Mathematics

CLC number: O156.1

A Note on a Class of Permutation Quadrinomials of F_q²

有限域上一类置换四项式的注记

¹ School of Mathematics and Big Data, Chongqing University of Education, Chongqing 400065, China
² School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China

^† Corresponding author. E-mail: yanl930@163.com

Received: 15 July 2024

Abstract

Constructing permutation polynomials is a hot topic in finite fields, and permutation polynomials have many applications in different areas. In this paper, by using monomials on the cosets of a subgroup to characterize the permutational property of rational functions on ${\mu }_{q+\mathrm{1}}$, we construct a class of permutation quadrinomials with the form $f_{r,a,b,c,s,t,u}(x)=x^r(\mathrm{1}+a{x}^{s(q-\mathrm{1})}+b{x}^{t(q-\mathrm{1})}+c{x}^{u(q-\mathrm{1})})$ of ${\mathrm{F}}_{q^{\mathrm{2}}}$.

摘要

构造置换多项式是有限域上的一个热点问题，置换多项式在诸多领域都有应用。本文利用${\mu }_{q+\mathrm{1}}$的子群陪集上的单项式来刻画${\mu }_{q+\mathrm{1}}$上有理函数的置换性质，构造了一类有限域${\mathrm{F}}_{q^{\mathrm{2}}}$上形如$f_{r,a,b,c,s,t,u}(x)=x^r(\mathrm{1}+a{x}^{s(q-\mathrm{1})}+b{x}^{t(q-\mathrm{1})}+c{x}^{u(q-\mathrm{1})})$的置换四项式。