# Factor Graph Optimization of Error-Correcting Codes for Belief Propagation Decoding

arXiv: https://arxiv.org/abs/2406.12900 · HTML: https://mlsanity.com/paper/2406.12900/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 68 references that are in the map.
- First posted on arXiv 9 Jun 2024.

## Builds on (most cited first, 23 of 23)

- [Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation](https://mlsanity.com/paper/1308.3432/) (2013)
- [XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks](https://mlsanity.com/paper/1603.05279/) (2016)
- [Understanding Straight-Through Estimator in Training Activation Quantized Neural Nets](https://mlsanity.com/paper/1903.05662/) (2019)
- [Sionna: An Open-Source Library for Next-Generation Physical Layer Research](https://mlsanity.com/paper/2203.11854/) (2022)
- [Deep Learning Methods for Improved Decoding of Linear Codes](https://mlsanity.com/paper/1706.07043/) (2017)
- [Learning to Decode Linear Codes Using Deep Learning](https://mlsanity.com/paper/1607.04793/) (2016)
- [Error Correction Code Transformer](https://mlsanity.com/paper/2203.14966/) (2022)
- [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)
- [Deep Learning for Decoding of Linear Codes - A Syndrome-Based Approach](https://mlsanity.com/paper/1802.04741/) (2018)
- [Neural Offset Min-Sum Decoding](https://mlsanity.com/paper/1701.05931/) (2017)
- [Denoising Diffusion Error Correction Codes](https://mlsanity.com/paper/2209.13533/) (2022)
- [Five Disruptive Technology Directions for 5G](https://mlsanity.com/paper/1312.0229/) (2013)
- [Autoregressive Belief Propagation for Decoding Block Codes](https://mlsanity.com/paper/2103.11780/) (2021)
- [Deep Quantum Error Correction](https://mlsanity.com/paper/2301.11930/) (2023)
- [Boosting Learning for LDPC Codes to Improve the Error-Floor Performance](https://mlsanity.com/paper/2310.07194/) (2023)
- [DeepTurbo: Deep Turbo Decoder](https://mlsanity.com/paper/1903.02295/) (2019)
- [Learning Linear Block Error Correction Codes](https://mlsanity.com/paper/2405.04050/) (2024)
- [A Branch-and-Bound Algorithm for MDL Learning Bayesian Networks](https://mlsanity.com/paper/1301.3897/) (2013)
- [perm2vec: Graph Permutation Selection for Decoding of Error Correction Codes using Self-Attention](https://mlsanity.com/paper/2002.02315/) (2020)
- [Coded Random Access: Applying Codes on Graphs to Design Random Access Protocols](https://mlsanity.com/paper/1405.4127/) (2014)

## Built on by (most cited first, 3 of 3)

- [Learning Linear Block Codes with Gradient Quantization](https://mlsanity.com/paper/2503.16169/) (2025)
- [Learning High-Dimensional Parity Functions with Product Networks using Gradient Descent](https://mlsanity.com/paper/2605.28612/) (2026)
- [ZeroCode: On-demand Error-Correcting Code Construction from the Zero Matrix via Reinforcement Learning](https://mlsanity.com/paper/2609.34265/) (2026)

## Abstract (arXiv)

The design of optimal linear block codes capable of being efficiently decoded is of major concern, especially for short block lengths. As near capacity-approaching codes, Low-Density Parity-Check (LDPC) codes possess several advantages over other families of codes, the most notable being its efficient decoding via Belief Propagation. While many LDPC code design methods exist, the development of efficient sparse codes that meet the constraints of modern short code lengths and accommodate new channel models remains a challenge. In this work, we propose for the first time a gradient-based data-driven approach for the design of sparse codes. We develop locally optimal codes with respect to Belief Propagation decoding via the learning of the Factor graph under channel noise simulations. This is performed via a novel complete graph tensor representation of the Belief Propagation algorithm, optimized over finite fields via backpropagation and coupled with an efficient line-search method. The proposed approach is shown to outperform the decoding performance of existing popular codes by orders of magnitude and demonstrates the power of data-driven approaches for code design.