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Complementary dual skew polycyclic codes and their applications to EAQECCs

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Abstract

For integers \(e\ge 2\), \(m\ge 1\) and a prime p, consider the ring \({\mathcal {A}}_e= {\mathbb {F}}_q + u{\mathbb {F}}_q+\cdots +u^{e-1}{\mathbb {F}}_q\), \(u^e=u\) where \(q=p^m\) and e satisfies \(q\equiv 1 \pmod {e-1}\). This work investigates skew polycyclic codes over \({\mathcal {A}}_e\). Using the idempotent decomposition technique, we present the generator polynomial and generator matrix of skew polycyclic codes over \({\mathcal {A}}_e\). Then we determine the necessary and sufficient conditions for these codes to be self-dual and derive some conditions under which skew polycyclic codes satisfy the complementary duality property. Further, we define a Gray map and investigate the Gray image of an LCD (or self-dual) code over \({\mathcal {A}}_e\). Besides, we also present a construction of LCD codes from self-dual codes (see Theorem 13). Finally, several examples of entanglement-assisted quantum error-correcting codes obtained from LCD codes are provided.

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The authors declare that [the/all other] data supporting the findings of this study are available within the article. Any clarification may be requested from the corresponding author, provided it is essential.

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Acknowledgements

The first and third authors thank the Department of Science and Technology, Govt. of India (under Project CRG/2020/005927, vide Diary No. SERB/F/6780/2020-2021 dated 31 December 2020) and second author thank the Council of Scientific & Industrial Research, Govt. of India (under grant No. 09/1023(0030)/2019-EMR-I), respectively, for financial support. Also, the authors would like to thank the anonymous referee(s) and the Editor of this journal for their valuable comments to improve the presentation of the paper.

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  1. Department of Mathematics, Indian Institute of Technology Patna, Patna, 801 106, India

    Shikha Yadav, Ashutosh Singh & Om Prakash

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  1. Shikha Yadav
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Correspondence to Om Prakash.

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Yadav, S., Singh, A. & Prakash, O. Complementary dual skew polycyclic codes and their applications to EAQECCs. Eur. Phys. J. Plus 138, 637 (2023). https://doi.org/10.1140/epjp/s13360-023-04253-1

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