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

TL;DR

This paper constructs infinite families of Euclidean and Hermitian self-dual cyclic codes with minimum distances that grow at least as fast as the square root of their length, advancing the understanding of code bounds.

Contribution

It introduces new infinite families of self-dual cyclic codes over various fields with square-root-like minimum distance bounds, surpassing previous limitations.

Findings

Constructed infinite Euclidean self-dual cyclic codes over ${f F}_{2^s}$ with square-root-like bounds.

Established infinite subfamilies with better-than-square-root bounds for $s \\geq 2$.

Presented infinite families of Hermitian self-dual cyclic codes with square-root-like bounds.

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.