A Proof of Threshold Saturation for Spatially-Coupled LDPC Codes on BMS Channels

TL;DR

This paper proves that spatially-coupled LDPC codes achieve threshold saturation on BMS channels, extending previous results from erasure channels and generalizing the proof technique to a broader class of graphical models.

Contribution

It generalizes the proof of threshold saturation for spatially-coupled LDPC codes from erasure channels to BMS channels, broadening the theoretical understanding.

Findings

Threshold saturation proven for BMS channels

Extension of potential function method to irregular LDPC ensembles

Broader applicability to graphical models with Bethe free energy

Abstract

Low-density parity-check (LDPC) convolutional codes have been shown to exhibit excellent performance under low-complexity belief-propagation decoding [1], [2]. This phenomenon is now termed threshold saturation via spatial coupling. The underlying principle behind this appears to be very general and spatially-coupled (SC) codes have been successfully applied in numerous areas. Recently, SC regular LDPC codes have been proven to achieve capacity universally, over the class of binary memoryless symmetric (BMS) channels, under belief-propagation decoding [3], [4]. In [5], [6], potential functions are used to prove that the BP threshold of SC irregular LDPC ensembles saturates, for the binary erasure channel, to the conjectured MAP threshold (known as the Maxwell threshold) of the underlying irregular ensembles. In this paper, that proof technique is generalized to BMS channels, thereby extending some results of [4] to irregular LDPC ensembles. We also believe that this approach can be expanded to cover a wide class of graphical models whose message-passing rules are associated with a Bethe free energy.