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Cyclic codes over a semi-local ring and their applications to QEC and EAQEC codes

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Abstract

Let \(R_{q,v}={\mathbb {F}}_q+v{\mathbb {F}}_q+ v^2{\mathbb {F}}_q\) where q is an odd prime power and \(v^3=v\). In this paper, we first provide structures of the Euclidean sums and hulls of cyclic codes of length n over \(R_{q,v}\). Then, we exhibit a method of constructing new quantum error-correcting (abbreviated to QEC) codes via the Euclidean sums of cyclic codes over \(R_{q,v}\) and CSS constructions. Finally, we construct two new classes of entanglement-assisted quantum error-correcting (abbreviated to EAQEC) codes by means of the Euclidean hulls of cyclic codes of length n over \(R_{q,v}\). In addition, to enrich the variety of available QEC and EAQEC codes, many new QEC and EAQEC codes are constructed to illustrate our results.

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Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.

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Acknowledgements

This work was supported by Research Funds of Hubei Province, Grant No. Q20174503.

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Authors and Affiliations

  1. School of Mathematics and Physics, Hubei Polytechnic University, Huangshi, 435003, Hubei, China

    Hui Li

  2. School of Science and Technology, College of Arts and Science, Hubei Normal University, Huangshi, 435109, Hubei, China

    Xiusheng Liu

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  1. Hui Li
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Hui Li and Xiusheng Liu discussed and come up with the intial idea. Xiusheng Liu developed the theory. Hui Li edited the text.

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Correspondence to Xiusheng Liu.

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Li, H., Liu, X. Cyclic codes over a semi-local ring and their applications to QEC and EAQEC codes. Quantum Inf Process 24, 49 (2025). https://doi.org/10.1007/s11128-025-04666-0

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