Abstract
Hadamard \(\mathbb{Z}_{2}\mathbb{Z}_{4}Q_{8}\)-codes are Hadamard binary codes coming from a subgroup of the direct product of \(\mathbb{Z}_{2}\), \(\mathbb{Z}_{4}\) and Q 8 groups, where Q 8 is the quaternionic group. We characterize Hadamard \(\mathbb{Z}_{2}\mathbb{Z}_{4}Q_{8}\)-codes as a quotient of a semidirect product of \(\mathbb{Z}_{2}\mathbb{Z}_{4}\)-linear codes and we show that all these codes can be represented in a standard form, from a set of generators. On the other hand, we show that there exist Hadamard \(\mathbb{Z}_{2}\mathbb{Z}_{4}Q_{8}\)-codes with any given pair of allowable parameters for the rank and dimension of the kernel.
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Acknowledgements
This work has been partially supported by the Spanish MICINN grant TIN2013-40524-P and the Catalan AGAUR grant 2014SGR-691.
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Montolio, P., Rifà, J. (2015). Hadamard \(\mathbb{Z}_{2}\mathbb{Z}_{4}Q_{8}\)-Codes: Rank and Kernel. In: Pinto, R., Rocha Malonek, P., Vettori, P. (eds) Coding Theory and Applications. CIM Series in Mathematical Sciences, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-319-17296-5_29
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DOI: https://doi.org/10.1007/978-3-319-17296-5_29
Publisher Name: Springer, Cham
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