Covering sets for limited-magnitude errors

TL;DR

This paper introduces explicit constructions of covering sets for limited-magnitude errors, achieving near-optimal sizes for most integers and optimal sizes for prime moduli, with applications in error-correcting codes.

Contribution

The paper provides explicit constructions of covering sets with sizes close to theoretical bounds for almost all integers and optimal for prime moduli, advancing coding theory.

Findings

Explicit construction of covering sets with size $q^{1+o(1)} imes ext{max}\{ ext{lambda}, ext{mu} ight ext{-}1/2$ for most integers.

Achieves optimal size $p imes ext{max}\{ ext{lambda}, ext{mu} ight ext{-}1$ for prime moduli.

Proves a bound on the covering number using character sum bounds, though non-constructively.

Abstract

For a set $\cM=\{-\mu,-\mu+1,\ldots, \lambda\}\setminus\{0\}$ with non-negative integers $\lambda,\mu<q$ not both 0, a subset $\cS$ of the residue class ring $\Z_q$ modulo an integer $q\ge 1$ is called a $(\lambda,\mu;q)$-\emph{covering set} if $$ \cM \cS=\{ms \bmod q : m\in \cM,\ s\in \cS\}=\Z_q. $$ Small covering sets play an important role in codes correcting limited-magnitude errors. We give an explicit construction of a $(\lambda,\mu;q)$-covering set $\cS$ which is of the size $q^{1 + o(1)}\max\{\lambda,\mu\}^{-1/2}$ for almost all integers $q\ge 1$ and of optimal size $p\max\{\lambda,\mu\}^{-1}$ if $q=p$ is prime. Furthermore, using a bound on the fourth moment of character sums of Cochrane and Shi we prove the bound $$\omega_{\lambda,\mu}(q)\le q^{1+o(1)}\max\{\lambda,\mu\}^{-1/2},$$ for any integer $q\ge 1$, however the proof of this bound is not constructive.