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\({\mathbb {F}}_qR\)-linear skew constacyclic codes and their application of constructing quantum codes

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Abstract

Let q be a prime power with \(\mathrm{gcd}(q,6)=1\). Let \(R={\mathbb {F}}_{q^2}+u{\mathbb {F}}_{q^2}+v{\mathbb {F}}_{q^2}+uv{\mathbb {F}}_{q^2}\), where \(u^2=u\), \(v^2=v\) and \(uv=vu\). In this paper, we give the definition of linear skew constacyclic codes over \({\mathbb {F}}_{q^2}R\). By the decomposition method, we study the structural properties and determine the generator polynomials and the minimal generating sets of linear skew constacyclic codes. We define a Gray map from \({\mathbb {F}}_{q^2}^{\alpha }\times R^{\beta }\) to \({\mathbb {F}}_{q^2}^{\alpha +4\beta }\) preserving the Hermitian orthogonality, where \(\alpha \) and \(\beta \) are positive integers. As an application, by Hermitian construction, we obtain some good quantum error-correcting codes.

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Acknowledgements

This research is supported by the 973 Program of China (Grant No. 2013CB834204), the National Natural Science Foundation of China (Grant Nos. 61571243, 11701336, 11626144 and 11671235), the Fundamental Research Funds for the Central Universities of China, the Scientific Research Fund of Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (Changsha University of Science & Technology) (Grant No. 2018MMAEZD04).

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Authors and Affiliations

  1. Chern Institute of Mathematics and LPMC, Nankai University, Tianjin, 300071, China

    Juan Li, Fang-Wei Fu & Fanghui Ma

  2. School of Mathematics and Statistics, Shandong University of Technology, Zibo, 255000, China

    Jian Gao

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  1. Juan Li
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Correspondence to Jian Gao.

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Li, J., Gao, J., Fu, FW. et al. \({\mathbb {F}}_qR\)-linear skew constacyclic codes and their application of constructing quantum codes. Quantum Inf Process 19, 193 (2020). https://doi.org/10.1007/s11128-020-02700-x

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