Galois hulls of constacyclic codes over affine algebra rings
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929625300467712 |
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| author | Debnath, Indibar Islam, Habibul Martínez-Moro, Edgar Prakash, Om |
| author_facet | Debnath, Indibar Islam, Habibul Martínez-Moro, Edgar Prakash, Om |
| contents | Let $\mathcal A$ the affine algebra given by the ring $\mathbb{F}_q[X_1,X_2,\ldots,X_\ell]/ I$, where $I$ is the ideal $\langle t_1(X_1), t_2(X_2), \ldots, t_\ell(X_\ell) \rangle$ with each $t_i(X_i)$, $1\leq i\leq \ell$, being a square-free polynomial over $\mathbb{F}_q$. This paper studies the $k$-Galois hulls of $λ$-constacyclic codes over $\mathcal A$ regarding their idempotent generators. For this, first, we define the $k$-Galois inner product over $\mathcal A$ and find the form of the generators of the $k$-Galois dual and the $k$-Galois hull of a $λ$-constacyclic code over $\mathcal A$. Then, we derive a formula for the $k$-Galois hull dimension of a $λ$-constacyclic code. Further, we provide a condition for a $λ$-constacyclic code to be $k$-Galois LCD. Finally, we give some examples of the use of these codes in constructing entanglement-assisted quantum error-correcting codes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_08512 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Galois hulls of constacyclic codes over affine algebra rings Debnath, Indibar Islam, Habibul Martínez-Moro, Edgar Prakash, Om Information Theory 94B05, 94B15, 94B60 Let $\mathcal A$ the affine algebra given by the ring $\mathbb{F}_q[X_1,X_2,\ldots,X_\ell]/ I$, where $I$ is the ideal $\langle t_1(X_1), t_2(X_2), \ldots, t_\ell(X_\ell) \rangle$ with each $t_i(X_i)$, $1\leq i\leq \ell$, being a square-free polynomial over $\mathbb{F}_q$. This paper studies the $k$-Galois hulls of $λ$-constacyclic codes over $\mathcal A$ regarding their idempotent generators. For this, first, we define the $k$-Galois inner product over $\mathcal A$ and find the form of the generators of the $k$-Galois dual and the $k$-Galois hull of a $λ$-constacyclic code over $\mathcal A$. Then, we derive a formula for the $k$-Galois hull dimension of a $λ$-constacyclic code. Further, we provide a condition for a $λ$-constacyclic code to be $k$-Galois LCD. Finally, we give some examples of the use of these codes in constructing entanglement-assisted quantum error-correcting codes. |
| title | Galois hulls of constacyclic codes over affine algebra rings |
| topic | Information Theory 94B05, 94B15, 94B60 |
| url | https://arxiv.org/abs/2412.08512 |