Hadamard \(\mathbb{Z}_{2}\mathbb{Z}_{4}Q_{8}\)-Codes: Rank and Kernel

  • Pere MontolioEmail author
  • Josep Rifà
Conference paper
Part of the CIM Series in Mathematical Sciences book series (CIMSMS, volume 3)


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 Q8 groups, where Q8 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.


Dimension of the kernel Error-correcting codes Hadamard codes Rank \(\mathbb{Z}_{2}\mathbb{Z}_{4}\)-linear codes \(\mathbb{Z}_{2}\mathbb{Z}_{4}Q_{8}\)-codes 



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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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Computing, Multimedia and Telecommunication StudiesUniversitat Oberta de CatalunyaBarcelonaSpain
  2. 2.Department of Information and Communications EngineeringUniversitat Autònoma de BarcelonaBarcelonaSpain

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