Cyclic Codes and Sequences from Kasami-Welch Functions

Abstract

Let $q=2^n$, $0\leq k\leq n-1$ and $k\neq n/2$. In this paper we determine the value distribution of following exponential sums \[\sum\limits_{x\in \bF_q}(-1)^{\Tra_1^n(αx^{2^{3k}+1}+βx^{2^k+1})}\quad(α,β\in \bF_{q})\] and \[\sum\limits_{x\in \bF_q}(-1)^{\Tra_1^n(αx^{2^{3k}+1}+βx^{2^k+1}+\ga x)}\quad(α,β,\ga\in \bF_{q})\] where $\Tra_1^n: \bF_{2^n}\ra \bF_2$ is the canonical trace mapping. As applications: (1). We determine the weight distribution of the binary cyclic codes $\cC_1$ and $\cC_2$ with parity-check polynomials $h_2(x)h_3(x)$ and $h_1(x)h_2(x)h_3(x)$ respectively where $h_1(x)$, $h_2(x)$ and $h_3(x)$ are the minimal polynomials of $π^{-1}$, $π^{-(2^k+1)}$ and $π^{-(2^{3k}+1)}$ respectively for a primitive element $π$ of $\bF_q$. (2). We determine the correlation distribution among a family of binary m-sequences.