Short coverings and matching in weighted graphs

Given a prime power q, the number c ( q ) denotes the minimum cardinality of a subset H of F q 3 which satisfies the following property: every element in this space differs in at most 1 coordinate from a scalar multiple of a vector in H . In this work, the upper bound on c ( q ) is improved when q is odd. The method is based on ω-partition, a combinatorial concept which can be also reformulated as a kind of matching in weighted graph.