Classification of some quadrinomials over finite fields of odd characteristic

---

Atıf İçin Kopyala

Özbudak F., Gulmez Temur B. o. T.

FINITE FIELDS AND THEIR APPLICATIONS, cilt.87, ss.102158, 2023 (SCI-Expanded)

• Yayın Türü: Makale / Tam Makale
• Cilt numarası: 87
• Basım Tarihi: 2023
• Doi Numarası: 10.1016/j.ffa.2022.102158
• Dergi Adı: FINITE FIELDS AND THEIR APPLICATIONS
• Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Applied Science & Technology Source, Compendex, Computer & Applied Sciences, MathSciNet, zbMATH
• Sayfa Sayıları: ss.102158
• Anahtar Kelimeler: Permutation polynomials, Finite fields, Absolutely irreducible, PERMUTATION POLYNOMIALS, TRINOMIALS, BINOMIALS
• Orta Doğu Teknik Üniversitesi Adresli: Evet

Özet

In this paper, we completely determine all necessary and sufficient conditions such that the polynomial f(x)=x3+axq+2+bx2q+1+cx3q" role="presentation" >$�(�)={�}^3+�{�}^{�+2}+�{�}^{2�+1}+�{�}^{3�}$, where a,b,c∈Fq⁎" role="presentation" >$�,�,�\in {\mathbb{�}}_{�}^{⁎}$, is a permutation quadrinomial of Fq2" role="presentation" >${\mathbb{�}}_{{�}^2}$ over any finite field of odd characteristic. This quadrinomial has been studied first in [25] by Tu, Zeng and Helleseth, later in [24] Tu, Liu and Zeng revisited these quadrinomials and they proposed a more comprehensive characterization of the coefficients that results with new permutation quadrinomials, where char(Fq)=2" role="presentation" >$�ℎ��({\mathbb{�}}_{�})=2$ and finally, in [16], Li, Qu, Li and Chen proved that the sufficient condition given in [24] is also necessary and thus completed the solution in even characteristic case. In [6] Gupta studied the permutation properties of the polynomial x3+axq+2+bx2q+1+cx3q" role="presentation" >${�}^3+�{�}^{�+2}+�{�}^{2�+1}+�{�}^{3�}$, where char(Fq)=3,5" role="presentation" >$�ℎ��({\mathbb{�}}_{�})=3,5$ and a,b,c∈Fq⁎" role="presentation" >$�,�,�\in {\mathbb{�}}_{�}^{⁎}$ and proposed some new classes of permutation quadrinomials of Fq2" role="presentation" >${\mathbb{�}}_{{�}^2}$.

In particular, in this paper we classify all permutation polynomials of Fq2" role="presentation" >${\mathbb{�}}_{{�}^2}$ of the form f(x)=x3+axq+2+bx2q+1+cx3q" role="presentation" >$�(�)={�}^3+�{�}^{�+2}+�{�}^{2�+1}+�{�}^{3�}$, where a,b,c∈Fq⁎" role="presentation" >$�,�,�\in {\mathbb{�}}_{�}^{⁎}$, over all finite fields of odd characteristic and obtain several new classes of such permutation quadrinomials.