On the E-Optimality of PBIB Designs with a Small Number of Blocks

Abstract

It is shown that several families of PBIB designs with relatively few blocks are $E$-optimal over the collection of all block designs. Among these are: the Partial Geometries with two associate classes; PBIB designs with $\lambda_1 = 1, \lambda_2 = 0$ and fewer blocks than varieties; PBIB designs with triangular schemes of size $n, \lambda_1 = 0, \lambda_2 = 1$ and block size $k \geq n/2$ (or $\lambda_1 = 1, \lambda_2 = 0$ and $k \geq n - 1)$; PBIB designs with $L_i$ schemes based on $\nu$ varieties with $\lambda_1 = 0, \lambda_2 = 1, k \geq \sqrt{\nu}$ (or $\lambda_1 = 1, \lambda_2 = 0$ and either $i - 1 \leq \sqrt{\nu} \leq k$ or $k \leq \sqrt{\nu} \leq i - 1$). The duals of these designs are also $E$-optimal. In certain settings Partial Geometries are the unique $E$-optimal designs.