Skip to main content
SpringerLink
Log in

New optimal asymmetric quantum codes and quantum convolutional codes derived from constacyclic codes

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

In this paper, some families of asymmetric quantum codes and quantum convolutional codes that satisfy the quantum Singleton bound are constructed by utilizing constacyclic codes with length \(n=\frac{q^2+1}{10h}\), where q is an odd prime power with the form \(q=10hm+t\) or \(q=10hm+10h-t\), where m is a positive integer, and both h and t are odd with \(10h=t^2+1\) and \(t\ge 3\). Compared with those codes constructed in the literature, the parameters of these constructed quantum codes in this paper are more general. Moreover, the distance \(d_z\) of optimal asymmetric quantum codes \([[n,k,d_z/d_x]]_{q^2}\) here is larger than most of the ones given in the literature.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aly, S.A.: Asymmetric quantum BCH codes. In: Proceedings International Conference on Computer Engineering System, pp. 157–162 (2008)

  2. Aly, S.A., Grassl, M., Klappenecker, A., Rötteler, M., Sarvepalli, P.K.: Quantum convolutional BCH codes. In: Proceedings of 10th Canadian Workshop on Information Theory, pp. 180–183 (2007)

  3. Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: Quantum convolutional codes derived from Reed–Solomon and Reed–Muller codes. arXiv:quant-ph/0701037

  4. Ashikhmin, A., Knill, E.: Non-binary quantum stabilizer codes. IEEE Trans. Inf. Theory 47(7), 3065–3072 (2001)

    Article  Google Scholar 

  5. 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)

    Article  MathSciNet  Google Scholar 

  6. Bakshi, G.K., Raka, M.: A class of constacyclic codes over a finite field. Finite Fields Appl. 18, 362–377 (2012)

    Article  MathSciNet  Google Scholar 

  7. Berlekamp, E.R.: Negacyclic codes for the Lee metric. In: Proceedings of Symposium in Combinatorial Mathematics and Its Applications, pp. 1–27 (1967)

  8. Blackford, T.: Negacyclic duadic codes. Finite Fields Appl. 14, 930–943 (2008)

    Article  MathSciNet  Google Scholar 

  9. 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)

    Article  MathSciNet  Google Scholar 

  10. Chen, B., Fan, Y., Lin, L., Liu, H.: Constacyclic codes over finite fields. Finite Fields Appl. 18, 1217–1231 (2012)

    Article  MathSciNet  Google Scholar 

  11. Chen, B., Ling, S., Zhang, G.: Application of constacyclic codes to quantum MDS codes. IEEE Trans. Inf. Theory 61(3), 1474–1484 (2015)

    Article  MathSciNet  Google Scholar 

  12. Chen, J., Li, J., Lin, J.: New optimal asymmetric quantum codes derived from negacyclic codes. Int. J. Theor. Phys. 53(1), 72–79 (2014)

    Article  MathSciNet  Google Scholar 

  13. Chen, J., Huang, Y., Feng, C., Chen, R.: Some families of optimal quantum codes derived from constacyclic codes. Linear Multilinear Algebra (2018). https://doi.org/10.1080/03081087.2018.1432544

  14. Chen, J., Li, J., Yang, F., Huang, Y.: Nonbinary quantum convolutional codes derived from negacyclic nodes. Int. J. Theor. Phys. 54(1), 198–209 (2015)

    Article  Google Scholar 

  15. Chen, J., Lin, J., Huang, Y.: Asymmetric quantum codes and quantum convolutional codes derived from nonprimitive non-narrow-sense BCH codes. IEICE Trans. Fund. Electr. 98(5), 1130–1135 (2015)

    Article  Google Scholar 

  16. Chen, J., Li, J., Yang, F., Lin, J.: Some families of asymmetric quantum codes and quantum convolutional codes from constacyclic codes. Linear Algebra Appl. 475, 186–199 (2015)

