Mandatory representation designs MRD (4, k; v) with k ≡ 1 mod 3

Abstract

A mandatory representation design MRD (K; v) is a pairwise balanced design on v points with block sizes from the set K in which for each k ∈ K there is at least one block in the design of size k. MRDs with K = {3, k} have been extensively studied by Mendelsohn and Rees [MR], Rees [R], and Grüttmüller [G1, G2]. In this paper, we consider MRDs with K = {4, k}, where k ≡ 1 mod 3. We show that the necessary conditions for existence are sufficient, with at most 56 exceptions.