Up to linear transformations, we obtain a classification of permutation polynomials (PPs) of degree 8 over ${\mathbb{F}}_{2^r}$ with $r>3$. By Bartoli et al. (2017) [1], a polynomial f of degree 8 over ${\mathbb{F}}_{2^r}$ is exceptional if and only if $f-f(0)$ is a linearized PP, which has already been classified. 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.

直到线性变换，我们获得了8阶上的置换多项式（PPs）的分类 ${\mathbb{F}}_{2^{\mathrm{[R}}}$ 与 $\mathrm{[R}>3$。由Bartoli等人撰写。（2017）[1]，阶数为8的多项式f${\mathbb{F}}_{2^{\mathrm{[R}}}$ 仅当且仅当是例外 $F-F（0）$是线性PP，已经分类。因此，只要搜索超过8级的非异常PP就足够了${\mathbb{F}}_{2^{\mathrm{[R}}}$，仅当 $\mathrm{[R}⩽9$根据先前的结果。这可以通过在个人计算机上运行的SageMath软件耗尽。为了方便计算，对多项式系数提供了线性变换和根据Hermite准则的显式方程后的一些要求。主要结果是8级以上的非异常PP f${\mathbb{F}}_{2^{\mathrm{[R}}}$ （与 $\mathrm{[R}>3$）存在且仅当 $\mathrm{[R}\in \{4，5，6\}$，并且明确列出了此类f直到线性变换。