$A$-optimal designs for comparing each of $\nu$ test treatments simultaneously with a control, in $b$ blocks of size $k$ each are considered. It is shown that several families of BIB designs in the test treatments augmented by $t$ replications of a control in each block are $A$-optimal. In particular these designs with $t = 1$ are optimal whenever $(k - 2)^2 + 1 \leq \nu \leq (k - 1)^2$ irrespective of the number of blocks. This includes BIB designs associated with finite projective and Euclidean geometries.
"Families of $A$-Optimal Block Designs for Comparing Test Treatments with a Control." Ann. Statist. 13 (2) 757 - 767, June, 1985. https://doi.org/10.1214/aos/1176349552