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New upper bounds for binary linear covering codes

Alexander A. Davydov, Stefano Marcugini, Fernanda Pambianco

TL;DR

This work advances binary covering codes by delivering new constructive upper bounds on the length function $\ell_2(r,R)$ for radii $R\in\{2,3,4\}$ through innovative binary $q^m$-concatenating constructions. It introduces refined partitions of parity-check matrices and new $(R,\ell)$-objects, enabling several infinite families with significantly improved asymptotic covering densities $\overline{\mu}(R)$ and smaller lengths than previously known. The main results provide explicit bounds such as $\ell_2(r,2)\le26\cdot 2^{r/2-4}-1$, $\ell_2(r,3)\le819\cdot 2^{(r-26)/3}-1$, and $\ell_2(r,4)\le2943\cdot 2^{r/4-10}-1$ for growing $r$ in specified congruence classes, along with new infinite families and sporadic codes. These constructions enhance practical code design for covering problems and offer a pathway toward extending the methodology to radii $R\ge5$.

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 $Δ(r,R)$ 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;\,Δ(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;\,Δ(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;\,Δ(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 $\overlineμ(2)\thickapprox1.3203$, $\overlineμ(3)\thickapprox1.3643$, $\overlineμ(4)\thickapprox2.8428$, provided by the codes of the new families, are smaller than the known ones.

New upper bounds for binary linear covering codes

TL;DR

This work advances binary covering codes by delivering new constructive upper bounds on the length function for radii through innovative binary -concatenating constructions. It introduces refined partitions of parity-check matrices and new -objects, enabling several infinite families with significantly improved asymptotic covering densities and smaller lengths than previously known. The main results provide explicit bounds such as , , and for growing in specified congruence classes, along with new infinite families and sporadic codes. These constructions enhance practical code design for covering problems and offer a pathway toward extending the methodology to radii .

Abstract

The length function is the smallest length of a binary linear code with codimension (redundancy) and covering radius . We obtain the following new upper bounds on , which yield a decrease 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;\,Δ(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;\,Δ(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;\,Δ(r,4)=2^{r/4-10}-1. \end{equation*} To obtain these bounds we construct new infinite code families, using distinct versions of the -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 , , , provided by the codes of the new families, are smaller than the known ones.

Paper Structure

This paper contains 16 sections, 30 theorems, 46 equations, 4 tables.

Table of Contents

  1. Introduction, notations, and background
  2. Covering codes, covering density, the length function
  3. Connection of covering codes with other areas of theory and practice
  4. The one-to-one correspondence between covering codes and saturating sets in projective spaces
  5. The short content of this paper
  6. The main results
  7. The $q^m$-concatenating constructions of covering codes for $q=2$
  8. Spherical $(R,\ell )$-capsules, $(R,\ell)$-objects, and partitions of the column set of a code parity check matrix
  9. Basic $q^m$-concatenating Construction QM for $\boldsymbol{q=2}$
  10. The known results on binary linear codes of covering radius 2
  11. New results on binary linear codes of covering radius 2
  12. The known results on binary linear codes of covering radius 3
  13. New results on binary linear codes of covering radius 3
  14. The known results on binary linear codes of covering radius 4
  15. New results on binary linear codes of covering radius 4
  16. ...and 1 more sections

Key Result

Theorem 2.1

There are new infinite families of covering $[n,n-r]_2R$ codes such that:

Theorems & Definitions (53)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 2.1
  • Theorem 2.2
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Theorem 4.1
  • ...and 43 more