New construction of minimal (v, 3,2)-coverings

Abstract

A (v, 3,2)-covering is a family of 3-subsets of a v-set, called blocks, such that any two elements of v-set appear in at least one of the blocks. In this paper, we propose new construction of (v, 3,2)-coverings with the minimum number of blocks. This construction represents a generalization of Bose's and Skolem's constructions of Steiner systems S(2,3,6n + 3) and S(2,3,6n + 1). Unlike the existing constructions, our construction is direct and it uses the set of base blocks and permutation p, so by applying it to the remaining blocks of (v, 3,2)-coverings are obtained.