Learning LDPC codes with quantized density evolution over relaxed protographs

2026-07-09

AI Summary

This work addresses the limitations of conventional LDPC code design under iterative decoding—namely, combinatorial optimization difficulty, high computational cost, and complex parameter tuning—by proposing a deterministic gradient-based end-to-end optimization framework. The approach operates in a relaxed protograph space, modeling protograph elements as probabilities of being one, and introduces for the first time a differentiable density evolution (DE) model whose average performance is equivalent to that of the corresponding ensemble of binary protographs. Using DE-based bit error rate as the loss function enables efficient training without Monte Carlo estimation or line search. Combined with quantized DE and gradient optimization, the method significantly enhances design efficiency. Under min-sum decoding, the resulting long-block-length protographs outperform 5G LDPC codes of the same dimension and exhibit fast, reliable convergence.

Abstract

We consider the design of low-density parity-check (LDPC) codes for a given iterative decoder. Despite tools such as direct simulation, density evolution (DE), and EXIT-chart analysis, selecting a parity-check matrix remains a difficult combinatorial optimization problem. Existing approaches often rely on population-based search, random mutations, genetic algorithms, or related heuristics, which require careful parameter tuning and may be computationally expensive. Recent gradient descent (GD)-based methods optimize relaxed parity-check matrices by differentiating through decoder simulations. However, such decoder-in-the-loop strategies rely on noisy Monte Carlo estimates, require line search over soft matrix representations, and remain costly for long LDPC codes. Moreover, although optimization is performed in a relaxed domain, the loss is typically evaluated only at integer-valued parity-check matrices. In this work, we focus on the design of long protograph-based LDPC codes and propose a deterministic GD-based framework that operates directly on a relaxed protograph representation. Each protograph entry is interpreted as the probability that the corresponding element is equal to one. The loss function is based on DE bit error rate (BER) performance and can be evaluated directly for relaxed protographs. To justify this relaxation, we associate the relaxed representation with an ensemble of binary protographs and show that the proposed relaxed DE gives the ensemble-averaged DE performance. The resulting optimization procedure is fully autonomous and uses standard GD methods. Owing to deterministic DE evaluation and informative gradients, the proposed approach provides fast and reliable convergence. Numerical experiments for the min-sum decoder show that the optimized protographs outperform 5G LDPC codes with the same protograph dimensions.

Problem

• LDPC codes
• protograph
• density evolution
• code design
• iterative decoding

Innovation

• quantized density evolution
• relaxed protographs
• gradient-based LDPC design
• ensemble-averaged performance
• deterministic optimization