Permutation binomials of the form $x^r(x^{q-1}+a)$ over $\mathbb F_{q^e}$

TL;DR

This paper investigates the existence of permutation binomials of a specific form over finite fields, providing comprehensive results and characterizations for various field extensions.

Contribution

It offers new existence and nonexistence results and fully characterizes such permutation binomials over several finite fields.

Findings

Complete characterization over _{q^2}, _{q^3}, _{q^4}, _{p^5}, _{p^6}

New existence and nonexistence results for permutation binomials

Conditions for permutation binomials over finite fields

Abstract

We present several existence and nonexistence results for permutation binomials of the form $x^r(x^{q-1}+a)$, where $e\geq 2$ and $a\in \mathbb{F}_{q^e}^*$. As a consequence, we obtain a complete characterization of such permutation binomials over $\mathbb{F}_{q^2}$, $\mathbb{F}_{q^3}$, $\mathbb{F}_{q^4}$, $\mathbb{F}_{p^5}$, and $\mathbb{F}_{p^6}$, where $p$ is an odd prime.