Balanced incomplete block designs and related designs

This paper introduced the concept of a partial geometry, which serves to unify and generalize certain theorems on embedding of nets, and uniqueness of association schemes of partially balanced designs, by Bruck, Connor, Shrikhande and others. Certain lemmas and theorems are direct generalizations of those proved by Bruck ;-57, for the case of nets, which are a special class of partial geometries.

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Strongly regular graphs, partial geometries and partially balanced designs.

Introduction to BIBDs.- Symmetric BIBDs.- Difference sets and automorphisms.- Hadamard matrices and designs.- Resolvable BIBDs.- Steiner triple systems.- Mutually orthogonal Latin squares.- Pairwise balanced designs.- t-designs.- Orthogonal arrays and codes.- Index.

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Combinatorial Designs: Constructions and Analysis

Construction and optimality of generalized Youden designs II

A table of binary constant weight codes of length nl28 is presented. Explicit constructions are given for most of the 600 codes in the table; the majority of these codes are new. The known techniques for constructing constant weight codes are surveyed, and a table of (unrestricted) binary codes of length nl28 is given

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A new table of constant weight codes

then it is known, MacNeish [16] and Mann [17] that there exists a set of at least n(v) mutually orthogonal Latin squares (m.o.l.s.) of order v. It seemed plausible that n(v) is also the maximum possible number of m.o.l.s. of order v. This would have implied the correctness of Euler's [13] conjecture about the nonexistence of two orthogonal Latin squares of order v when v =2 (mod 4), since n(v) -1 in this case. If we denote by N(v) the maximum possible number of mutually orthogonal Latin squares of order v, then Parker [18 ] showed that in certain cases N(v) >n(v) by proving that if there exists a balanced incomplete block (BIB) design with v treatments, X = 1, and block size k which is a prime power then N(v) > k -2. This result cast a doubt on the validity of Euler's conjecture. In this paper we generalize Parker's method of constructing m.o.l.s., by showing that a very general class of designs, which we have called pairwise balanced designs of index unity, can be used for the construction of sets of m.o.l.s. Various applications of this method have been made. In particular we show that Euler's conjecture is false for an infinity of values of v, including all values of the form 36m+22. We also provide a table for all values of v n(v). The smallest number for which we have been able to demonstrate the falsity of Euler's conjecture is 22. Two orthogonal squares of order 22 constructed by our method are given in the Appendix. Several attempts, necessarily erroneous, have been made in the past to prove Euler's conjecture, e.g., Peterson [19], Wernicke [20] and MacNeish [16]. Levi [14, p. 14] points

https://www.ams.org/tran/1960-095-02/S0002-9947-1960-0111695-3/S0002-9947-1960-0111695-3.pdf

On the construction of sets of mutually orthogonal Latin squares and the falsity of a conjecture of Euler

https://www.pnas.org/content/pnas/45/5/734.full.pdf

ON THE FALSITY OF EULER'S CONJECTURE ABOUT THE NON-EXISTENCE OF TWO ORTHOGONAL LATIN SQUARES OF ORDER 4t + 2.

It has been shown in [1] that all partially balanced incomplete block (PBIB) designs with two associate classes, can be divided into a small number of types according to the nature of the association relations among the treatments. One simple and important type is the group divisible (GD). The combinatorial properties of GD designs have been studied in [2] and the analysis along with that for other types is given in [1]. Here we give methods of constructing GD designs. These designs are likely to prove useful in agricultural, genetic and industrial experiments.

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On the construction of group divisible incomplete block designs

This note establishes a connection between Hadamard matrices H4t and the maximal binary codes M(4t, 2t; 8t), M(4t — 1, 2t; 4t) and M(4t — 2, 2t; 2t) in two symbols 0 and 1, where by M(n, d; m) we mean a set of m n-place sequences with 0 and 1 such that the Hamming distance between any two sequences is greater than or equal to d. The structure of these maximal codes is also studied in this paper.

S. S. Shrikhande

Papers

On the construction of sets of mutually orthogonal Latin squares and the falsity of a conjecture of Euler

ON THE FALSITY OF EULER'S CONJECTURE ABOUT THE NON-EXISTENCE OF TWO ORTHOGONAL LATIN SQUARES OF ORDER 4t + 2.

On the construction of group divisible incomplete block designs

A note on a result in the theory of code construction

Affine resolvable balanced incomplete block designs: A survey