Design and Analysis of Nonbinary LDPC Codes for Arbitrary Discrete-Memoryless Channels

TL;DR

This paper analyzes nonbinary LDPC codes over arbitrary discrete-memoryless channels, introducing a permutation-invariance property that simplifies density analysis and demonstrates their superior performance at short block lengths and near Shannon capacity.

Contribution

It introduces a novel analysis method for nonbinary LDPC codes, including permutation-invariance and Gaussian approximation, and demonstrates their effectiveness with shaping gains and near-capacity performance.

Findings

Codes outperform multilevel codes at short block lengths

Achieve within 0.56 dB of Shannon limit at 6 bits/s/Hz

Developed EXIT charts for nonbinary LDPC codes

Abstract

We present an analysis, under iterative decoding, of coset LDPC codes over GF(q), designed for use over arbitrary discrete-memoryless channels (particularly nonbinary and asymmetric channels). We use a random-coset analysis to produce an effect that is similar to output-symmetry with binary channels. We show that the random selection of the nonzero elements of the GF(q) parity-check matrix induces a permutation-invariance property on the densities of the decoder messages, which simplifies their analysis and approximation. We generalize several properties, including symmetry and stability from the analysis of binary LDPC codes. We show that under a Gaussian approximation, the entire q-1 dimensional distribution of the vector messages is described by a single scalar parameter (like the distributions of binary LDPC messages). We apply this property to develop EXIT charts for our codes. We use appropriately designed signal constellations to obtain substantial shaping gains. Simulation results indicate that our codes outperform multilevel codes at short block lengths. We also present simulation results for the AWGN channel, including results within 0.56 dB of the unconstrained Shannon limit (i.e. not restricted to any signal constellation) at a spectral efficiency of 6 bits/s/Hz.