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On the duals of quasi-cyclic codes and their application to quantum codes

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Abstract

This paper presents some results on 1-generator quasi-cyclic (QC) codes and their duals over finite fields. We have explicitly obtained a set of generators for the dual of a 1-generator QC code C from the generator of C, for some restricted cases. From the form of the generators of the dual code \(C^\perp \), we have derived upper bounds on the minimum distance of \(C^\perp \). For their applications in constructing quantum codes, we have presented some criteria for 1-generator QC codes to be self-orthogonal or self-dual. Then using the CSS construction, we have obtained some new and better quantum codes with the help of self-orthogonal 1-generator QC codes.

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Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and suggestions that greatly improved the presentation of the paper. This research is partially supported by Science and Engineering Research Board (SERB), India, under Grant No. MTR/2022/000542. The first author would like to thank Ministry of Human Resource Development (MHRD), India, for providing financial support.

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  1. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, 247667, India

    Shivanshu Benjwal & Maheshanand Bhaintwal

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  1. Shivanshu Benjwal
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  2. Maheshanand Bhaintwal
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Both the authors contributed to the manuscript equally.

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Correspondence to Maheshanand Bhaintwal.

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Benjwal, S., Bhaintwal, M. On the duals of quasi-cyclic codes and their application to quantum codes. Quantum Inf Process 23, 113 (2024). https://doi.org/10.1007/s11128-024-04318-9

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