Invariant sets under permutation, extremal graphs, and covering codes

Let cq(n,R) denote the minimum cardinality of a subset H in such that every word in this space differs in at most R coordinates from a multiple of a vector in H, where q is a prime or a prime power. In order to explore the symmetries of such coverings, we investigate a few algebraic properties of invariant sets under permutation. Extremal problems arising from invariant sets are also studied on a graph theoretical viewpoint. As an application, a new class of upper bounds on cq(n,R) is constructed.