On the size of partial block designs with large blocks

A t − (n, k, ) design is a k-uniform hypergraph with the property that every set of t vertices is contained in exactly of the edges (blocks). A partial t − (n, k, ) design is a k-uniform hypergraph with the property that every set of t vertices is contained in at most edges; or equivalently the intersection of every set of + 1 blocks contains fewer than t elements. Let us denote by f (n, k, t) the maximum size of a partial t− (n, k, ) design. We determine f (n, k, t) as a fundamental problem in design theory and in coding theory. In this paper we provide some new bounds for f (n, k, t). © 2005 Elsevier B.V. All rights reserved.