The completely symmetric designs with blocksize three

Abstract The proper block design setting consists of v treatments to be assigned to b blocks, each block consisting of k experimental units. A design for this setting is said to be a completely symmetric design, or CSD, if its bottom stratum information matrix for estimation of treatment contrasts is constant on the diagonal and constant off the diagonal. Write bk=vr+p for some integer p with 0⩽p⩽v−1. If p=0, the CSDs of maximal conceivable trace are the balanced incomplete block designs. This paper constructs all maximal trace CSDs for k=3 and p=1,2,3. All of these designs are shown to be optimal under the E-criterion, and all but two under the MV-criterion.