 1 O -h 5 C Lattice. 73-3 10 9 o -8 10 c a: 2;2. - _..._.._---c —--..~ .—.._....'~-'—.-—a-..—a.-—-~<~.~~s TABLE IV (continued) Scheme#93,L2,n=10,v=100, 8 9 216 g=2,n1=18:P1= ,P2=1 , I II 9 , n21: 81, _9 72 6 61. RN' 8 rk(r - zl)18(r - 22)81 . # r k b >i )2 21 22 Remarks - 93-1 9 3 300 0 1 —8 2 93-2 6 h 150 0 1 -8 2 93-3 9 5 180 0 2-16 1: 93-4 9 9 100 0 h-32 8 93-5 __2 1o_ “29- o‘_1_-.-8__2_ _ c Lattice. 93—6 z; 10 1.0 o 2 -16 I. ‘8: 535‘.” 93-7 6 10 60 0 3 -26 6 R: 5:5;5. 93-8 8 10 80 0 h -32 8 R: 5535”. 93-9 9 10 90 1 0 9 -1 93-10 10 10 100 0 5 ~60 10 R. 5;5;5;5;5. Scheme #91. . No designs possible with r S. 10. Scheme #95 ,L3 , n=10.,v=100, 10 1 6 21 8 = 3 2 “1 = 27 2 P1 = 2 P2 = 2 {-8.112372, 6 2 = _ 27 _ 72 NN' rk(r 21) (r 22) . -————.__.--_._. -_— # r k b )1 )2 zl 22 Remarks .-~~~—_ -..-‘4. .-.. 95-1 9 I: 225 1 95-2 9 9 100 0 95-3 3 10 30 1 ~7 3 C Lattice. 95-1. 6 10 60 2 -11: 6 C R: 3:3. %fi__ _8 m m 0 3 95-6 9 10’96“"' dwoowo 21 9. ’ C" 1153333.. trq‘cFTV-w' ' A.3. Construction 2; Two Particular Designs; Impossibilitz'Proofs 2; Particular Designs. CONSTRUCTION OF DESIGNS #6-7 and 6-13 Reference is made to these designs in Section 2.3 and Table IV. The construction or these designs involved a good.deal of enumeration of possible blocks and will not be described in detail. Both designs de- pend on negative Latin square association scheme #6, which is con- structed in Section 3.2 and is reproduced here for reference. Treatment First associates m R U n B M m D u n R U n R M 10 13 15 m R M n B U h 7 9 6 8 6 n 7 m 6 U 7 u 6 H 7 M 0HHOHOHOO‘QP’WNWOH H H “FuSBSemqamrmeo wmmuuwummoomomom mrmrrmm Design #6—7 has parameter values v=16,r=5,k-'-5,b-'-'16, A =2, 1 =1. The blocks of the design are the following: 280 0 1 2 12 13 2 3 7 14 15 0 1 3 10 11 2 2 8 9 13 0 A 8 12 1h 2 h 5 8 10 O 5 9 10 15 2 6 10 11 16 0 6 7 8 15 3 h 11 12 15 1 h 6 9 lb 3 5 6 9 12 1 5 13 lb 15 h 5 7 11 13 1 7 8 911 6 7101213 __1 a Design #6-13 has parameter values ; I! v-16,r=5,k=8,b=1o,11=1,12-3. L; The blocks of the design are the f6110wing: 0 1 2 3 A 5 6 7 0 2 h 6 9 11 13 15 0 2 5 7 9 11 12 1h 0 3 h 7 9 10 13 16 0 3 5 6 8 11 13 16 1 2 h 7 9 10 12 15 1 2 5 6 8 11 12 15 1 3 h 6 8 10 13 15 1 3 5 7 8 10 12 lb 8 9 10 11 12 13 16 15 PROOF OF IMPOSSIBILITY 0F DESIGN #20-2 This design has parameter values v=36,r=5,k=3,b=60, 21:1: A2=o. The design is based on an L2 association scheme which by Theorem h.2 is unique and may be assumed to be defined by the flollowing array, treatments occurring in the same row or the same column.being taken as first associates. 281 1 23 A 5 6' 7 89101112 13 14 15 16 17 18 19 20 21 22 23 21 25 26 27 28 29 3o 31 32 33 31: 35 36 It follows from the values of Al and A 2 that each block containing treatment 1 must contain a pair of its first associates which are first associates of each other. Notation may be chosen so that two of the b1ocks are 1 2 3 art! 1 l. 5 . The pair of first associates 1 , 6 must then occur together in a block. It is impossible to choose a third treatment for this block which is a cannon first associate of treatments 1 and 6 and has not already been used in a block with treatment 1. Therefore the design cannot be constructed. PROOF OF IMPOSSIBILITY 0? DESIGN #30-2 This design has parameter values v-h9,r=h,k=h,b=h9, AIM, 12=0. The design is based on an L2 association scheme Which by Theorem l..2 is unique ani may be «em-be- assumed to be defined by the following array, treatments occurring in'the same row of the same column being taken as first associates. 