Skip to main content
SpringerLink
Log in

Three classes of optimal Hermitian dual-containing codes and quantum codes

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

Hermitian dual-containing codes play an important role in constructing quantum codes, constructing Hermitian dual-containing codes with optimal parameters is an interesting topic. The objective of this paper is to construct several classes of quantum codes with optimal parameters. Using cyclic codes, three classes of optimal Hermitian dual-containing codes are constructed. By employing the Hermitian construction, three classes of quantum codes with optimal parameters are obtained. These codes are new in the sense that their parameters are not covered by the codes 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

Data Availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study

References

  1. Abdel-Aty, A.H., Zakaria, N., Cheong, L.Y., et al.: Entanglement and teleportation via partial entangled-state quantum network. J. Comput. Theor. Nanosci. 12(9), 2213–2220 (2015). https://doi.org/10.1166/jctn.2015.4010

    Article  Google Scholar 

  2. Ahmed, A.H., Cheong, L.Y., Zakaria, N., et al.: Dynamics of information coded in a single cooper pair box. Int. J. Theor. Phys. 52(6), 1979–1988 (2013). https://doi.org/10.1007/s10773-012-1399-9

    Article  MathSciNet  Google Scholar 

  3. Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: On quantum and classical BCH codes. IEEE Trans. Inf. Theory 53(3), 1183–1188 (2007). https://doi.org/10.1109/TIT.2006.890730

    Article  MathSciNet  MATH  Google Scholar 

  4. Cafaro, C., van Loock, P.: Approximate quantum error correction for generalized amplitude-damping errors. Phys. Rev. A 89(2), :022316 (2014). https://doi.org/10.1103/PhysRevA.89.022316

    Article  ADS  Google Scholar 

  5. 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). https://doi.org/10.1109/18.681315

    Article  MathSciNet  MATH  Google Scholar 

  6. Calderbank, A.R., Shor, P.W.: Good quantum error-correcting codes exist. Phys. Rev. A 54(2), 1098–1105 (1996). https://doi.org/10.1103/PhysRevA.54.1098

    Article  ADS  Google Scholar 

  7. Chen, B., Ling, S., Zhang, G.: Application of constacyclic codes to quantum MDS codes. IEEE Trans. Inf. Theory 61(3), 1474–1484 (2015). https://doi.org/10.1109/TIT.2015.2388576

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, G., Li, R.: Ternary self-orthogonal codes of dual distance three and ternary quantum codes of distance three. Des. Codes Cryptogr. 69, 53–63 (2013). https://doi.org/10.1007/s10623-012-9620-7

    Article  MathSciNet  MATH  Google Scholar 

  9. Ding, C., Helleseth, T.: Optimal ternary cyclic codes from monomials. IEEE Trans. Inf. Theory 59(9), 5898–5904 (2013). https://doi.org/10.1109/TIT.2013.2260795

    Article  MathSciNet  MATH  Google Scholar 

  10. Djordjevic, I.: Quantum Information Processing, Quantum Computing, and Quantum Error Correction: An Engineering Approach. Academic Press, London (2021)

    MATH  Google Scholar 

  11. Fang, W., Wen, J., Fu, F.: A \(q\)-polynomial approach to constacyclic codes. Finite Fields Their Appl. 47, 161–182 (2017). https://doi.org/10.1016/j.ffa.2017.06.009

    Article  MathSciNet  MATH  Google Scholar 

  12. Grassl, M., Beth, T.: Cyclic quantum error-correcting codes and quantum shift registers. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 456(2003), 2689–2706 (2000). https://doi.org/10.1098/rspa.2000.0633

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Grassl, M., Rötteler, M., Beth, T.: Efficient quantum circuits for non-qubit quantum error-correcting codes. Int. J. Found. Comput. Sci. 14(5), 757–775 (2003). https://doi.org/10.1142/S0129054103002011

    Article  MathSciNet  MATH  Google Scholar 

  14. Grassl, M., Kong, L., Wei, Z., et al.: Quantum error-correcting codes for qudit amplitude damping. IEEE Trans. Inf. Theory 64(6), 4674–4685 (2018). https://doi.org/10.1109/TIT.2018.2790423

    Article  MathSciNet  MATH  Google Scholar 

  15. Kodaira, K., Shibuya, T.: On the condition on classical codes to construct quantum error correcting codes for amplitude damping channel. ISITA 2014, Melbourne, Australia, October, 26–29 (2014)

  16. Ketkar, A., Klappenecker, A., Kumar, S., Sarvepalli, P.K.: Nonbinary stabilizer codes over finite fields. IEEE Trans. Inf. Theory 52(11), 4892–4914 (2006). https://doi.org/10.1109/TIT.2006.883612

    Article  MathSciNet  MATH  Google Scholar 

  17. Liang, F.: Self-orthogonal codes with dual distance three and quantum codes with distance three over \(\mathbb{F} _5\). Quantum Inf. Process. 12, 3617–3623 (2013). https://doi.org/10.1007/s11128-013-0620-y

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Liao, D., Kai, X., Zhu, S., Li, P.: A class of optimal cyclic codes with two zero. IEEE Commun. Lett. 23(8), 1293–1296 (2019). https://doi.org/10.1109/LCOMM.2019.2921330

    Article  Google Scholar 

  19. Li, C., Ding, C., Li, S.: LCD cyclic codes over finite fields. IEEE Trans. Inf. Theory 63(7), 4344–4356 (2017). https://doi.org/10.1109/TIT.2017.2672961

