Mixed block designs

Let V i ( i = 1, 2) be a set of size v i . Let D be a collection of ordered pairs ( b 1 , b 2 ) where b i is a k i ‐element subset of V i . We say that D is a mixed t ‐design if there exist constants λ ( j , j 2) , (0 ≤ j i ≤ k i , j 1 + j 2 ≤ t ) such that, for every choice of a j 1 ‐element subset S 1 of V 1 and every choice of a j 2 ‐element subset S 2 of V 2 , there exist exactly λ ( j 1, j 2) ordered pairs ( b 1 , b 2 ) in D satisfying S 1 ⊆ b 1 and S 2 ⊆ b 2 . 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 | ≥ v 1 + v 2 − 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 | = v 1 + v 2 − 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