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

Abstract

We present several existence and nonexistence results for permutation binomials of the form $x^r(x^{q-1}+a)$, where $e\ge 2$ and $a\in {\mathbb{F}}_{q^e}^{\ast }$. 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.

arXiv commentRevised version with a new title