In this paper, all the possibilities for the value distribution of a perfect nonlinear function from $\mathbb{F}_{p^m}$ to $\mathbb{F}_p$ are determined, where $p$ is an odd prime number and $m\in\mathbb{N}_+$. As an application, we determine the weight distributions of two classes of linear codes over $\mathbb{F}_p$ constructed from perfect nonlinear functions.

翻译：本文确定了从$\mathbb{F}_{p^m}$到$\mathbb{F}_p$的完全非线性函数的所有可能值分布，其中$p$为奇素数，$m\in\mathbb{N}_+$。作为应用，我们确定了基于完全非线性函数构造的两类$\mathbb{F}_p$上线性码的重量分布。