### The Multicovering Radii of Codes

IEEE Transactions on Information Theory ** 43 ** (1997) 1372-1377.
Authors:

Andrew Klapper, 779A Anderson Hall, Dept. of Computer Science,
University of Kentucky, Lexington, KY, 40506-0046, klapper at cs.uky.edu.

**Abstract**
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.

**Index Terms --**
Covering radius, coding theory, Hamming distance, binary vectors.