Proving Threshold Saturation for Nonbinary SC-LDPC Codes on the Binary Erasure Channel

TL;DR

This paper proves that belief propagation decoding for nonbinary SC-LDPC codes on the binary erasure channel achieves the potential threshold, extending the understanding of threshold saturation in coding theory.

Contribution

It generalizes the proof of threshold saturation to nonbinary codes on the binary erasure channel using potential functions.

Findings

Proof of threshold saturation for nonbinary SC-LDPC codes

Existence conditions for potential functions in vector systems

Density evolution and simulation results confirming theoretical predictions

Abstract

We analyze nonbinary spatially-coupled low-density parity-check (SC-LDPC) codes built on the general linear group for transmission over the binary erasure channel. We prove threshold saturation of the belief propagation decoding to the potential threshold, by generalizing the proof technique based on potential functions recently introduced by Yedla et al.. The existence of the potential function is also discussed for a vector sparse system in the general case, and some existence conditions are developed. We finally give density evolution and simulation results for several nonbinary SC-LDPC code ensembles.