Construction of Near-Capacity Protograph LDPC Code Sequences with Block-Error Thresholds

TL;DR

This paper develops methods to construct protograph LDPC codes with near-capacity performance, ensuring double-exponential error decay and matching block-error thresholds to bit-error thresholds for reliable communication.

Contribution

It introduces conditions for the stability of density evolution, constructs large-girth protograph sequences with optimal thresholds, and optimizes small protographs for near-capacity performance.

Findings

Density evolution shows double-exponential error decay below threshold.

Constructed large-girth protographs achieve block-error thresholds equal to bit-error thresholds.

Optimized small protographs attain near-capacity thresholds for key channels.

Abstract

Density evolution for protograph Low-Density Parity-Check (LDPC) codes is considered, and it is shown that the message-error rate falls double-exponentially with iterations whenever the degree-2 subgraph of the protograph is cycle-free and noise level is below threshold. Conditions for stability of protograph density evolution are established and related to the structure of the protograph. Using large-girth graphs, sequences of protograph LDPC codes with block-error threshold equal to bit-error threshold and block-error rate falling near-exponentially with blocklength are constructed deterministically. Small-sized protographs are optimized to obtain thresholds near capacity for binary erasure and binary-input Gaussian channels.