%0 Book Section %1 vicente03 %A Vicente, Renato %A Saad, David %A Kabashima, Yoshiyuki %B Advances in Imaging and Electron Physics %D 2003 %E Hawkes, Peter W. %E Kazan, Benjamin %E Mulvey, Tom %I Elsevier %K codes error-correcting-codes ldpc physics %P 231 -- 353 %R http://dx.doi.org/10.1016/S1076-5670(02)80018-0 %T Low-Density Parity-Check Codes--A Statistical Physics Perspective %U http://www.sciencedirect.com/science/article/pii/S1076567002800180 %V 125 %X Publisher Summary This chapter analyzes the error-correcting codes based on very sparse matrices by mapping them onto spin systems of the statistical physics. The equivalence between coding concepts and statistical physics is summarized in a tabulated form. In the statistical physics framework, random parity-check matrices (or generator matrices as in the case of Sourlas codes), random messages, and noise are treated as quenched disorder and the replica method is employed to compute the free-energy. A phase transition occurs between successful and failure states that coincides with the information theory upper bounds in most cases, the exception being MacKay–Neal (MN) codes with K = 2 (and to some extent K = 1) where the transition is below the upper bound. An analysis of replica symmetric quantities reveals unphysical behavior for low noise levels with the appearance of negative entropies. Despite the difficulties in the replica symmetric analysis, threshold noise values observed in simulations using probability propagation (PP) decoding agree with the noise level where metastable states (or spinodal points) appear in the replica symmetric free-energy.

@incollection{vicente03, abstract = {Publisher Summary This chapter analyzes the error-correcting codes based on very sparse matrices by mapping them onto spin systems of the statistical physics. The equivalence between coding concepts and statistical physics is summarized in a tabulated form. In the statistical physics framework, random parity-check matrices (or generator matrices as in the case of Sourlas codes), random messages, and noise are treated as quenched disorder and the replica method is employed to compute the free-energy. A phase transition occurs between successful and failure states that coincides with the information theory upper bounds in most cases, the exception being MacKay–Neal (MN) codes with K = 2 (and to some extent K = 1) where the transition is below the upper bound. An analysis of replica symmetric quantities reveals unphysical behavior for low noise levels with the appearance of negative entropies. Despite the difficulties in the replica symmetric analysis, threshold noise values observed in simulations using probability propagation (PP) decoding agree with the noise level where metastable states (or spinodal points) appear in the replica symmetric free-energy. }, added-at = {2015-04-22T10:22:34.000+0200}, author = {Vicente, Renato and Saad, David and Kabashima, Yoshiyuki}, biburl = {https://www.bibsonomy.org/bibtex/21979c3d03822d706380620cd1b357272/ytyoun}, doi = {http://dx.doi.org/10.1016/S1076-5670(02)80018-0}, editor = {Hawkes, Peter W. and Kazan, Benjamin and Mulvey, Tom}, interhash = {0e01ac3dd1ae96f332e41e023d5a182d}, intrahash = {1979c3d03822d706380620cd1b357272}, issn = {1076-5670}, keywords = {codes error-correcting-codes ldpc physics}, pages = {231 -- 353}, publisher = {Elsevier}, series = {Advances in Imaging and Electron Physics }, timestamp = {2015-11-09T12:35:19.000+0100}, title = {Low-Density Parity-Check Codes--A Statistical Physics Perspective }, url = {http://www.sciencedirect.com/science/article/pii/S1076567002800180}, volume = 125, year = 2003 }