GDDs with two associate classes and with three groups of sizes 1, n, n and λ1 < λ2

Abstract

A group divisible design GDD(v = 1 + n + n, 3, 3, λ1, λ2) is an ordered pair (V, B) where V is an (1 + n + n)-set of symbols and B is a collection of 3-subsets (called blocks) of V satisfying the following properties: the (1 + n + n)-set is divided into 3 groups of sizes 1, n and n; each pair of symbols from the same group occurs in exactly λ1 blocks in B; and each pair of symbols from different groups occurs in exactly λ2 blocks in B. Let λ1, λ2 be positive integers. Then the spectrum of λ1, λ2, denoted by Spec(λ 1, λ2), is defined by Spec(λ1, λ2) = {n ∈ ℕ : a GDD(v = 1 + n + n, 3, 3, λ1, λ2) exists}. We found in [10] the spectrum Spec(λ1, λ2) provided that λ1 ≥ λ2 in all situations. We find in this paper Spec(λ1, λ2) when λ1 < λ2 in all situations. © 2013 Springer-Verlag.