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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2512.24301 |
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| _version_ | 1866918266230800384 |
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| author | Li, Shuang Yuan, Pingzhi |
| author_facet | Li, Shuang Yuan, Pingzhi |
| contents | A subspace $U$ of $\mathbb{F}_q^n$ is called \textit{cyclically covering} if the whole space $\mathbb{F}_q^n$ is the union of the cyclic shifts of $U$. The case $\mathbb{F}_q^n$ itself is the only covering subspace, is of particular interest. Recently, Huang solved this problem completely under the condition $\gcd(n, q)=1$ using primitive idempotents and trace functions, and explicitly posed the non-coprime case as an open question.
This paper provides a complete answer to Huang's question. We prove that if $n = p^k m$ where $p = \operatorname{char}(\mathbb{F}_q)$ and $\gcd(m, p)=1$, then $h_q(p^k m) = 0$ if and only if $h_q(m) = 0$. This result fully reduces the non-coprime case to the coprime case settled by Huang. Our proof employs the structure theory of cyclic group algebras in modular characteristic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_24301 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Trivial Cyclically Covering Subspaces of $\mathbb{F}_q^n$ in Non-Coprime Characteristic Li, Shuang Yuan, Pingzhi Number Theory A subspace $U$ of $\mathbb{F}_q^n$ is called \textit{cyclically covering} if the whole space $\mathbb{F}_q^n$ is the union of the cyclic shifts of $U$. The case $\mathbb{F}_q^n$ itself is the only covering subspace, is of particular interest. Recently, Huang solved this problem completely under the condition $\gcd(n, q)=1$ using primitive idempotents and trace functions, and explicitly posed the non-coprime case as an open question. This paper provides a complete answer to Huang's question. We prove that if $n = p^k m$ where $p = \operatorname{char}(\mathbb{F}_q)$ and $\gcd(m, p)=1$, then $h_q(p^k m) = 0$ if and only if $h_q(m) = 0$. This result fully reduces the non-coprime case to the coprime case settled by Huang. Our proof employs the structure theory of cyclic group algebras in modular characteristic. |
| title | On Trivial Cyclically Covering Subspaces of $\mathbb{F}_q^n$ in Non-Coprime Characteristic |
| topic | Number Theory |
| url | https://arxiv.org/abs/2512.24301 |