# Autoregressive Belief Propagation for Decoding Block Codes

arXiv: https://arxiv.org/abs/2103.11780 · HTML: https://mlsanity.com/paper/2103.11780/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 53 references that are in the map.
- First posted on arXiv 23 Jan 2021.

## Builds on (most cited first, 19 of 19)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [HyperNetworks](https://mlsanity.com/paper/1609.09106/) (2016)
- [Deep Learning Methods for Improved Decoding of Linear Codes](https://mlsanity.com/paper/1706.07043/) (2017)
- [Learning to Detect](https://mlsanity.com/paper/1805.07631/) (2018)
- [Learning to Decode Linear Codes Using Deep Learning](https://mlsanity.com/paper/1607.04793/) (2016)
- [Neural Enhanced Belief Propagation on Factor Graphs](https://mlsanity.com/paper/2003.01998/) (2020)
- [Turbo Autoencoder: Deep learning based channel codes for point-to-point communication channels](https://mlsanity.com/paper/1911.03038/) (2019)
- [Communication Algorithms via Deep Learning](https://mlsanity.com/paper/1805.09317/) (2018)
- [Deepcode: Feedback Codes via Deep Learning](https://mlsanity.com/paper/1807.00801/) (2018)
- [Hyper-Graph-Network Decoders for Block Codes](https://mlsanity.com/paper/1909.09036/) (2019)
- [Neural Offset Min-Sum Decoding](https://mlsanity.com/paper/1701.05931/) (2017)
- [Fast Deep Learning for Automatic Modulation Classification](https://mlsanity.com/paper/1901.05850/) (2019)
- [Unsupervised Linear and Nonlinear Channel Equalization and Decoding using Variational Autoencoders](https://mlsanity.com/paper/1905.08795/) (2019)
- [WGAN-based Autoencoder Training Over-the-air](https://mlsanity.com/paper/2003.02744/) (2020)
- [Reinforcement Learning for Channel Coding: Learned Bit-Flipping Decoding](https://mlsanity.com/paper/1906.04448/) (2019)
- [Blind Channel Equalization using Variational Autoencoders](https://mlsanity.com/paper/1803.01526/) (2018)
- [What Can Machine Learning Teach Us about Communications?](https://mlsanity.com/paper/1901.07592/) (2019)
- [Ensemble Wrapper Subsampling for Deep Modulation Classification](https://mlsanity.com/paper/2005.04586/) (2020)
- [Deep Ensemble of Weighted Viterbi Decoders for Tail-Biting Convolutional Codes](https://mlsanity.com/paper/2009.02591/) (2020)

## Built on by (most cited first, 15 of 15)

- [Error Correction Code Transformer](https://mlsanity.com/paper/2203.14966/) (2022)
- [Denoising Diffusion Error Correction Codes](https://mlsanity.com/paper/2209.13533/) (2022)
- [CrossMPT: Cross-attention Message-Passing Transformer for Error Correcting Codes](https://mlsanity.com/paper/2405.01033/) (2024)
- [Boosting Learning for LDPC Codes to Improve the Error-Floor Performance](https://mlsanity.com/paper/2310.07194/) (2023)
- [Learning Linear Block Error Correction Codes](https://mlsanity.com/paper/2405.04050/) (2024)
- [How to Mask in Error Correction Code Transformer: Systematic and Double Masking](https://mlsanity.com/paper/2308.08128/) (2023)
- [Hybrid Mamba-Transformer Decoder for Error-Correcting Codes](https://mlsanity.com/paper/2505.17834/) (2025)
- [Factor Graph Optimization of Error-Correcting Codes for Belief Propagation Decoding](https://mlsanity.com/paper/2406.12900/) (2024)
- [Hybrid Neural Coded Modulation: Design and Training Methods](https://mlsanity.com/paper/2202.01972/) (2022)
- [Toward Uncertainty-Aware and Generalizable Neural Decoding for Quantum LDPC Codes](https://mlsanity.com/paper/2510.06257/) (2025)
- [Unified Error Correction Code Transformer with Low Complexity](https://mlsanity.com/paper/2410.03364/) (2024)
- [Cross-Attention Message-Passing Transformers for Code-Agnostic Decoding in 6G Networks](https://mlsanity.com/paper/2507.01038/) (2025)
- [Syndrome-Flow Consistency Model Achieves One-step Denoising Error Correction Codes](https://mlsanity.com/paper/2512.01389/) (2025)
- [In-Context Source and Channel Coding](https://mlsanity.com/paper/2601.10267/) (2026)
- [Score Based Error Correcting Code Decoder](https://mlsanity.com/paper/2605.28358/) (2026)

## Abstract (arXiv)

We revisit recent methods that employ graph neural networks for decoding error correcting codes and employ messages that are computed in an autoregressive manner. The outgoing messages of the variable nodes are conditioned not only on the incoming messages, but also on an estimation of the SNR and on the inferred codeword and on two downstream computations: (i) an extended vector of parity check outcomes, (ii) the mismatch between the inferred codeword and the re-encoding of the information bits of this codeword. Unlike most learned methods in the field, our method violates the symmetry conditions that enable the other methods to train exclusively with the zero-word. Despite not having the luxury of training on a single word, and the inability to train on more than a small fraction of the relevant sample space, we demonstrate effective training. The new method obtains a bit error rate that outperforms the latest methods by a sizable margin.