A Triple-Error-Correcting Cyclic Code from the Gold and Kasami-Welch APN Power Functions

Details A Triple-Error-Correcting Cyclic Code from the Gold and Kasami-Welch APN Power Functions A Triple-Error-Correcting Cyclic Code from the Gold and Kasami-Welch APN Power Functions

2010-03-31

arxiv.org/abs/1003.5993v1

Computer Science

Discrete Mathematics

29 pages

Scientific paper

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 $\alpha$, $\alpha^3$, and $\alpha^{13}$ of length $2^m-1$ and the weight divisibility of its dual code are studied, where $m\geq 5$ is odd and $\alpha$ 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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