Designs in Product Association Schemes

Let (Y; A) be an association scheme with primitive idempotents E 0 ; E 1 ;. .. ; E d. For T f1;. .. ; dg, a Delsarte T-design in (Y; A) is a subset D of Y whose characteristic vector is annihilated by the idempotents E j (j 2 T). The case most studied is that in which (Y; A) is Q-polynomial and T = f1;. .. ; tg. For many such examples, a combina-torial characterization is known, giving an equivalence between Delsarte T-designs and poset t-designs in what we call here \Q-posets". For example, combinatorial t-designs (i.e., block designs) can be described via the truncated Boolean lattice while orthogonal arrays can be described via the Hamming lattice. For 1 i m, let (Y i ; A i) be a Q-polynomial association scheme. Assume that Delsarte t-designs in each (Y i ; A i) are characterised as poset t-designs in a Q-poset P i attached to that scheme. With these assumptions, we consider the product association scheme (Y i ; A). The primitive idempotents for this scheme naturally inherit the partial order structure of a product of chains. Our main result, Theorem 2.3, characterises Delsarte T-designs in (Y i ; A) as poset designs in the product poset P i where T is any downset in the product of chains. Using this characterisation, we immediately obtain linear programming bounds for a wide variety of combinatorial objects. On the other hand, if we assume each component scheme is Q-polynomial, we obtain bounds on the size and degree of such a T-design analogous to Delsarte's bounds for t-designs in Q-polynomial association schemes. 1 Overview and Background We are interested in applications of the theory of association schemes to problems in coding theory and design theory. This investigation is motivated by two applications. In 10],