Short coverings in tridimensional spaces arising from sum-free sets

Abstract

Given a prime power q , define c ( q ) as the minimum cardinality of a subset H of the tridimensional space F q 3 which satisfies the following property: every vector in this space differs in at most 1 coordinate from a multiple of a vector in H . On the basis of suitable actions of group, there is established a connection between sum-free sets and corresponding coverings. As an application of our method, there is constructed a class of short coverings which yields c ( q ) ≤ 3 ( q + 4 ) / 4 , improving the earlier upper bound c ( q ) ≤ q + 1 .