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Information, Physics, and Computation$
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Marc Mézard and Andrea Montanari

Print publication date: 2009

Print ISBN-13: 9780198570837

Published to Oxford Scholarship Online: September 2009

DOI: 10.1093/acprof:oso/9780198570837.001.0001

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Glassy states in coding theory

Glassy states in coding theory

Chapter:
(p.493) 21 Glassy states in coding theory
Source:
Information, Physics, and Computation
Author(s):

Marc Mézard

Andrea Montanari

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198570837.003.0021

In the limit of large block-length, iteratively decoded low density parity check (LDPC) codes exhibit two phase transitions. At low noise, the bit error rate under belief propagation decoding vanishes. In a second regime belief propagation decoding fails but maximum a posteriori probability (MAP) decoding succeeds. Finally above a second noise threshold decoding is impossible even with unbounded computational power. This chapter develops a common approach to these two transitions, through the study of ‘metastable’ configurations of the bits that are not codewords. It identifies the belief propagation phase transition with the onset of a dynamical glass phase, detected through the one step replica symmetry breaking approach. This is a structural phenomenon that spoils the performance of a large variety of decoders, from general iterative message passing schemes to simulated annealing.

Keywords:   decoding, low density parity check, code, LDPC, belief propagation, maximum a posteriori probability, MAP, glass, metastable

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