### Improved Multicovering Bounds from Linear Inequalities and Supercodes

IEEE Transactions on Information Theory, ** 50** (2004) 532-536.
Authors:

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

**Abstract**
The multicovering radii of a code are natural generalizations of the covering
radius in which the goal is to cover all m-tuples of vectors for some m
as cheaply as possible. In this paper describe several techniques for
obtaining lower bounds on the sizes of codes achieving a given multicovering
radius. Included among these methods is a generalization of the method of
linear inequalitie, based on refined weight distributions
of the code. For m=2, we also obtain a linear upper bound on the
m-covering radius. We further study bounds on the sizes of codes with a
given multicovering radius that are subcodes of a fixed code. We find, for
example, constraints on parity checks that can hold for codes with small
ordinary covering radius.

**Index Terms --**
Covering radius, error correcting codes, weight distribution, linear
inequalities, supercodes.