    Article  MathSciNet  MATH  Google Scholar 

  20. Li, H., Li, R., Yao, Y., Fu, Q.: Some quantum error-correcting codes with \(d=5\). J. Phys. Conf. Ser. 1684, 012078 (2020). https://doi.org/10.1088/1742-6596/1684/1/012078

    Article  Google Scholar 

  21. Ling, S., Luo, J., Xing, C.: Generalization of Steane’s enlargement construction of quantum codes and applications. IEEE Trans. Inf. Theory 56(8), 4080–4084 (2010). https://doi.org/10.1109/TIT.2010.2050828

    Article  MathSciNet  MATH  Google Scholar 

  22. Li, R., Chen, G.: Quantum codes of minimum distance three constructed from binary codes of odd length. In: 2010 Second International Conference on Future Networks, Sanya, China, pp. 395-398 (2010). https://doi.org/10.1109/ICFN.2010.20

  23. Li, R., Li, X.: Binary construction of quantum codes of minimum distance three and four. IEEE Trans. Inf. Theory 50(6), 1331–1335 (2004). https://doi.org/10.1109/TIT.2004.828149

    Article  MathSciNet  MATH  Google Scholar 

  24. Li, R., Li, X.: Binary construction of quantum codes of minimum distances five and six. Discrete Math. 308(9), 1603–1611 (2008). https://doi.org/10.1016/j.disc.2007.04.016

    Article  MathSciNet  MATH  Google Scholar 

  25. Liu, J.: Ternary quantum codes of minimum distance three. Int. J. Quantum Inf. 08(07), 1179–1186 (2010). https://doi.org/10.1142/S0219749910006137

    Article  MATH  Google Scholar 

  26. Liu, Y., Cao, X.: Optimal \(p\)-ary cyclic codes with two zeros. arXiv:1908.03070v1 [cs.IT] 8 Aug (2019)

  27. Liu, Y., Cao, X.: Four classes of optimal quinary cyclic codes. IEEE Commun. Lett. 24(7), 1387–1390 (2020). https://doi.org/10.1109/LCOMM.2020.2983373

    Article  ADS  Google Scholar 

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

    MATH  Google Scholar 

  29. Montanaro, A.: Quantum algorithms: an overview. NPJ Quantum Inf. 2, 15023 (2016). https://doi.org/10.1038/npjqi.2015.23

    Article  ADS  Google Scholar 

  30. Rouayheb, S.Y.E., Georghiades, C.N., Soljanin, E., Sprintson, A.: Bounds on codes based on graph theory. In: 2007 IEEE International Symposium on Information Theory, Nice, France, pp. 1876–1879 (2007). https://doi.org/10.1109/ISIT.2007.4557151

  31. Shor, P.W.: Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52(4), R2493–R2496 (1995). https://doi.org/10.1103/physreva.52.r2493

    Article  ADS  Google Scholar 

  32. Steane, A.M.: Multiple-particle interference and quantum error correction. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 452, 2551–2577 (1996). https://doi.org/10.1098/rspa.1996.0136

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Steane, A.M.: Error correcting codes in quantum theory. Phys. Rev. Lett. 77(5), 793–797 (1996). https://doi.org/10.1103/PhysRevLett.77.793

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. Steane, A.M.: Enlargement of Calderbank–Shor–Steane quantum codes. IEEE Trans. Inf. Theory 45(7), 2492–2495 (1999). https://doi.org/10.1109/18.796388

    Article  MathSciNet  MATH  Google Scholar 

  35. Xu, G., Cao, X., Xu, S.: Optimal \(p\)-ary cyclic codes with minimum distance four from monomials. Cryptogr. Commun. 8, 541–554 (2016). https://doi.org/10.1007/s12095-015-0159-0

    Article  MathSciNet  MATH  Google Scholar 

  36. Yu, S., Bierbrauer, J., Dong, Y., Chen, Q., Oh, C.H.: All the stabilizer codes of distance \(3\). IEEE Trans. Inf. Theory 59(8), 5179–5185 (2013). https://doi.org/10.1109/TIT.2013.2259138

    Article  MathSciNet  MATH  Google Scholar 

  37. Zidan, M., Abdel-Aty, A., Younes, A., et al.: A novel algorithm based on entanglement measurement for improving speed of quantum algorithms. Appl. Math. Inf. Sci. 12(1), 265–269 (2018). https://doi.org/10.18576/amis/120127

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We thank the anonymous referees and the associate editor Raphael Pooser for their valuable comments and suggestions that helped to improve greatly the quality of this paper. This research work is supported by the Key Projects of Natural Science Research of Universities in Anhui Province under Grant No.KJ2021A1469, and the National Natural Science Foundation of China under Grant Nos. U21A20428, 12171134 and 61972126, and the Fundamental Research Funds for the Central Universities No.JZ2022HGTB0264.

Author information

Authors and Affiliations

  1. Department of Information Management, Anhui Vocational College of Police Officers, Hefei, 230031, China

    Shan Huang

  2. School of Mathematics, Hefei University of Technology, Hefei, 230601, Anhui, China

    Shixin Zhu & Jin Li

Authors
  1. Shan Huang
    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

  3. Jin Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shan Huang.

Ethics declarations

Ethical approval

All the procedures performed in this study were in accordance with the ethical standards of the Institutional and/or National Research Committee and with the 1964 Helsinki Declaration and its later amendments or comparable ethical standards.

Informed consent

Informed consent was obtained from all individual participants included in the study.

Conflict of interest

All the authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, S., Zhu, S. & Li, J. Three classes of optimal Hermitian dual-containing codes and quantum codes. Quantum Inf Process 22, 45 (2023). https://doi.org/10.1007/s11128-022-03791-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-022-03791-4

Keywords

Mathematics Subject Classification

Search

Navigation