The Weight Distributions of Two Classes of Linear Codes From Perfect Nonlinear Functions

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.