    Article  MathSciNet  Google Scholar 

  17. Chen, X., Zhu, S., Kai, X.: Two classes of new optimal asymmetric quantum codes. Int. J. Theor. Phys. 57(6), 1829–1838 (2018)

    Article  MathSciNet  Google Scholar 

  18. de Almeida, A.C.A., Palazzo, R. Jr.: A concatenated [(4,1,3)] quantum convolutional code. In: Proceedings of Information Theory Workshop, pp. 28–33 (2004)

  19. Huang, Y., Chen, J., Feng, C., Chen, R.: Some families of asymmetric quantum MDS codes constructed from constacyclic codes. Int. J. Theor. Phys. 57(2), 453–464 (2018)

    Article  MathSciNet  Google Scholar 

  20. Huffman, W.C., Pless, V.: Fundamentals of Error-Correcting Codes. University Press, Cambridge (2003)

    Book  Google Scholar 

  21. Kai, X., Zhu, S.: New quantum MDS codes from negacyclic codes. IEEE Trans. Inf. Theory 59(2), 1193–1197 (2013)

    Article  MathSciNet  Google Scholar 

  22. Kai, X., Zhu, S., Tang, Y.: Quantum negacyclic codes. Phys. Rev. A 88(1), 012326 (2013)

    Article  ADS  Google Scholar 

  23. Kai, X., Zhu, S., Li, P.: Constacyclic codes and some new quantum MDS codes. IEEE Trans. Inf. Theory 60(4), 2080–2086 (2014)

    Article  MathSciNet  Google Scholar 

  24. Ketkar, A., Klappenecker, A., Kumar, S., Sarvepalli, P.K.: Nonbinary stabilizer codes over finite fields. IEEE Trans. Inf. Theory 52(11), 4892–4914 (2006)

    Article  MathSciNet  Google Scholar 

  25. Krishna, A., Sarwate, D.V.: Pseudocyclic maximum-distance-separable codes. IEEE Trans. Inf. Theory 36(4), 880–884 (1990)

    Article  MathSciNet  Google Scholar 

  26. La Guardia, G.G.: Constructions of new families of nonbinary quantum codes. Phys. Rev. A 80(4), 042331 (2009)

    Article  ADS  Google Scholar 

  27. La Guardia, G.G.: New quantum MDS codes. IEEE Trans. Inf. Theory 57(8), 5551–5554 (2011)

    Article  MathSciNet  Google Scholar 

  28. La Guardia, G.G.: New families of asymmetric quantum BCH codes. Quantum Inf. Comput. 11(3), 239–252 (2011)

    MathSciNet  MATH  Google Scholar 

  29. La Guardia, G.G.: Asymmetric quantum Reed-Solomon and generalized Reed–Solomon codes. Quantum Inf. Process. 11(2), 591–604 (2012)

    Article  MathSciNet  Google Scholar 

  30. La Guardia, G.G.: Asymmetric quantum product codes. Int. J. Quantum Inf. 10(1), 1250005 (2012)

    Article  MathSciNet  Google Scholar 

  31. La Guardia, G.G.: On nonbinary quantum convolutional BCH codes. Quantum Inf. Comput. 12(9–10), 820–842 (2012)

    MathSciNet  MATH  Google Scholar 

  32. La Guardia, G.G.: Asymmetric quantum codes: new codes from old. Quantum Inf. Process. 12(8), 2771–2790 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  33. La Guardia, G.G.: On the construction of asymmetric quantum codes. Int. J. Theor. Phys. 53(7), 2312–2322 (2014)

    MathSciNet  MATH  Google Scholar 

  34. La Guardia, G.G.: On classical and quantum MDS-convolutional BCH codes. IEEE Trans. Inf. Theory 60(1), 304–312 (2014)

