On many-to-one mappings over finite fields
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929452625166336 |
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| author | Zheng, Yanbin Ding, Yanjin Zhang, Meiying Yuan, Pingzhi Wang, Qiang |
| author_facet | Zheng, Yanbin Ding, Yanjin Zhang, Meiying Yuan, Pingzhi Wang, Qiang |
| contents | The definition of many-to-one mapping, or $m$-to-$1$ mapping for short, between two finite sets is introduced in this paper, which unifies and generalizes the definitions of $2$-to-$1$ mappings and $n$-to-$1$ mappings. A generalized local criterion is given, which is an abstract criterion for a mapping to be $m$-to-$1$. By employing the generalized local criterion, three constructions of $m$-to-$1$ mapping are proposed, which unify and generalize all the previous constructions of $2$-to-$1$ mappings and $n$-to-$1$ mappings. Then the $m$-to-$1$ property of polynomials $f(x) = x^r h(x^s)$ on $\mathbb{F}_{q}^{*}$ is studied by using these three constructions. A series of explicit conditions for~$f$ to be an $m$-to-$1$ mapping on $\mathbb{F}_{q}^{*}$ are found through the detailed discussion of the parameters $m$, $s$, $q$ and the polynomial $h$. These results extend many conclusions in the literature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_04218 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On many-to-one mappings over finite fields Zheng, Yanbin Ding, Yanjin Zhang, Meiying Yuan, Pingzhi Wang, Qiang Information Theory The definition of many-to-one mapping, or $m$-to-$1$ mapping for short, between two finite sets is introduced in this paper, which unifies and generalizes the definitions of $2$-to-$1$ mappings and $n$-to-$1$ mappings. A generalized local criterion is given, which is an abstract criterion for a mapping to be $m$-to-$1$. By employing the generalized local criterion, three constructions of $m$-to-$1$ mapping are proposed, which unify and generalize all the previous constructions of $2$-to-$1$ mappings and $n$-to-$1$ mappings. Then the $m$-to-$1$ property of polynomials $f(x) = x^r h(x^s)$ on $\mathbb{F}_{q}^{*}$ is studied by using these three constructions. A series of explicit conditions for~$f$ to be an $m$-to-$1$ mapping on $\mathbb{F}_{q}^{*}$ are found through the detailed discussion of the parameters $m$, $s$, $q$ and the polynomial $h$. These results extend many conclusions in the literature. |
| title | On many-to-one mappings over finite fields |
| topic | Information Theory |
| url | https://arxiv.org/abs/2408.04218 |