Several new classes of optimal ternary cyclic codes with two or three zeros

TL;DR

This paper introduces four new classes of optimal ternary cyclic codes with two or three zeros, expanding the known set of such codes by analyzing solutions over finite fields and their polynomial factorizations.

Contribution

The paper presents novel classes of optimal ternary cyclic codes with specific parameters, demonstrating their inequivalence to existing codes through finite field analysis.

Findings

Four new classes of optimal ternary cyclic codes with parameters [3^m-1, 3^m - 3m/2 - 2, 4]

Four new classes of optimal ternary cyclic codes with parameters [3^m-1, 3^m - 2m - 1, 4]

Codes are proven to be inequivalent to known codes.

Abstract

Cyclic codes are a subclass of linear codes and have wide applications in data storage systems, communication systems and consumer electronics due to their efficient encoding and decoding algorithms. Let $\alpha $ be a generator of $\mathbb{F}_{3^m}^*$, where $m$ is a positive integer. Denote by $\mathcal{C}_{(i_1,i_2,\cdots, i_t)}$ the cyclic code with generator polynomial $m_{\alpha^{i_1}}(x)m_{\alpha^{i_2}}(x)\cdots m_{\alpha^{i_t}}(x)$, where ${{m}_{\alpha^{i}}}(x)$ is the minimal polynomial of ${{\alpha }^{i}}$ over ${{\mathbb{F}}_{3}}$. In this paper, by analyzing the solutions of certain equations over finite fields, we present four classes of optimal ternary cyclic codes $\mathcal{C}_{(0,1,e)}$ and $\mathcal{C}_{(1,e,s)}$ with parameters $[3^m-1,3^m-\frac{3m}{2}-2,4]$, where $s=\frac{3^m-1}{2}$. In addition, by determining the solutions of certain equations and analyzing the irreducible factors of certain polynomials over $\mathbb{F}_{3^m}$, we present four classes of optimal ternary cyclic codes $\mathcal{C}_{(2,e)}$ and $\mathcal{C}_{(1,e)}$ with parameters $[3^m-1,3^m-2m-1,4]$. We show that our new optimal cyclic codes are inequivalent to the known ones.