تقرير
$m$-adic residue codes over $\mathbb{F}_q[v]/(v^s-v)$ and their application to quantum codes
| العنوان: | $m$-adic residue codes over $\mathbb{F}_q[v]/(v^s-v)$ and their application to quantum codes |
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| المؤلفون: | Kuruz, Ferhat, Sarı, Mustafa, Koroglu, Mehmet E. |
| المصدر: | K\"{u}r\"{u}z, F., Sar\i , M., \& K\"{o}ro\u{g}lu, M.E. (2022). $m$-adic residue codes over $\mathbb{F}_{q}[v]/(v^{s}-v)$ and their application to quantum codes, Quantum Information \& Computation, 22(5-6), 427-439 |
| سنة النشر: | 2018 |
| المجموعة: | Computer Science Mathematics |
| مصطلحات موضوعية: | Computer Science - Information Theory |
| الوصف: | 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. |
| نوع الوثيقة: | Working Paper |
| DOI: | 10.26421/qic22.5-6-4 |
| URL الوصول: | http://arxiv.org/abs/1810.11826 |
| رقم الأكسشن: | edsarx.1810.11826 |
| قاعدة البيانات: | arXiv |
| DOI: | 10.26421/qic22.5-6-4 |
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