Relations between belief propagation on erasure and symmetric channels

An upper bound on the performance of the belief propagation algorithm in decoding a code over a binary-input output-symmetric channel in terms of the decoding threshold of the code over the erasure channel is presented in this paper. Using this upper bound, we obtain the overhead of fountain codes on the erasure channel, provided that they are capacity-achieving on a symmetric channel. The upper bound is similar to a lower bound proved by Khandekar. The lower bound will be used to bound from above the reception overhead of fountain codes on symmetric channels.