On the size of partial block designs with large blocks

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