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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
Article
  • 64 Downloads
Part of the following topical collections:
  1. Special Issue: Coding and Cryptography

Abstract

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.

Keywords

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

Mathematics Subject Classification

94B25 94B60 

Notes

Acknowledgements

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

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Copyright information

© 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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