$m$-adic residue codes over $\mathbb{F}_q[v]/(v^s-v)$ and their application to quantum codes
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2018
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| _version_ | 1866916259894919168 |
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| author | Kuruz, Ferhat Sarı, Mustafa Koroglu, Mehmet E. |
| author_facet | Kuruz, Ferhat Sarı, Mustafa Koroglu, Mehmet E. |
| contents | Due to their rich algebraic structure, cyclic codes have a great deal of significance amongst linear codes. Duadic codes are the generalization of the quadratic residue codes, a special case of cyclic codes. The $m$-adic residue codes are the generalization of the duadic codes. The aim of this paper is to study the structure of the $m$-adic residue codes over the quotient ring $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}$. We determine the idempotent generators of the $m$-adic residue codes over $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}$. We obtain some parameters of optimal $m$-adic residue codes over $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}$ with respect to Griesmer bound for rings. Furthermore, we derive a condition for $m$-adic residue codes over $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}$ to contain their dual. By making use of a preserving-orthogonality Gray map, we construct a family of quantum error correcting codes from the Gray images of dual-containing $m$-adic residue codes over $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}$ and give some examples to illustrate our findings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1810_11826 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | $m$-adic residue codes over $\mathbb{F}_q[v]/(v^s-v)$ and their application to quantum codes Kuruz, Ferhat Sarı, Mustafa Koroglu, Mehmet E. Information Theory Due to their rich algebraic structure, cyclic codes have a great deal of significance amongst linear codes. Duadic codes are the generalization of the quadratic residue codes, a special case of cyclic codes. The $m$-adic residue codes are the generalization of the duadic codes. The aim of this paper is to study the structure of the $m$-adic residue codes over the quotient ring $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}$. We determine the idempotent generators of the $m$-adic residue codes over $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}$. We obtain some parameters of optimal $m$-adic residue codes over $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}$ with respect to Griesmer bound for rings. Furthermore, we derive a condition for $m$-adic residue codes over $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}$ to contain their dual. By making use of a preserving-orthogonality Gray map, we construct a family of quantum error correcting codes from the Gray images of dual-containing $m$-adic residue codes over $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}$ and give some examples to illustrate our findings. |
| title | $m$-adic residue codes over $\mathbb{F}_q[v]/(v^s-v)$ and their application to quantum codes |
| topic | Information Theory |
| url | https://arxiv.org/abs/1810.11826 |