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