Mixed block designs

✍ Scribed by William J. Martin

• Publisher: John Wiley and Sons
• Year: 1998
• Tongue: English
• Weight: 281 KB
• Volume: 6
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/(sici)1520-6610(1998)6:2<151::aid-jcd5>3.0.co;2-e

✦ Synopsis

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 1 ,j 2 ) (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 (j 1 ,j 2 ) 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.