Cyclic Codes and Sequences from Kasami-Welch Functions

Let $q=2^n$, $0\le k\le n-1$ and $k\mathrm{\ne }n\mathrm{/}2$. In this paper we determine the value distribution of following exponential sums $$\sum _{x\in {\text{bF}}_q}(-1{)}^{{\text{Tra}}_1^n(\alpha x^{2^{3k}+1}+\beta x^{2^k+1})}(\alpha ,\beta \in {\text{bF}}_q)$$ and $$\sum _{x\in {\text{bF}}_q}(-1{)}^{{\text{Tra}}_1^n(\alpha x^{2^{3k}+1}+\beta x^{2^k+1}+\text{ga}x)}(\alpha ,\beta ,\text{ga}\in {\text{bF}}_q)$$ where ${\text{Tra}}_1^n:{\text{bF}}_{2^n}\text{ra}{\text{bF}}_2$ is the canonical trace mapping. As applications: (1). We determine the weight distribution of the binary cyclic codes ${\text{cC}}_1$ and ${\text{cC}}_2$ with parity-check polynomials $h_2(x)h_3(x)$ and $h_1(x)h_2(x)h_3(x)$ respectively where $h_1(x)$, $h_2(x)$ and $h_3(x)$ are the minimal polynomials of ${\pi }^{-1}$, ${\pi }^{-(2^k+1)}$ and ${\pi }^{-(2^{3k}+1)}$ respectively for a primitive element $\pi$ of ${\text{bF}}_q$. (2). We determine the correlation distribution among a family of binary m-sequences.