Group divisible designs with block size four and group type g_u m₁

Non-uniform group divisible designs have been studied by numerous researchers in the past two decades due to their vital applications in the constructions for other types of designs. Much progress has been made for the existence of {4}-GDDs of type g_u m₁, especially when gu is even. The corresponding problem for block size three had been solved by Colbourn et al. (J Comb Theory Ser A 59:73–89, 1992). In this paper, we consider the entire existence problem for such {4}-GDDs. We show that, for each given g, up to a small number of undetermined cases of u, the necessary conditions on (u,m) for the existence of a {4}-GDD of type g_u m₁ are also sufficient.