Cyclic Codes and Sequences: the Generalized Kasami Case

Let $q=2^n$ with $n=2m$ . Let $1\leq k\leq n-1$ and $k\neq m$. In this paper we determine the value distribution of following exponential sums $$\sum\limits_{x\in \bF_q}(-1)^{\Tra_1^m (\alpha x^{2^{m}+1})+\Tra_1^n(\beta x^{2^k+1})}\quad(\alpha\in \bF_{2^m},\beta\in \bF_{q})$$ and $$\sum\limits_{x\in \bF_q}(-1)^{\Tra_1^m (\alpha x^{2^{m}+1})+\Tra_1^n(\beta x^{2^k+1}+\ga x)}\quad(\alpha\in \bF_{2^m},\beta,\ga\in \bF_{q})$$ where $\Tra_1^n: \bF_q\ra \bF_2$ and $\Tra_1^m: \bF_{p^m}\ra\bF_2$ are the canonical trace mappings. As applications: (1). We determine the weight distribution of the binary cyclic codes $\cC_1$ and $\cC_2$ 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 $\pi^{-(2^m+1)}$ over $\bF_{2}$ respectively for a primitive element $\pi$ of $\bF_q$. (2). We determine the correlation distribution among a family of m-sequences. This paper is the binary version of Luo, Tang and Wang\cite{Luo Tan} and extends the results in Kasami\cite{Kasa1}, Van der Vlugt\cite{Vand2} and Zeng, Liu and Hu\cite{Zen Liu}.