Diallel cross experiments for comparing p test lines with a control in the set up of block designs and completely randomized designs are investigated. Complete diallel crosses including all p(p+1)/2 distinct crosses are considered. Families of A-optimal and efficient type S0 block designs for p=2,3, and for p>=4, k=2 are obtained, and the construction methods are given. For p>=k>=3, and p>=4, a sufficient condition for type S0 block designs with the control line appearing tb times, where t>=1 is an integer, to be A-optimal is obtained, and families of A-optimal type S0 block designs are provided. The A-optimality of type S designs under the model of completely randomized designs when 2<=p<=6 is also tudied, and some examples are given.