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 codes based on an irregular bipartite graph. Index Terms Low density parity check codes, expander graph, belief propagation, iterative decoding.