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This paper provides simple lower bounds on the number of iterations which isrequired for successful message-passing decoding of some important families ofgraph-based code ensembles (including low-density parity-check codes andvariations of repeat-accumulate codes). The transmission of the code ensemblesis assumed to take place over a binary erasure channel, and the bounds refer tothe asymptotic case where we let the block length tend to infinity. Thesimplicity of the bounds derived in this paper stems from the fact that theyare easily evaluated and are expressed in terms of some basic parameters of theensemble which include the fraction of degree-2 variable nodes, the target biterasure probability and the gap between the channel capacity and the designrate of the ensemble. This paper demonstrates that the number of iterationswhich is required for successful message-passing decoding scales at least likethe inverse of the gap (in rate) to capacity, provided that the fraction ofdegree-2 variable nodes of these turbo-like ensembles does not vanish (hence,the number of iterations becomes unbounded as the gap to capacity vanishes).

Bounds on the Number of Iterations for Turbo-Like Ensembles over the Binary Erasure Channe