Differential uniformity and costacyclic code from some power mapping

TL;DR

This paper analyzes the differential properties and $c$-differential uniformity of a specific power mapping over finite fields, and constructs six-weight constacyclic codes with explicit weight distributions.

Contribution

It determines the differential spectrum and $c$-differential uniformity of the power mapping $x^d$, and constructs a new class of six-weight constacyclic codes with known weight distribution.

Findings

Complete differential spectrum of $x^d$ over finite fields.

Explicit $c$-differential uniformity of the power mapping.

Construction of six-weight constacyclic codes with known weight distribution.

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.