Construction of some sets of mutually orthogonal latin squares

Abstract

H. F. MacNeish [1 ] demonstrated constructively the existence of a set of t mutually orthogonal latin squares of each order n, where t is one less than the smallest factor of the prime-power decomposition of n. The construction was generalized somewhat and put on an algebraic foundation by H. B. Mann [2; 3, p. 105]. MacNeish [1] conjectured that t is the maximum number for each n. Had this conjecture been established, answers to two major questions would have been corollaries. These are: (1) the famous conjecture of Euler, dating from 1782, that there exists no pair of orthogonal latin squares of order= 2 (mod 4); (2) the conjecture that all finite projective planes are of prime-power orders-for an affine plane of order n is equivalent to a set of n -1 mutually orthogonal latin squares of order n. The purpose of this paper is to develop a construction yielding some new sets of mutually orthogonal latin squares. The general result is Theorem 1. For a few orders (possibly infinitely many distributed sparsely among the positive integers), Theorem 2 establishes the existence of sets of more than t mutually orthogonal latin squares; thus MacNeish's conjecture is disproved. Theorem 1 likely yields more than t for orders other than those covered by Theorem 2, but the author has found no example. The following lemma is familiar to some, but is apparently not in the literature.