Life Above Threshold: From List Decoding to Area Theorem and MSE

We consider communication over mem- oryless channels using low-density parity-check code ensembles above the iterative (belief propagation) threshold. What is the computational complexity of decoding (i.e., of reconstructing all the typical in- put codewords for a given channel output) in this regime? We define an algorithm accomplishing this task and analyze its typical performance. The behav- ior of the new algorithm can be expressed in purely information-theoretical terms. Its analysis provides an alternative proof of the area theorem for the bi- nary erasure channel. Finally, we explain how the area theorem is generalized to arbitrary memoryless channels. We note that the recently discovered rela- tion between mutual information and minimal square error is an instance of the area theorem in the setting of Gaussian channels.