Differential uniformity and costacyclic code from some power mapping

Abstract

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.