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New Galois Hulls of GRS Codes and Application to EAQECCs

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Abstract

Galois hulls of linear codes have important applications in quantum coding theory. In this paper, we construct some new classes of (extended) generalized Reed-Solomon (GRS) codes with Galois hulls of arbitrary dimensions. We also propose a general method on constructing GRS codes with Galois hulls of arbitrary dimensions from special Euclidean orthogonal GRS codes. Finally, we construct several new families of entanglement-assisted quantum error-correcting codes (EAQECCs) and MDS EAQECCs by utilizing the above results.

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References

  1. Blaum, M., Roth, R.M.: On lowest density MDS codes. IEEE Trans. Inf. Theory 45(1), 46–59 (1999)

    Article  MathSciNet  Google Scholar 

  2. Brun, T., Devetak, I., Hsieh, M.H.: Correcting quantum errors with entanglement. Science 314, 436–439 (2006)

    Article  MathSciNet  Google Scholar 

  3. Cao, M.: Galois hulls of MDS codes and their quantum error correcttion. arXiv:2002.12892v2 [cs.IT] (2020)

  4. Carlet, C., Guilley, S. In: Pinto, ER (ed.) : Complementary dual codes for counter-measures to side-channel attacks. Coding Theory and Applications (CIM Series in Mathematical Sciences), vol. 3, pp 97–105. Springer, Berlin (2014)

  5. Carlet, C., Mesnager, S., Tang, C., Qi, Y.: Euclidean and Hermitian LCD MDS codes. Des. Codes Cryptogr. 86(11), 2605–2618 (2018)

    Article  MathSciNet  Google Scholar 

  6. Carlet, C., Mesnager, S., Tang, C., Qi, Y., Pellikaan, R.: Linear codes over \(\mathbb {F}_{q}\) which are equivalent to LCD codes for q > 3. IEEE Trans. Inf. Theory 64(4), 3010–3017 (2018)

    Article  Google Scholar 

  7. Fan, Y., Zhang, L.: Galois self-dual constacyclic codes. Des. Codes Cryptogr. 84(11), 473–492 (2017)

    Article  MathSciNet  Google Scholar 

  8. Fang, W., Fu, F. -W., Li, L., Zhu, S.: Euclidean and Hermitian hulls of MDS codes and their applications to EAQECCs. IEEE Trans. Inf. Theory 66(6), 3527–3537 (2020)

    Article  MathSciNet  Google Scholar 

  9. Fang, W., Xia, S. -T., Fu, F. -W.: Construction of MDS euclidean Self-Dual codes via two subsets. IEEE Trans. Inf. Theory 67(8), 5005–5015 (2021)

    Article  MathSciNet  Google Scholar 

  10. Fang, W., Zhang, J., Xia, S. -T., Fu, F. -W.: A Note on Self-Dual Generalized Reed-Solomon Codes. arXiv:2005.11732v1 [cs.IT] (2020)

  11. Fang, X., Liu, M., Luo, J.: On euclidean hulls of MDS codes. Cryptogr. Commun. 13(1), 1–14 (2021)

    Article  MathSciNet  Google Scholar 

  12. Fang, X., Liu, M., Luo, J.: New MDS Euclidean Self-orthogonal Codes. IEEE Trans. Inf. Theory 67(1), 130–137 (2021)

    Article  MathSciNet  Google Scholar 

  13. Grassl, M., Gulliver, T.A.: On Self-Dual MDS Codes In: Proceedings of ISIT, pp. 1954–1957 (2008)

  14. Guenda, K., Jitman, S., Gulliver, T.A.: Constructions of good entanglement-assisted quantum error correcting codes. Des. Codes Cryptogr. 86, 121–136 (2018)

    Article  MathSciNet  Google Scholar 

  15. Hu, P., Liu, X.: A new method for constructing EAQEC MDS codes. arXiv:2003.00843v1 [cs.IT] (2020)

  16. Jin, L., Xing, C.: New MDS self-dual codes from generalized Reed-Solomon codes. IEEE Trans. Inf. Theory 63(3), 1434–1438 (2017)

    Article  MathSciNet  Google Scholar 

  17. Lai, C.Y., Ashikhmin, A.: Linear programming boundes entanglement-assisted quantum error-correcting codes by split weight enumerators. IEEE Trans. Inf. Theory 64(1), 622–639 (2018)

    Article  Google Scholar 

  18. Luo, G., Cao, X.: Two new families of entanglement-assisted quantum MDS codes from generalized Reed-Solomon codes. Quantum Inf. Process. 18(3), 89 (2019)

    Article  MathSciNet  Google Scholar 

  19. Luo, G., Cao, X., Chen, X.: MDS Codes with hulls of arbitrary dimensions and their quantum error correction. IEEE Trans. Inf. Theory 65(5), 2944–2952 (2019)

    Article  MathSciNet  Google Scholar 

  20. Markus, G., Felix, H., Andreas, W.: Entropic proofs of Singleton bounds for quantum error-correcting codes. arXiv:2010.07902v1 [cs.IT] (2020)

  21. MacWilliams, F.J., Sloane, N.J.A.: The theory of Error-Correcting codes, The Netherlands (1977)

  22. Qian, L., Cao, X., Wu, X., Lu, W.: Entanglement-assisted quantum codes from l-Galois hulls MDS codes of arbitrary dimensions preprint (2019)

  23. Shor, P., Laflamme, R.: Quantum Analog of the MacWilliams Identities for Classical Coding Theory. Phys. Rev. Lett. 78(8), 1600–1602 (1997)

    Article  Google Scholar 

  24. Suh, C., Ramchandran, K.: Exact-repair MDS code construction using interference alignment. IEEE Trans. Inf. Theory 57(3), 1425–1442 (2011)

    Article  MathSciNet  Google Scholar 

  25. Wilde, M.M., Brun, T.A.: Optimal entanglement formulas for entanglement-assisted quantum coding. Phys. Rev. 77(6) (2008)

  26. Yan, H.: A note on the construction of MDS self-dual codes. Cryptogr. Commun. 11(2), 259–268 (2019)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors thank anonymous reviewers, the editor and the associate editor for their suggestions and comments to improve the readability of this paper. This research is supported by National Natural Science Foundation of China under Grant 11871025 and 61977036 and the Fundamental Research Funds for the Central Universities of CCNU under grant CCNU20TD002.

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Authors and Affiliations

  1. School of Mathematics and Computer Science, Wuhan Polytechnic University, Wuhan, 430048, China

    Xiaolei Fang

  2. College of Liberal Arts and Sciences, National University of Defense Technology, Changsha, 410073, China

    Renjie Jin

  3. School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, China

    Jinquan Luo & Wen Ma

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  1. Xiaolei Fang
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  2. Renjie Jin
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  3. Jinquan Luo
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  4. Wen Ma
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Correspondence to Jinquan Luo.

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Fang, X., Jin, R., Luo, J. et al. New Galois Hulls of GRS Codes and Application to EAQECCs. Cryptogr. Commun. 14, 145–159 (2022). https://doi.org/10.1007/s12095-021-00525-8

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  • DOI: https://doi.org/10.1007/s12095-021-00525-8

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