On Binary Cyclic Codes with Five Nonzero Weights

Let $q=2^n$, $0leq kleq n-1$, $n/gcd(n,k)$ be odd and $k eq n/3, 2n/3$. In this paper the value distribution of following exponential sums [sumlimits_{xin bF_q}(-1)^{mathrm{Tr}_1^n(alpha x^{2^{2k}+1}+beta x^{2^k+1}+ga x)}quad(alpha,beta,gain 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.