Designs, Codes and Cryptography

, Volume 87, Issue 2–3, pp 417–435 | Cite as

On \(\mathbb {Z}_{2^s}\)-linear Hadamard codes: kernel and partial classification

  • Cristina Fernández-Córdoba
  • Carlos VelaEmail author
  • Mercè Villanueva
Part of the following topical collections:
  1. Special Issue: Coding and Cryptography


The \(\mathbb {Z}_{2^s}\)-additive codes are subgroups of \(\mathbb {Z}^n_{2^s}\), and can be seen as a generalization of linear codes over \(\mathbb {Z}_2\) and \(\mathbb {Z}_4\). A \(\mathbb {Z}_{2^s}\)-linear Hadamard code is a binary Hadamard code which is the Gray map image of a \(\mathbb {Z}_{2^s}\)-additive code. It is known that the dimension of the kernel can be used to give a complete classification of the \(\mathbb {Z}_4\)-linear Hadamard codes. In this paper, the kernel of \(\mathbb {Z}_{2^s}\)-linear Hadamard codes of length \(2^t\) and its dimension are established for \(s > 2\). Moreover, we prove that this invariant only provides a complete classification for some values of t and s. The exact amount of nonequivalent such codes are given up to \(t=11\) for any \(s\ge 2\), by using also the rank.


Kernel Hadamard code \(\mathbb {Z}_{2^s}\)-linear code \(\mathbb {Z}_{2^s}\)-additive code Gray map Classification 

Mathematics Subject Classification

94B25 94B60 



This work has been partially supported by the Spanish MINECO under Grants TIN2016-77918-P (AEI/FEDER, UE) and MTM2015-69138-REDT.


  1. 1.
    Assmus E.F., Jr., Key J.D.: Designs and Their Codes. Cambridge University Press, Great Britain (1992).Google Scholar
  2. 2.
    Bauer H., Ganter B., Hergert F.: Algebraic techniques for nonlinear codes. Combinatorica 3(1), 21–33 (1983).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Blake I.F.: Codes over integer residue rings. Inf. Control 29, 295–300 (1975).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Borges J., Fernández C., Rifà J.: Every \(\mathbb{Z}_{2k}\)-code is a binary propelinear code. Electron. Notes Discret. Math. 10, 100–102 (2001).CrossRefzbMATHGoogle 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. Des. Codes Cryptogr. 54(2), 167–179 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bosma W., Cannon, J.J., Fieker, C., Steel, A.: Handbook of Magma functions, Edn 2.22 (2016).
  7. 7.
    Carlet C.: \(\mathbb{Z}_{2^k}\)-linear codes. IEEE Trans. Inf. Theory 44(4), 1543–1547 (1998).CrossRefzbMATHGoogle Scholar
  8. 8.
    Dougherty S.T., Fernández-Córdoba C.: Codes over \(\mathbb{Z}_{2^ k}\), gray map and self-dual codes. Adv. Math. Commun. 5(4), 571–588 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fernández-Córdoba C., Vela C., Villanueva M.: Construction and classification of the \(\mathbb{Z}_{2^s}\)-linear Hadamard codes. Electron. Notes Discret. Math. 54, 247–252 (2016).CrossRefzbMATHGoogle Scholar
  10. 10.
    Fernández-Córdoba C., Vela C., Villanueva M.: On the Kernel of \(\mathbb{Z}_{2^{s}}\)-linear Hadamard Codes. In: Coding Theory and Applications, ICMCTA 2017. Lecture Notes in Computer Science, vol. 10495, pp. 107–117 (2017).Google Scholar
  11. 11.
    Gupta M.K., Bhandari M.C., Lal A.K.: On linear codes over \(\mathbb{Z}_{2^s}\). Des. Codes Cryptogr. 36(3), 227–244 (2005).MathSciNetCrossRefGoogle Scholar
  12. 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. Inf. Theory 40(2), 301–319 (1994).CrossRefzbMATHGoogle Scholar
  13. 13.
    Krotov D.S.: \(\mathbb{Z}_4\)-linear Hadamard and extended perfect codes. International workshop on coding and cryptography, ser. Electron. Notes Discret. Math. 6, 107–112 (2001).CrossRefGoogle Scholar
  14. 14.
    Krotov D.S.: On \(\mathbb{Z}_{2^k}\)-dual binary codes. IEEE Trans. Inf. Theory 53(4), 1532–1537 (2007).CrossRefzbMATHGoogle Scholar
  15. 15.
    Krotov D.S., Villanueva M.: Classification of the \(\mathbb{Z}_2\mathbb{Z}_4\)-linear Hadamard codes and their automorphism groups. IEEE Trans. Inf. Theory 61(2), 887–894 (2015).CrossRefzbMATHGoogle Scholar
  16. 16.
    MacWilliams F.J., Sloane N.J.A.: The Theory of Error-correcting Codes, vol. 16. Elsevier, Amsterdam (1977).zbMATHGoogle Scholar
  17. 17.
    Nechaev A.A., Khonol’d T.: Weighted modules and representations of codes. Probl. Inf. Transm. 35(3), 205–223 (1999).MathSciNetGoogle Scholar
  18. 18.
    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. Inf. Theory 52(1), 316–319 (2006).CrossRefzbMATHGoogle Scholar
  19. 19.
    Shankar P.: On BCH codes over arbitrary integer rings. IEEE Trans. Inf. Theory 25(4), 480–483 (1979).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Shi M., Sepasdar Z., Alahmadi A., Solé P.: On two-weight \(\mathbb{Z}_{2^k}\)-codes. Des. Codes Cryptogr. 86(6), 1201–1209 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Tapia-Recillas H., Vega G.: On \(\mathbb{Z}_{2^k}\)-linear and quaternary codes. SIAM J. Discret. Math. 17(1), 103–113 (2003).CrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Information and Communications EngineeringUniversitat Autònoma de BarcelonaCerdanyola del VallèsSpain

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