Skip to main content
Log in

1-generator generalized quasi-cyclic codes over \(\mathbb {Z}_{4}\)

  • Published:
Cryptography and Communications Aims and scope Submit manuscript

Abstract

In this short paper, we determine the minimal generating set of 1-generator generalized quasi-cyclic codes over \(\mathbb {Z}_{4}\). We also determine their rank and introduce a lower bound for the minimum distance of free 1-generator generalized quasi-cyclic codes. Further, we construct some new \(\mathbb {Z}_{4}\)-linear codes and we obtain some good binary nonlinear codes using the usual Gray map.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Aydin, N., Asamov, T.: The \(\mathbb {Z}_{4}\) Database [Online]. Available: http://z4codes.info/. accessed on 28.08.2015

  2. Aydin, N., Ray-Chaudhuri, D.K.: Quasi cyclic codes over \(\mathbb {Z}_{4}\) and some new binary codes. IEEE Trans. Inform. Theory 48, 2065–2069 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bosma, W., Cannon, J., Playoust: The Magma algebra system. J. Symb. Comput. 24, 235–265 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cao, Y.: Generalized quasi-cyclic codes over Galois rings: structural properties and enumeration. Appl. Algebra Eng. Commun. Comput. 22, 219–233 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cao, Y.: Structural properties and enumeration of 1-generator generalized quasi-cyclic codes. Des. Codes Cryptogr. 60, 67–79 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Esmaeili, M., Yari, S.: Generalized quasi-cyclic codes: Structural properties and codes construction. Appl. Algebra Eng. Commun. Comput. 20, 159–173 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gao, J., Shi, M., Wu, T., Fu, F.-W.: On double cyclic codes over \(\mathbb {Z}_{4}\). arXivpreprint, arXiv:1501.01360v1 (2015)

  8. Gao, J., Fu, F.-W., Shen, L., Ren, W.: Some results on generalized quasi-cyclic codes over \(\mathbb {F}_{q}+u\mathbb {F}_{q}\). IEICE Trans. Fundamentals 97, 1005–1011 (2014)

    Article  Google Scholar 

  9. Grassl, M.: Table of Bounds on Linear Codes [Online]. Available: http://www.codetables.de/. accessed on 14.05.2015

  10. Hammons, J., Kumar, P.V., Calderbank, A.R., Sloane, N.J., Sol é, P.: The \(\mathbb {Z}_{4}\)-linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inform. Theory 40(2), 301–319 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Van Lint, J.H.: Introduction to Coding Theory. Springer-Verlag, Berlin (1999)

    Book  MATH  Google Scholar 

  12. Pless, V., Qian, Z.: Cyclic codes and quadratic residue codes over \(\mathbb {Z}_{4}\). IEEE Trans. Inform. Theory 42, 1594–1600 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  13. Siap, I., Abualrub, T., Aydin, N.: Quaternary quasi-cyclic codes with even length components. Ars Combinatoria 101, 425–434 (2011)

    MathSciNet  MATH  Google Scholar 

  14. Siap, I., Kulhan, N.: The structure of generalized quasi-cyclic codes. Appl. Math. E-Notes 5, 24–30 (2005)

    MathSciNet  MATH  Google Scholar 

  15. Wan, Z.-X.: Quaternary Codes. World Scientific, Singapore (1997)

    Book  MATH  Google Scholar 

Download references

Acknowledgments

This research is supported by the National Key Basic Research Program of China (973 Program Grant No. 2013CB834204), the National Natural Science Foundation of China (Nos. 61571243, 61171082 and 61301137).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jian Gao.

Additional information

The first two authors contributed equally to this work.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, T., Gao, J. & Fu, FW. 1-generator generalized quasi-cyclic codes over \(\mathbb {Z}_{4}\) . Cryptogr. Commun. 9, 291–299 (2017). https://doi.org/10.1007/s12095-015-0175-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12095-015-0175-0

Keywords

Mathematics Subject Classification (2010)

Navigation