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**Optimality of some two-associate-class partially balanced incomplete- block designs.**
*(English)*
Zbl 0741.62071

Summary: 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 \(E\) criteria. The dual of \(d^*\) is also optimal over \({\mathcal D}_{b,v,r}\) with respect to the same criteria, where \(r=bk/v\).

The result can be applied to many designs which were not previously known to be optimal. In another application, the second author’s [J. Stat. Plann. Inference 18, 299-312 (1988; Zbl 0639.62066)] conjecture on the optimality of Trojan squares over semi-Latin squares is confirmed.

The result can be applied to many designs which were not previously known to be optimal. In another application, the second author’s [J. Stat. Plann. Inference 18, 299-312 (1988; Zbl 0639.62066)] conjecture on the optimality of Trojan squares over semi-Latin squares is confirmed.

### MSC:

62K05 | Optimal statistical designs |

62K10 | Statistical block designs |

05B05 | Combinatorial aspects of block designs |