On Binary Cyclic Codes with Five Nonzero Weights

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 \bFq}(-1)^{{\mathrm{Tr}_1n(\alpha} x^{{2{2k}+1}+\beta} x^{2k+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 $h1(x)h2(x)h3(x)$ where $h1(x)$, $h2(x)$ and $h3(x)$ are the minimal polynomials of $\pi^{-1}$, $\pi^{-(2k+1)}$ and $\pi^{{-(2{2k}+1)}$} respectively for a primitive element $\pi$ of $\bFq$, is also determined.