Exponential Sums, Cyclic Codes and Sequences: the Odd Characteristic Kasami Case

Let $q=p^n$ with $n=2m$ and $p$ be an odd prime. Let $0\leq k\leq n-1$ and $k\neq m$. In this paper we determine the value distribution of following exponential(character) sums [\sum\limits{x\in \bFq}\zetap^{\Tra1^m (\alpha x^{{p{m}+1})+\Tra_1n(\beta} x^{{pk+1})}\quad(\alpha\in} \bF{p^m},\beta\in \bF{q})] and [\sum\limits{x\in \bFq}\zetap^{\Tra1^m (\alpha x^{{p{m}+1})+\Tra_1n(\beta} x^{pk+1}+\ga x)}\quad(\alpha\in \bF{p^{m},\beta,\ga\in} \bF{q})] where $\Tra1^n: \bFq\ra \bFp$ and $\Tra1^m: \bF{p^m}\ra\bFp$ are the canonical trace mappings and $\zetap=e^{\frac{2\pi i}{p}}$ is a primitive $p$-th root of unity. As applications: (1). We determine the weight distribution of the cyclic codes $\cC1$ and $\cC2$ over $\bF{p^t}$ with parity-check polynomials $h2(x)h3(x)$ and $h1(x)h2(x)h3(x)$ respectively where $t$ is a divisor of $d=\gcd(m,k)$, and $h1(x)$, $h2(x)$ and $h3(x)$ are the minimal polynomials of $\pi^{-1}$, $\pi^{-(pk+1)}$ and $\pi^{-(pm+1)}$ over $\bF{p^t}$ respectively for a primitive element $\pi$ of $\bFq$. (2). We determine the correlation distribution among a family of m-sequences. This paper extends the results in \cite{Zen Li}.