Extending the T-design Concept

Let B be a family of k-subsets of a v-set V , with 1 less-than-or-equal-to k less-than-or-equal-to v/2. Given only the inner distribution of B i.e., the number of pairs of blocks that meet in j points (with j = 0, 1, ... , k), we are able to completely describe the regularity with which B meets an arbitrary t-subset of V , for each order t (with 1 less-than-or-equal-to t less-than-or-equal-to v/2). This description makes use of a linear transform based on a system of dual Hahn polynomials with parameters v, k, t. The main regularity parameter is the dimension of a well-defined subspace of R(t+1), called the t-form space of B. (This subspace coincides with R(t+1) if and only if B is a t-design.) We show that the t-form space has the structure of an ideal, and we explain how to compute its canonical generator