New Construction of Minimal (v, 3, 2)-Coverings

Nebojša Nikolić

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.