Quantum Information Processing

, 18:292 | Cite as

Entanglement-assisted quantum error correction codes with length \(n=q^2+1\)

  • Junli Wang
  • Ruihu LiEmail author
  • Jingjie Lv
  • Guanmin Guo
  • Yang Liu


In this paper, by investigating \(q^2\)-cyclotomic coset modulo rn in detail, where q is a prime power, \(n=q^2+1\) and \(r\mid (q+1)\), series of entanglement-assisted quantum error correction (EAQEC) codes with flexible parameters of length n are constructed from constacyclic codes (including cyclic codes). Most of our EAQEC codes are new and have large minimum distance. As to EAQEC codes constructed from cyclic codes, their all possible parameters are determined completely. When minimum distance \(d\le \frac{n+2}{2}\), all of our constructed EAQEC codes are entanglement-assisted quantum MDS (EAQMDS) codes. Those previously known EAQMDS codes with the same length in Fan et al. (Quantum Inf Comput 16:423–434, 2016), Chen et al. (Quantum Inf Process 16(303):1–22, 2017), Lu et al. (Finite Fields Their Appl 53:309–325, 2018), Mustafa and Emre (Comput Appl Math 38(75):1–13, 2019) and Qian and Zhang (Quantum Inf Process 18(71):1–12, 2019) are special cases of ours. Besides, some maximum entanglement EAQEC codes and maximum entanglement EAQMDS codes are derived as well.


Constacyclic code Cyclotomic coset EAQEC code EAQMDS code 



  1. 1.
    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
  2. 2.
    Grassl, M., Beth, T.: Quantum BCH codes. In: Proceedings X. International Symposium on Theoretical Electrical Engineering Magdeburg, pp. 207–212 (1999)Google Scholar
  3. 3.
    Ashikhim, A., Knill, E.: Non-binary quantum stabilizer codes. IEEE. Trans. Inf. Theory 47, 3065–3072 (2001)CrossRefGoogle Scholar
  4. 4.
    Li, R., Li, X.: Binary construction of quantum codes of minimum distance three and four. IEEE. Trans. Inf. Theory 50, 1331–1336 (2004)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ketkar, A., Klappenecker, A., Kumar, S.: Nonbinary stablizer codes over finite fields. IEEE. Trans. Inf. Theory 52, 4892–4914 (2006)CrossRefGoogle Scholar
  6. 6.
    Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: On quantum and classical BCH codes. IEEE. Trans. Inf. Theory 53, 1183–1188 (2007)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Li, R., Zuo, F., Liu, Y., Xu, Z.: Hermitian dual-containing BCH codes and construction of new quantum codes. Quantum Inf. Comput. 12, 0021–0035 (2013)MathSciNetGoogle Scholar
  8. 8.
    Liu, Y., Li, R., Lv, L., Ma, Y.: A class of constacyclic BCH codes and new quantum codes. Quantum Inf. Process. 16(66), 1–16 (2017)ADSMathSciNetzbMATHGoogle Scholar
  9. 9.
    Song, H., Li, R., Wang, J., Liu, Y.: Two classes of BCH codes and new quantum codes. Quantum Inf. Process. 17(270), 1–24 (2018)ADSzbMATHGoogle Scholar
  10. 10.
    Li, R., Wang, J., Liu, Y., Guo, G.: New quantum constacyclic codes. Quantum Inf. Process. 18(127), 1–23 (2019)ADSMathSciNetGoogle Scholar
  11. 11.
    Jin, L., Ling, S., Luo, J., Xing, C.: Application of classical Hermitian self-orthogonal MDS codes to quantum MDS codes. IEEE. Trans. Inf. Theory 56, 4735–4740 (2010)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Guardia, G.G.L.: New quantum MDS codes. IEEE. Trans. Inf. Theory 57, 5551–5554 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kai, X., Zhu, S.: New quantum MDS codes from negacyclic codes. IEEE. Trans. Inf. Theory 59, 1193–1197 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Jin, L., Xing, C.: A construction of new quantum MDS codes. IEEE. Trans. Inf. Theory 60, 2921–2925 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Grassl, M., Rtteler, M.: Quantum MDS codes over small fields. In: IEEE IEEE International Symposium on Information Theory, pp. 1104–1108(2015)Google Scholar
  16. 16.
    Brun, T., Devetak, I., Hsieh, M.: Correcting quantum errors with entanglement. Science 314, 436–439 (2006)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Grassl, M.: Entanglement-assisted quantum communication beating the quantum singleton bound. In: AQIS, Taiwan (2016)Google Scholar
  18. 18.
    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, 423–434 (2016)MathSciNetGoogle Scholar
  19. 19.
    Chen, J., Huang, Y., Feng, C., Chen, R.: Entanglement-assisted quantum MDS codes constructed from negacyclic codes. Quantum Inf. Process. 16(303), 1–22 (2017)ADSMathSciNetzbMATHGoogle Scholar
  20. 20.
    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
  21. 21.
    Mustafa, S., Emre, K.: An application of constacyclic codes to entanglement-assisted quantum MDS codes. Comput. Appl. Math. 38(75), 1–13 (2019)MathSciNetGoogle Scholar
  22. 22.
    Qian, J., Zhang, L.: On MDS linear complementary dual codes and entanglement-assisted quantum codes. Des. Codes Cryptogr. 86, 1565–1572 (2018)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Qian, J., Zhang, L.: Constructions of new entanglement-assisted quantum MDS codes and almost MDS codes. Quantum Inf. Process. 18(71), 1–12 (2019)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Li, R., Guo, G., Song, H., Liu, Y.: New constructions of entanglement-assisted quantum MDS codes from negacyclic codes. Int. J. Quantum Inf. 17(1), 1950022 (2019)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Liu, Y., Li, R., Lv, L., Ma, Y.: Application of constacyclic codes to entanglement-assisted quantum maximum diatance separable codes. Quantum Inf. Process. 17(210), 1–19 (2018) ADSCrossRefGoogle Scholar
  26. 26.
    Fang, W., Fu, F., Li, L., Zhu, S.: Euclidean and Hermitian hulls of MDS codes and their applications to EAQECCs (2018)Google Scholar
  27. 27.
    Wilde, M., Burn, T.: Optimal entanglement formulas for entanglement-assisted quantum coding. Phys. Rev. A 77, 064302 (2008)ADSCrossRefGoogle Scholar
  28. 28.
    Macwilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland Publishing Company, Amsterdam (1977)zbMATHGoogle Scholar
  29. 29.
    Huffman, W.C., Pless, V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003)CrossRefGoogle Scholar
  30. 30.
    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
  31. 31.
    Krishna, A., Sarwate, D.V.: Pseudo-cyclic maximum-distance separable codes. IEEE. Trans. Inf. Theory 36, 880–884 (1990)CrossRefGoogle Scholar
  32. 32.
    Lü, L., Li, R.: Entanglement-assisted quantum codes constructed from primitive quaternary BCH codes. Int. J. Quantum Inf. 12(3), 1450015 (2014)MathSciNetCrossRefGoogle Scholar

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

Authors and Affiliations

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

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