Simple Non-trivial Designs with an Arbitrary Automorphism Group

For positive integers t; v; k and l satisfying k > t and v > k; a t-design D with parameters ðv; k; lÞ is a set X of v points, together with a collectionB of subsets, called blocks, such that each block contains exactly k points and each t-tuple of points is contained in exactly l blocks. A t ðv; k; lÞ design is simple if repeated blocks are not allowed and is non-trivial if not all k-sets of points are blocks. In this paper, all the designs are supposed simple and nontrivial. Let D 1⁄4 ðX ;BÞ be a t ðv; k; lÞ design. It is well known that for all 14i4t; D is also a i ðv; k; lðiÞÞ design with lðiÞ 1⁄4 l v i t i = k i t i : Let Y be a set of m points ð04m4tÞ; we define