William D. Palmer

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 @l=|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.

Generalized Bhaskar Rao designs and dihedral groups

We solve the existence problem for generalized Bhaskar Rao designs of block size 3 for an infinite family of non-abelian groups, the dihedral groups Dn, of order 2n. In our main result we show that for n ≥ 1 and v ≥ 3 the following set of conditions is necessary and sufficient for the existence of a GBRD(v, 3, λ Dn): 1. λ ≡ 0 (mod 2n): 2. λ v(v - 1) ≡ 0 (mod 24).

GBRDs over groups of orders ≤100 or of order p q with p, q primes

There are well-known necessary conditions for the existence of a generalized Bhaskar Rao design over a group G, with block size k=3. It has been conjectured that these necessary conditions are indeed sufficient. We prove that they are sufficient for groups G of order pq where p,q are primes and for groups of all orders @?100 except possibly 32, 36, 48, 54, 60, 64, 72, 96.

GBRDs over supersolvable groups and solvable groups of order prime to 3

We show that the established necessary conditions for a GBRD

Bhaskar Rao designs over small groups

{(v,3,\lambda; \mathbb {G})}

Bhaskar Rao designs and the alternating group A 4 .

are sufficient (i) when

Partial generalized Bhaskar Rao designs over abelian groups.

{\mathbb {G}}