Existence of generalized Bhaskar Rao designs with block size 3

摘要

There are well-known necessary conditions for the existence of a generalized Bhaskar Rao design over a group G, with block size k = 3. The recently proved Hall-Paige conjecture shows that these are sufficient when v = 3 and X = vertical bar G vertical bar. We prove these conditions are sufficient in general when v = 3, and also when vertical bar G vertical bar is small, or when G is dicyclic. We summarize known results supporting the conjecture that these necessary conditions are always sufficient when k = 3.