Fuchsian codes with arbitrarily high code rates

• I. Blanco-Chacón ^[1] ; C. Hollanti ^[2] ; M. Alsina ^[3] ; D. Remón ^[4]
  1. [1] University College Dublin

     University College Dublin

     Irlanda

     
  2. [2] Aalto University

     Aalto University

     Helsinki, Finlandia

     
  3. [3] Universitat Politècnica de Catalunya

     Universitat Politècnica de Catalunya

     Barcelona, España

     
  4. [4] Universitat de Barcelona

     Universitat de Barcelona

     Barcelona, España

     

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• Localización: Journal of pure and applied algebra, ISSN 0022-4049, Vol. 220, Nº 1 (January 2016), 2016, págs. 180-196
• Idioma: inglés
• DOI: 10.1016/j.jpaa.2015.06.005
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• Resumen

  • Recently, Fuchsian codes have been proposed in Blanco-Chacón et al. (2014) [2] for communication over channels subject to additive white Gaussian noise (AWGN). The two main advantages of Fuchsian codes are their ability to compress information, i.e., high code rate, and their logarithmic decoding complexity. In this paper, we improve the first property further by constructing Fuchsian codes with arbitrarily high code rates while maintaining logarithmic decoding complexity. Namely, in the case of Fuchsian groups derived from quaternion algebras over totally real fields we obtain a code rate that is proportional to the degree of the base field. In particular, we consider arithmetic Fuchsian groups of signature (1;e) to construct explicit codes having code rate six, meaning that we can transmit six independent integers during one channel use.