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.

翻译：循环码是线性码的一个子类，因其高效的编码与译码算法，在数据存储系统、通信系统和消费电子领域具有广泛应用。设 $\alpha$ 为 $\mathbb{F}_{3^m}^*$ 的生成元，其中 $m$ 为正整数。记 $\mathcal{C}_{(i_1,i_2,\cdots, i_t)}$ 为生成多项式是 $m_{\alpha^{i_1}}(x)m_{\alpha^{i_2}}(x)\cdots m_{\alpha^{i_t}}(x)$ 的循环码，其中 ${{m}_{\alpha^{i}}}(x)$ 是 ${{\alpha }^{i}}$ 在 ${{\mathbb{F}}_{3}}$ 上的极小多项式。本文通过分析有限域上特定方程的解，提出了四类具有参数 $[3^m-1,3^m-\frac{3m}{2}-2,4]$ 的最优三元循环码 $\mathcal{C}_{(0,1,e)}$ 和 $\mathcal{C}_{(1,e,s)}$，其中 $s=\frac{3^m-1}{2}$。此外，通过确定特定方程的解并分析 $\mathbb{F}_{3^m}$ 上某些多项式的不可约因子，我们提出了四类具有参数 $[3^m-1,3^m-2m-1,4]$ 的最优三元循环码 $\mathcal{C}_{(2,e)}$ 和 $\mathcal{C}_{(1,e)}$。我们证明了这些新的最优循环码与已知码类不等价。