Spatially Coupled Ensembles Universally Achieve Capacity under Belief Propagation

TL;DR

This paper demonstrates that spatially coupled code ensembles can universally approach channel capacity under belief propagation decoding across a broad class of binary-input memoryless channels, explaining their exceptional performance.

Contribution

It extends the capacity-achieving property of spatially coupled ensembles from erasure channels to general binary-input symmetric channels, providing a universal construction.

Findings

Spatial coupling raises belief propagation thresholds to near the maximum a-priori threshold.

Most codes in the ensemble meet the desired error probability and capacity gap.

The proof shows these ensembles achieve the area threshold of the underlying codes.

Abstract

We investigate spatially coupled code ensembles. For transmission over the binary erasure channel, it was recently shown that spatial coupling increases the belief propagation threshold of the ensemble to essentially the maximum a-priori threshold of the underlying component ensemble. This explains why convolutional LDPC ensembles, originally introduced by Felstrom and Zigangirov, perform so well over this channel. We show that the equivalent result holds true for transmission over general binary-input memoryless output-symmetric channels. More precisely, given a desired error probability and a gap to capacity, we can construct a spatially coupled ensemble which fulfills these constraints universally on this class of channels under belief propagation decoding. In fact, most codes in that ensemble have that property. The quantifier universal refers to the single ensemble/code which is good for all channels but we assume that the channel is known at the receiver. The key technical result is a proof that under belief propagation decoding spatially coupled ensembles achieve essentially the area threshold of the underlying uncoupled ensemble. We conclude by discussing some interesting open problems.