Existence of primitive normal pairs over finite fields with prescribed subtrace
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866908327312621568 |
|---|---|
| author | Chatterjee, K. Kapetanakis, G. Sharma, H. Tiwari, S. K. |
| author_facet | Chatterjee, K. Kapetanakis, G. Sharma, H. Tiwari, S. K. |
| contents | Given positive integers $q,n,m$ and $a\in\mathbb{F}_{q}$, where $q$ is an odd prime power and $n\geq 5$, we investigate the existence of a primitive normal pair $(ε,f(ε))$ in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ such that $\mathrm{STr}_{q^n/q}(ε)=a$, where $f(x)=\frac{f_{1}(x)}{f_{2}(x)}\in\mathbb{F}_{q^n}(x)$ is a rational function together with deg$(f_{1})+$deg$(f_{2})=m$ and $\mathrm{STr}_{q^n/q}(ε) = \sum_{0\leq i<j\leq n-1}^{}ε^{q^i+q^j}$. Finally, we conclude that for $m=2$, $n\geq 6$ and $q=7^k$; $k\in\mathbb{N}$, such a pair will exist certainly for all $(q,n)$ except at most $11$ choices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_11463 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence of primitive normal pairs over finite fields with prescribed subtrace Chatterjee, K. Kapetanakis, G. Sharma, H. Tiwari, S. K. Number Theory 11T23, 12E20 Given positive integers $q,n,m$ and $a\in\mathbb{F}_{q}$, where $q$ is an odd prime power and $n\geq 5$, we investigate the existence of a primitive normal pair $(ε,f(ε))$ in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ such that $\mathrm{STr}_{q^n/q}(ε)=a$, where $f(x)=\frac{f_{1}(x)}{f_{2}(x)}\in\mathbb{F}_{q^n}(x)$ is a rational function together with deg$(f_{1})+$deg$(f_{2})=m$ and $\mathrm{STr}_{q^n/q}(ε) = \sum_{0\leq i<j\leq n-1}^{}ε^{q^i+q^j}$. Finally, we conclude that for $m=2$, $n\geq 6$ and $q=7^k$; $k\in\mathbb{N}$, such a pair will exist certainly for all $(q,n)$ except at most $11$ choices. |
| title | Existence of primitive normal pairs over finite fields with prescribed subtrace |
| topic | Number Theory 11T23, 12E20 |
| url | https://arxiv.org/abs/2405.11463 |