International Journal of Theoretical Physics

, Volume 58, Issue 4, pp 1088–1107 | Cite as

Quantum Cyclic Codes Over \( {\mathbb{Z}}_m \)

  • Nianqi Tang
  • Zhuo LiEmail author
  • Lijuan Xing
  • Ming Zhang


Quantum codes over finite rings have the advantage of being able to adapt to quantum physical systems with arbitrary order. Furthermore, operations are much easier to execute in finite rings than they are in fields. This paper discusses quantum cyclic codes over the modulo m residue class ring \( {\mathbb{Z}}_m \). A connection is established between the stabilizer codes over \( {\mathbb{Z}}_m \) and the additive codes over an extension ring of \( {\mathbb{Z}}_m \) that generalizes the well-known relationship between the stabilizer codes over \( {\mathbb{F}}_q \) and the additive codes over \( {\mathbb{F}}_{q^2} \). We prove that if the irreducible polynomial is selected according to a simple criterion, the additive codes which are self-orthogonal with respect to the conjugate inner product correspond to the stabilizer codes. The structure of cyclic stabilizer codes is developed, and some simple conditions for finding them are presented. We also define the quantum Bose-Chaudhuri-Hocquenghem (BCH) and quantum Reed-Solomon (RS) codes over \( {\mathbb{Z}}_m \). Finally, new quantum cyclic codes over \( {\mathbb{Z}}_m \) are given.


Stabilizer codes Cyclic codes Bose-Chaudhuri-Hocquenghem (BCH) codes Reed-Solomon (RS) codes 



  1. 1.
    Shor, P.W.: Scheme for reducing decoherence in quantum memory. Phys. Rev. A 52, 2493–2496 (1995)ADSCrossRefGoogle Scholar
  2. 2.
    Calderbank, A.R., Shor, P.W.: Good quantum error correction codes exist. Phys. Rev. A 54, 1098–1105 (1996)ADSCrossRefGoogle Scholar
  3. 3.
    Gottesman, D.: Class of quantum error-correcting codes saturating the quantum Hamming bound. Phys. Rev. A 54, 1862–1868 (1996)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Calderbank, A.R., Rains, E.M., Shor, P.W., Sloane, N.J.A.: Quantum error correction via codes over G F(4). IEEE Trans. Inf. Theory 44, 1369–1387 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ketkar, A., Klappenecker, A., Kumar, S., Sarvepalli, P.K.: Nonbinary stabilizer codes over finite fields. IEEE Trans. Inf. Theory 52, 4892–4914 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Grassl, M., Beth, T.: Quantum BCH codes. Quantum Physics 10, 207–212 (1999)Google Scholar
  7. 7.
    Li, R., Li, X.: Quantum codes constructed from binary cyclic codes. Int. J. Quantum Inf. 2, 265–272 (2004)CrossRefzbMATHGoogle Scholar
  8. 8.
    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
  9. 9.
    Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: On quantum and classical BCH codes. IEEE. Trans. Inf. Theory 53, 1183–1188 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Guardia, G.G.: Constructions of new families of nonbinary quantum codes. Phys. Rev. A 80, 042331 (2009)ADSCrossRefGoogle Scholar
  11. 11.
    Kai, X., Zhu, S.: Quantum negacyclic codes. Phys. Rev. A 88, 012326 (2013)ADSCrossRefGoogle Scholar
  12. 12.
    Chen, B., Ling, S., Zhang, G.: Application of constacyclic codes to quantum MDS codes. IEEE Trans. Inf. Theory 61, 1474–1484 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Wang, L., Zhu, S.: New quantum MDS codes derived from constacyclic codes. Quantum Inf. Process. 14, 881–889 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Zhang, T., Ge, G.: Some new class of quantum MDS codes from constacyclic codes. IEEE Trans. Inf. Theory 61, 5224–5228 (2015)ADSCrossRefzbMATHGoogle Scholar
  15. 15.
    Guardia, G.G.: New quantum MDS codes. IEEE Trans. Inf. Theory 57, 5551–5554 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Qian, J., Ma, W., Gou, W.: Quantum codes from cyclic codes over finite ring. Int. J. Quantum Inf. 7, 1277–1283 (2009)CrossRefzbMATHGoogle Scholar
  17. 17.
    Ashraf, M., Mohammad, G.: Quantum codes from cyclic codes over \( {\mathbb{F}}_q+u{\mathbb{F}}_q+v{\mathbb{F}}_q+ uv{\mathbb{F}}_q \). Quantum Inf. Process. 15, 4089–4098 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gao, J., Wang, Y.: u-Constacyclic codes over \( {\mathbb{F}}_p+u{\mathbb{F}}_p \) and their applications of constructing new non-binary quantum codes. Quantum Inf. Process. (2018)Google Scholar
  19. 19.
    Liu, X., Liu, H.: Quantum codes from linear codes over finite chain rings. Quantum Inf. Process. (2017)
  20. 20.
    Qian, J., Zhang, L.: Improved constructions for nonbinary quantum BCH codes. Int. J. Theor. Phys. 56, 1355–1363 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Tang, Y., Zhu, S., Kai, X., Ding, J.: New quantum codes from dual-containing cyclic codes over finite rings. Quantum Inf. process. 15(11), 4489–4500 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Norton, G.H., Salagean, A.: On the hamming distance of linear codes over a finite chain ring. IEEE Trans. Inf. Theory 46, 1060–1067 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Artin, M.: Algebra, 2nd edn. Pearson Education, London (2011)zbMATHGoogle Scholar
  24. 24.
    McDonald, B.R.: Finite Rings with Identity. Marcel Dekker, New York (1974)zbMATHGoogle Scholar
  25. 25.
    Wan, Z.X.: Lectures on finite fields and galois rings. World Science Publishing, Singapore (2003)CrossRefzbMATHGoogle Scholar
  26. 26.
    Nadella, S., Klappenecker, A.: Stabilizer codes over Frobenius rings. In: Proceedings of IEEE Symposium and Information Theory (2012)Google Scholar
  27. 27.
    Norton, G.H., Salagean, A.: On the structure of linear and cyclic codes over a finite chain ring. Applicable Alg. Eng. Commun. Comput. 10 (2000)Google Scholar
  28. 28.
    Shankar, P.: On BCH codes over arbitrary integer rings. IEEE Trans. Inf. Theory 25 (1979)Google Scholar
  29. 29.
    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
  30. 30.
    Qian, J., Zhang, L.: Improved construction for nonbinary quantum BCH codes. Int. J. Theor. Phys. 56, 1355–1363 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

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

Authors and Affiliations

  1. 1.State Key Laboratory of Integrated Services NetworksXidian UniversityXi’anChina

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