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arXiv:2610.33559

Joint End-to-End Training of Weighted BP Decoder Ensembles

Felix Krieg, Paul Bezner, Stephan ten Brink

Published Oct 24, 2026• 0 citations
arXivHTMLPDF

Abstract

Ensemble decoding improves decoding reliability by combining component decoders with complementary failure events. Existing approaches typically obtain this diversity from code-specific structure, such as code automorphisms, multiple parity-check representations, or targeted modifications of the message-passing graph. In this work, we instead learn the diversity of an ensemble of weighted belief propagation decoders by jointly optimizing trainable Tanner-graph edge weights. A central design question is how the training objective should distribute gradient among the ensemble members to promote complementary decoding behavior. To this end, we compare four ensemble objectives under direct gradient-based optimization and identify best-member training as the most effective approach. For a (64,32) progressive edge-growth (PEG) low-density parity-check (LDPC) code, the resulting ensemble outperforms the considered trained and untrained belief-propagation-based ensemble baselines. With 10 members, it provides a gain of approximately 0.66 dB over a single belief propagation decoder at a target frame error rate (FER) of $10^{-4}$. Similar behavior is observed for a (128,64) CCSDS LDPC code, indicating that the learned complementarity is not specific to PEG. Although the ensemble performance improves substantially, all trained members show a higher standalone FER. The gain therefore results from learned specialization and complementary failure events rather than uniformly improved component decoders.

