Designs in Product Association Schemes

A number of important families of association schemes—such as the Hamming and Johnson schemes—enjoy the property that, in each member of the family, Delsarte t-designs can be characterised combinatorially as designs in a certain partially ordered set attached to the scheme. In this paper, we extend this characterisation to designs in a product association scheme each of whose components admits a characterisation of the above type. As a consequence of our main result, we immediately obtain linear programming bounds for a wide variety of combinatorial objects as well as bounds on the size and degree of such designs analogous to Delsarte's bounds for t-designs in Q-polynomial association schemes.