On Binary Cyclic Codes with Five Nonzero Weights

TL;DR

This paper determines the value distribution of specific exponential sums over finite fields and applies these results to find the weight distribution of certain binary cyclic codes with five nonzero weights.

Contribution

It provides a detailed analysis of exponential sums related to binary cyclic codes and derives their weight distribution, which was previously unknown for these code parameters.

Findings

Explicit value distribution of exponential sums is obtained.

Weight distribution of the associated binary cyclic code is determined.

Conditions on parameters for the distribution are established.

Abstract

Let $q=2^n$, $0\leq k\leq n-1$, $n/\gcd(n,k)$ be odd and $k\neq n/3, 2n/3$. In this paper the value distribution of following exponential sums \[\sum\limits_{x\in \bF_q}(-1)^{\mathrm{Tr}_1^n(\alpha x^{2^{2k}+1}+\beta x^{2^k+1}+\ga x)}\quad(\alpha,\beta,\ga\in \bF_{q})\] is determined. As an application, the weight distribution of the binary cyclic code $\cC$, with parity-check polynomial $h_1(x)h_2(x)h_3(x)$ where $h_1(x)$, $h_2(x)$ and $h_3(x)$ are the minimal polynomials of $\pi^{-1}$, $\pi^{-(2^k+1)}$ and $\pi^{-(2^{2k}+1)}$ respectively for a primitive element $\pi$ of $\bF_q$, is also determined.