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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2501.01708 |
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| _version_ | 1866915102976901120 |
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| author | Akanksha Bhagat, Anuj Kumar Sarma, Ritumoni |
| author_facet | Akanksha Bhagat, Anuj Kumar Sarma, Ritumoni |
| contents | In this article, for a finite field $\mathbb{F}_q$ and a natural number $l,$ let $\mathcal{R}$ denote the product ring $\mathbb{F}_q^l.$ Firstly, for an automorphism $Θ$ of $\mathcal{R},$ a $Θ$-derivation $Δ_Θ$ of $\mathcal{R}$ and for a unit $\mathbf{a}$ in $\mathcal{R},$ we study $(Θ, Δ_Θ, \mathbf{a})$-cyclic codes over $\mathcal{R}.$ In this direction, we give an algebraic characterization of a $(Θ, Δ_Θ, \mathbf{a})$-cyclic code over $\mathcal{R}$, determine its generator polynomial, and find its decomposition over $\mathbb{F}_q.$ Secondly, we give a necessary and sufficient condition for a $(Θ, 0, \mathbf{a})$-cyclic code to be Euclidean dual-containing code over $\mathcal{R}.$ Thirdly, we study Gray maps and obtain several MDS and optimal linear codes over $\mathbb{F}_q$ as Gray images of $(Θ, Δ_Θ, \mathbf{a})$-cyclic codes over $\mathcal{R}.$ Moreover, we determine orthogonality preserving Gray maps and construct Euclidean dual-containing codes with good parameters. Lastly, as an application, we construct MDS and almost MDS quantum codes by employing the Euclidean dual-containing and annihilator dual-containing CSS constructions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_01708 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $(Θ, Δ_Θ, \mathbf{a})$-cyclic codes over $\mathbb{F}_q^l$ and their applications in the construction of quantum codes Akanksha Bhagat, Anuj Kumar Sarma, Ritumoni Information Theory 94B60, 94B15, 94B05, 16Z05 In this article, for a finite field $\mathbb{F}_q$ and a natural number $l,$ let $\mathcal{R}$ denote the product ring $\mathbb{F}_q^l.$ Firstly, for an automorphism $Θ$ of $\mathcal{R},$ a $Θ$-derivation $Δ_Θ$ of $\mathcal{R}$ and for a unit $\mathbf{a}$ in $\mathcal{R},$ we study $(Θ, Δ_Θ, \mathbf{a})$-cyclic codes over $\mathcal{R}.$ In this direction, we give an algebraic characterization of a $(Θ, Δ_Θ, \mathbf{a})$-cyclic code over $\mathcal{R}$, determine its generator polynomial, and find its decomposition over $\mathbb{F}_q.$ Secondly, we give a necessary and sufficient condition for a $(Θ, 0, \mathbf{a})$-cyclic code to be Euclidean dual-containing code over $\mathcal{R}.$ Thirdly, we study Gray maps and obtain several MDS and optimal linear codes over $\mathbb{F}_q$ as Gray images of $(Θ, Δ_Θ, \mathbf{a})$-cyclic codes over $\mathcal{R}.$ Moreover, we determine orthogonality preserving Gray maps and construct Euclidean dual-containing codes with good parameters. Lastly, as an application, we construct MDS and almost MDS quantum codes by employing the Euclidean dual-containing and annihilator dual-containing CSS constructions. |
| title | $(Θ, Δ_Θ, \mathbf{a})$-cyclic codes over $\mathbb{F}_q^l$ and their applications in the construction of quantum codes |
| topic | Information Theory 94B60, 94B15, 94B05, 16Z05 |
| url | https://arxiv.org/abs/2501.01708 |