In this paper, we study the differential properties of $x^d$ over $\mathbb{F}_{p^n}$ with $d=p^{2l}-p^{l}+1$. By studying the differential equation of $x^d$ and the number of rational points on some curves over finite fields, we completely determine differential spectrum of $x^{d}$. Then we investigate the $c$-differential uniformity of $x^{d}$. We also calculate the value distribution of a class of exponential sum related to $x^d$. In addition, we obtain a class of six-weight consta-cyclic codes, whose weight distribution is explicitly determined. Part of our results is a complement of the works shown in [\ref{H1}, \ref{H2}] which mainly focus on cross correlations.

翻译：本文研究了定义在有限域$\mathbb{F}_{p^n}$上的幂函数$x^d$的差分性质，其中$d=p^{2l}-p^{l}+1$。通过分析$x^d$的差分方程以及有限域上某些曲线的有理点个数，我们完整确定了$x^{d}$的差分谱。进一步，我们探讨了$x^{d}$的$c$-差分均匀性，并计算了与$x^d$相关的一类指数和的值分布。此外，我们构造了一类六重量的常循环码，并精确给出了其重量分布。本文部分结果是对文献[\ref{H1}, \ref{H2}]中主要关注互相关函数的研究的补充。