On generalised tt-designs and their parameters

Recently, P.J. Cameron studied a class of block designs which generalises the classes of tt-designs, αα-resolved 2-designs, orthogonal arrays, and other classes of combinatorial designs. In fact, Cameron’s generalisation of tt-designs (when there are no repeated blocks) is essentially a special case of the “poset tt-designs” in product association schemes studied ten years earlier by W.J. Martin, who further studied the special case of “mixed block designs”. In this paper, we study Cameron’s generalisation of tt-designs from the point of view of classical tt-design theory, in particular investigating the parameters of these generalised tt-designs. We show that the tt-design constants λiλi (the number of blocks containing an ii-subset of the points, where i≤ti≤t) and λij (the number of blocks containing an ii-subset II of the points and disjoint from a jj-subset JJ of the points, where I∩J=0̸I∩J=0̸ and i+j≤ti+j≤t) have very natural counterparts for generalised tt-designs. Our main result places strong restrictions on the block structure of Cameron’s tt-(v,k,λ) designs, an important subclass of generalised tt-designs. We also generalise N.S. Mendelsohn’s concept of “intersection numbers of order rr” for tt-designs, and show that analogous equations to those of Mendelsohn hold for generalised tt-designs.