Optimality of some two-associate-class partially balanced incomplete-block designs

Let \mathcal{D}_{v,b,k} be the set of all the binary equireplicate incomplete-block designs for v treatments in b blocks of size k. It is shown that if \mathcal{D}_{v,b,k} contains a connected two-associate-class partially balanced design d^* with \lambda_2 = \lambda_1 \pm 1 which has a singular concurrence matrix, then it is optimal over \mathcal{D}_{v,b,k} with respect to a large class of criteria including the A,D and E critieria. The dual of d^* is also optimal over \mathcal{D}_{b,v,r} with respect to the same criteria, where r = bv/k. The result can be applied to many designs which were not previously known to be optimal. In another application, Bailey's (1988) conjecture on the optimality of Trojan squares over semi-Latin squares is confirmed.