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 the 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 (they don't 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. 1 Introduction -- Basic Concepts The covering radius of a block code C is the smallest radius such that the set of balls of that radius covers the ambient space. More precisely, if C has length n, it is the smallest integer t such that every vector of length n has distance at most t from at least one code word. This concept has been the subject of hu..
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