Relations Among the Blocks of the Kronecker Product of Designs

Abstract

In the case of some incomplete block designs, interesting relations among their blocks have been discovered. For example, Fisher [1] has shown that in the case of a symmetrical BIB (Balanced Incomplete Block) design with parameters $v = b, r = k, \lambda$, any two blocks have exactly $\lambda$ treatments in common. Similarly, Bose [2] has shown that in the case of an affine resolvable BIB design with parameters $v = nk = n^2\{(n - 1)t + 1\}, \quad b = nr = n\{n^2t + n + 1\}, \quad \lambda = nt + 1,$ the blocks can be divided into sets of $n$ blocks, such that each set is a complete replication and any two blocks have $(k^2)/v = (nt - t + 1)$ or 0 treatments in common according as they belong to different groups or the same group. Also see Connor [3] and Bose and Connor [4] for similar results. Confining our attention to PBIB (Partially Balanced Incomplete Block) designs with two or three associate classes, we wish to see how this type of information for blocks of BIB designs can be used to obtain similar information for the blocks of their Kronecker product. In the next section are given a few general properties of the Kronecker product of designs. In Section 3 the main theorems of the paper are proved and their important particular cases are discussed. Some observations on the interconnection between these results and the theorems on inversion of designs (cf. Roy [5], Shrikhande [6]) are made in Section 4.