Mixed block designs

Abstract

Let Vi (i = 1, 2) be a set of size vi. Let D be a collection of ordered pairs (b1, b2) where bi is a ki-element subset of Vi. We say that D is a mixed t-design if there exist constants λ (j,j2), (0 ≤ ji ≤ ki, j1 + j2 ≤ t) such that, for every choice of a j1-element subset S1 of V1 and every choice of a j2-element subset S2 of V2, there exist exactly λ(j1,j2) ordered pairs (b1, b2) in D satisfying S1 ⊆ b1 and S2 ⊆ b2. In W. J. Martin [Designs in product association schemes, submitted for publication], Delsarte's theory of designs in association schemes is extended to products of Q-polynomial association schemes. Mixed t-designs arise as a particularly interesting case. These include symmetric designs with a distinguished block and α-resolvable balanced incomplete block designs as examples. The theory in the above-mentioned paper yields results on mixed t-designs analogous to those known for ordinary t-designs, such as the Ray-Chaudhuri/Wilson bound. For example, the analogue of Fisher's inequality gives |D| ≥ v1 + v2 − 1 for mixed 2-designs with Bose's condition on resolvable designs as a special case. Partial results are obtained toward a classification of those mixed 2-designs D with |D| = v1 + v2 − 1. The central result of this article is Theorem 3.1, an analogue of the Assmus–Mattson theorem which allows us to construct mixed (t + 1 − s)-designs from any t-design with s distinct block intersection numbers. © 1998 John Wiley & Sons, Inc. J Combin Designs 6:151–163, 1998