Optimality of Some Two-Associate-Class Partially Balanced Incomplete-Block Designs

Abstract

Let $\mathscr{D}_{\upsilon,b,k}$ be the set of all the binary equireplicate incomplete-block designs for $\upsilon$ treatments in $b$ blocks of size $k$. It is shown that if $\mathscr{D}_{\upsilon,b,k}$ contains a connected two-associate-class partially balanced design $d^\ast$ with $\lambda_2 = \lambda_1 \pm 1$ which has a singular concurrence matrix, then it is optimal over $\mathscr{D}_{\upsilon,b,k}$ with respect to a large class of criteria including the $A,D$ and $E$ criteria. The dual of $d^\ast$ is also optimal over $\mathscr{D}_{b,\upsilon,r}$ with respect to the same criteria, where $r = bk/\upsilon$. 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.