Iterative decoding of LDPC codes: Some bounds and properties

We consider iterative message passing algorithms for decoding low-density parity-check codes, when applied to an arbitrary binary-input symmetric-output channel, and review some bounds and properties of these algorithms that we recently derived. We show that expander graph arguments may be used to prove that message passing algorithms can correct a linear number of erroneous messages. The implication of this result is that when the block length is sufficiently large, once a message passing algorithm has corrected a sufficiently large fraction of the errors, it will eventually correct all errors. The results hold for various message passing algorithms, including Gallager's hard decision and soft decision (with clipping) decoding algorithms. We also discuss some other properties of the iterative algorithm, such as the gap between maximum likelihood and iterative decoding as the connectivity of the parity-check matrix increases.