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Two new families of entanglement-assisted quantum MDS codes from generalized Reed–Solomon codes

  • Gaojun LuoEmail author
  • Xiwang Cao
Article
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Abstract

Entanglement-assisted quantum error-correcting codes (EAQECCs) make use of preexisting entanglement between the sender and receiver to boost the rate of transmission. It is possible to construct an EAQECC by any classical linear code. In this paper, we propose two constructions of generalized Reed–Solomon codes and calculate the dimension of their hulls. With these generalized Reed–Solomon codes, we present two new infinite families of EAQECCs, which are optimal with respect to the Singleton bound for EAQECCs. Notably, the parameters of our EAQECCs are new and flexible.

Keywords

Hull Generalized Reed–Solomon code Entanglement-assisted quantum error-correcting code (EAQECC) 

Notes

Acknowledgements

We are grateful to the anonymous referees and the associate editor for useful comments and suggestions that improved the presentation and quality of this paper.

References

  1. 1.
    Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: On quantum and classical BCH codes. IEEE Trans. Inf. Theory 53(3), 1183–1188 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Assmus, E.F., Key, J.: Designs and Their Codes. Cambridge Tracts in Mathematics, vol. 103. Cambridge University Press, Cambridge (1992). (Second printing with corrections, 1993)CrossRefGoogle Scholar
  3. 3.
    Brun, T., Devetak, I., Hsieh, M.H.: Correcting quantum errors with entanglement. Science 314, 436–439 (2006)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Brun, T., Devetak, I., Hsieh, M.H.: Catalytic quantum error correction. IEEE Trans. Inf. Theory 60(6), 3073–3089 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Calderbank, A.R., Rains, E.M., Shor, P.W., Sloane, N.J.A.: Quantum error correction via codes over GF(4). IEEE Trans. Inf. Theory 44(4), 1369–1387 (1998)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen, B., Liu, H.: New constructions of MDS codes with complementary duals. IEEE Trans. Inf. Theory 64(8), 5776–5782 (2018)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chen, J., Huang, Y., Feng, C., Chen, R.: Entanglement-assisted quantum MDS codes constructed from negacyclic codes. Quantum Inf. Process. 16(12), 303 (2017)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Chen, X., Zhu, S., Kai, X.: Two classes of new optimal asymmetric quantum codes. Int. J. Theor. Phys. 57(6), 1829–1838 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chen, X., Zhu, S., Kai, X.: Entanglement-assisted quantum MDS codes constructed from constacyclic codes. Quantum Inf. Process. 17, 273 (2018)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Grassl, M.: Entanglement-Assisted Quantum Communication Beating the Quantum Singleton Bound. AQIS, Taiwan (2016)Google Scholar
  11. 11.
    Grassl, M., Beth, T., Rötteler, M.: On optimal quantum codes. Int. J. Quantum Inf. 2(1), 55–64 (2004)CrossRefGoogle Scholar
  12. 12.
    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
  13. 13.
    Jin, L.: Construction of MDS codes with complementary duals. IEEE Trans. Inf. Theory 63(5), 2843–2847 (2017)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Ketkar, A., Klappenecker, A., Kumar, S., Sarvepalli, P.K.: Nonbinary stabilizer codes over finite fields. IEEE Trans. Inf. Theory 52(11), 4892–4914 (2006)MathSciNetCrossRefGoogle Scholar
  15. 15.
    La Guardia, G.: New families of asymmetric quantum BCH codes. Quantum Inf. Comput. 11, 239–252 (2011)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Lai, C.Y., 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
  17. 17.
    Lai, C., Brun, T.: Entanglement increases the error-correcting ability of quantum error-correcting codes. Phys. Rev. A 88, 012320 (2013)ADSCrossRefGoogle Scholar
  18. 18.
    Lu, L., Li, R., Ma, W., Li, R., Ma, Y., Liu, Y., Cao, H.: Entanglement-assisted quantum MDS codes from constacyclic codes with large minimum distance. arXiv:1803.04168
  19. 19.
    Lu, L., Li, R., Guo, L., Fu, Q.: Maximal entanglement-assisted quantum codes constructed from linear codes. Quantum Inf. Process. 14(1), 165–182 (2015)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    MacWilliams, F.J., Sloane, N.J.A.: The theory of Error-Correcting Codes. North-Holland, Amsterdam (1977)zbMATHGoogle Scholar
  21. 21.
    Qian, J., Zhang, L.: On MDS linear complementary dual codes and entanglement-assisted quantum codes. Des. Codes Cryptogr. 86(7), 1565–1572 (2018)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Shor, P.W.: Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52(4), 2493–2496 (1995)ADSCrossRefGoogle Scholar
  23. 23.
    Steane, A.M.: Error correcting codes in quantum theory. Phys. Rev. Lett 77, 793–797 (1996)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Wang, L., Feng, K.Q., Ling, S., Xing, C.P.: Asymmetric quantum codes: characterization and constructions. IEEE Trans. Inf. Theory 56(6), 2938–2945 (2010)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Wilde, M., Brun, T.: Optimal entanglement formulas for entanglement-assisted quantum coding. Phys. Rev. A 77, 064302 (2008)ADSCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsNanjing University of Aeronautics and AstronauticsNanjingChina
  2. 2.State Key Laboratory of Information Security, Institute of Information EngineeringChinese Academy of SciencesBeijingChina

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