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Kernels and p-Kernels of pr-ary 1-Perfect Codes

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Abstract

The rank of a q-ary code C is the dimension of the subspace spanned by C. The kernel of a q-ary code C of length n can be defined as the set of all translations leaving C invariant. Some relations between the rank and the dimension of the kernel of q-ary 1-perfect codes, over \(\mathbb{F}_{q} = GF(q)\) as well as over the prime field \(\mathbb{F}_{p}\), are established. Q-ary 1-perfect codes of length n=(qm − 1)/(q − 1) with different kernel dimensions using switching constructions are constructed and some upper and lower bounds for the dimension of the kernel, once the rank is given, are established.

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Authors and Affiliations

  1. Discrete and Statistical Sciences, Auburn University, Auburn, Al, 36849-5307, USA

    K. T. Phelps

  2. Information and Communications Engineering, Universitat Autònoma de Barcelona, 08193, Bellaterra, Spain

    J. Rifà

  3. Information and Communications Engineering, Universitat Autònoma de Barcelona, 08193, Bellaterra, Spain

    M. Villanueva

Authors
  1. K. T. Phelps
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  2. J. Rifà
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  3. M. Villanueva
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Correspondence to M. Villanueva.

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Communicated by: I.F. Blake

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Phelps, K.T., Rifà, J. & Villanueva, M. Kernels and p-Kernels of pr-ary 1-Perfect Codes. Des Codes Crypt 37, 243–261 (2005). https://doi.org/10.1007/s10623-004-3989-x

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  • DOI: https://doi.org/10.1007/s10623-004-3989-x

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