New upper bounds for binary linear covering codes

TL;DR

This paper establishes new upper bounds on the length function of binary linear covering codes for specific parameters, improving previous bounds through novel code constructions and partitions, and analyzes their asymptotic densities.

Contribution

The paper introduces new upper bounds for the length function of binary linear covering codes using innovative code families and partition techniques.

Findings

New upper bounds on ll_2(r,R) for R=2,3,4.

Construction of infinite code families with improved properties.

Asymptotic covering densities are reduced compared to previous results.

Abstract

The length function $\ell_2(r,R)$ is the smallest length of a binary linear code with codimension (redundancy) $r$ and covering radius $R$. We obtain the following new upper bounds on $\ell_2(r,R)$, which yield a decrease $\Delta(r,R)$ compared to the best previously known upper bounds: \begin{equation*} R=2,\,r=2t,\,r=18,20,\text{ and }r\ge28,\,\ell_2(r,2)\le26\cdot2^{r/2-4}-1;\,\Delta(r,2)=2^{r/2-4}. \end{equation*} \begin{equation*} R=3,\,r=3t-1,\,r=26\text{ and }r\ge44,\,\ell_2(r,3)\le819\cdot2^{(r-26)/3}-1;\,\Delta(r,3)=2^{(r-23)/3}. \end{equation*} \begin{equation*} R=4,\,r=4t,\,r=40\text{ and }r\ge68,\,\ell_2(r,4)\le2943\cdot2^{r/4-10}-1;\,\Delta(r,4)=2^{r/4-10}-1. \end{equation*} To obtain these bounds we construct new infinite code families, using distinct versions of the $q^m$-concatenating constructions of covering codes; some of these versions are proposed in this paper. We also introduce new useful partitions of column sets of parity check matrices of some codes. The asymptotic covering densities $\overline{\mu}(2)\thickapprox1.3203$, $\overline{\mu}(3)\thickapprox1.3643$, $\overline{\mu}(4)\thickapprox2.8428$, provided by the codes of the new families, are smaller than the known ones.