The Concept of a (t,r)-Regular Design as an Extension of the Classical Concept of a t-Design.

Let font 4 B be a family of d-subsets of an n-set sP, with 1 = d = n/2. Given only the inner distribution of font 4 B, i.e., the number of pairs of blocks in font 4 B that meet in i points (for i = 0, 1, ..., d), we are able to completely describe the regularity with which font 4 B meets an arbitrary t-subset of font 4 P , for each order t (with 1 = t = n/2). This description makes use of a linear transform based on a system of dual Hahn polynomials with parameters t, d, n. The t-regularity properties of font 4 B involve a well-defined integer r, called t-degree (with 0 = r = t), which can be computed from the inner distribution; the family font 4 B is referred to as a (t, r)-regular r design. In particular, a (t, 0)-regular r design is a classical t-design. In general, the smaller the value of the t-degree, the more regular the design font 4 B (for a given order t).