基于Gold和Kasami-Welch APN幂函数的三重纠错循环码
A Triple-Error-Correcting Cyclic Code from the Gold and Kasami-Welch APN Power Functions
摘要：
基于Hollmann和Xiang提出的构造三重纠错码的一个充分条件，研究了长度为$2^m-1$的二进制循环码$\mathcal{C}_{1,3,13}$的最小距离及其对偶码的重量可除性，其中$m\geq 5$是奇数，$α$是有限域$\mathbb{F}_{2^m}$的本原元.证明了该码$\mathcal{C}_{1,3,13}$与相同长度的二进制三重纠错基元BCH码$\mathcal{C}_{1,3,5}$具有相同的权值分布。
英文摘要：
Based on a sufficient condition proposed by Hollmann and Xiang for constructing triple-error-correcting codes, the minimum distance of a binary cyclic code $\mathcal{C}_{1,3,13}$ with three zeros $α$, $α^3$, and $α^{13}$ of length $2^m-1$ and the weight divisibility of its dual code are studied, where $m\geq 5$ is odd and $α$ is a primitive element of the finite field $\mathbb{F}_{2^m}$. The code $\mathcal{C}_{1,3,13}$ is proven to have the same weight distribution as the binary triple-error-correcting primitive BCH code $\mathcal{C}_{1,3,5}$ of the same length.
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