New permutation polynomials over $\mathbb{F}_{q^2}$

TL;DR

This paper introduces a novel method for constructing permutation polynomials over finite fields, extending known classes and providing alternative approaches for specific polynomial forms, enhancing the toolkit for finite field applications.

Contribution

It presents a new general method to generate permutation polynomials over _{q^2} and extends existing classes, including alternative constructions for specific polynomial forms.

Findings

Extended many known permutation polynomials using the new method.

Provided an alternative construction for permutation polynomials involving trace functions.

Enhanced the understanding of permutation polynomial structures over finite fields.

Abstract

In this paper, we propose a new method to obtain new permutation polynomials over $\mathbb{F}_{q^2}$. Using this method, we extend many known permutation polynomials, which take the form $\sum_i(x^q-x+\delta)^{s_i}+L(x)$, where $L(x)$ is a $q$-polynomial over $\mathbb{F}_q$ and $\delta\in\mathbb{F}_{q^2}$. We also present an alternative approach for constructing permutation polynomials of the form $x+\gamma Tr_q^{q^d}(x^{q+1}+x^{2q+2})$ for the cases where $q=2^m$, $2\nmid d$ and $ Tr_q^{q^d}(x)=x+x^q+\dots+x^{q^{d-1}}$.