Linearity and classification of Z2Z4Z8-linear Hadamard codes
The Z2Z4Z8-additive codes are subgroups of Z2α1 × Z4α2 × Z8α3. A Z2Z4Z8-linear Hadamard code is a Hadamard code which is the Gray map image of a Z2Z4Z8-additive code. A recursive construction of Z2Z4Z8-additive Hadamard codes of type (α1, α2, α3; t1, t2, t3) with α1 ≠ 0, α2 ≠ 0, α3 ≠ 0, t1 ≥ 1, t2 ≥...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2025 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:318754 |
| Acceso en línea: | https://ddd.uab.cat/record/318754 https://dx.doi.org/urn:doi:10.1007/s10623-025-01696-8 |
| Access Level: | acceso abierto |
| Palabra clave: | Hadamard code Gray map Z2Z4Z8-linear code Z2Z4Z8-additive code Kernel Rank Classification |
| Sumario: | The Z2Z4Z8-additive codes are subgroups of Z2α1 × Z4α2 × Z8α3. A Z2Z4Z8-linear Hadamard code is a Hadamard code which is the Gray map image of a Z2Z4Z8-additive code. A recursive construction of Z2Z4Z8-additive Hadamard codes of type (α1, α2, α3; t1, t2, t3) with α1 ≠ 0, α2 ≠ 0, α3 ≠ 0, t1 ≥ 1, t2 ≥ 0, and t3 ≥ 1 is known. In this paper, we generalize some known results for Z2Z4-linear Hadamard codes to Z2Z4Z8-linear Hadamard codes with α1 ≠ 0, α2 ≠ 0, and α3 ≠ 0. First, we show for which types the corresponding Z2Z4Z8-linear Hadamard codes of length 2t are nonlinear. For these codes, we compute the kernel and its dimension, which allows us to give a partial classification of these codes. Moreover, for 3 ≤ t ≤ 11, we give a complete classification by providing the exact amount of nonequivalent such codes. We also prove the existence of several families of infinite such nonlinear Z2Z4Z8-linear Hadamard codes, which are not equivalent to any other constructed Z2Z4Z8-linear Hadamard code, nor to any Z2Z4-linear Hadamard code, nor to any previously constructed Z2s-linear Hadamard code with s ≥ 2, with the same length 2t. |
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