Short covering codes arising from matchings in weighted graphs

The concept of embedded matching in a weighted graph is introduced, and the maximum cardinality of an embedded matching is computed. On the other hand, consider the following problem induced by a short covering. Given a prime power q, the number c(q) denotes the minimum cardinality of a subset H of Fq 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. As another goal, a connection between embedded matching and short covering code is established. Moreover, this link is applied to improve the upper bound on c(q) for every odd prime power q.