Nonuniform Fuchsian codes for noisy channels |
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| Authors: | Iván Blanco-Chacón Dionís Remón Camilla Hollanti Montserrat Alsina |
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| Institution: | 1. Aalto University, Department of Mathematics and Systems Analysis, P.O. Box 11100, FI-00076 AALTO, Helsinki, Finland;2. University of Barcelona, Faculty of Mathematics. Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain;3. Universitat Politècnica de Catalunya - BarcelonaTech, Dept. Applied Mathematics III - EPSEM, Av. Bases de Manresa 61-73, 08242 Manresa, Spain |
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| Abstract: | 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. |
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