Linearity of ℤ_2^L-Linear Codes via Schur Product

2023/09/21 by Gustavo Terra Bastos, Maiara F. Bollauf, Bastos, Gustavo T. +4
Computer Science · Engineering · #Coding theory and cryptography #Error Correcting Code Techniques #FOS: Computer and information sciences #Information Theory (cs.IT) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2309.12291

openalex publication_date 2023/09/21 · openalex created_date 2023/09/23 · openalex updated_date 2026/07/28

Abstract

We propose an innovative approach to investigating the linearity of ℤ_2^L-linear codes derived from ℤ_2^L-additive codes using the generalized Gray map. To achieve this, we define two related binary codes: the associated and the decomposition codes. By considering the Schur product between codewords, we can determine the linearity of the respective ℤ_2^L-linear code. As a result, we establish a connection between the linearity of the ℤ_2^L-linear codes with the linearity of the decomposition code for ℤ₄ and ℤ₈-additive codes. Furthermore, we construct ℤ_2^L-additive codes from nested binary codes, resulting in linear ℤ_2^L-linear codes. This construction involves multiple layers of binary codes, where a code in one layer is the square of the code in the previous layer. We also present a sufficient condition that allows checking nonlinearity of the ℤ_2^L-linear codes by simple binary operations in their respective associated codes. Finally, we employ our arguments to verify the linearity of well-known ℤ_2^L-linear code constructions, including the Hadamard, simplex, and MacDonald codes.

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