On Binary Cyclic Codes with Five Nonzero Weights
Abstract
Description
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 \[\sum\limits_{x\in \bF_q}(-1)^{\mathrm{Tr}_1^n(αx^{2^{2k}+1}+βx^{2^k+1}+\ga x)}\quad(α,β,\ga\in \bF_{q})\] 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 $π^{-1}$, $π^{-(2^k+1)}$ and $π^{-(2^{2k}+1)}$ respectively for a primitive element $π$ of $\bF_q$, is also determined.