A holey perfect Mendelsohn design of type h n 1 1 h n 2 2 h n k k (HPMD(h n 1 1 h n 2 2 h n k k) with block size four is equivalent to a frame idempotent quasigroup satisfying Stein's third law with the same type, where a frame quasigroup of type h n 1

It is proved in this paper that for m ≡ 0; 2; 6; 10 (mod 12) there exists a resolvable group divisible design of order v, block size 4 and group size m if and only v ≡ 0 (mod 4), v ≡ 0 (modm), v − m ≡ 0 (mod 3), except when (3; 12) and except possibly when (3; 264); (3; 372); (8; 80); (8; 104), (9; 396) (40; 400) or (40; 520). c © 2002 Elsevier Science B.V. All rights reserved.

We find the 302 graphs of valency 8 with the property that any two nonadjacent vertices have 0 or 2 common neighbours, and, in particular, the 104 semibiplanes of block size 8.

A transversal design of order n and block size k, or TD(k; n), is a triple (X, G, B ), where X is a set of kn elements. G 5 hG1 , . . . , Gkj is a partition of X into k sets each of size n; each class of the partition is a group. B is a set of k-subsets of X, with the property that each B [ B satisfies uB > Giu 5 1 for each 1 # i # k; sets in B are blocks. Finally, each unordered pair of elemen...

A defining set of a t-(v, k, λ) design is a subcollection of its blocks which is contained in no other t-design with the given parameters, on the same point set. A minimal defining set is a defining set, none of whose proper subcollections is a defining set. The spectrum of minimal defining sets of a design D is the set {|M | | M is a minimal defining set of D}. We show that if a t-(v, k, λ) de...

We present constructions and results about GDDs with two groups and block size five in which each block has Configuration (s, t), that is, in which each block has exactly s points from one of the two groups and t points from the other. After some results for a general k, s and t, we consider the (2, 3) case for block size 5. We give new necessary conditions for this family of GDDs and give mini...

Let Vn(q) denote a vector space of dimension n over the field with q elements. A set P of subspaces of Vn(q) is a partition of Vn(q) if every nonzero vector in Vn(q) is contained in exactly one subspace in P. A uniformly resolvable design is a pairwise balanced design whose blocks can be resolved in such a way that all blocks in a given parallel class have the same size. A partition of Vn(q) co...

We complete the solution of the existence problem for generalized Bhaskar Rao designs of block size 3 over groups of order 12. In particular we prove that if G is a group of order 12 which is cyclic or dicyclic, then a generalized Bhaskar Rao design, GBRD(v, 3, λ = 12t;G) exists for all v ≥ 3 when t is even and for all v ≥ 4 when t is odd.