$A$-optimality of block designs for control versus test comparisons 
in diallel crosses is investigated. 
A sufficient condition for designs to be $A$-optimal is derived.  
Type $S_0$ designs are defined and A-optimal type $S_0$ designs are 
characterized.  A lower bound to the $A$-efficiency of type $S_0$ 
designs is also given.  Using the lower bound to $A$-efficiency, 
type $S_0$ designs are shown to yield efficient designs for test versus 
control comparisons.