The multicovering radii of codes

The covering radius of a code is the least r such that the set of balls of radius r around codewords covers the entire ambient space. We introduce a generalization of the notion of covering radius. The m-covering radius of a code is the least radius such that the set of balls of that radius covers all m-tuples of elements in the ambient space. We investigate basic properties of m-covering radii. We investigate whether codes exist with given m-covering radii (not always). We derive bounds on the size of the smallest code with a given m-covering radius, based on generalizations of the sphere bound and the method of counting excesses.