Designs in product association schemes (Q1299901)

The concept of a \(Q\)-poset attached to an association scheme is introduced. The main result is Theorem 2.2 which uses these partially ordered sets to characterise the Delsarte \({\mathcal T}^*\)-designs in a product association scheme where each component scheme is assumed to have an attached \(Q\)-poset and where \(\mathcal T\) is a downset in a product of chains. The balance of the paper explores bounds on \(\mathcal T\)-designs in product schemes where each component is a \(Q\)-polynomial. The results include a linear programming bound, a Delsarte bound analogous to the Rao bound for orthogonal arrays, and a degree bound.