The Minimal Binomial Multiples of Polynomials over Finite Fields
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915567609315328 |
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| author | Zhu, Li Wu, Hongfeng |
| author_facet | Zhu, Li Wu, Hongfeng |
| contents | Let $f(X)$ be a nonconstant polynomial over $\mathbb{F}_{q}$, with a nonzero constant term. The order of $f(X)$ is a classical notion in the theory of polynomials over finite fields, and recently the definition of freeness of binomials of $f(X)$ was given in \cite{Martínez}. Generalizing these two notions, we introduce the definition of the minimal binomial multiple of $f(X)$ in this paper, which is the monic binomial with the lowest degree among the binomials over $\mathbb{F}_{q}$ divided by $f(X)$. Based on the equivalent characterization of binomials via the defining sets of their radicals, we prove that a series of properties of the classical order can be naturally generalized to this case. In particular, the minimal binomial multiple of $f(X)$ is presented explicitly in terms of the defining set of the radical of $f(X)$. And a criterion for $f(X)$ being free of binomials is given. As an application, for any positive integer $N$ and nonzero element $λ$ in $\mathbb{F}_{q}$, the $λ$-constacyclic codes of length $N$ with minimal distance $2$ are determined. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_18624 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Minimal Binomial Multiples of Polynomials over Finite Fields Zhu, Li Wu, Hongfeng Number Theory Information Theory Let $f(X)$ be a nonconstant polynomial over $\mathbb{F}_{q}$, with a nonzero constant term. The order of $f(X)$ is a classical notion in the theory of polynomials over finite fields, and recently the definition of freeness of binomials of $f(X)$ was given in \cite{Martínez}. Generalizing these two notions, we introduce the definition of the minimal binomial multiple of $f(X)$ in this paper, which is the monic binomial with the lowest degree among the binomials over $\mathbb{F}_{q}$ divided by $f(X)$. Based on the equivalent characterization of binomials via the defining sets of their radicals, we prove that a series of properties of the classical order can be naturally generalized to this case. In particular, the minimal binomial multiple of $f(X)$ is presented explicitly in terms of the defining set of the radical of $f(X)$. And a criterion for $f(X)$ being free of binomials is given. As an application, for any positive integer $N$ and nonzero element $λ$ in $\mathbb{F}_{q}$, the $λ$-constacyclic codes of length $N$ with minimal distance $2$ are determined. |
| title | The Minimal Binomial Multiples of Polynomials over Finite Fields |
| topic | Number Theory Information Theory |
| url | https://arxiv.org/abs/2510.18624 |