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

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 $\hat{I}\pm$ be a generator of ${\mathbb{F}}_{3^m}^{\ast }$, where $m$ is a positive integer. Denote by ${\mathcal{C}}_{(i_1,i_2,\cdots ,i_t)}$ the cyclic code with generator polynomial $m_{\hat{I}{\pm }^{i_1}}(x)m_{\hat{I}{\pm }^{i_2}}(x)\cdots m_{\hat{I}{\pm }^{i_t}}(x)$, where $m_{\hat{I}{\pm }^i}(x)$ is the minimal polynomial of ${\hat{I}\pm }^i$ 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.

16 pages