Exchange of limits: why iterative decoding works
Satish Babu Korada Ruediger Urbanke
Proceedings of the IEEE International Symposium on Information Theory, Toronto, ON, Canada, July 2008

We consider communication over a family of binary-input memoryless output-symmetric channels using low-density parity-check codes under message passing  decoding . The asymptotic (in the length) performance of such a combination for a fixed number of iterations is given by density evolution. It is customary to define the threshold of density evolution as the maximum channel parameter for which the bit error probability under density evolution converges to zero as a function of the iteration number. In practice we often  work  with short codes and perform a large number of iterations. It is therefore interesting to consider what happens if in the standard analysis we exchange  the order in which the blocklength and the number of iterations diverge to infinity. In particular, we can ask whether both  limits  give the same threshold. Although empirical observations strongly suggest that the  exchange  of  limits  is valid for all channel parameters, we  limit  our discussion to channel parameters below the density evolution threshold. Specifically, we show that under some suitable technical conditions the bit error probability vanishes below the density evolution threshold regardless of how the  limit  is taken.