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
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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.