282 1 2 3 h 5 6 7 8 9 10 11 12 13 ll. 15 16 17 18 19 20 21 22 23 21 25 26 27 28 29 30 31 32 33 3h 35 36 37 38 39 to M £2 #3 65 k5 #6 h? h8 #9 Treatments 1 to 7 are pairwise first associates, and no two treatments l“ of this set have common first associates which are not in the set. It 1 follows from the values of 11 and X 2 that the four treatmaits in i't a block must be pairwise first associates and that no two first associates . can occur tagether in more than one block. Notation can be chosen so that two of the blocks containing treatment 1 are 1 2 3 k and 1 5 6 7 . The pair of first associates 2 , 5 must then occur to- gather in a block. It is impossible to choose further treatments for this block mid: are common first treatments of treatments 2 and 5 and have not already been used in a block with one of them. Therefore the design cannot be constructed. PROOF OF IMPOSSIBILITY 0F DESIGN #93-1 This design has parameter values v=100,r=9,k=3,b=300,)\1=1, 12:0. The design is based on an L2 association scheme which by Theorem 11.2 18 unique and may be assumed to be definedby a 10 x 10 array of the int'Ogers from 1 to 100 , treatments occurring in the same row or the same column being taken as first associates. The first row may be taken to contain the numbers from 1 to 10; these treatments will then be 283 pairwise first associates and no two of them will have any common first associates not in this set. It follows from the values of A] and A 2 that each treatment containing treatment 1 met contain a pair of its first associates which are first associates of each other. Notation may be chosen so that four of the blocks are l 2 3 , 11.5,167 andl89. Thepairoffirstassociatesland L 10 must then occur together in a block. It is impossible to choose a :- Vrc .‘ i _. - third treatment. for this block which is a connon first associate of fit- .’ 1‘, treatment 1 and 10 and has not already been used in a block with treatment 1. Therefore the design cannot be constructed. DISCUSSION OF DESIGNS #7-3 AND 12-2 Design #7-3 has parameter values . v=16,r=3,k=3,b-16,A1=1, 12:0. The design may be based on an L 2 association scheme defined by the array WOW!“ pl“ CON ...: “that; HP O‘NCDP and if so is easily shown to be impossible, using the method applied to designs #20-2 and #93-1 . However, it is shown in Section 11.1 that the association scheme is not unique and a reference is given in Table IV to a published design which differs from this one only in the designation of first and second associates. 284 Design #12—2 has parameter values v=25,r=h,k=h,b=25,)\1=l, Azzo. The association scheme of this design has L3 parameter values and may be defined by a 5 x.5 array of the numbers from 1 to 25 , super- imposed on a 5 x.5 Latin square. A proof’that the design is imp possible with this association scheme must include an investigation of different possible Latin squares, and is longer than the preceding proofs of impossibility. The proof can be carried out but will not be presented here. It is shown in Section 4.1 that this L3 association scheme is not unique, and is pointed out in Section 2.3 that this proof of impossibility is therefore not conclusive. 285 A.L.. -I_.i_g_t_ of Negative 9.23.“. 