Several new classes of optimal ternary cyclic codes with two or three zeros
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914864630333440 |
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| author | Wu, Gaofei You, Zhuohui Zha, Zhengbang Zhang, Yuqing |
| author_facet | Wu, Gaofei You, Zhuohui Zha, Zhengbang Zhang, Yuqing |
| contents | 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 $α$ 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 $m_{α^{i_1}}(x)m_{α^{i_2}}(x)\cdots m_{α^{i_t}}(x)$, where ${{m}_{α^{i}}}(x)$ is the minimal polynomial of ${{α}^{i}}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_07332 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Several new classes of optimal ternary cyclic codes with two or three zeros Wu, Gaofei You, Zhuohui Zha, Zhengbang Zhang, Yuqing Information Theory 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 $α$ 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 $m_{α^{i_1}}(x)m_{α^{i_2}}(x)\cdots m_{α^{i_t}}(x)$, where ${{m}_{α^{i}}}(x)$ is the minimal polynomial of ${{α}^{i}}$ 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. |
| title | Several new classes of optimal ternary cyclic codes with two or three zeros |
| topic | Information Theory |
| url | https://arxiv.org/abs/2407.07332 |