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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2401.14799 |
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| _version_ | 1866914654112972800 |
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| author | Bhunia, Dipak K. Fernández-Córdoba, Cristina Villanueva, Mercè |
| author_facet | Bhunia, Dipak K. Fernández-Córdoba, Cristina Villanueva, Mercè |
| contents | The $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive codes are subgroups of $\mathbb{Z}_2^{α_1} \times \mathbb{Z}_4^{α_2} \times \mathbb{Z}_8^{α_3}$. A $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard code is a Hadamard code which is the Gray map image of a $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive code. A recursive construction of $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive Hadamard codes of type $(α_1,α_2, α_3;t_1,t_2, t_3)$ with $α_1 \neq 0$, $α_2 \neq 0$, $α_3 \neq 0$, $t_1\geq 1$, $t_2 \geq 0$, and $t_3\geq 1$ is known. In this paper, we generalize some known results for $\mathbb{Z}_2\mathbb{Z}_4$-linear Hadamard codes to $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes with $α_1 \neq 0$, $α_2 \neq 0$, and $α_3 \neq 0$. First, we show for which types the corresponding $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes of length $2^t$ 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 $3 \leq t \leq 11$, 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 $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes, which are not equivalent to any other constructed $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard code, nor to any $\mathbb{Z}_2\mathbb{Z}_4$-linear Hadamard code, nor to any previously constructed $\mathbb{Z}_{2^s}$-linear Hadamard code with $s\geq 2$, with the same length $2^t$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_14799 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Linearity and Classification of $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-Linear Hadamard Codes Bhunia, Dipak K. Fernández-Córdoba, Cristina Villanueva, Mercè Information Theory The $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive codes are subgroups of $\mathbb{Z}_2^{α_1} \times \mathbb{Z}_4^{α_2} \times \mathbb{Z}_8^{α_3}$. A $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard code is a Hadamard code which is the Gray map image of a $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive code. A recursive construction of $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive Hadamard codes of type $(α_1,α_2, α_3;t_1,t_2, t_3)$ with $α_1 \neq 0$, $α_2 \neq 0$, $α_3 \neq 0$, $t_1\geq 1$, $t_2 \geq 0$, and $t_3\geq 1$ is known. In this paper, we generalize some known results for $\mathbb{Z}_2\mathbb{Z}_4$-linear Hadamard codes to $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes with $α_1 \neq 0$, $α_2 \neq 0$, and $α_3 \neq 0$. First, we show for which types the corresponding $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes of length $2^t$ 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 $3 \leq t \leq 11$, 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 $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes, which are not equivalent to any other constructed $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard code, nor to any $\mathbb{Z}_2\mathbb{Z}_4$-linear Hadamard code, nor to any previously constructed $\mathbb{Z}_{2^s}$-linear Hadamard code with $s\geq 2$, with the same length $2^t$. |
| title | Linearity and Classification of $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-Linear Hadamard Codes |
| topic | Information Theory |
| url | https://arxiv.org/abs/2401.14799 |