Expander graph arguments for message-passing algorithms

We show how expander-based 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. This result is then combined with known results on the ability of message-passing algorithms to reduce the number of errors to an arbitrarily small fraction for relatively high transmission rates. The results hold for various message-passing algorithms, including Gallager's hard-decision and soft-decision (with clipping) decoding algorithms. Our results assume low-density parity-check (LDPC) codes based on an irregular bipartite graph.