New Entanglement-Assisted Quantum MDS Codes Derived From Generalized Reed-Solomon Codes

  • Guanmin Guo
  • Ruihu LiEmail author


Entanglement-assisted quantum error-correcting codes (abbreviate to EAQECCs) expand the usual paradigm of quantum error correction by allowing two parties to make use of pre-shared entanglement. It is well-known that we can construct an EAQECC from arbitrary classical linear code. In this paper, we construct several classes of entanglement-assisted quantum MDS (EAQMDS) codes by utilizing generalized Reed-Solomon (GRS) codes. The main contribution of the paper is extend the code length of EAQMDS in the literature (Guo et al. 2019). Consequently, the results show that almost all of these EAQMDS codes are new in the sense that the parameters of these codes are not covered by the previously known ones.


Entanglement-assisted quantum error-correcting codes Generalized Reed-Solomon codes MDS codes Rank 

Mathematics Subject Classification (2010)

81P45 81P70 94B05 



This work is supported by the National Natural Science Foundation of China under Grant Nos. 11801564 and 11471011.


  1. 1.
    Brun, T.A., Devetak, I., Hsieh, M. -H.: Correcting quantum errors with entanglement. Science 314, 436–439 (2006)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Grassl, M.: Entanglement-assisted quantum communication beating the quantum Singleton bound. Talk at AQIS, Taiwan, China (2016)Google Scholar
  3. 3.
    Lai, C., Ashikhmin, A.: Linear programming bounds for entanglement-assisted quantum error-correcting codes by split weight enumerators. IEEE Trans. Inf. Theory 64(1), 622–639 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Hsich, M. -H., Devetak, I., Brun, T.A.: General entanglement-assisted quantum error-correcting codes. Phys. Rev. A 76, 062313 (2007)ADSCrossRefGoogle Scholar
  5. 5.
    Wilde, M.M., Brun, T.A.: Optimal entanglement formulas for entanglement-assisted quantum coding. Phys. Rev. A 77, 064302 (2008)ADSCrossRefGoogle Scholar
  6. 6.
    Lai, C.Y., Brun, T.A.: Entanglement increases the error-correcting ability of quantum error-correcting codes. Phys. Rev. A 88, 012320 (2013)ADSCrossRefGoogle Scholar
  7. 7.
    Lai, C.Y., Brun, T.A., Wilde, M.M.: Duality in entanglement-assisted quantum error correction. IEEE Trans. Inf. Theory 59, 4020–4024 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Fan, J., Chen, H., Xu, J.: Construction of q-ary entanglement-assisted quantum MDS codes with minimum distance greater than q + 1. Quantum Inf. Comput. 16, 0423–0434 (2016)MathSciNetGoogle Scholar
  9. 9.
    Li, L., Zhu, S., Liu, L., Kai, X.: Entanglement-assisted quantum MDS codes from generalized Reed-Solomon codes. Quantum Inf. Process. 18, 153 (2019)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Li, R., Guo, L., Xu, Z.: Entanglement-assisted quantum codes achieving the quantum Singleton bound but violating the quantum Hamming bound. Quantum Inf. Comput. 14, 1107–1116 (2014)MathSciNetGoogle Scholar
  11. 11.
    Guenda, K., Jitman, S., Gulliver, T.A.: Constructions of good entanglement-assisted quantum error correcting codes. Des. Codes Cryptogr. 86, 121–136 (2018)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Guo, L., Li, R.: Linear Plotkin bound for entanglement-assisted quantum codes. Phys. Rev. A 87, 032309 (2013)ADSCrossRefGoogle Scholar
  13. 13.
    Lv, L., Li, R., Guo, L., Ma, Y., Liu, Y.: Entanglement-assisted quantum MDS codes from negacyclic codes. Quantum Inf. Process. 17, 69 (2018)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Qian, J., Zhang, L.: Entanglement-assisted quantum codes from arbitrary binary linear codes. Des. Codes Cryptogr. 77, 193–202 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lv, L., Ma, W., Li, R., Ma, Y., Liu, Y., Cao, H.: Entanglement-assisted quantum MDS codes from constacyclic codes with large minimum distance. Finite Fields Appl. 53, 309–325 (2018)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Chen, J., Huang, Y., Feng, C., Chen, R.: Entanglement-assisted quantum MDS codes constructed from negacyclic codes. Quantum Inf Process. 16, 303 (2017)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Liu, Y., Li, R., Lv, L., Ma, Y.: Application of constacyclic codes to entanglement-assisted quantum maximum distance separable codes. Quantum Inf. Process. 17, 210 (2018)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Qian, J., Zhang, L.: Constructions of new entanglement-assisted quantum MDS and almost MDS codes. Quantum Inf. Process. 18, 71 (2019)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Li, R., Guo, G., Song, H., Liu, Y.: New constructions of entanglement-assisted quantum MDS codes from negacyclic codes. Int. J. Quantum Inf. 17(3), 1950022 (2019)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Luo, G., Cao, X.: MDS codes with hulls of arbitrary dimensions and their quantum error correction. IEEE. Trans. Inf. Theory 65(5), 2944–2952 (2019)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Luo, G., Cao, X.: Two new families of entanglement-assisted quantum MDS codes from generalized Reed-Solomon codes. Quantum Inf. Process. 18, 89 (2019)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Sari, M., Kolotoğlu, M.: An application of constacyclic codes to entanglement-assisted quantum MDS codes. E. Comp. Appl. Math. 38, 75 (2019)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Fang, W., Fu, F., Li, L., Zhu, S.: Euclidean and hermitian hulls of MDS codes and their applications to EAQECCs. arXiv:1812.09019 (2019)
  24. 24.
    Guo, G., Li, R., Liu, Y., Wang, J.: Some construction of entanglement-assisted quantum MDS codes. (submitted) (2019)Google Scholar
  25. 25.
    Fang, W., Fu, F.: Some new constructions of quantum MDS codes. IEEE Transactions on Information Theory, (2019)MathSciNetCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2020

Authors and Affiliations

  1. 1.Department of Basic SciencesAir Force Engineering UniversityXi’anPeople’s Republic of China

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