5.99.11! Association Schemes. Five association schemes of Negative Latin square type are constructed in Chapter III. The first of these, #6 of Table II, is described fully in Section 3.2 and repeated in Appendix A. #3. Another , #91; of Table II , is described in Section 3.3 and may be constructed from data given there. A simple method of constructing the remaining schemes will now be given. ASSOCIATION scum; # 51 The parameter values of scheme #51 include v 3 6L and a1 = 27 . The treatments will be represented by pairs (at , y) of marks of the finite field of order 8 . The addition table of the field follows. qqmrumwoo o~VsNWOHH “F’QQHOW N10 Pmadowmwu WNHOQOVIJPL‘ MWOHO-Qrvcvi Hoummrdmo ownws-moxsijq «TONUIFWNHO Addition of pairs of marks is defined by (xsy)+(z .w) : (DH-2 ,y+w) . The first associates of any treatment (x , y) are obtained by adding (1 , y) to each of the first associates of (0 , O) . The 27 first associates of (O , O) are listed below, with commas and parentheses omitted for brevity. 'vflmaurw-ni- ...—.- _— .' . #- 2% 01 02 05 06 07 13 11 22 21 26 27 32 31 36 37 10 1.7 50 57 61 61 6 5 6 7 7 O 7 2 7 3 7 6 ASSOCIATION SCHEMES #68 and #70 Each of these schemes is based on 81 treatments, which will be . ! represented by pairs (at , y) of marks of the finite field of order 9 . The addition table of the inLd follows. ‘Ffll’lm -' u-.. .(:.rv- a '1' 4012315678 0012315678 1120153786 2201531867 3315678012 1153786120 5531867201 6 6 7 8 O 1 2 3 h 5 7786120153 _8__8_6_7_2_Q_1_5._3_ 1., Addition of pairs of marks is defined as for Scheme #51. Each treatment in scheme #68 has 111 _-_ 20 first associates. The first associates of any treatment (x , y) are obtained by adding (2: , y) to each of the first associates of (0 , O) , which are given in the following list. 0 l O 2 0 A O 8 1 5 1 7 2 5 2 7 3 1 3 7 h 1 h 7 5 2 5 6 6 2 6 5 7 1 7 3 8 2 8 5 Each treatment in scheme #70 has nl : 30 first associates. The first associates of any treatment (x 5 Y) are obtained by adding (at. , y) to each of the first associates of (O , 0) , which are given in the following list. 0 1 2 1 h 6 7 6 02 23 53 78 03 25 5h 80 06 27 56 83 12 37 57 87 15 Lo 65 88 16 AA 73 287 17 15 75 ‘111815- :1 ‘51..La_." ‘I 2% res with S cial Orthogonalitl Properties. Fflums A05. square arrays which have a common AwtmofmeRHWMghxh number in their identifying symbols [1,3/ are orthogonal. That is, every ordered pair of the letters if the two squares are superimposed A , B , C and D occurs in exactly one of the sixteen positions. The construction of these squares is described in Section 3.3, ”91W. DDDD CCCC BBBB AAAA DCAB CDBA BACD ABDC DABC CBAD BCDA ADCB DBCA CADB BDAC ACBD ABCD ABCD ABCD ABCD my 1115/ as 1112/ DCBA CDAB BADC ABCD. DDCC CCDD BBAA AABB DBDB CACA BDBD ACAC DAAD CBBC BCCB ADDA 816/ 12.17 ACDB BDCA BDCA ACDB BDBD ACAC BDBD ACAC BBDD BBDD AACC AACC BDAC AGED BDAC ACBD ACCA BDDB BDDB. ACCA BADC BADC ABCD ABCD 1516/ 10. 11. 12. 13. lb. 289 LIST OF REFERENCES Aitken, A. C., Determinants and Matrices, ed. 5, Oliver and Boyd, London, 1918. Berman, Gerald, Finite Projective Geometries, Can. J. Math., 1(1952). 302-313. Bose, R. C., On the Application of the PrOperties of Galois .L‘ Fields to the Problem of Construction of Hyper-Graeco-Latin Squares, Sankhya, 2 (1938), 323-338. On the Construction of Balanced Incomplete Block Designs, = 1 Ann. of Eugenics, 2 (1939), 353-399. , l , An Affine Analogue of Singer's Theorem, J. Indian Math. 8°C., (N.S.) .6- (19262)’ l'lSe Bose, R. C., w. H. Clatworthy and S. S. Shrikhande, Tables of Partially Balanced Incomplete Block Designs with Two Associate Classes, Technical Bulletin No. 107, Inst. of Stat., Univ. of N. Carolina, 1951. Bose, R. C., and W. S. Connor, Jr., Combinatorial Properties of Group Divisible Incomplete Block Designs, Ann. Math. Stat., 22 (1952), 367-383. Bose, R. C., and K. R. Nair, Partially Balanced Incomplete Block Designs, Sankhya, g (1939), 337-372. 