Group divisible designs with two associate classes and (λ 1, λ 2) = (1, 2)

Abstract

A group divisible design GDD(v = v 1 + v 2 + -+v g, g, k; λ 1, λ 2) is an ordered pair (V, B) where V is a v-set of symbols and B is a collection of k-subsets (called blocks) of V satisfying the following properties: the v-set is divided into g groups of sizes v 1, v 2,...,V g; each pair of symbols from the same group occurs in exactly λ 1 blocks in B; and each pair of symbols from different groups occurs in exactly λ 2 blocks in B. In this paper we give necessary conditions on m and n for the existence of a GDD(v = m + n, 2, 3; 1, 2), along with sufficient conditions for each m ≤ n/2. Furthermore, we introduce some construction techniques to construct some GDD(v = m + n, 2, 3, l, 2)s when m>%n/2, namely, a GDD(v = 9+15, 2, 3; 1, 2) and a GDD(v = 25 + 33, 2, 3; 1, 2).