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A study of QECCs and EAQECCs construction from cyclic codes over the ring \({\mathbb {F}}_q+v_1{\mathbb {F}}_q+v_2{\mathbb {F}}_q+\cdots +v_s{\mathbb {F}}_q\)

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Abstract

In this paper, we present a construction of quantum error-correcting codes (QECCs) codes and entanglement-assisted quantum error-correcting (EAQECCs) using Euclidean hulls and sums of cyclic codes of length n over a family of ring \(R_s = {\mathbb {F}}_q+v_1{\mathbb {F}}_q+v_2{\mathbb {F}}_q+\cdots +v_s{\mathbb {F}}_q\), where q is an odd prime power and \(v_i ^2=v_i\), \(v_iv_j=v_jv_i=0\), for \(i,j= 1,2,3,\cdots ,s\) and \(i \ne j\). The study delves into various aspects of this construction. We explore the generator polynomials, the dimension of both Euclidean hulls and the sums of cyclic codes over the ring \(R_s\). Further, we determine several new QECCs and EAQECCs. This paper claims that our obtained codes have improved parameters (e.g. higher minimum distance or greater dimension) than the existing quantum codes. Moreover, we present some detailed examples that effectively illustrate our findings.

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Acknowledgements

The authors are grateful for the considerate and helpful comments given by the anonymous reviewers. These comments have played a crucial role in improving the quality of the manuscript. The first, third and fourth authors are, respectively, their gratitude to the DST-INSPIRE, the University Grant Commission (UGC), Govt. of India and NBHM, Department of Atomic Energy, for their financial support. Also, AKU expresses gratitude to SERB-DST, India, for their financial assistance via project No. MTR/2020/000006 within the MATRICS framework. The authors would like to express their gratitude to Dr. Devendra Kumar Mishra, Associate Professor, Department of Physics, BHU for dedicating his time to the discussion on entangled bits and their importance in quantum physics.

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Authors and Affiliations

  1. Department of Mathematics, Banaras Hindu University, Varanasi, Uttar Pradesh, 221005, India

    Om Prakash Pandey, Awadhesh Kumar Shukla, Vipul Mishra & Ashish Kumar Upadhyay

  2. Department of Mathematics and Basic Sciences, NIIT University, Neemrana, 301705, India

    Sachin Pathak

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  1. Om Prakash Pandey
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  2. Sachin Pathak
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  3. Awadhesh Kumar Shukla
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  4. Vipul Mishra
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  5. Ashish Kumar Upadhyay
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OPP and SP wrote the main manuscript text. All authors reviewed the manuscript and prepared accordingly.

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Correspondence to Sachin Pathak.

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Pandey, O.P., Pathak, S., Shukla, A.K. et al. A study of QECCs and EAQECCs construction from cyclic codes over the ring \({\mathbb {F}}_q+v_1{\mathbb {F}}_q+v_2{\mathbb {F}}_q+\cdots +v_s{\mathbb {F}}_q\). Quantum Inf Process 23, 31 (2024). https://doi.org/10.1007/s11128-023-04240-6

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