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On quantum and LCD Codes Over The Ring \(F_q+vF_q+v^2F_q\)

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Abstract

Let \(m \ge 1\) be a fixed integer and q be an odd prime such that \(q~=~p^m\). The aim of this paper is to study cyclic codes over a non-chain ring \(R~=~F_q+vF_q+v^2F_q\), where \(v^3~=~v\). Precisely, we describe better quantum error-correcting codes than the previously known quantum error-correcting codes over \(F_q\). Moreover, as an application, we construct MDS LCD codes and prove that the Gray image of an LCD code of length n over R is also an LCD code of length 3n over \(F_q\).

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Acknowledgements

The authors are grateful to the anonymous reviewers who have given us very thoughtful and helpful comments to improve the manuscript. The research of first named author is supported by SERB-DST MATRIC Project (Grant No.: MTR/2019/000603), India.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Aligarh Muslim University, Aligarh, India

    S. Ali, G. Mohammad, N. Khan & P. Sharma

  2. Department of Computer Science, Faculty of Science, Aligarh Muslim University, Aligarh, India

    Jeelani

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  1. S. Ali
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  4. N. Khan
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Correspondence to S. Ali.

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Ali, S., Mohammad, G., Jeelani et al. On quantum and LCD Codes Over The Ring \(F_q+vF_q+v^2F_q\). Quantum Inf Process 21, 306 (2022). https://doi.org/10.1007/s11128-022-03654-y

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