Covering codes and extremal problems from invariant sets under permutations

Let c q ( n , R ) denote the minimum cardinality of a subset H in F q n 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 c q ( n , R ) are obtained, extending previous results. Let K q ( 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 K q ( n , R ) is improved under certain conditions. Moreover, two extremal problems are discussed by using tools from graph theory.