A class of constacyclic BCH codes and new quantum codes

  • Yang liu
  • Ruihu Li
  • Liangdong Lv
  • Yuena Ma


Constacyclic BCH codes have been widely studied in the literature and have been used to construct quantum codes in latest years. However, for the class of quantum codes of length \(n=q^{2m}+1\) over \(F_{q^2}\) with q an odd prime power, there are only the ones of distance \(\delta \le 2q^2\) are obtained in the literature. In this paper, by a detailed analysis of properties of \(q^{2}\)-ary cyclotomic cosets, maximum designed distance \(\delta _\mathrm{{max}}\) of a class of Hermitian dual-containing constacyclic BCH codes with length \(n=q^{2m}+1\) are determined, this class of constacyclic codes has some characteristic analog to that of primitive BCH codes over \(F_{q^2}\). Then we can obtain a sequence of dual-containing constacyclic codes of designed distances \(2q^2<\delta \le \delta _\mathrm{{max}}\). Consequently, new quantum codes with distance \(d > 2q^2\) can be constructed from these dual-containing codes via Hermitian Construction. These newly obtained quantum codes have better code rate compared with those constructed from primitive BCH codes.


Negacyclic code Constacyclic code Quantum code Cyclotomic coset 



This work was supported by National Natural Science Foundation of China under Grant No.11471011 and Natural Science Foundation of Shaanxi under Grant No.2015JM1023.


  1. 1.
    Shor, P.W.: Scheme for reducing decoherence in quantum computing memory. Phys. Rev. A 52, R2493 (1995)ADSCrossRefGoogle Scholar
  2. 2.
    Steane, A.M.: Error correcting codes in quantum theory. Phys. Rev. Lett. 77, 793–797 (1996)ADSMathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    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)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Gottesman, D.: Stabilizer codes and quantum error correction. Ph.D. Thesis, California Institute of Technology (1997)Google Scholar
  5. 5.
    Steane, A.M.: Enlargement of Calderbank-Shor-Steane quantum codes. IEEE. Trans. Inf. Theory 45, 2492–2495 (1999)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Li, R., Li, X.: Binary construction of quantum codes of minimum distance three and four. IEEE. Trans. Inf. Theory 50, 1331–1336 (2004)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Rains, E.M.: Non-binary quantum codes. IEEE. Trans. Inf. Theory 45, 1827–1832 (1999)CrossRefMATHGoogle Scholar
  8. 8.
    Ashikhim, A., Knill, E.: Non-binary quantum stabilizer codes. IEEE. Trans. Inf. Theory 47, 3065–3072 (2001)CrossRefMATHGoogle Scholar
  9. 9.
    Ketkar, A., Klappenecker, A., Kumar, S.: Nonbinary stabilizer codes over finite fields. IEEE. Trans. Inf. Theory 52, 4892–4914 (2006)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ling, S., Luo, J., Xing, C.: Generalization of Steane’s enlargement construction of quantum codes and applications. IEEE Trans. Inf. Theory 56, 4080–4084 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hamada, M.: Concatenated quantum codes constructible in polynomial time: efficient decoding and error correction. IEEE. Trans. Inf. Theory 54, 5689–5704 (2008)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Grassl, M., Beth, T.: Quantum BCH codes. Proc. X. int’l. Symp. Theoretical. Electrical Engineering Magdeburg, 207-212 (1999)Google Scholar
  13. 13.
    Li, R., Li, X.: Quantum codes constructed from binary cyclic codes. Int. J. Quantum Inf. 2, 265–272 (2004)CrossRefMATHGoogle Scholar
  14. 14.
    Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: Primitive quantum BCH codes over finite fields. Proc. Int. Symp. Inf. Theory, 1114-1118 (2006)Google Scholar
  15. 15.
    Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: On quantum and classical BCH codes. IEEE. Trans. Inf. Theory 53, 1183–1188 (2007)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Guardia, G.G.La: Constructions of new families of nonbinary quantum codes. Phys. Rev. A 80, 042331 (2009)ADSCrossRefGoogle Scholar
  17. 17.
    Li, R., Zuo, F., Liu, Y.: A study of skew symmetric \(q^2\)-cyclotomic coset and its application. J. Air Force Eng. Univ. 12(1), 87–89 (2011)Google Scholar
  18. 18.
    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
  19. 19.
    Kai, X., Zhu, S.: Quantum negacyclic codes. Phys. Rev. A 88, 012326 (2013)ADSCrossRefGoogle Scholar
  20. 20.
    Kai, X., Zhu, S., Li, P.: Constacyclic codes and some new quantum MDS codes. IEEE Trans. Inf. Theory 60, 2080–2086 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Hu, X., Zhang, G., Chen, B.: Constructions of new nonbinary quantum codes. Int. J. Theory Phys. 54, 92–99 (2015)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Chen, B., Ling, S., Zhang, G.: Application of constacyclic codes to quantum MDS codes. IEEE Trans. Inf. Theory 61, 1474–1484 (2015)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Guardia, G.G.La: On optimal constacyclic codes. Linear Algebra Appl. 496, 594–610 (2016)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Wang, L., Zhu, S.: New quantum MDS codes derived from constacyclic codes. Quantum Inf. Processing 14 3, 881-889(2015). See also arXiv:1405.5421v1
  25. 25.
    Zhang, T., Ge, G.: Some new class of quantum MDS codes from constacyclic codes. IEEE Trans. Inf. Theory 61, 5224–5228 (2015)CrossRefGoogle Scholar
  26. 26.
    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)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Krishna, A., Sarwate, D.V.: Pseudo-cyclic maximum-distance separable codes. IEEE Trans. Inf. Theory 36, 880–884 (1990)CrossRefMATHGoogle Scholar
  28. 28.
    Peterson, W.W., Weldon, E.J.: Error-correcting codes. The M.I.T. Press, Cambridge (1972)MATHGoogle Scholar
  29. 29.
    Macwilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland Publishing Company, Amsterdam (1977)MATHGoogle Scholar
  30. 30.
    Guardia, G.G.La: New quantum MDS codes. IEEE Trans. Inf. Theory 57, 5551–5554 (2011)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Huffman, W.C., Pless, V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003)CrossRefMATHGoogle Scholar
  32. 32.
    Sloane, N.J.A., Thompson, J.G.: Cyclic self-dual codes. IEEE Trans. Inf. Theory 29, 364–366 (1983)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.School of ScienceAir Force Engineering UniversityXi’anPeople’s Republic of China
  2. 2.The First Aeronautical College of Air ForceXinyangPeople’s Republic of China

Personalised recommendations