DOI
10.48550/arxiv.2610.33559

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Ensemble decoding improves decoding reliability by combining component decoders with complementary failure events. Existing approaches typically obtain this diversity from code-specific structure, such as code automorphisms, multiple parity-check representations, or targeted modifications of the message-passing graph. In this work, we instead learn the diversity of an ensemble of weighted belief propagation decoders by jointly optimizing trainable Tanner-graph edge weights. A central design question is how the training objective should distribute gradient among the ensemble members to promote complementary decoding behavior. To this end, we compare four ensemble objectives under direct gradient-based optimization and identify best-member training as the most effective approach. For a (64,32) progressive edge-growth (PEG) low-density parity-check (LDPC) code, the resulting ensemble outperforms the considered trained and untrained belief-propagation-based ensemble baselines. With 10 members, it provides a gain of approximately 0.66 dB over a single belief propagation decoder at a target frame error rate (FER) of 10−410^{-4}10−4. Similar behavior is observed for a (128,64) CCSDS LDPC code, indicating that the learned complementarity is not specific to PEG. Although the ensemble performance improves substantially, all trained members show a higher standalone FER. The gain therefore results from learned specialization and complementary failure events rather than uniformly improved component decoders.
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We consider automorphism ensemble decoding (AED) of quasi-cyclic (QC) low-density parity-check (LDPC) codes. Belief propagation (BP) decoding on the conventional factor graph is equivariant to the quasi-cyclic automorphisms and therefore prevents gains by AED. However, by applying small modifications to the parity-check matrix at the receiver side, we can break the symmetry without changing the code at the transmitter. This way, we can leverage a gain in error-correcting performance using an ensemble of identical BP decoders, without increasing the worst-case decoding latency. The proposed method is demonstrated using LDPC codes from the CCSDS, 802.11n and 5G standards and produces gains of 0.2 to 0.3 dB over conventional BP decoding. Compared to the similarly performing saturated BP (SBP), the proposed algorithm reduces the average decoding latency by more than eight times.
A low-density parity-check (LDPC) code is a linear block code described by a sparse parity-check matrix, which can be efficiently represented by a bipartite Tanner graph. The standard iterative decoding algorithm, known as belief propagation, passes messages along the edges of this Tanner graph. Density evolution is an efficient method to analyze the performance of the belief propagation decoding algorithm for a particular LDPC code ensemble, enabling the determination of a decoding threshold. The basic problem addressed in this work is how to optimize the Tanner graph so that the decoding threshold is as large as possible. We introduce a new code optimization technique which involves the search space range which can be thought of as minimizing randomness in differential evolution or limiting the search range in exhaustive search. This technique is applied to the design of good irregular LDPC codes and multi-edge type LDPC codes.
Weighted belief propagation (WBP) for the decoding of linear block codes is considered. In WBP, the Tanner graph of the code is unrolled with respect to the iterations of the belief propagation decoder. Then, weights are assigned to the edges of the resulting recurrent network and optimized offline using a training dataset. The main contribution of this paper is an adaptive WBP where the weights of the decoder are determined for each received word. Two variants of this decoder are investigated. In the parallel WBP decoders, the weights take values in a discrete set. A number of WBP decoders are run in parallel to search for the best sequence- of weights in real time. In the two-stage decoder, a small neural network is used to dynamically determine the weights of the WBP decoder for each received word. The proposed adaptive decoders demonstrate significant improvements over the static counterparts in two applications. In the first application, Bose-Chaudhuri-Hocquenghem, polar and quasi-cyclic low-density parity-check (QC-LDPC) codes are used over an additive white Gaussian noise channel. The results indicate that the adaptive WBP achieves bit error rates (BERs) up to an order of magnitude less than the BERs of the static WBP at about the same decoding complexity, depending on the code, its rate, and the signal-to-noise ratio. The second application is a concatenated code designed for a long-haul nonlinear optical fiber channel where the inner code is a QC-LDPC code and the outer code is a spatially coupled LDPC code. In this case, the inner code is decoded using an adaptive WBP, while the outer code is decoded using the sliding window decoder and static belief propagation. The results show that the adaptive WBP provides a coding gain of 0.8 dB compared to the neural normalized min-sum decoder, with about the same computational complexity and decoding latency.
In this letter we propose a novel framework for designing decoders, for Low-Density Parity Check (LDPC) codes, that surpasses the frame error rate performance of Belief-Propagation (BP) decoding on binary symmetric channels. Its key component is the adaptation method, based on the genetic optimization algorithm, that is incorporated into the recently proposed Gradient Descent Bit-Flipping Decoding with Momentum (GDBF-w/M). We show that the resulting decoder outperforms all state-of-the-art probabilistic bit-flipping decoders and, additionally, it can be trained to perform beyond BP decoding, which is verified by numerical examples that include codes used in IEEE 802.3an and 5GNR standards. The proposed framework provides a systematic method for decoder optimization without requiring knowledge of trapping sets. Moreover, it is applicable to both regular and irregular LDPC codes.
This thesis investigates the optimization of Low-Density Parity-Check (LDPC) decoding through machine learning techniques, with a specific focus on 5G communication systems. Traditional LDPC decoding algorithms face a critical trade-off: Belief Propagation (BP) offers excellent error-correction performance but with high computational complexity, while Min-Sum (MS) reduces complexity at the cost of degraded performance. To address this limitation, a hybrid neural network architecture is proposed that incorporates learnable normalization and offset weights into the message-passing framework of LDPC decoders. This approach integrates multiplicative (normalization) and additive (offset) weights into the Tanner graph structure, allowing the decoder to adapt to specific channel conditions. A novel weighted binary cross-entropy loss function is developed that prioritizes entire block recovery by assigning different importance to information and parity bits during training. Experimental results demonstrate that ML-assisted decoder consistently outperforms conventional Min-Sum decoding, approaching the error-correction capability of Belief Propagation while maintaining comparable computational efficiency to Min-Sum. Comprehensive evaluations across different weight configurations (scalar and vector-based) reveal that increased parametrization generally improves decoding performance. Analysis of the impact of iteration counts shows that the optimal number of decoding iterations varies between different code configurations with studied Basegraph 1 case requiring 15 iterations for optimal performance while studied Basegraph 2 case benefits significantly from 20 iterations. The proposed ML-assisted LDPC decoder represents a significant advancement for 5G systems, offering improved error correction at lower signal strengths while balancing latency and computational requirements. This research contributes to the ongoing evolution of wireless technology by demonstrating how machine learning can effectively enhance traditional decoding algorithms, providing a practical pathway toward more efficient and adaptive error correction in modern communication systems.