Nonuniform Fuchsian codes for noisy channels

We develop a new transmission scheme for additive white Gaussian noisy (AWGN) channels based on Fuchsian groups from rational quaternion algebras. The structure of the proposed Fuchsian codes is nonlinear and nonuniform, hence conventional decoding methods based on linearity and symmetry do not appl...

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Bibliographic Details
Published in:Journal of the Franklin Institute 2014-11, Vol.351 (11), p.5076-5098
Main Authors: Blanco-Chacón, Iván, Remón, Dionís, Hollanti, Camilla, Alsina, Montserrat
Format: Article
Language:English
Subjects:
20 Group theory and generalizations
Classificació AMS
Discontinuous groups
Fuchsian groups
Grups discontinus
Grups, Teoria de
Matemàtiques i estadística
Noisy channel
Teoria de grups
Àlgebra
Àrees temàtiques de la UPC
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Summary:We develop a new transmission scheme for additive white Gaussian noisy (AWGN) channels based on Fuchsian groups from rational quaternion algebras. The structure of the proposed Fuchsian codes is nonlinear and nonuniform, hence conventional decoding methods based on linearity and symmetry do not apply. Previously, only brute force decoding methods with complexity that is linear in the code size exist for general nonuniform codes. However, the properly discontinuous character of the action of the Fuchsian groups on the complex upper half-plane translates into decoding complexity that is logarithmic in the code size via a recently introduced point reduction algorithm.
ISSN:0016-0032
1879-2693
DOI:10.1016/j.jfranklin.2014.08.012