Linearity of $\mathbb{Z}_{2^L}$-Linear Codes via Schur Product
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866918135341252608 |
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| author | Bastos, Gustavo T. Bollauf, Maiara F. Ferrari, Agnaldo J. Ytrehus, Øyvind |
| author_facet | Bastos, Gustavo T. Bollauf, Maiara F. Ferrari, Agnaldo J. Ytrehus, Øyvind |
| contents | We propose an innovative approach to investigating the linearity of $\mathbb{Z}_{2^L}$-linear codes derived from $\mathbb{Z}_{2^L}$-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 $\mathbb{Z}_{2^L}$-linear code. As a result, we establish a connection between the linearity of the $\mathbb{Z}_{2^L}$-linear codes with the linearity of the decomposition code for $\mathbb{Z}_4$ and $\mathbb{Z}_8$-additive codes. Furthermore, we construct $\mathbb{Z}_{2^L}$-additive codes from nested binary codes, resulting in linear $\mathbb{Z}_{2^L}$-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 $\mathbb{Z}_{2^L}$-linear codes by simple binary operations in their respective associated codes. Finally, we employ our arguments to verify the linearity of well-known $\mathbb{Z}_{2^L}$-linear code constructions, including the Hadamard, simplex, and MacDonald codes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_12291 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Linearity of $\mathbb{Z}_{2^L}$-Linear Codes via Schur Product Bastos, Gustavo T. Bollauf, Maiara F. Ferrari, Agnaldo J. Ytrehus, Øyvind Information Theory We propose an innovative approach to investigating the linearity of $\mathbb{Z}_{2^L}$-linear codes derived from $\mathbb{Z}_{2^L}$-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 $\mathbb{Z}_{2^L}$-linear code. As a result, we establish a connection between the linearity of the $\mathbb{Z}_{2^L}$-linear codes with the linearity of the decomposition code for $\mathbb{Z}_4$ and $\mathbb{Z}_8$-additive codes. Furthermore, we construct $\mathbb{Z}_{2^L}$-additive codes from nested binary codes, resulting in linear $\mathbb{Z}_{2^L}$-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 $\mathbb{Z}_{2^L}$-linear codes by simple binary operations in their respective associated codes. Finally, we employ our arguments to verify the linearity of well-known $\mathbb{Z}_{2^L}$-linear code constructions, including the Hadamard, simplex, and MacDonald codes. |
| title | Linearity of $\mathbb{Z}_{2^L}$-Linear Codes via Schur Product |
| topic | Information Theory |
| url | https://arxiv.org/abs/2309.12291 |