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Weighted codes in Lee metrics

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Abstract

Perfect weighted coverings of radius one have been often studied in the Hamming metric. In this paper, we study these codes in the Lee metric. To simplify the notation, we use a slightly different description, yet equivalent. Given two integers a and b, an (a, b)-code is a set of vertices such that vertices in the code have a neighbours in the code and other vertices have b neighbours in the code. An (a, b)-code is exactly a perfect weighted covering of radius one with weight \({(\frac{b-a}{b},\frac{1}{b})}\). In this paper, we prove results of existence as well as of non-existence for (a, b)-codes on the multidimensional grid graphs.

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Authors and Affiliations

  1. ERTÉ “Maths à modeler”, Laboratoire de Recherche en Informatique, Université Paris Sud, Orsay, 91405, France

    Paul Dorbec

  2. ERTÉ “Maths à modeler”, Institut Fourier, 100 rue des maths, BP74, 38402, Saint Martin d’Hères, France

    Sylvain Gravier & Michel Mollard

  3. Department of Mathematics, University of Turku, 20014, Turku, Finland

    Iiro Honkala

Authors
  1. Paul Dorbec
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  2. Sylvain Gravier
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  3. Iiro Honkala
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  4. Michel Mollard
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Corresponding author

Correspondence to Paul Dorbec.

Additional information

Communicated by D. Ghinelli.

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Dorbec, P., Gravier, S., Honkala, I. et al. Weighted codes in Lee metrics. Des. Codes Cryptogr. 52, 209–218 (2009). https://doi.org/10.1007/s10623-009-9277-z

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  • DOI: https://doi.org/10.1007/s10623-009-9277-z

Keywords

Mathematics Subject Classifications (2000)