Designs, Codes and Cryptography

, Volume 54, Issue 2, pp 167–179 | Cite as

\({{{\mathbb Z}_2}{{\mathbb Z}_4}}\) -linear codes: generator matrices and duality

  • J. BorgesEmail author
  • C. Fernández-Córdoba
  • J. Pujol
  • J. Rifà
  • M. Villanueva


A code \({{\mathcal C}}\) is \({{{\mathbb Z}_2}{{\mathbb Z}_4}}\) -additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of \({{\mathcal C}}\) by deleting the coordinates outside X (respectively, Y) is a binary linear code (respectively, a quaternary linear code). In this paper \({{{\mathbb Z}_2}{{\mathbb Z}_4}}\) -additive codes are studied. Their corresponding binary images, via the Gray map, are \({{{\mathbb Z}_2}{{\mathbb Z}_4}}\) -linear codes, which seem to be a very distinguished class of binary group codes. As for binary and quaternary linear codes, for these codes the fundamental parameters are found and standard forms for generator and parity-check matrices are given. In order to do this, the appropriate concept of duality for \({{{\mathbb Z}_2}{{\mathbb Z}_4}}\) -additive codes is defined and the parameters of their dual codes are computed.

Mathematics Subject Classification (2000)

94B60 94B25 


Binary linear codes Duality Quaternary linear codes \({{{\mathbb Z}_2}{{\mathbb Z}_4}}\) -additive codes \({{{\mathbb Z}_2}{{\mathbb Z}_4}}\) -linear codes 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bierbrauer J.: Introduction to Coding Theory. Chapman & Hall/CRC, Boca Raton, FL (2005)zbMATHGoogle Scholar
  2. Borges J., Fernández C., Phelps K.T.: Quaternary Reed–Muller codes. IEEE Trans. Inform. Theory 51(7), 2686–2691 (2005)CrossRefMathSciNetGoogle Scholar
  3. Borges J., Fernández-Córdoba C., Phelps K.T.: ZRM codes. IEEE Trans. Inform. Theory 54(1), 380–386 (2008)CrossRefMathSciNetGoogle Scholar
  4. Borges J., Fernández C., Pujol J., Rifà J., Villanueva M.: On \({{{\mathbb Z}_2}{{\mathbb Z}_4}}\) -linear codes and duality. VJMDA, pp. 171–177, Ciencias, 23. Secr. Publ. Intercamb. Ed., Valladolid (2006).Google Scholar
  5. Borges J., Fernández-Córdoba C., Pujol J., Rifà J., Villanueva M.: \({{{\mathbb Z}_2}{{\mathbb Z}_4}}\) -Additive Codes. A Magma Package. Autonomous University of Barcelona (UAB), Bellaterra, Barcelona (2007). Accessed July 2009.
  6. Borges J., Fernández C., Rifà J.: Every \({{\mathbb Z}_{2k}}\) -code is a binary propelinear code. In: COMB’01. Electronic Notes in Discrete Mathematics, vol. 10, pp. 100–102. Elsevier Science, Amsterdam, November (2001).Google Scholar
  7. Borges J., Phelps K.T., Rifà J.: The rank and kernel of extended 1-perfect \({{\mathbb Z}_4}\) -linear and additive non-\({{\mathbb Z}_4}\) -linear codes. IEEE Trans. Inform. Theory 49(8), 2028–2034 (2003)CrossRefMathSciNetGoogle Scholar
  8. Borges J., Rifà J.: A characterization of 1-perfect additive codes. IEEE Trans. Inform. Theory 45(5), 1688–1697 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  9. Bosma W., Cannon J., Playoust C.: The MAGMA algebra system I: the user language. J. Symb. Comput. 24(3–4), 235–265 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  10. Delsarte P.: An algebraic approach to the association schemes of coding theory. Philips Research Rep. Suppl. 10, vi + 97 (1973).Google Scholar
  11. Delsarte P., Levenshtein V.: Association schemes and coding theory. IEEE Trans. Inform. Theory 44(6), 2477–2504 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 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. Inform. Theory 40, 301–319 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  13. Heden O.: A new construction of group and nongroup perfect codes. Inform. Control 34, 314–323 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  14. Huffman W.C., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003)zbMATHGoogle Scholar
  15. Krotov D.S.: \({{\mathbb Z}_4}\) -linear Hadamard and extended perfect codes. Electron. Notes Discrete Math. 6, 107–112 (2001)CrossRefMathSciNetGoogle Scholar
  16. Ledermann W.: Introduction to Group Characters. Cambridge University Press, Cambridge (1977)zbMATHGoogle Scholar
  17. Lindström B.: Group partitions and mixed perfect codes. Can. Math. Bull. 18, 57–60 (1975)zbMATHGoogle Scholar
  18. MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland Publishing Company, Amsterdam, New York, Oxford (1977)zbMATHGoogle Scholar
  19. Phelps K.T., Rifà J., Villanueva M.: On the additive (\({{\mathbb Z}_4}\) -linear and non-\({{\mathbb Z}_4}\) -linear) Hadamard codes: Rank and Kernel. IEEE Trans. Inform. Theory 52(1), 316–319 (2006)CrossRefMathSciNetGoogle Scholar
  20. Pujol J., Rifà J.: Translation invariant propelinear codes. IEEE Trans. Inform. Theory 43, 590–598 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  21. Pujol J., Rifà J., Solov’eva F.: Construction of \({{\mathbb Z}_4}\) -linear Reed–Muller codes. IEEE Trans. Inform. Theory 55(1), 99–104 (2009)CrossRefGoogle Scholar
  22. Rifà J., Basart J.M., Huguet L.: On completely regular propelinear codes. In: Proceedings of 6th International Conference, AAECC-6. LNCS, vol. 357, pp. 341–355. Springer, Berlin (1989).Google Scholar
  23. Rifa J., Phelps K.T.: On binary 1-perfect additive codes: some structural properties. IEEE Trans. Inform. Theory 48(9), 2587–2592 (2002)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • J. Borges
    • 1
    Email author
  • C. Fernández-Córdoba
    • 1
  • J. Pujol
    • 1
  • J. Rifà
    • 1
  • M. Villanueva
    • 1
  1. 1.Department of Information and Communications EngineeringUniversitat Autònoma de BarcelonaBellaterraSpain

Personalised recommendations