### Resolvable bibd and sols

Ronald D. Baker
1983 Discrete Mathematics
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 J(k -2) mutually orthogonal SOLS of order n, with symmetric mate, then there exists a
more » ... esolvable BIBD with block size k on u = kn points of index A, where A = k -1 if k = 0 (mod 4) and A = 2(k -1) if k = 2 (mod 41. The technique is illustrated for k =4, A = 3 and k =6, A = 10, in which cases u = 0 (mod k) is shown to be a necessary and sufficient condition (NASC) for the existence of a resolvable BIBD on u points. The pair (k, A) = (6,lO) thus becomes only the fifth pair for which NASC are known, the other pairs being (3,l). (4,1), (3,2) and (4,3) .