一类Dembowski-Ostrom函数的循环码和序列
Cyclic Codes and Sequences from a Class of Dembowski-Ostrom Functions
摘要：
设$q=p^n$和$p$是奇素数。设$0\leq k\leq n-1$和$k\neq n/2$。本文确定了如下指数（特征）和\[\sum\limits_{x\in \bF_q}ζ_p^{\Tra_1^n(αx^{p^{3k}+1}+βx^{p^k+1})}\quad(α\in \bF_{p^m},β\in \bF_{q})\]与\[\sum\limits_{x\in \bF_q}ζ_p^{\Tra_1^n(αx^{p^{3k}+1}+βx^{p^k+1}+\ga x)}\quad(α\in \bF_{p^m},β,\ga\in \bF_{q})\]的值分布，其中$\Tra_1^n: \bF_q\ra \bF_p$与$\Tra_1^m: \bF_{p^m}\ra\bF_p$是典型迹映射，$ζ_p=e^{\frac{2πi}{p}}$是本原$p$ -次单位根。作为应用：（1）。我们分别用奇偶校验多项式$h_2(x)h_3(x)$和$h_1(x)h_2(x)h_3(x)$确定了$\bF_{p^t}$上循环码$\cC_1$和$\cC_2$的重量分布，其中$t$是$d=\gcd(n,k)$的除数，$h_1(x)$，$h_2(x)$和$h_3(x)$分别是$π^{-1}$，$π^{-(p^k+1)}$和$π^{-(p^{3k}+1)}$在$\bF_{p^t}$上的极小多项式，对于$\bF_q$的本原元$π$.（二）、我们确定了一族m序列之间的相关分布。
英文摘要：
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 \[\sum\limits_{x\in \bF_q}ζ_p^{\Tra_1^n(αx^{p^{3k}+1}+βx^{p^k+1})}\quad(α\in \bF_{p^m},β\in \bF_{q})\] and \[\sum\limits_{x\in \bF_q}ζ_p^{\Tra_1^n(αx^{p^{3k}+1}+βx^{p^k+1}+\ga x)}\quad(α\in \bF_{p^m},β,\ga\in \bF_{q})\] 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 $ζ_p=e^{\frac{2πi}{p}}$ 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 $π^{-1}$, $π^{-(p^k+1)}$ and $π^{-(p^{3k}+1)}$ over $\bF_{p^t}$ respectively for a primitive element $π$ of $\bF_q$. (2). We determine the correlation distribution among a family of m-sequences.
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