Permutation polynomials of degree 8 over finite fields of characteristic 2

Up to linear transformations, we obtain a classification of permutation polynomials (PPs) of degree $8$ over ${\mathbb{F}}_{2^r}$ with $r>3$. By [J. Number Theory 176 (2017) 466-66], a polynomial $f$ of degree $8$ over ${\mathbb{F}}_{2^r}$ is exceptional if and only if $f-f(0)$ is a linearized PP. So it suffices to search for non-exceptional PPs of degree $8$ over ${\mathbb{F}}_{2^r}$, which exist only when $r⩽9$ by a previous result. This can be exhausted by the SageMath software running on a personal computer. To facilitate the computation, some requirements after linear transformations and explicit equations by Hermite’s criterion are provided for the polynomial coefficients. The main result is that a non-exceptional PP $f$ of degree $8$ over ${\mathbb{F}}_{2^r}$ (with $r>3$) exists if and only if $r\in \{4,5,6\}$, and such $f$ is explicitly listed up to linear transformations.