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Advances in imaging and electron physics

Low-density parity-check codes—A statistical physics perspective

Renato Vicente, David Saad, Yoshiyuki Kabashima

Published Jan 1, 2003• 16 citations
Publisher
DOI
10.1016/s1076-5670(02)80018-0

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arXiv:cond-mat/0010073•Europhysics Letters (EPL)•Nov 1, 2001•44 citations
Low-density parity-check (LDPC) codes on symmetric memoryless channels have been analyzed using statistical physics by several authors. In this paper, statistical mechanical analysis of LDPC codes is performed for asymmetric memoryless channels and general Markov channels. It is shown that the saddle point equations of the replica symmetric solution for a Markov channel is equivalent to the density evolution of the belief propagation on the factor graph representing LDPC codes on the Markov channel. The derivation uses the method of types for Markov chain.
A variation of low-density parity check (LDPC) error-correcting codes defined over Galois fields ( GF ( q )) is investigated using statistical physics. A code of this type is characterised by a sparse random parity check matrix composed of C non-zero elements per column. We examine the dependence of the code performance on the value of q , for finite and infinite C values, both in terms of the thermodynamical transition point and the practical decoding phase characterised by the existence of a unique (ferromagnetic) solution. We find different q -dependence in the cases of C = 2 and C ⩾ 3; the analytical solutions are in agreement with simulation results, providing a quantitative measure to the improvement in performance obtained using non-binary alphabets.