Skip to main content
SpringerLink
Log in

New quantum MDS codes derived from constacyclic codes

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

Quantum maximum-distance-separable (MDS) codes form an important class of quantum codes. It is very hard to construct quantum MDS codes with relatively large minimum distance. In this paper, based on classical constacyclic codes, we construct two classes of quantum MDS codes with parameters

$$\begin{aligned}{}[[\lambda (q-1),\lambda (q-1)-2d+2,d]]_q \end{aligned}$$

where \(2\le d\le (q+1)/2+\lambda -1\), and \(q+1=\lambda r\) with \(r\) even, and

$$\begin{aligned}{}[[\lambda (q-1),\lambda (q-1)-2d+2,d]]_q \end{aligned}$$

where \(2\le d\le (q+1)/2+\lambda /2-1\), and \(q+1=\lambda r\) with \(r\) odd. The quantum MDS codes exhibited here have parameters better than the ones available 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. Ashikhmin, A., Kill, E.: Nonbinary quantum stabilizer codes. IEEE Trans. Inf. Theory 47(7), 3065–3072 (2001)

    Article  MATH  Google Scholar 

  2. Aydin, N., Siap, I., Ray-Chaudhuri, D.J.: The structure of 1-generator quasi-twisted codes and new linear codes. Des. Codes Cryptogr. 24, 313–326 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bierbrauer, J., Edel, Y.: Quantum twisted codes. J. Comb. Des. 8(3), 174–188 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  5. Fan, J., Chen, H.: Construction of pure asymmetric quantum alternant codes based on subclasses of alternant codes (2014). arXiv:1401.3215v2

  6. Feng, K.: Quantum codes \([[6,2,3]]_p\) and \([[7,3,3]]_p\) \(p\ge 3\) exist. IEEE Trans. Inf. Theory 48(8), 2384–2391 (2002)

    Article  MATH  Google Scholar 

  7. Grassl, M., Beth, T., Rötteler, M.: On optimal quantum codes. Int. J. Quantum Inf. 2(1), 757–775 (2004)

    Article  Google Scholar 

  8. Grassl, M., Rötteler, M., Beth, T.: On quantum MDS codes. In: Proceedings of the International Symposium on Information, Chicago, USA, p. 356 (2004)

  9. Hu, D., Tang, W., Zhao, M., Chen, Q., Yu, S., Oh, C.H.: Graphical nonbinary quantum error-correcting codes. Phys. Rev. A. 78(1), 012306(1–11) (2008)

    Article  ADS  Google Scholar 

  10. 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(9), 4735–4740 (2010)

    Article  MathSciNet  Google Scholar 

  11. Jin, L., Xing, C.: A construction of new quantum MDS codes. IEEE Trans. Inf. Theory 60(5), 2921–2925 (2014)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. Kai, X., Zhu, S., Li, P.: Constacyclic codes and some new quantum MDS codes. IEEE Trans. Inf. Theory 60(4), 2080–2085 (2014)

    Article  MathSciNet  Google Scholar 

  14. 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  MATH  MathSciNet  Google Scholar 

  15. Knill, E., Laflamme, R.: Theory of quantum error-correcting codes. Phys. Rev. A 55(2), 900–911 (1997)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  18. Laflamme, R., Miquel, C., Paz, J.P., Zurek, W.H.: Perfect quantum error correcting code. Phys. Rev. Lett. 77(1), 198–201 (1996)

    Article  ADS  Google Scholar 

  19. Li, Z., Xing, L.J., Wang, X.M.: Quantum generalized Reed–Solomon codes: unified framework for quantum MDS codes. Phys. Rev. A 77(1), 012306(1–4) (2008)

  20. Li, R., Xu, Z.: Construction of \([[n, n-4,3]]_q\) quantum codes for odd prime power \(q\). Phys. Rev. A. 82(5), 052316(1–4) (2010)

    ADS  Google Scholar 

Download references

Acknowledgments

We would like to thank the referees for their invaluable comments and a very meticulous reading of the manuscript. This research is supported by the National Natural Science Foundation of China under Grant No. 61370089, and the Fundamental Research Funds for the Central Universities under Grant No. 2013HGCH0024.

Author information

Authors and Affiliations

  1. School of Mathematics, Hefei University of Technology, Hefei, 230009, Anhui, People’s Republic of China

    Liqi Wang & Shixin Zhu

Authors
  1. Liqi Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shixin Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Liqi Wang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, L., Zhu, S. New quantum MDS codes derived from constacyclic codes. Quantum Inf Process 14, 881–889 (2015). https://doi.org/10.1007/s11128-014-0903-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11128-014-0903-y

Keywords

Search

Navigation