New entanglement-assisted MDS quantum codes from constacyclic codes

  • Mehmet E. KorogluEmail author


Construction of good quantum codes via classical codes is an important task for quantum information and quantum computing. In this work, by virtue of a decomposition of the defining set of constacyclic codes we have constructed eight new classes of entanglement-assisted quantum maximum-distance-separable codes.


Entanglement-assisted quantum error-correcting codes Constacyclic codes MDS codes 

Mathematics Subject Classification

94B05 94B15 81P70 81P45 



The author is very grateful to the reviewers for their comments and suggestions that improved the presentation and quality of this paper.


  1. 1.
    Ashikhmin, A., Litsyn, S., Tsfasman, M.A.: Asymptotically good quantum codes. Phys. Rev. A 63, 032311 (2001)ADSCrossRefGoogle Scholar
  2. 2.
    Aydin, N., Siap, I., Ray-Chaudhuri, D.K.: The structure of 1-generator quasi-twisted codes and new linear codes. Des. Codes Cryptogr. 24, 313–326 (2001)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brun, T., Devetak, I., Hsieh, M.H.: Correcting quantum errors with entanglement. Science 52, 436 (2006)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Calderbank, A.R., Shor, P.W.: Good quantum error-correcting codes exist. Phys. Rev. A 54, 1098 (1996)ADSCrossRefGoogle 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, 1369–1387 (1998)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen, B., Ling, S., Zhang, G.: Application of constacyclic codes to quantum MDS codes. IEEE Trans. Inf. Theory 61, 1474–1484 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chen, H.: Some good quantum error-correcting codes from algebraic-geometric codes. IEEE Trans. Inf. Theory 47, 2059–2061 (2001)MathSciNetCrossRefGoogle Scholar
  8. 8.
    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
  9. 9.
    Fan, J., Chen, H., Xu, J.: Constructions of \(q\)-ary entanglement-assisted quantum MDS codes with minimum distance greater than \(q+1\). Quantum Inf. Comput. 16, 0423–0434 (2016)MathSciNetGoogle Scholar
  10. 10.
    Fujiwara, Y., Clark, D., Vandendriessche, P., Boeck, M.D., Tonchev, V.D.: Entanglement-assisted quantum low-density parity-check codes. Phys. Rev. A 82, 042338 (2010)ADSCrossRefGoogle Scholar
  11. 11.
    Grassl, M.: Entanglement-assisted quantum communication beating the quantum singleton bound. In: AQIS, Taiwan (2016)Google 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.
    Hsieh, M.H., Devetak, I., Brun, T.A.: General entanglement-assisted quantum error-correcting codes. Phys. Rev. A 76, 062313 (2007)ADSCrossRefGoogle Scholar
  14. 14.
    Hsieh, M.H., Yen, W.T., Hsu, L.Y.: High performance entanglement-assisted quantum LDPC codes need little entanglement. IEEE Trans. Inf. Theory 57, 1761–1769 (2011)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kai, X., Zhu, S.: New quantum MDS codes from negacyclic codes. IEEE Trans. Inf. Theory 59, 1193–1197 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kai, X., Zhu, S., Li, P.: Constacyclic codes and some new quantum MDS codes. IEEE Trans. Inf. Theory 60, 2080–2086 (2014)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ketkar, A., Klappenecker, A., Kumar, S., Sarvepalli, P.K.: Nonbinary stabilizer codes over finite fields. IEEE Trans. Inf. Theory 52, 4892–4914 (2006)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Krishna, A., Sarwate, D.V.: Pseudocyclic maximum-distance-separable codes. IEEE Trans. Inf. Theory 36, 880–884 (1990)MathSciNetCrossRefGoogle Scholar
  19. 19.
    La Guardia, G.G.: On optimal constacyclic codes. Linear Algebra Appl. 496, 594–610 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    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
  21. 21.
    Li, R., Li, X., Guo, L.: On entanglement-assisted quantum codes achieving the entanglement-assisted Griesmer bound. Quantum Inf. Process. 14, 4427–4447 (2015)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Li, S., Xiong, M., Ge, G.: Pseudo-cyclic codes and the construction of quantum MDS codes. IEEE Trans. Inf. Theory 62, 1703–1710 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Lu, L., Li, R.: Entanglement-assisted quantum codes constructed from primitive quaternary BCH codes. Int. J. Quantum Inf. 12, 1450015 (2014)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Lu, 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
  25. 25.
    Lu, 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 Their Appl. 53, 309–325 (2018)MathSciNetCrossRefGoogle Scholar
  26. 26.
    MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. Elsevier, Amsterdam (1977)zbMATHGoogle Scholar
  27. 27.
    Qian, J., Zhang, L.: Nonbinary quantum codes derived from group character codes. Int. J. Quantum Inf. 10, 1250042 (2012)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Qian, J., Zhang, L.: New optimal subsystem codes. Discrete Math. 313, 2451–2455 (2013)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Qian, J., Zhang, L.: On MDS linear complementary dual codes and entanglement-assisted quantum codes. Des. Codes Cryptogr. 86, 1565–1572 (2017)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Shor, P.W.: Scheme for reducing decoherence in quantum memory. Phys. Rev. A 52, 2493–2496 (1995)ADSCrossRefGoogle Scholar
  31. 31.
    Steane, A.M.: Simple quantum error-correcting codes. Phys. Rev. A 54, 4741 (1996)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Wilde, M.M., Brun, T.A.: Optimal entanglement formulas for entanglement-assisted quantum coding. Phys. Rev. A 77, 064302 (2008)ADSCrossRefGoogle Scholar
  33. 33.
    Xiaoyan, L.: Quantum cyclic and constacyclic codes. IEEE Trans. Inf. Theory 50, 547–549 (2004)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Zhang, T., Ge, G.: Some new classes of quantum MDS codes from constacyclic codes. IEEE Trans. Inf. Theory 61, 5224–5228 (2015)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Art and SciencesYıldız Technical UniversityEsenler, IstanbulTurkey

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