Fuchsian codes with arbitrarily high code rates

• Published in: Journal of pure and applied algebra Vol. 220; no. 1; pp. 180 - 196
• Main Authors: Blanco-Chacón, I., Hollanti, C., Alsina, M., Remón, D.
• Format: Journal Article Publication
• Language: English
• Published: Elsevier B.V 01.01.2016
• Subjects: 20 Group theory and generalizations; Classificació AMS; Codificació, Teoria de la; Coding theory; Domains; Fuchsian groups; Fuchsian, Grups de; Matemàtiques i estadística; Teoria de grups; Time Block-Codes; Àlgebra; Àrees temàtiques de la UPC
• ISSN: 0022-4049; 1873-1376
• DOI: 10.1016/j.jpaa.2015.06.005

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.