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

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. It was shown by Grüttmüller and Rees (Utilitas Math. 60 (2001) 153–180.) that MRDs with K={4,k} and k≡1 mod 3 exist with at most 56 possible exceptions. In this paper, we continue to investigate the spectrum for such MRDs. It is shown that the necessary conditions for existence are sufficient, with at most 10 possible exceptions. As a preliminary result, some new 4-GDDs of type gum1 are also constructed.