In this paper, we study the differential properties of \(x^d\) over \({\mathbb {F}}_{p^n}\) with \(d=p^{2l}-p^{l}+1\) and \(n=4l\) . By studying the differential equation of \(x^d\) and the number of rational points on some curves over finite fields, we completely determine the differential spectrum 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 [17, 18] which mainly focus on cross correlations.
