Cyclic Codes and Sequences from a Class of Dembowski-Ostrom Functions

Let $q=p^n$ with $p$ be an odd prime. Let $0\leq k\leq n-1$ and $k\neq n/2$. In this paper we determine the value distribution of following exponential(character) sums $$\sum\limits_{x\in \bF_q}\zeta_p^{\Tra_1^n(\alpha x^{p^{3k}+1}+\beta x^{p^k+1})}\quad(\alpha\in \bF_{p^m},\beta\in \bF_{q})$$ and $$\sum\limits_{x\in \bF_q}\zeta_p^{\Tra_1^n(\alpha x^{p^{3k}+1}+\beta x^{p^k+1}+\ga x)}\quad(\alpha\in \bF_{p^m},\beta,\ga\in \bF_{q})$$ where $\Tra_1^n: \bF_q\ra \bF_p$ and $\Tra_1^m: \bF_{p^m}\ra\bF_p$ are the canonical trace mappings and $\zeta_p=e^{\frac{2\pi i}{p}}$ is a primitive $p$-th root of unity. As applications: (1). We determine the weight distribution of the cyclic codes $\cC_1$ and $\cC_2$ over $\bF_{p^t}$ with parity-check polynomials $h_2(x)h_3(x)$ and $h_1(x)h_2(x)h_3(x)$ respectively where $t$ is a divisor of $d=\gcd(n,k)$, and $h_1(x)$, $h_2(x)$ and $h_3(x)$ are the minimal polynomials of $\pi^{-1}$, $\pi^{-(p^k+1)}$ and $\pi^{-(p^{3k}+1)}$ over $\bF_{p^t}$ respectively for a primitive element $\pi$ of $\bF_q$. (2). We determine the correlation distribution among a family of m-sequences.