Projective covering designs

Abstract

A (2, k, v) covering design is a pair (X, F) such that X is a v-element set and F is a family of k-element subsets, called blocks, of X with the property that every pair of distinct elements of X is contained in at least one block. Let C(2, k, v) denote the minimum number of blocks in a (2, k, v) covering design. We construct in this paper a class of (2, k, v) covering designs using number theoretic means, and determine completely the functions C(2,6,6n · 28) for all n ≥ 0, and C(2,6,6n · 28 − 5) for all n ≥ 1. Our covering designs have interesting combinatorial properties.