Further results on permutation pentanomials over ${\mathbb F}_{q^3}$ in characteristic two
Fuente:
arXiv
Saved in:
| Main Authors: | , , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909466649165824 |
|---|---|
| author | Zhang, Tongliang Zheng, Lijing Wang, Hengtai Peng, Jie Li, Yanjun |
| author_facet | Zhang, Tongliang Zheng, Lijing Wang, Hengtai Peng, Jie Li, Yanjun |
| contents | Let $q=2^m.$ In a recent paper \cite{Zhang3}, Zhang and Zheng investigated several classes of permutation pentanomials of the form $ε_0x^{d_0}+L(ε_{1}x^{d_1}+ε_{2}x^{d_2})$ over ${\mathbb F}_{q^3}~(d_0=1,2,4)$ from some certain linearized polynomial $L(x)$ by using multivariate method and some techniques to determine the number of the solutions of some equations. They proposed an open problem that there are still some permutation pentanomials of that form have not been proven. In this paper, inspired by the idea of \cite{LiKK}, we further characterize the permutation property of the pentanomials with the above form over ${\mathbb F}_{q^3}~(d_0=1,2,4)$. The techniques in this paper would be useful to investigate more new
classes of permutation pentanomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_15741 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Further results on permutation pentanomials over ${\mathbb F}_{q^3}$ in characteristic two Zhang, Tongliang Zheng, Lijing Wang, Hengtai Peng, Jie Li, Yanjun Number Theory Let $q=2^m.$ In a recent paper \cite{Zhang3}, Zhang and Zheng investigated several classes of permutation pentanomials of the form $ε_0x^{d_0}+L(ε_{1}x^{d_1}+ε_{2}x^{d_2})$ over ${\mathbb F}_{q^3}~(d_0=1,2,4)$ from some certain linearized polynomial $L(x)$ by using multivariate method and some techniques to determine the number of the solutions of some equations. They proposed an open problem that there are still some permutation pentanomials of that form have not been proven. In this paper, inspired by the idea of \cite{LiKK}, we further characterize the permutation property of the pentanomials with the above form over ${\mathbb F}_{q^3}~(d_0=1,2,4)$. The techniques in this paper would be useful to investigate more new classes of permutation pentanomials. |
| title | Further results on permutation pentanomials over ${\mathbb F}_{q^3}$ in characteristic two |
| topic | Number Theory |
| url | https://arxiv.org/abs/2501.15741 |