Cyclic Codes and Sequences from Kasami-Welch Functions

TL;DR

This paper analyzes exponential sums related to Kasami-Welch functions over finite fields, determines the weight distributions of certain cyclic codes, and studies correlation properties of binary m-sequences, advancing coding theory and sequence design.

Contribution

It provides explicit value distributions of exponential sums and applies these results to characterize the weight distributions of new cyclic codes and the correlation distribution of binary m-sequences.

Findings

Explicit value distribution of exponential sums involving Kasami-Welch functions.

Determination of weight distributions for two classes of binary cyclic codes.

Analysis of correlation distribution among a family of binary m-sequences.

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 \[\sum\limits_{x\in \bF_q}(-1)^{\Tra_1^n(\alpha x^{2^{3k}+1}+\beta x^{2^k+1})}\quad(\alpha,\beta\in \bF_{q})\] and \[\sum\limits_{x\in \bF_q}(-1)^{\Tra_1^n(\alpha x^{2^{3k}+1}+\beta x^{2^k+1}+\ga x)}\quad(\alpha,\beta,\ga\in \bF_{q})\] 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 $\pi^{-1}$, $\pi^{-(2^k+1)}$ and $\pi^{-(2^{3k}+1)}$ respectively for a primitive element $\pi$ of $\bF_q$. (2). We determine the correlation distribution among a family of binary m-sequences.