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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)

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 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.

Keywords

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 

Notes

Acknowledgements

This work has been partially supported by the Spanish MICINN grant TIN2013-40524-P and the Catalan AGAUR grant 2014SGR-691.

References

  1. 1.
    Flannery, D.: Cocyclic Hadamard matrices and Hadamard groups are equivalent. J. Algebra 192(2), 749–779 (1997). http://dx.doi.org/http://dx.doi.org/10.1006/jabr.1996.6949 doi:http://dx.doi.org/10.1006/jabr.1996.6949
  2. 2.
    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). http://dx.doi.org/http://dx.doi.org/10.1109/18.312154 doi:http://dx.doi.org/10.1109/18.312154
  3. 3.
    Horadam, K.J.: Hadamard matrices and their applications: progress 2007–2010. Cryptogr. Commun. 2(2), 129–154 (2010). http://dx.doi.org/http://dx.doi.org/10.1007/s12095-010-0032-0 doi:http://dx.doi.org/10.1007/s12095-010-0032-0
  4. 4.
    Ito, N.: On Hadamard groups. J. Algebra 168(3), 981–987 (1994). http://dx.doi.org/http://dx.doi.org/10.1006/jabr.1994.1266 doi:http://dx.doi.org/10.1006/jabr.1994.1266
  5. 5.
    Phelps, K.T., Rifà, J., Villanueva, M.: On the additive (\(\mathbb z_4\)-linear and non-\(\mathbb Z_2\mathbb Z_4Q_8\)-linear) Hadamard codes: rank and kernel. IEEE Trans. Inf. Theory 52(1), 316–319 (2006). http://dx.doi.org/http://dx.doi.org/10.1109/TIT.2005.860464 doi:http://dx.doi.org/10.1109/TIT.2005.860464
  6. 6.
    Rifà, J., Pujol, J.: Translation-invariant propelinear codes. IEEE Trans. Inf. Theory 43(2), 590–598 (1997). http://dx.doi.org/http://dx.doi.org/10.1109/18.556115 doi:http://dx.doi.org/10.1109/18.556115
  7. 7.
    de Launey, W., Flannery, D.L., Horadam, K.J.: Cocyclic Hadamard matrices and difference sets. Discret. Appl. Math. 102(1–2), 47–61 (2000). http://dx.doi.org/http://dx.doi.org/10.1016/S0166-218X(99)00230-9 doi:http://dx.doi.org/10.1016/S0166-218X(99)00230-9
  8. 8.
    del Río, Á., Rifà, J.: Families of Hadamard \(\mathbb{Z}_{2}\mathbb{Z}_{4}Q_{8}\)-codes. IEEE Trans. Inf. Theory 59(8), 5140–5151 (2013). http://dx.doi.org/http://dx.doi.org/10.1109/TIT.2013.2258373 doi:http://dx.doi.org/10.1109/TIT.2013.2258373

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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