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Constacyclic codes over the ring \({\mathbb {F}}_q+v{\mathbb {F}}_q+v^{2}{\mathbb {F}}_q\) and their applications of constructing new non-binary quantum codes

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Abstract

Let \(R={\mathbb {F}}_q+v{\mathbb {F}}_q+v^{2}{\mathbb {F}}_q\) be a finite non-chain ring, where q is an odd prime power and \(v^3=v\). In this paper, we propose two methods of constructing quantum codes from \((\alpha +\beta v+\gamma v^{2})\)-constacyclic codes over R. The first one is obtained via the Gray map and the Calderbank–Shor–Steane construction from Euclidean dual-containing \((\alpha +\beta v+\gamma v^{2})\)-constacyclic codes over R. The second one is obtained via the Gray map and the Hermitian construction from Hermitian dual-containing \((\alpha +\beta v+\gamma v^{2})\)-constacyclic codes over R. As an application, some new non-binary quantum codes are obtained.

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Acknowledgements

Part of this work was done when Gao was visiting the Chern Institute of Mathematics, Nankai University. Gao would like to thank the kindly invitation. This research is supported by the National Key Basic Research Program of China (Grant No. 2013CB834204) and the National Natural Science Foundation of China (Grant Nos. 61171082, 61571243, 11701336, 11626144 and 11671235).

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Authors and Affiliations

  1. Chern Institute of Mathematics and LPMC, Nankai University, Tianjin, 300071, People’s Republic of China

    Fanghui Ma & Fang-Wei Fu

  2. School of Mathematics and Statistics, Shandong University of Technology, Zibo, 255000, People’s Republic of China

    Jian Gao

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  1. Fanghui Ma
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  2. Jian Gao
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  3. Fang-Wei Fu
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Correspondence to Jian Gao.

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Ma, F., Gao, J. & Fu, FW. Constacyclic codes over the ring \({\mathbb {F}}_q+v{\mathbb {F}}_q+v^{2}{\mathbb {F}}_q\) and their applications of constructing new non-binary quantum codes. Quantum Inf Process 17, 122 (2018). https://doi.org/10.1007/s11128-018-1898-6

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  • DOI: https://doi.org/10.1007/s11128-018-1898-6

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