On Binary Cyclic Codes with Five Nonzero Weights

Abstract

Let $q=2^n$, $0\le k\le n-1$, $n/\gcd (n,k)$ be odd and $k\ne n/3,2n/3$. In this paper the value distribution of following exponential sums \[\sum\limits_{x\in \bF_q}(-1)^{\mathrm{Tr}_1^n(αx^{2^{2k}+1}+βx^{2^k+1}+\ga x)}\quad(α,β,\ga\in \bF_{q})\] 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.