Short coverings in tridimensional spaces arising from sum-free sets

Given a prime power qq, define c(q)c(q) as the minimum cardinality of a subset HH of the tridimensional space Fq3 which satisfies the following property: every vector in this space differs in at most 1 coordinate from a multiple of a vector in HH. 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)/4c(q)≤3(q+4)/4, improving the earlier upper bound c(q)≤q+1c(q)≤q+1.