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On quantum codes derived from quasi-cyclic codes over a non-chain ring

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Abstract

This paper presents a study on the structure of 1-generator quasi-cyclic (QC) codes over the non-chain ring \(R=\mathbb {F}_{q}+u\mathbb {F}_{q}+v\mathbb {F}_{q}+uv\mathbb {F}_{q}\), where \(u^2=v^2=0,~ uv=vu\), and \(\mathbb {F}_q\) is a finite field of cardinality \(q=p^r\); p is a prime. A minimal spanning set and size of these codes are determined. A sufficient condition for 1-generator QC codes over R to be free is given. BCH-type bounds on the minimum distance of free QC codes over R are also presented. Some optimal linear codes over \(\mathbb {F}_q\) are obtained as the Gray images of quasi-cyclic codes over R. Some characterizations of the Gray images of QC codes over R in \(\mathbb {F}_q\) and \(\mathbb {F}_q+u\mathbb {F}_q~(u^2=0)\) are done. As an application, we consider self-orthogonal subcodes of the Gray images of QC codes over R to obtain new and better quantum codes than those are available in the literature.

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Acknowledgements

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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Authors and Affiliations

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

    Shivanshu Benjwal & Maheshanand Bhaintwal

  2. Department of Mathematics, Government Nehru Memorial College, Hanumangarh, Rajasthan, 335513, India

    Raj Kumar

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  1. Shivanshu Benjwal
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  2. Maheshanand Bhaintwal
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  3. Raj Kumar
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Benjwal, S., Bhaintwal, M. & Kumar, R. On quantum codes derived from quasi-cyclic codes over a non-chain ring. Quantum Inf Process 23, 309 (2024). https://doi.org/10.1007/s11128-024-04514-7

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