Short coverings in tridimensional spaces arising from sum-free sets

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 I 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 + I. (c) 2007 Elsevier Ltd. All rights reserved.