Skip to main content
SpringerLink
Log in

A new construction of quantum codes from quasi-cyclic codes over finite fields

  • Original Research
  • Published:
Indian Journal of Pure and Applied Mathematics Aims and scope Submit manuscript

Abstract

In this paper we present a construction of quantum codes from 1-generator quasi-cyclic (QC) codes of index 2 over a finite field \(\mathbb {F}_q\). We have studied QC codes of index 2 as a special case of \(\mathbb {F}_q\)-double cyclic codes. We have determined the structure of the duals of such QC codes and presented a necessary and sufficient condition for them to be self-orthogonal. A construction of 1-generator QC codes with good minimum distance is also presented. To obtain quantum codes from QC codes, we use the Calderbank-Shor-Steane (CSS) construction. Few examples have been given to demonstrate this construction. Also, we present two tables of quantum codes with good parameters obtained from QC codes over \(\mathbb {F}_q\).

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. P. W. Shor, Scheme for reducung decoherence in quantum computer memory, Phys. Rev. A, 52(1995), 2493–2496.

    Article  Google Scholar 

  2. A. M. Steane, Error correcting codes in quantum theory, Phys. Rev. Lett., 77(5)(1996), 793–797.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, Quantum error correction via codes over \(GF(4)\), IEEE Trans. Inf. Theory, 44(1998), 1369–1387.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. G. G. La Guardia, R. Jr. Palazzo, Construction of new families of nonbinary CSS codes, Discrete Math, 310(2010), 2935–2945 .

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

    Article  MathSciNet  MATH  Google Scholar 

  7. G. G. La Guardia, On the construction of nonbinary quantum BCH codes, IEEE Trans. Inf. Theory, 60(3)(2014), 1528–1535.

    Article  MathSciNet  MATH  Google Scholar 

  8. K. Guenda, S. Jitman, T. A. Gulliver, Constructions of good entanglement-assisted quantum error correcting codes, Des. Codes Cryptogr., 86(1)(2018), 121–136.

    Article  MathSciNet  MATH  Google Scholar 

  9. S. A. Aly, A. Klappenecker and P. K. Sarvepalli, On quantum and classical BCH codes, IEEE Trans. Inf. Theory, 53(3)(2007), 1183–1188.

    Article  MathSciNet  MATH  Google Scholar 

  10. G. G. La Guardia, Quantum codes derived from cyclic codes, Int. J. Theor. Phys., 56(8)(2017), 2479–2484 .

    Article  MATH  Google Scholar 

  11. J. Gao, Y. Wang, Quantum codes derived from negacyclic codes, Int. J. Theor. Phys., 57(3)(2018), 682–686.

    Article  MathSciNet  MATH  Google Scholar 

  12. H. Islam, O. Prakash and D. K. Bhunia, Quantum codes obtained from constacyclic codes, Int. J. Theor. Phys., 58(11)(2019), 3945–3951.

    Article  MathSciNet  MATH  Google Scholar 

  13. J. Conan, , G. Séguin, Stuctural properties and enumeration of quasicyclic codes, Appl. Algebra Eng. Commun. Comput., 4(1993), 25–39.

    Article  MATH  Google Scholar 

  14. K. Lally, P. Fitzpatrick, Algebraic structure of quasicyclic codes, Discrete Appl. Math, 111(1-2)(2001), 157–175.

    Article  MathSciNet  MATH  Google Scholar 

  15. J. Gao, Q. Kong, 1-Generator quasi-cyclic codes over \({{\mathbb{F}} _{p^m}}+u{{\mathbb{F}} _{p^m}}+\cdots +u^{s-1}{{\mathbb{F}} _{p^m}}\), Journal of the Franklin Institute, 350(2013), 3260–3276.

    Article  Google Scholar 

  16. N. Aydin, D. K. Ray-Chaudhuri, Quasi-cyclic codes over \({{\mathbb{Z}}_4}\) and some new binary codes, IEEE Trans. and Inf. Theory, 48(7)(2002), 2065–2069.

    Article  MATH  Google Scholar 

  17. T. Kasami, A Gilbert-Varshamov bound for quasi-cyclic codes of rate \(1/2\), IEEE Trans. Inf. Theory, 20(5)(1974), 679–679.

    Article  MathSciNet  MATH  Google Scholar 

  18. J. Borges, C. Fernández-Córdoba and R. Ten-Valls, \({{\mathbb{Z}} _2}\)-double cyclic codes, Des. Codes Cryptogr., 86(3)(2018), 463–479.

    Article  MathSciNet  MATH  Google Scholar 

  19. C. Galindo, F. Hernando and R. Mastsumoto, Quasi-cyclic constructions of quantum codes, Finite Fields Appl., 52(2018), 261–280.

    Article  MathSciNet  MATH  Google Scholar 

  20. J. Lv, R. Li and J. Wang, Quantum codes derived from one generator quasi-cyclic codes with Hermitian inner product, Int. J. Theor. Phys. , 59(2020), 300–312.

    Article  MathSciNet  MATH  Google Scholar 

  21. J. Lv, R. Li and J. Wang, New binary quantum codes derived from one-generator quasi-cyclic codes, IEEE Access, 7(2019), 85782–85785.

    Article  Google Scholar 

  22. J. Qian, W. Ma and X. Wang, Quantum error-correcting codes from quasi-cyclic codes, Int. J. Quantum Inf., 6(6)(2008), 1263–1269.

    Article  MATH  Google Scholar 

  23. A. Ashikhmin, E. Knill, Nonbinary quantum stabilizer codes, IEEE Trans. Inf. Theory, 47(7)(2001), 3065–3072.

    Article  MathSciNet  MATH  Google Scholar 

  24. S. Ling, C. Xing, Coding Theory: A First Course, Cambridge University Press, New York (2004).

    Book  Google Scholar 

  25. T. Abualrub, I. Siap and N. Aydin, \({{{\mathbb{Z}}}_2} {{{\mathbb{Z}}}_4}\)-additive cyclic codes, IEEE Trans. Inf. Theory, 60(3)(2014), 1508–1514.

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  27. W. Bosma, J. Cannon, Handbook of Magma Functions, Univ. of Sydney (1995).

Download references

Acknowledgements

The authors would like to thank the anonymous referee(s) and the editor for their valuable comments and suggestions that greatly improved the presentation of the paper. The first author would like to thank Ministry of Human Resource Development (MHRD), India for providing financial support.

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India

    Soumak Biswas & Maheshanand Bhaintwal

Authors
  1. Soumak Biswas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Maheshanand Bhaintwal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Soumak Biswas.

Additional information

Communicated by Bakshi Gurmeet Kaur.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Biswas, S., Bhaintwal, M. A new construction of quantum codes from quasi-cyclic codes over finite fields. Indian J Pure Appl Math 54, 375–388 (2023). https://doi.org/10.1007/s13226-022-00259-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13226-022-00259-0

Keywords

Mathematics Subject Classification

Search

Navigation