The Multiple Equal-Difference Structure of Cyclotomic Cosets
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915467080237056 |
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| author | Zhu, Li Zhou, Juncheng Liu, Jinle Wu, Hongfeng |
| author_facet | Zhu, Li Zhou, Juncheng Liu, Jinle Wu, Hongfeng |
| contents | In this paper we introduce the definition of equal-difference cyclotomic coset, and prove that in general any cyclotomic coset can be decomposed into a disjoint union of equal-difference subsets. Among the equal-difference decompositions of a cyclotomic coset, an important class consists of those in the form of cyclotomic decompositions, called the multiple equal-difference representations of the coset. There is an equivalent correspondence between the multiple equal-difference representations of $q$-cyclotomic cosets modulo $n$ and the irreducible factorizations of $X^{n}-1$ in binomial form over finite extension fields of $\mathbb{F}_{q}$. We give an explicit characterization of the multiple equal-difference representations of any $q$-cyclotomic coset modulo $n$, through which a criterion for $X^{n}-1$ factoring into irreducible binomials is obtained. In addition, we present an algorithm to simplify the computation of the leaders of cyclotomic cosets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_03516 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Multiple Equal-Difference Structure of Cyclotomic Cosets Zhu, Li Zhou, Juncheng Liu, Jinle Wu, Hongfeng Number Theory Information Theory In this paper we introduce the definition of equal-difference cyclotomic coset, and prove that in general any cyclotomic coset can be decomposed into a disjoint union of equal-difference subsets. Among the equal-difference decompositions of a cyclotomic coset, an important class consists of those in the form of cyclotomic decompositions, called the multiple equal-difference representations of the coset. There is an equivalent correspondence between the multiple equal-difference representations of $q$-cyclotomic cosets modulo $n$ and the irreducible factorizations of $X^{n}-1$ in binomial form over finite extension fields of $\mathbb{F}_{q}$. We give an explicit characterization of the multiple equal-difference representations of any $q$-cyclotomic coset modulo $n$, through which a criterion for $X^{n}-1$ factoring into irreducible binomials is obtained. In addition, we present an algorithm to simplify the computation of the leaders of cyclotomic cosets. |
| title | The Multiple Equal-Difference Structure of Cyclotomic Cosets |
| topic | Number Theory Information Theory |
| url | https://arxiv.org/abs/2501.03516 |