Existence of generalized Bhaskar Rao designs with block size 3

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 λ = | G | . We prove these conditions are sufficient in general when v = 3 , and also when | G | is small, or when G is dicyclic. We summarize known results supporting the conjecture that these necessary conditions are always sufficient when k = 3 .