331-335. Bose, R. C., and T. Shimsmoto, Classification and Analysis of PBIB Designs with Two Associate Classes, J. Amer. Stat. Assoc., 51 (1952)) 151-18he Bose, R. C., S. S. Shrikhande and K. N. Bhattycharya, On the Construction of Group Divisible Incomplete Block Designs, Ann. Math. Stat., x g), (1953). 167-195. On Complete Sets of Latin Squares, Sankhya, 2 (1911), Bruck, R. H., and H. J. Ryser, The Nonexistence of Certain Finite Projective Planes, Can. Math” 1 (19119), 88-93. Bush, K. A., Orthogonal Arrays of Index Unity, Ann. Math. Stat., 32 (1952), 1126-1631:». Cochran, W. C., and G. M. Cox, Experimental Designs, Wiley, New York, 1950. l5. l6. 17. 18. 19. 20. 22. 23. 21. 25. 26. 27 . 28. 29, 30. 31. 290 Connor, W. 8., Jr., 011 the Structure of Balanced Incomplete Block Designs, Ann. Math. Stat., _22 (1952), 57-71. , Some Relations among the Blocks of Symmetric Group Divis- ible Designs, Ann. Math. Stat., 22 (1952), 602-609. Connor, W. 3., Jr., and W. H. Clatworthy, Same Theorems for Par- tially Balanced Designs, Ann. Math. Stat., 22 (1951), 100-112. Euler, L., Recherches eur une EspSce de Carrés Nagique, Commenta- ciones Arithmeticae Collectae, Vol. II (1819), 302-361. Fisher, R. A., The Design of Experiments, 0d. 11, Hefner Publishing Co., New York, 1917. Fisher, R. A., and F. Yates, The 6 x 6 Latin Squares, Proc. Camb. Phil. Soc., 29 (19311), 1192-507. Statistical Tables, 1933. Hall, P., On Representatives of Subsets, J. Loud. Math. Soc., ed. 1, Oliver and Boyd, London, Hoffman, A. J., Cyclic Affine Planes, Can. J. Math” _1 (1952), 295-301e Hussein, Q. 14. Structure of Some Incomplete Block Designs, San- khya. 8 (19185, 381-383. - Katz, Leo, An Application of Matrix Algebra to the Study of Human Relations within Organizations, Himeograph Series, Inst. of Stat. , Univ. of N. Carolina, 1950. MacNeish, H. F., Euler's Squares, Ann. of Math., 32 (1922), 221-227. Mann, H. B., and R. J. Ryser, Distinct Representatives of Subsets, Amer. Math. Monthly, _6_o_ (1953), 397-101. Nair, K. R., and C. R. Rao, A Note on Partially Balanced Incom- plete Block Designs, Science and Culture, 1 (192.2), 568-569. Ross, Ian C., and Frank Harary, 0n the Determination of Redundan- cies in Sociometric Chains, Peychometrica, _11 (1952), 195-208. Shrikhande, S. 3., The Impossibility of Certain Symmetric Balanced Incomplete Block Designs, Ann. Math. Stat., 21; (1950), 106-111. , On the Dual of Some Balanced Incomplete Block Designs, Biometrics, -8_ (1952), 66-72. ,m‘wu- —e- .-s _.__ ‘1 . . _ . ',. he 1 .er- 32. 33. 3h. 35. 36. 37. 38. 39. AD. 291 Singer, J., A Theorem in Finite Projective Geometry and Some Appli- cations to Number Theory, Trans. Amer. Math. Soc., '51 (1938), 377-385. Sprott, D. A. A Note on Balanced Incomplete Block Designs, Can. J. Math., 6(1951), 311-316. . , Some Series of Partially Balanced Incomplete Block Designs, emthe, Z (1955), 369-3810 Terry, C., Le ProblEme de 36 Officiers, Compte Rendu de l'Associa- tion Frangaise pour 1'Avancement de Science Naturel, 1_(1900), 122-123, 2 (1901), 170-203. Turan, P., Mat... .8 Physikti BM, 5.; (191.1), 1936.1052e 0n the Theory of Graphs, Colloquium Mathematicum, 3 Yates, F., Incomplete Randomized Blocks, Ann. of Eugenics, :1 (1936), 121-110. Youden, ‘W. J., Linked Blocks: A New Class of Incomplete Block De- signs, Biometrics, Z.(1951), 121 (Abstract). Zelen, M., A Note on Partially Balanced Designs, Ann. Math. Stat., _5 (1951), 599-602. ill-u. MICHIGAN STATE UNIV. LIBRRRIES mIIIWIIIUIIIWIN”WWW“IIWIWIWI‘HI 31293107857728