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

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 d...

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Bibliographic Details
Published in:arXiv.org 2010-03
Main Authors: Zeng, Xiangyong, Shan, Jinyong, Hu, Lei
Format: Article
Language:English
Subjects:
Binary codes
Codes
Error correcting codes
Error correction
Fields (mathematics)
Gold
Weight
Online Access:Get full text
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Summary: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.
ISSN:2331-8422