GDDs WITH TWO ASSOCIATE CLASSES AND WITH THREE GROUPS OF SIZES 3, n AND n

Abstract

A group divisible design GDD(v = 3+n+n, 3, 3, λ1, λ2) is an ordered pair (V, B) where V is an (3 + n + n)-set of symbols and B is a collection of 3-subsets (called blocks) of V satisfying the following properties: the (3 + n + n)-set is divided into 3 groups of sizes 3, 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(λ1, λ2), is defined by Spec(λ1, λ2) = {n ∈ N : a GDD(v =3+ n + n, 3, 3, λ1, λ2) exists}.
We find the spectrum Spec(λ1, λ2) for all λ1 ≥ λ2.