We consider the problem of comparing test treatments with a control in a proper block design. We give conditions on the parameters of both R-type and S-type designs that guarantee their $A$-optimality, and demonstrate how these conditions can be used to obtain families of $A$-optimal designs. We give an example for the construction of the desired S-type designs. A table with optimal R-type designs $(3 \leq k \leq 10, k \leq v \leq 30)$ is also given.
"$A$-Optimal Block Designs for Comparing Test Treatments with a Control." Ann. Statist. 15 (4) 1629 - 1638, December, 1987. https://doi.org/10.1214/aos/1176350614