An Entropy-based Proof of Threshold Saturation for Nonbinary SC-LDPC Ensembles on the BEC

TL;DR

This paper proves threshold saturation for nonbinary SC-LDPC ensembles over the BEC using an entropy-based approach, extending binary results to nonbinary codes with explicit potential functions and duality rules.

Contribution

It introduces a novel entropy duality rule and constructs explicit potential functions for nonbinary density evolution, establishing threshold saturation for nonbinary SC-LDPC codes.

Findings

Proves threshold saturation for nonbinary SC-LDPC on BEC.

Establishes entropy duality for nonbinary variable and check nodes.

Shows potential functions have monotonicity properties similar to binary cases.

Abstract

In this paper we are concerned with the asymptotic analysis of nonbinary spatially-coupled low-density parity-check (SC-LDPC) ensembles defined over GL$\left(2^{m}\right)$ (the general linear group of degree $m$ over GF$\left(2\right)$). Our purpose is to prove threshold saturation when the transmission takes place on the binary erasure channel (BEC). To this end, we establish the duality rule for entropy for nonbinary variable-node (VN) and check-node (CN) convolutional operators to accommodate the nonbinary density evolution (DE) analysis. Based on this, we construct the explicit forms of the potential functions for uncoupled and coupled DE recursions. In addition, we show that these functions exhibit similar monotonicity properties as those for binary LDPC and SC-LDPC ensembles over general binary memoryless symmetric (BMS) channels. This leads to the threshold saturation theorem and its converse for nonbinary SC-LDPC ensembles on the BEC, following the proof technique developed by S. Kumar et al.