Covering codes and extremal problems from invariant sets under permutations

Let cq(n,R) denote the minimum cardinality of a subset H in Fqn such that every word in this space differs in at most R coordinates from a scalar multiple of a vector in H, where q is a prime power. In order to explore symmetries of such coverings, a few properties of invariant sets under certain permutations are investigated. New classes of upper bounds on cq(n,R) are obtained, extending previous results. Let Kq(n,R) denote the minimum cardinality of an R-covering code in the n-dimensional space over an alphabet with q symbols. As another application, a very-known upper bound on Kq(n,R) is improved under certain conditions. Moreover, two extremal problems are discussed by using tools from graph theory.