\({\mathbb {Z}}_{2}{\mathbb {Z}}_{4}{\mathbb {Z}}_{8}\)-Cyclic codes

  • Ismail AydogduEmail author
  • Fatmanur Gursoy
Original Research


In this paper we study \({{\mathbb {Z}}_{2}{\mathbb {Z}}_{4}{\mathbb {Z}}_{8}}\)-additive codes, which are extensions of recently introduced \(\mathbb {Z}_{2}\mathbb {Z}_{4}\)-additive codes. We determine the standard forms of the generator and parity-check matrices of \({{\mathbb {Z}}_{2}{\mathbb {Z}}_{4}{\mathbb {Z}}_{8}}\)-additive codes. Moreover, we investigate \({{\mathbb {Z}}_{2}{\mathbb {Z}}_{4}{\mathbb {Z}}_{8}}\)-cyclic codes and give their generator polynomials and spanning sets. We also give some illustrative examples of both \({{\mathbb {Z}}_{2}{\mathbb {Z}}_{4}{\mathbb {Z}}_{8}}\)-additive codes and \({{\mathbb {Z}}_{2}{\mathbb {Z}}_{4}{\mathbb {Z}}_{8}}\)-cyclic codes.


Generator matrix Parity-check matrix Cyclic codes \({{\mathbb {Z}}_{2}{\mathbb {Z}}_{4}{\mathbb {Z}}_{8}}\)-additive codes 

Mathematics Subject Classification

94B05 94B60 



The authors would like to thank the anonymous reviewers for their valuable remarks that led to an improved version of the paper.


  1. 1.
    Abualrub, T., Siap, I., Aydin, N.: \({\mathbb{Z}}_2{\mathbb{Z}}_4\)-additive cyclic codes. IEEE Trans. Info. Theory 60(3), 1508–1514 (2014)CrossRefzbMATHGoogle Scholar
  2. 2.
    Aydogdu, I., Siap, I.: The structure of \({\mathbb{Z}}_{2}{\mathbb{Z}}_{2^s}\)-additive codes: bounds on the minimum distance. Appl. Math. Inf. Sci. (AMIS) 7(6), 2271–2278 (2013)CrossRefGoogle Scholar
  3. 3.
    Aydogdu, I., Siap, I.: On \({\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^s}\)-additive codes. Linear Multilinear Algebra 63(10), 2089–2102 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bannai, E., Dougherty, S.T., Harada, M., Oura, M.: Type II codes, even unimodular lattices, and invariant rings. IEEE Trans. Info. Theory 45(4), 1194–1205 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Borges, J., Fernández-Córdoba, C., Pujol, J., Rifà, J., Villanueva, M.: \(\mathbb{Z}_{2}\mathbb{Z}_{4}\)-linear codes: generator matrices and duality. Designs, Codes Cryptogr. 54(2), 167–179 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Borges, J., Fernández-Córdoba, C., Ten-Valls, R.: \(\mathbb{Z}_{2}\mathbb{Z}_{4}\)-additive cyclic codes, generator polynomials and dual codes. IEEE Trans. Info. Theory 62(11), 6348–6354 (2016)CrossRefzbMATHGoogle Scholar
  7. 7.
    Calderbank, A.R., Sloane, N.J.A.: Modular and \(p\)-adic cyclic codes. Designs Codes Cryptogr. 6, 21–35 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dougherty, S.T., Fernández-Córdoba, C.: Codes over \(\mathbb{Z}_{2^{k}}\). Gray Map Self-Dual Codes Adv. Math. Commun. 5(4), 571–588 (2011)zbMATHGoogle Scholar
  9. 9.
    Greferath, M., Schmidt, S.E.: Gray isometries for finite chain rings. IEEE Trans. Info. Theory 45(7), 2522–2524 (1999)CrossRefzbMATHGoogle Scholar
  10. 10.
    Hammons, A.R., Kumar, P.V., Calderbank, A.R., Sloane, N.J.A., Solé, P.: The \({\mathbb{Z}}_{4}\)-linearity of Kerdock, Preparata, Goethals and related codes. IEEE Trans. Inf. Theory 40(2), 301–319 (1994)CrossRefzbMATHGoogle Scholar
  11. 11.
    Honold, T., Landjev, I.: Linear codes over finite chain rings. In: Optimal Codes and Related Topics, pp. 116–126. Sozopol, Bulgaria (1998)Google Scholar
  12. 12.
    Rifà-Pous, H., Rifà, J., Ronquillo, L.: \({\mathbb{Z}}_{2}{\mathbb{Z}}_{4}\)-additive perfect codes in steganography. Adv. Math. Commun. 5(3), 425–433 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2018

Authors and Affiliations

  1. 1.Department of MathematicsYildiz Technical UniversityIstanbulTurkey

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