Explicit Representatives and Sizes of Cyclotomic Cosets and their Application to Cyclic Codes over Finite Fields
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| Format: | Preprint |
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2024
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| _version_ | 1866918022781861888 |
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| author | Zhu, Li Liu, Jinle Wu, Hongfeng |
| author_facet | Zhu, Li Liu, Jinle Wu, Hongfeng |
| contents | Cyclotomic coset is a classical notion in the theory of finite field which has wide applications in various computation problems. Let $q$ be a prime power, and $n$ be a positive integer coprime to $q$. In this paper we determine explicitly the representatives and the sizes of all $q$-cyclotomic cosets modulo $n$ in the general settings. We introduce the definition of $2$-adic cyclotomic system, which is a profinite space consists of certain compatible sequences of cyclotomic cosets. A precise characterization of the structure of the $2$-adic cyclotomic system is given, which reveals the general formula for representatives of cyclotomic cosets. With the representatives and the sizes of $q$-cyclotomic cosets modulo $n$, we improve the formulas for the factorizations of $X^{n}-1$ and of $Φ_{n}(X)$ over $\mathbb{F}_{q}$ given in \cite{Graner}. As a consequence, we classify the cyclic codes over finite fields via giving their generator polynomials. Moreover, the self-dual cyclic codes are determined and enumerated. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_12122 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Explicit Representatives and Sizes of Cyclotomic Cosets and their Application to Cyclic Codes over Finite Fields Zhu, Li Liu, Jinle Wu, Hongfeng Information Theory Number Theory Cyclotomic coset is a classical notion in the theory of finite field which has wide applications in various computation problems. Let $q$ be a prime power, and $n$ be a positive integer coprime to $q$. In this paper we determine explicitly the representatives and the sizes of all $q$-cyclotomic cosets modulo $n$ in the general settings. We introduce the definition of $2$-adic cyclotomic system, which is a profinite space consists of certain compatible sequences of cyclotomic cosets. A precise characterization of the structure of the $2$-adic cyclotomic system is given, which reveals the general formula for representatives of cyclotomic cosets. With the representatives and the sizes of $q$-cyclotomic cosets modulo $n$, we improve the formulas for the factorizations of $X^{n}-1$ and of $Φ_{n}(X)$ over $\mathbb{F}_{q}$ given in \cite{Graner}. As a consequence, we classify the cyclic codes over finite fields via giving their generator polynomials. Moreover, the self-dual cyclic codes are determined and enumerated. |
| title | Explicit Representatives and Sizes of Cyclotomic Cosets and their Application to Cyclic Codes over Finite Fields |
| topic | Information Theory Number Theory |
| url | https://arxiv.org/abs/2410.12122 |