Learning LDPC codes with density evolution over relaxed protographs
Abstract
We consider the design of low-density parity-check (LDPC) codes for a given iterative decoder. While LDPC performance can be evaluated using simulation, density evolution (DE), or EXIT-chart analysis, selecting a parity-check matrix (PCM) remains a difficult combinatorial optimization problem. Existing approaches often rely on population-based search, random mutations, or genetic algorithms, which require careful tuning and incur high computational cost. Recent gradient descent (GD)-based methods optimize relaxed PCMs by differentiating through decoder simulations, but rely on noisy Monte Carlo estimates, line searches over soft matrix representations, and remain costly for long codes. Moreover, the loss is typically evaluated only at integer-valued PCMs. We focus on long protograph-based LDPC codes and propose a deterministic GD-based framework operating directly on a relaxed protograph representation. The loss is based on DE bit error rate (BER) and can be evaluated directly for relaxed protographs. To justify this relaxation, we associate the relaxed representation with an ensemble of binary PCMs and show that the proposed relaxed DE yields the ensemble-averaged DE performance. The resulting procedure supports standard GD optimization and achieves fast, reliable convergence through deterministic DE evaluation and informative gradients. Numerical results show that the optimized protographs outperform 5G LDPC codes with matching dimensions.
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BibTeX
@misc{shutkov2026learning,
title = {{Learning LDPC codes with density evolution over relaxed protographs}},
author = {Gennady Shutkov and Dmitry Artemasov and Alexey Frolov and Pavel Rybin and Kirill Andreev},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.08484},
eprint = {2607.08484},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.08484},
}
RIS
TY - UNPB
TI - Learning LDPC codes with density evolution over relaxed protographs
AU - Gennady Shutkov
AU - Dmitry Artemasov
AU - Alexey Frolov
AU - Pavel Rybin
AU - Kirill Andreev
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2607.08484
UR - https://arxiv.org/abs/2607.08484
AB - We consider the design of low-density parity-check (LDPC) codes for a given iterative decoder. While LDPC performance can be evaluated using simulation, density evolution (DE), or EXIT-chart analysis, selecting a parity-check matrix (PCM) remains a difficult combinatorial optimization problem. Existing approaches often rely on population-based search, random mutations, or genetic algorithms, which require careful tuning and incur high computational cost. Recent gradient descent (GD)-based methods optimize relaxed PCMs by differentiating through decoder simulations, but rely on noisy Monte Carlo estimates, line searches over soft matrix representations, and remain costly for long codes. Moreover, the loss is typically evaluated only at integer-valued PCMs. We focus on long protograph-based LDPC codes and propose a deterministic GD-based framework operating directly on a relaxed protograph representation. The loss is based on DE bit error rate (BER) and can be evaluated directly for relaxed protographs. To justify this relaxation, we associate the relaxed representation with an ensemble of binary PCMs and show that the proposed relaxed DE yields the ensemble-averaged DE performance. The resulting procedure supports standard GD optimization and achieves fast, reliable convergence through deterministic DE evaluation and informative gradients. Numerical results show that the optimized protographs outperform 5G LDPC codes with matching dimensions.
ER -