Tight incomplete block designs Original Research Article

An incomplete t-wise balanced design (ItBD) of type t-(v,h,K,λ) is a triple (X,H,B) where X is a v-element set of points, H is an h-element subset H⊆X called the hole, and B is a collection of subsets of X called blocks, such that the size of every block B∈B is in K and every t-element subset of X is either in the hole or in exactly λ blocks, but not both. Kreher and Rees (Codes an Designs, Ohio state University Research Institute Publication, 10 (2002) 179) derived an upper bound on the size of the hole, which is given here in Theorem 3. An ItBD meeting this bound is called a tight incomplete block design. In this paper we study the existence of tight incomplete block designs whose automorphism group is as large as possible. In particular, we obtain a characterization of those tight ItBDs (X,H,B) of prime-power index λ admitting Sym(H)×Sym(X⧹H) as an automorphism group.