Let ${\mathcal{C}}_{(u,v)}$ denote the ternary cyclic code with two nonzeros ${\alpha }^u$ and ${\alpha }^v$, where α is a generator of ${\mathbb{F}}_{3^m}^{⁎}$ and $0\le u,v\le 3^m-2$. In this paper, we present a sufficient condition such that ${\mathcal{C}}_{(u,v)}$ is an optimal ternary cyclic code. Based on this condition, we get several classes of optimal ternary cyclic codes by choosing the proper u and v. Moreover, we show that ${\mathcal{C}}_{(\frac{3^m+1}{2},\frac{3^m-1}{2}+v)}$ and ${\mathcal{C}}_{(1,v)}$ have the same optimality.

中文翻译：

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最优三进制循环码的进一步结果

让 ${\mathcal{C}}_{(你,v)}$ 用两个非零表示三元循环码 ${\alpha }^{你}$ 和 ${\alpha }^v$, 其中α是${\mathbb{F}}_{3^{米}}^{⁎}$ 和 $0\le 你,v\le 3^{米}-2$. 在本文中，我们提出一个充分条件使得${\mathcal{C}}_{(你,v)}$是最优的三进制循环码。基于这个条件，我们通过选择合适的u和v得到几类最优的三进制循环码。此外，我们证明${\mathcal{C}}_{(\frac{3^{米}+1}{2},\frac{3^{米}-1}{2}+v)}$ 和 ${\mathcal{C}}_{(1,v)}$ 具有相同的最优性。