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 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.