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Infinitely many families of distance-optimal binary linear codes with respect to the sphere packing bound

Hao Chen, Conghui Xie, Cunsheng Ding

TL;DR

This work addresses the long-standing question of whether there exist infinite families of distance-optimal binary linear codes with arbitrarily large minimum distance $d$ relative to the sphere packing bound. It achieves this by constructing and analyzing BCH codes of carefully chosen lengths, including $n=(2^{2s}+1)(2^s-1)$, $n=2^{2s}+2^s+1$, and $n=(2^s-1)/\lambda$, along with their related and extended codes, and then proving distance-optimality via sphere-packing arguments and BCH bounds. The main contribution is the first known infinite family of distance-optimal binary cyclic codes with unbounded $d$ with respect to the sphere packing bound; additional by-products include infinite families with small $d$ and two infinite families of binary five-weight codes. These results significantly advance the catalog of distance-optimal codes, offering new avenues for theory and applications in coding theory and communications.

Abstract

R. W. Hamming published the Hamming codes and the sphere packing bound in 1950. In the past 75 years, infinite families of distance-optimal linear codes over finite fields with minimum distance at most 8 with respect to the sphere packing bound have been reported in the literature. However, it is a 75-year-old open problem in coding theory whether there is an infinite family of distance-optimal linear codes over finite fields with arbitrarily large minimum distance with respect to the sphere packing bound. This main objective of this paper is to settle this long-standing open problem in coding theory. As by-products, several infinite families of distance-optimal binary codes with small minimum distances are presented. Two infinite families of binary five-weight codes are reported. Some open problems are also proposed.

Infinitely many families of distance-optimal binary linear codes with respect to the sphere packing bound

TL;DR

This work addresses the long-standing question of whether there exist infinite families of distance-optimal binary linear codes with arbitrarily large minimum distance relative to the sphere packing bound. It achieves this by constructing and analyzing BCH codes of carefully chosen lengths, including , , and , along with their related and extended codes, and then proving distance-optimality via sphere-packing arguments and BCH bounds. The main contribution is the first known infinite family of distance-optimal binary cyclic codes with unbounded with respect to the sphere packing bound; additional by-products include infinite families with small and two infinite families of binary five-weight codes. These results significantly advance the catalog of distance-optimal codes, offering new avenues for theory and applications in coding theory and communications.

Abstract

R. W. Hamming published the Hamming codes and the sphere packing bound in 1950. In the past 75 years, infinite families of distance-optimal linear codes over finite fields with minimum distance at most 8 with respect to the sphere packing bound have been reported in the literature. However, it is a 75-year-old open problem in coding theory whether there is an infinite family of distance-optimal linear codes over finite fields with arbitrarily large minimum distance with respect to the sphere packing bound. This main objective of this paper is to settle this long-standing open problem in coding theory. As by-products, several infinite families of distance-optimal binary codes with small minimum distances are presented. Two infinite families of binary five-weight codes are reported. Some open problems are also proposed.
Paper Structure (23 sections, 16 theorems, 55 equations, 1 table)

This paper contains 23 sections, 16 theorems, 55 equations, 1 table.

Table of Contents

  1. Introduction
  2. Linear codes, cyclic codes and BCH codes
  3. The motivation of this paper
  4. The objectives and methodology of this paper
  5. The achievements of this paper
  6. The organization of this paper
  7. Known infinite families of distance-optimal binary linear codes
  8. Known infinite families of distance-optimal binary linear codes with respect to the sphere packing bound
  9. Known infinite families of distance-optimal binary linear codes with respect to the Griesmer bound
  10. Preliminaries
  11. The first type of BCH binary codes and their related codes
  12. The parameters of the codes ${\mathcal{C}}_{(2,n,3,1)}$ and their duals
  13. The extended codes $\overline{{\mathcal{C}}_{(2,n, 3, 1)}}$ of the codes ${\mathcal{C}}_{(2,n, 3, 1)}$
  14. The distance-optimal codes ${\mathcal{C}}_{(2,n, 6, 0)}$
  15. The codes ${\mathcal{C}}_{(2,n, 5, 1)}$ and their extended codes
  16. ...and 8 more sections

Key Result

Theorem 3.1

Let ${\mathcal{C}}$ be a cyclic code of length $n$ over ${\mathrm{GF}}(q)$ with defining set $T$ and minimum distance $d$. Assume that $T$ contains $\delta -1$ consecutive integers for some integer $\delta$. Then $d \geq \delta$.

Theorems & Definitions (48)

  • Theorem 3.1: BCH bound
  • Theorem 3.2
  • Lemma 3.3
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • proof
  • Example 4.3
  • Example 4.4
  • Theorem 4.5
  • ...and 38 more