Improved low-density parity-check codes using irregular graphs

TLDR

The authors aim to construct and improve low‑density parity‑check codes by introducing irregular parity‑check matrices and a rigorous analysis of hard‑decision decoding. They develop efficient methods for designing good irregular structures, employ martingale‑based analysis, and evaluate performance under hard‑decision decoding and belief propagation. Experiments show that for rate‑1/4 codes over a binary‑symmetric channel, the new irregular codes correct over 17% errors—about 1% better than previous LDPC codes—and approach turbo‑code performance.

Abstract

We construct new families of error-correcting codes based on Gallager's (1973) low-density parity-check codes. We improve on Gallager's results by introducing irregular parity-check matrices and a new rigorous analysis of hard-decision decoding of these codes. We also provide efficient methods for finding good irregular structures for such decoding algorithms. Our rigorous analysis based on martingales, our methodology for constructing good irregular codes, and the demonstration that irregular structure improves performance constitute key points of our contribution. We also consider irregular codes under belief propagation. We report the results of experiments testing the efficacy of irregular codes on both binary-symmetric and Gaussian channels. For example, using belief propagation, for rate 1/4 codes on 16000 bits over a binary-symmetric channel, previous low-density parity-check codes can correct up to approximately 16% errors, while our codes correct over 17%. In some cases our results come very close to reported results for turbo codes, suggesting that variations of irregular low density parity-check codes may be able to match or beat turbo code performance.