Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Self-Dual Cyclic Codes with Square-Root-Like Lower Bounds on Their Minimum Distances

Hao Chen, Cunsheng Ding

Abstract

Binary self-dual cyclic codes have been studied since the classical work of Sloane and Thompson published in IEEE Trans. Inf. Theory, vol. 29, 1983. Twenty five years later, an infinite family of binary self-dual cyclic codes with lengths $n_i$ and minimum distances $d_i \geq \frac{1}{2} \sqrt{n_i+2}$ was presented in a paper of IEEE Trans. Inf. Theory, vol. 55, 2009. However, no infinite family of Euclidean self-dual binary cyclic codes whose minimum distances have the square-root lower bound and no infinite family of Euclidean self-dual nonbinary cyclic codes whose minimum distances have a lower bound better than the square-root lower bound are known in the literature. In this paper, an infinite family of Euclidean self-dual cyclic codes over the fields ${\bf F}_{2^s}$ with a square-root-like lower bound is constructed. An infinite subfamily of this family consists of self-dual binary cyclic codes with the square-root lower bound. Another infinite subfamily of this family consists of self-dual cyclic codes over the fields ${\bf F}_{2^s}$ with a lower bound better than the square-root bound for $s \geq 2$. Consequently, two breakthroughs in coding theory are made in this paper. An infinite family of self-dual binary cyclic codes with a square-root-like lower bound is also presented in this paper. An infinite family of Hermitian self-dual cyclic codes over the fields ${\bf F}_{2^{2s}}$ with a square-root-like lower bound and an infinite family of Euclidean self-dual linear codes over ${\bf F}_{q}$ with $q \equiv 1 \pmod{4}$ with a square-root-like lower bound are also constructed in this paper.

Self-Dual Cyclic Codes with Square-Root-Like Lower Bounds on Their Minimum Distances

Abstract

Binary self-dual cyclic codes have been studied since the classical work of Sloane and Thompson published in IEEE Trans. Inf. Theory, vol. 29, 1983. Twenty five years later, an infinite family of binary self-dual cyclic codes with lengths and minimum distances was presented in a paper of IEEE Trans. Inf. Theory, vol. 55, 2009. However, no infinite family of Euclidean self-dual binary cyclic codes whose minimum distances have the square-root lower bound and no infinite family of Euclidean self-dual nonbinary cyclic codes whose minimum distances have a lower bound better than the square-root lower bound are known in the literature. In this paper, an infinite family of Euclidean self-dual cyclic codes over the fields with a square-root-like lower bound is constructed. An infinite subfamily of this family consists of self-dual binary cyclic codes with the square-root lower bound. Another infinite subfamily of this family consists of self-dual cyclic codes over the fields with a lower bound better than the square-root bound for . Consequently, two breakthroughs in coding theory are made in this paper. An infinite family of self-dual binary cyclic codes with a square-root-like lower bound is also presented in this paper. An infinite family of Hermitian self-dual cyclic codes over the fields with a square-root-like lower bound and an infinite family of Euclidean self-dual linear codes over with with a square-root-like lower bound are also constructed in this paper.

Paper Structure

This paper contains 16 sections, 12 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Basics of linear codes
  3. The importance of cyclic codes in theory and practice
  4. The motivations and objectives of this paper
  5. The organization of this paper
  6. A generalized van Lint theorem
  7. The $[{\bf u}|{\bf u}+{\bf v}]$ construction of Euclidean and Hermitian self-dual cyclic codes over ${\mathbf{{F}}}_{2^s}$
  8. Euclidean and Hermitian dual-containing BCH codes
  9. A useful lemma
  10. BCH codes
  11. Some Euclidean and Hermitian dual-containing BCH codes
  12. Self-dual cyclic codes over ${\bf F}_{2^s}$ with a square-root-like lower bound
  13. Self-dual linear codes over ${\bf F}_q$ with a square-root-like lower bound
  14. A family of self-dual binary cyclic codes with a square-root-like lower bound
  15. Some other known self-dual linear codes with a square-root-like lower bound
  16. ...and 1 more sections

Key Result

Theorem 1

Let $q$ be a power of $2$ and $n$ be an odd positive integer. Let ${\bf C}_1 \subseteq {\bf F}_q^n$ be a cyclic code with generator polynomial $g_1(x) \in {\bf F}_q[x]$ and ${\bf C}_2 \subseteq {\bf F}_q^n$ be a cyclic code generated by the polynomial $g_1(x)g_2(x) \in {\bf F}_q[x]$, where $g_2(x)$

Theorems & Definitions (24)

  • Theorem 1: The generalized van Lint theorem
  • proof
  • Theorem 2
  • proof
  • Lemma 3
  • Lemma 4
  • Theorem 5
  • proof
  • Theorem 6
  • Theorem 7
  • ...and 14 more