On the Properties of Proper $(M, S)$ Optimal Block Designs

Properties of designs which are $(M, S)$ optimal within various classes of proper block designs are studied. The classes of designs considered are not restricted to connected designs. Connectedness is shown to be a property generally possessed by designs which are $(M, S)$ optimal within these more general classes of designs. In addition, we show that the complement of any proper binary $(M, S)$ optimal design is $(M, S)$ optimal within an appropriate class of complementary designs and that the dual of any proper equireplicated $(M, S)$ optimal design is $(M, S)$ optimal within an appropriate class of dual designs.