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Hermitian LCD codes over \(\mathbb {F}_{q^{2}}+u \mathbb {F}_{q^{2}}\) and their applications to maximal entanglement EAQECCs

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Let \(R=\mathbb {F}_{q^{2}}+u \mathbb {F}_{q^{2}},\) where \(\mathbb {F}_{q^{2}}\) is the finite field with q2 elements, q is a power of a prime p, and u2 = 0. In this paper, a class of maximal entanglement entanglement-assisted quantum error-correcting codes (EAQECCs) is obtained by employing (1 − u)-constacyclic Hermitian linear complementary dual (LCD) codes of length n over R. First, we give a sufficient condition for a linear code C of length n over R to be a Hermitian LCD code and claim that there does not exist a non-free Hermitian LCD code of length n over R. Also, assume that \(\gcd (n, q)=1\), and γ is a unit in R, we obtain all γ-constacyclic Hermitian LCD codes. Finally, we derive symplectic LCD codes of length 2n over \(\mathbb {F}_{q^{2}}\) as Gray images of linear and constacyclic codes of length n over R. By using the explicit symplectic method in Galindo et al. (Quantum Inf. Process. 18(4), 116 9), we get the desired maximal entanglement EAQECCs.

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Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable comments which help to improve the presentation of this manuscript. This work was supported by the National Natural Science Foundation of China under Grant(61572168) and the Excellent Young Talents Fund Program of Higher Education Institutions of Anhui Province (CN)(gxyqZD2016228).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Hefei Normal University, Hefei, 230601, China

    Heqian Xu & Wei Du

  2. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230027, China

    Heqian Xu

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  1. Heqian Xu
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  2. Wei Du
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Correspondence to Heqian Xu.

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This article belongs to the Topical Collection: Sequences and Their Applications III Guest Editors: Chunlei Li, Tor Helleseth and Zhengchun Zhou

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Xu, H., Du, W. Hermitian LCD codes over \(\mathbb {F}_{q^{2}}+u \mathbb {F}_{q^{2}}\) and their applications to maximal entanglement EAQECCs. Cryptogr. Commun. 14, 259–269 (2022). https://doi.org/10.1007/s12095-021-00510-1

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  • DOI: https://doi.org/10.1007/s12095-021-00510-1

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