    Article  MathSciNet  Google Scholar 

  35. La Guardia, G.G.: On MDS-convolutional codes. Linear Algebra Appl. 448, 85–96 (2014)

    Article  MathSciNet  Google Scholar 

  36. La Guardia, G.G.: On optimal constacyclic codes. Linear Algebra Appl. 496, 594–610 (2016)

    Article  MathSciNet  Google Scholar 

  37. Leng, R.G., Ma, Z.: Constructions of new families of nonbinary asymmetric quantum BCH codes and subsystem BCH codes. Sci. China Phys. Mech. 55(3), 465–469 (2012)

    Article  Google Scholar 

  38. Li, F., Yue, Q.: New quantum MDS-convolutional codes derived from constacyclic codes. Mod. Phys. Lett. B 29, 1550252 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  39. Li, R., Xu, G., Guo, L.: On two problems of asymmetric quantum codes. Int. J. Mod. Phys. B 28(6), 1450017 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  40. Lü, L., Ma, W., Li, R., Ma, Y., Guo, L.: New quantum MDS codes constructed from constacyclic codes. arXiv:1803.07927

  41. MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-correcting Codes. North-Holland, Amsterdam (1977)

    MATH  Google Scholar 

  42. Qian, J., Zhang, L.: New optimal asymmetric quantum codes. Mod. Phys. Lett. B 27(2), 1350010 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  43. Qian, J., Zhang, L.: Improved constructions for quantum maximum distance separable codes. Quantum Inf. Process. 16(1), 20 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  44. Steane, A.M.: Enlargement of Calderbank–Shor–Steane quantum codes. IEEE Trans. Inf. Theory 45(7), 2492–2495 (1999)

    Article  MathSciNet  Google Scholar 

  45. Wang, L., Zhu, S.: New quantum MDS codes derived from constacyclic codes. Quantum Inf. Process. 14(3), 881–889 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  46. Wang, L., Zhu, S.: On the construction of optimal asymmetric quantum codes. Int. J. Quantum Inf. 12(3), 1450017 (2014)

    Article  MathSciNet  Google Scholar 

  47. Xu, G., Li, R., Guo, L., Lü, L.: New optimal asymmetric quantum codes constructed from constacyclic codes. Int. J. Mod. Phys. B 31(5), 1750030 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  48. Yan, T., Huang, X., Tang, Y.: Quantum convolutional codes derived from constacyclic codes. Mod. Phys. Lett. B 28(31), 1450241 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  49. Zhang, G., Chen, B., Li, L.: New optimal asymmetric quantum codes from constacyclic codes. Mod. Phys. Lett. B 28(15), 1450126 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  50. Zhang, G., Chen, B., Li, L.: A construction of MDS quantum convolutional codes. Int. J. Theor. Phys. 54(9), 3182–3194 (2015)

    Article  MathSciNet  Google Scholar 

  51. Zhu, S., Wang, L., Kai, X.: New optimal quantum convolutional codes. Int. J. Quantum Inf. 13(3), 1550019 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The research was supported by the Natural Science Foundation of China (No. 61802064) and the Natural Science Foundation of Fujian Province, China (Nos. 2016J01281, 2016J01278). We are indebted to anonymous reviewers who have made constructive suggestions for the improvement of this manuscript.

Author information

Authors and Affiliations

  1. College of Computer and Information Sciences, Fujian Agriculture and Forestry University, Fuzhou, 350002, China

    Jianzhang Chen & Chunhui Feng

  2. State Key Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing, Wuhan University, Wuhan, 430079, China

    Youqin Chen

  3. Department of Network Engineering, Chengdu University of Information Technology, Chengdu, 610225, China

    Yuanyuan Huang

Authors
  1. Jianzhang Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Youqin Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yuanyuan Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Chunhui Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuanyuan Huang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, J., Chen, Y., Huang, Y. et al. New optimal asymmetric quantum codes and quantum convolutional codes derived from constacyclic codes. Quantum Inf Process 18, 40 (2019). https://doi.org/10.1007/s11128-018-2156-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-018-2156-7

Keywords

Search

Navigation