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Hadamard full propelinear codes of type Q; rank and kernel

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Abstract

Hadamard full propelinear codes (\(\mathrm{HFP}\)-codes) are introduced and their equivalence with Hadamard groups is proven; on the other hand, the equivalence of Hadamard groups, relative (4n, 2, 4n, 2n)-difference sets in a group, and cocyclic Hadamard matrices, is already known. We compute the available values for the rank and dimension of the kernel of \(\mathrm{HFP}\)-codes of type Q and we show that the dimension of the kernel is always 1 or 2. We also show that when the dimension of the kernel is 2 then the dimension of the kernel of the transposed code is 1 (so, both codes are not equivalent). Finally, we give a construction method such that from an \(\mathrm{HFP}\)-code of length 4n, dimension of the kernel \(k=2\), and maximum rank \(r=2n\), we obtain an \(\mathrm{HFP}\)-code of double length 8n, dimension of the kernel \(k=2\), and maximum rank \(r=4n\).

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Acknowledgements

The authors are grateful to the anonymous referees for their helpful comments, which have improved the presentation of the results of this paper. This work has been partially supported by the Spanish MICINN Grants TIN2016-77918-P, MTM2015-69138-REDT and the Catalan AGAUR Grant 2014SGR-691.

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Authors and Affiliations

  1. Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, 08193, Bellaterra, Spain

    J. Rifà & Emilio Suárez Canedo

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  1. J. Rifà
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  2. Emilio Suárez Canedo
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Correspondence to J. Rifà.

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Communicated by J. D. Key.

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Rifà, J., Suárez Canedo, E. Hadamard full propelinear codes of type Q; rank and kernel. Des. Codes Cryptogr. 86, 1905–1921 (2018). https://doi.org/10.1007/s10623-017-0429-2

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  • DOI: https://doi.org/10.1007/s10623-017-0429-2

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