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Link to original content: https://doi.org/10.1007/s11128-023-04133-8
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Optimal quaternary linear codes with one-dimensional Hermitian hull and related EAQECCs

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Abstract

Linear codes with small hulls over finite fields have been extensively studied due to their practical applications in computational complexity and information protection. In this paper, we develop a general method to determine the exact value of \(D_4^H(n,k,1)\) for \(n\le 12\) or \(k\in \{1,2,3,n-1,n-2,n-3\}\), where \(D_4^H(n,k,1)\) denotes the largest minimum distance among all quaternary linear [n, k] codes with one-dimensional Hermitian hull. As a consequence, we solve a conjecture proposed by Mankean and Jitman on the largest minimum distance of a quaternary linear code with one-dimensional Hermitian hull. As an application, we construct some binary entanglement-assisted quantum error-correcting codes (EAQECCs) from quaternary linear codes with one-dimensional Hermitian hull. Some of these EAQECCs are optimal codes, and some of them are better than previously known ones.

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Acknowledgements

This research is supported by Natural Science Foundation of China (12071001 and 12201170) and the Natural Science Foundation of Anhui Province (2108085QA03). The authors would also like to thank the editor and the anonymous referees for helpful comments which have highly improved the quality of the paper.

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Authors and Affiliations

  1. Key Laboratory of Intelligent Computing and Signal Processing, Ministry of Education, School of Mathematical Sciences, Anhui University, Hefei, 230601, China

    Shitao Li & Minjia Shi

  2. The State Key Laboratory of Integrated Service Networks, Xidian University, Xi’an, 710071, China

    Shitao Li & Minjia Shi

  3. State Grid Anhui Electric Power Co., Ltd., Hefei, China

    Huizhou Liu

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  1. Shitao Li
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  3. Huizhou Liu
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Correspondence to Minjia Shi.

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This research is supported by Natural Science Foundation of China (12071001 and 12201170) and the Natural Science Foundation of Anhui Province (2108085QA03).

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Li, S., Shi, M. & Liu, H. Optimal quaternary linear codes with one-dimensional Hermitian hull and related EAQECCs. Quantum Inf Process 22, 388 (2023). https://doi.org/10.1007/s11128-023-04133-8

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