Let 픅 be a family of k-subsets of a υ-set V, with 1 ≤ k ≤ υ/2. Given only the inner distribution of 픅, 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 픅 meets an arbitrary t-subset of V, for each order t (with 1 ≤ t ≤ υ/2). This description makes use of a linear transform based on a system of dual Hahn polynomials with parameters υ, k, t. The main regularity parameter is the dimension of a well-defined subspace of ℝt+1, called the t-form space of 픅. (This subspace coincides with ℝt+1 if and only if 픅 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. © 1993 American Mathematical Society.