Several new classes of optimal ternary cyclic codes with two or three zeros (2407.07332v1)

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.