On Binary Cyclic Codes with Five Nonzero Weights

Let $q=2^n$, $0\le k\le n-1$, $n\mathrm{/}\gcd (n,k)$ be odd and $k\mathrm{\ne }n\mathrm{/}3,2n\mathrm{/}3$. In this paper the value distribution of following exponential sums $$\sum _{x\in {\text{bF}}_q}(-1{)}^{{\mathrm{T}\mathrm{r}}_1^n(\alpha x^{2^{2k}+1}+\beta x^{2^k+1}+\text{ga}x)}(\alpha ,\beta ,\text{ga}\in {\text{bF}}_q)$$ is determined. As an application, the weight distribution of the binary cyclic code $\text{cC}$, 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 ${\text{bF}}_q$, is also determined.