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

TL;DR

This paper explores decoding complexity above the belief propagation threshold for LDPC codes, introduces an algorithm with information-theoretic analysis, and generalizes the area theorem to all memoryless channels.

Contribution

It presents a new decoding algorithm applicable above the BP threshold and provides an information-theoretic analysis, offering an alternative proof of the area theorem and its generalization.

Findings

Algorithm achieves typical performance above the threshold

Provides an information-theoretic proof of the area theorem

Generalizes the area theorem to all memoryless channels

Abstract

We consider communication over memoryless 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 input codewords for a given channel output) in this regime? We define an algorithm accomplishing this task and analyze its typical performance. The behavior of the new algorithm can be expressed in purely information-theoretical terms. Its analysis provides an alternative proof of the area theorem for the binary erasure channel. Finally, we explain how the area theorem is generalized to arbitrary memoryless channels. We note that the recently discovered relation between mutual information and minimal square error is an instance of the area theorem in the setting of Gaussian channels.