New Results on Permutation Binomials of Finite Fields

TL;DR

This paper investigates permutation binomials over finite fields, introduces an equivalence notion, and proves new nonexistence results for specific binomials, partially confirming a recent conjecture and extending prior work.

Contribution

It introduces an equivalence framework for permutation binomials and establishes new nonexistence results for certain binomials over finite fields.

Findings

Proves nonexistence of certain permutation binomials over large finite fields.

Partially confirms a recent conjecture by Tu et al.

Extends previous results to new classes of binomials.

Abstract

After a brief review of existing results on permutation binomials of finite fields, we introduce the notion of equivalence among permutation binomials (PBs) and describe how to bring a PB to its canonical form under equivalence. We then focus on PBs of $\Bbb F_{q^2}$ of the form $X^n(X^{d(q-1)}+a)$, where $n$ and $d$ are positive integers and $a\in\Bbb F_{q^2}^*$. Our contributions include two nonexistence results: (1) If $q$ is even and sufficiently large and $a^{q+1}\ne 1$, then $X^n(X^{3(q-1)}+a)$ is not a PB of $\Bbb F_{q^2}$. (2) If $2\le d\mid q+1$, $q$ is sufficiently large and $a^{q+1}\ne 1$, then $X^n(X^{d(q-1)}+a)$ is not a PB of $\Bbb F_{q^2}$ under certain additional conditions. (1) partially confirms a recent conjecture by Tu et al. (2) is an extension of a previous result with $n=1$.