Kasami-Welch函数的循环码和循环序列
Cyclic Codes and Sequences from Kasami-Welch Functions
摘要：
设$q=2^n$、$0\leq k\leq n-1$和$k\neq n/2$。本文确定了如下指数和\[\sum\limits_{x\in \bF_q}(-1)^{\Tra_1^n(αx^{2^{3k}+1}+βx^{2^k+1})}\quad(α,β\in \bF_{q})\]和\[\sum\limits_{x\in \bF_q}(-1)^{\Tra_1^n(αx^{2^{3k}+1}+βx^{2^k+1}+\ga x)}\quad(α,β,\ga\in \bF_{q})\]的值分布，其中$\Tra_1^n: \bF_{2^n}\ra \bF_2$是典型迹映射.作为应用：（1）。本文确定了二元循环码$\cC_1$和$\cC_2$分别具有奇偶校验多项式$h_2(x)h_3(x)$和$h_1(x)h_2(x)h_3(x)$的重量分布，其中$h_1(x)$，$h_2(x)$和$h_3(x)$分别是$π^{-1}$，$π^{-(2^k+1)}$和$π^{-(2^{3k}+1)}$的最小多项式，对于$\bF_q$的本原元$π$.（二）、我们确定了一族二元m序列之间的相关分布。
英文摘要：
Let $q=2^n$, $0\leq k\leq n-1$ and $k\neq n/2$. In this paper we determine the value distribution of following exponential sums \[\sum\limits_{x\in \bF_q}(-1)^{\Tra_1^n(αx^{2^{3k}+1}+βx^{2^k+1})}\quad(α,β\in \bF_{q})\] and \[\sum\limits_{x\in \bF_q}(-1)^{\Tra_1^n(αx^{2^{3k}+1}+βx^{2^k+1}+\ga x)}\quad(α,β,\ga\in \bF_{q})\] where $\Tra_1^n: \bF_{2^n}\ra \bF_2$ is the canonical trace mapping. 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 $π^{-1}$, $π^{-(2^k+1)}$ and $π^{-(2^{3k}+1)}$ respectively for a primitive element $π$ of $\bF_q$. (2). We determine the correlation distribution among a family of binary m-sequences.
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