Some new generalised Kirkman systems

Constructions are given for some generalised Kirkman designs and other perfect Graeco-Latin balanced incomplete block designs. In these designs, each of two sets of treatments is at-ranged with respect to the blocks as a balanced incomplete block design (BIBD) with parameters (v, b, r, k, lambda), each treatment set is arranged with respect to the other as in a symmetric BIBD with parameters (v, v, r, r, mu), and there is adjusted orthogonality between the two sets. Resolvable designs with v = s(3)+s(2)+s+1 and lambda = 1 are constructed from PG(3,s) for s = 2,3,4 and 8, and a further design is constructed from PG(5,2). Resolvable designs with v = 3(s), k = 3 and lambda = 1 are constructed for s = 3 and 5. Relationships with self-orthogonal Latin squares and tournament designs are discussed.