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 $\alpha$ be a generator of $\mathbb{F}<em>{3<sup>m}<sup>*$, where $m$ is a positive integer. Denote by $\mathcal{C}</em>{(i_1,i_2,\cdots, i_t)}$ the cyclic code with generator polynomial $$m_{\alpha^{{i_1}}(x)m_{\alpha^{{i_2}}(x)\cdots}} m_{\alpha<sup>{i_t}}(x)$$, where ${{m}<em>{\alpha<sup>{i}}}(x)$ is the minimal polynomial of ${{\alpha }<sup>{i}}$ over ${{\mathbb{F}}</em>{3}}$. In this paper, by analyzing the solutions of certain equations over finite fields, we present four classes of optimal ternary cyclic codes $\mathcal{C}<em>{(0,1,e)}$ and $\mathcal{C}</em>{(1,e,s)}$ with parameters $[3<sup>m-1,3<sup>m-\frac{3m}{2}-2,4]$, where $s=\frac{3<sup>m-1}{2}$. In addition, by determining the solutions of certain equations and analyzing the irreducible factors of certain polynomials over $\mathbb{F}<em>{3<sup>m}$, we present four classes of optimal ternary cyclic codes $\mathcal{C}</em>{(2,e)}$ and $\mathcal{C}_{(1,e)}$ with parameters $[3<sup>m-1,3<sup>m-2m-1,4]$. We show that our new optimal cyclic codes are inequivalent to the known ones.