Resolvable bibd and sols

Abstract

The ideas of a base factorization and a resolvable grid system are introduced, and a construction of a resolvable balanced incomplete block design (BIBD) from these structures is given. Resolvable grid systems can be constructed from mutually orthogonal self-orthogonal latin squares (SOLS) with symmetric mate. Together these results prove, as a special case, that: if k−1 is an odd prime power and there exist 1 2 (k−2) mutually orthogonal SOLS of order n, with symmetric mate, then there exists a resolvable BIBD with block size k on v = kn points of index λ, where λ = k−1 if k = 0 (mod 4) and λ = 2( k−1) if k = 2 (mod 4). The technique is illustrated for k = 4, λ = 3 and k = 6, λ = 10, in which cases v = 0 (mod k) is shown to be a necessary and sufficient condition (NASC) for the existence of a resolvable BIBD on v points. The pair ( k, λ) = (6,10) thus becomes only the fifth pair for which NASC are known, the other pairs being (3,1), (4,1), (3,2), and (4,3).