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

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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.

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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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Authors and Affiliations

  1. Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, 08193, Cerdanyola del Vallès, Spain

    Cristina Fernández-Córdoba, Carlos Vela & Mercè Villanueva

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  1. Cristina Fernández-Córdoba
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  2. Carlos Vela
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  3. Mercè Villanueva
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Correspondence to Carlos Vela.

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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.

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Fernández-Córdoba, C., Vela, C. & Villanueva, M. On \(\mathbb {Z}_{2^s}\)-linear Hadamard codes: kernel and partial classification. Des. Codes Cryptogr. 87, 417–435 (2019). https://doi.org/10.1007/s10623-018-0546-6

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