On the Kernel of \Z_2^s-Linear Simplex and MacDonald Codes

2019/10/16 by Cristina Fernández-Córdoba, Fernández-Córdoba, Cristina, Carlos Vela +3
Computer Science · Engineering · #Coding theory and cryptography #Cooperative Communication and Network Coding #FOS: Computer and information sciences #Information Theory (cs.IT) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.1910.07911

openalex publication_date 2019/10/16 · openalex created_date 2019/10/25 · openalex updated_date 2026/07/28

Abstract

The \Z_2^s-additive codes are subgroups of \Zⁿ_2^s, and can be seen as a generalization of linear codes over \Z₂ and \Z₄. A \Z_2^s-linear code is a binary code which is the Gray map image of a \Z_2^s-additive code. We consider \Z_2^s-additive simplex codes of type α and β, which are a generalization over \Z_2^s of the binary simplex codes. These \Z_2^s-additive simplex codes are related to the \Z_2^s-additive Hadamard codes. In this paper, we use this relationship to establish the kernel of their binary images, under the Gray map, the \Z_2^s-linear simplex codes. Similar results can be obtained for the binary Gray map image of \Z_2^s-additive MacDonald codes.

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