Entanglement-Assisted Quantum Negacyclic BCH Codes

  • Xiaojing Chen
  • Shixin ZhuEmail author
  • Xiaoshan Kai


The entanglement-assisted quantum error correcting codes (EAQECCs) are a simple and important class of quantum codes. The entanglement-assisted formalism can transform arbitrary classical linear codes into EAQECCs by using pre-shared entanglement between the sender and the receiver. In this paper, by decomposing the defining set of negacyclic BCH codes, we construct a class of new EAQECCs with length \(n=\frac {q^{4m}-1}{q^{2}-1}\).


Negacyclic codes BCH codes EAQECCs 



  1. 1.
    Shor, P.W.: Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52(4), 2493–2496 (1995)ADSCrossRefGoogle Scholar
  2. 2.
    Steane, A.M.: Error correcting codes in quantum theory. Phys. Rev. Lett. 77, 793–797 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Steane, A.M.: Simple quantum error-correcting codes. Phys. Rev. A 54(6), 4741–4751 (1996)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Grassl, M., Beth, T., Rötteler, M.: On optimal quantum codes. Int. J. Quantum Inf. 2(1), 55–64 (2004)CrossRefzbMATHGoogle Scholar
  6. 6.
    Ketkar, A., Klappenecker, A., Kumar, S., Sarvepalli, P.K.: Nonbinary stabilizer codes over finite fields. IEEE Trans. Inf. Theory 52(11), 4892–4914 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    La Guardia, G.G.: Constructions of new families of nonbinary quantum codes. Phys. Rev. A 80(1-11), 042331 (2009)ADSCrossRefGoogle Scholar
  8. 8.
    Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: On quantum and classical BCH codes. IEEE Trans. Inf. Theory 53(3), 1183–1188 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Brun, T., Devetak, I., Hsieh, M.-H.: Correcting quantum errors with entanglement. Science 314(5798), 436–439 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Wilde, M.M., Brun, T.A.: Optimal entanglement formulas for entanglement-assisted quantum coding. Phys. Rev. A 77(1-4), 064302 (2008)ADSCrossRefGoogle Scholar
  11. 11.
    Lai, C.-Y., Brun, T.A.: Entanglement increases the error-correcting ability of quantum error-correcting codes. Phys. Rev. A 88(1-10), 012320 (2013)ADSCrossRefGoogle Scholar
  12. 12.
    Lü, L., Li, R., Guo, L., Fu, Q.: Maximal entanglement entanglement-assisted quantum codes constructed from linear codes. Quantum Inf. Process. 14, 165–182 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Qian, J., Zhang, L.: On MDS linear complementary dual codes and entanglement-assisted quantum codes. Des. Codes Cryptogr. 86, 1565–1572 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Guenda, K., Jitman, S., Gulliver, T.A.: Constructions of good entanglement-assisted quanutm error correcting codes. Des. Codes Cryptogr. 86, 121–136 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hsieh, M.-H., Brun, T.A., Devetak, I.: Entanglement-assisted quantum quasicyclic low-density parity-check codes. Phys. Rev. A 79(1-7), 032340 (2009)ADSCrossRefGoogle Scholar
  16. 16.
    Fujiwara, Y., Clark, D., Vandendriessche, P., Boeck, M.D., Tonchev, V.D.: Entanglement-assisted quantum low-density parity-check codes. Phys. Rev. A 82 (1-19), 042338 (2010)ADSCrossRefGoogle Scholar
  17. 17.
    Li, R., Zuo, F., Liu, Y.: A study of skew symmetric q 2-cyclotomic coset and its application, vol. 12. (in Chinese) (2011)Google Scholar
  18. 18.
    Li, R., Xu, G., Lü, L.: Decomposition of defining sets of BCH codes and its applications, vol. 14. (in Chinese) (2013)Google Scholar
  19. 19.
    Lü, L., Li, R.: Entanglement-assisted quantum codes constructed from primitive quaternary BCH codes. Int. J. Quantum Inf. 12(3), 1450015(1-14) (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Chen, J., Huang, Y., Feng, C., Chen, R.: Entanglement-assisted quantum MDS codes constructed from negacyclic codes. Quantum Inf. Process. 16(1-22), 303 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Lü, L., Li, R., Guo, L., Ma, Y., Liu, Y.: Entanglement-assisted quantum MDS codes from negacyclic codes. Quantum Inf. Process. 69(1-23), 17 (2018)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Kai, X., Zhu, S., Li, P.: Constacyclic codes and some new quantum MDS codes. IEEE Trans. Inf. Theory 60(4), 2080–2086 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Zhu, S., Sun, Z., Li, P.: A class of negacyclic BCH codes and its application to quantum codes. Des. Codes Cryptogr. 86(10), 2139–2165 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Krishna, A., Sarwate, D.V.: Pseudocyclic maximum-distance-separable codes. IEEE Trans. Inf. Theory 36(4), 880–884 (1990)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.School of MathematicsHefei University of TechnologyHefeiPeople’s Republic of China

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