Cryptography and Communications

, Volume 9, Issue 2, pp 291–299 | Cite as

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

  • Tingting Wu
  • Jian GaoEmail author
  • Fang-Wei Fu


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.


1-generator generalized quasi-cyclic codes New \(\mathbb {Z}_{4}\)-linear codes Gray map Good binary nonlinear codes 

Mathematics Subject Classification (2010)

94B05 94B15 



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).


  1. 1.
    Aydin, N., Asamov, T.: The \(\mathbb {Z}_{4}\) Database [Online]. Available: accessed on 28.08.2015
  2. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bosma, W., Cannon, J., Playoust: The Magma algebra system. J. Symb. Comput. 24, 235–265 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cao, Y.: Generalized quasi-cyclic codes over Galois rings: structural properties and enumeration. Appl. Algebra Eng. Commun. Comput. 22, 219–233 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cao, Y.: Structural properties and enumeration of 1-generator generalized quasi-cyclic codes. Des. Codes Cryptogr. 60, 67–79 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Esmaeili, M., Yari, S.: Generalized quasi-cyclic codes: Structural properties and codes construction. Appl. Algebra Eng. Commun. Comput. 20, 159–173 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gao, J., Shi, M., Wu, T., Fu, F.-W.: On double cyclic codes over \(\mathbb {Z}_{4}\). arXivpreprint, arXiv:1501.01360v1 (2015)
  8. 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)CrossRefGoogle Scholar
  9. 9.
    Grassl, M.: Table of Bounds on Linear Codes [Online]. Available: accessed on 14.05.2015
  10. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Van Lint, J.H.: Introduction to Coding Theory. Springer-Verlag, Berlin (1999)CrossRefzbMATHGoogle Scholar
  12. 12.
    Pless, V., Qian, Z.: Cyclic codes and quadratic residue codes over \(\mathbb {Z}_{4}\). IEEE Trans. Inform. Theory 42, 1594–1600 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Siap, I., Abualrub, T., Aydin, N.: Quaternary quasi-cyclic codes with even length components. Ars Combinatoria 101, 425–434 (2011)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Siap, I., Kulhan, N.: The structure of generalized quasi-cyclic codes. Appl. Math. E-Notes 5, 24–30 (2005)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Wan, Z.-X.: Quaternary Codes. World Scientific, Singapore (1997)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Chern Institute of Mathematics and LPMCNankai UniversityTianjinPeople’s Republic of China
  2. 2.School of ScienceShandong University of TechnologyZiboPeople’s Republic of China

Personalised recommendations