# Denoising Diffusion Error Correction Codes

arXiv: https://arxiv.org/abs/2209.13533 · HTML: https://mlsanity.com/paper/2209.13533/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion for compression and wireless, one of 14 directions in Diffusion models.
- Builds on 18 of its 32 references that are in the map.
- First posted on arXiv 16 Sep 2022.

## Placed in

- Diffusion models / [Diffusion for compression and wireless](https://mlsanity.com/diffusion/diffusion-for-compression-and-wireless/)

## Builds on (most cited first, 18 of 18)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Image Super-Resolution via Iterative Refinement](https://mlsanity.com/paper/2104.07636/) (2021)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [On Layer Normalization in the Transformer Architecture](https://mlsanity.com/paper/2002.04745/) (2020)
- [WaveGrad: Estimating Gradients for Waveform Generation](https://mlsanity.com/paper/2009.00713/) (2020)
- [Noise Estimation for Generative Diffusion Models](https://mlsanity.com/paper/2104.02600/) (2021)
- [Error Correction Code Transformer](https://mlsanity.com/paper/2203.14966/) (2022)
- [Communication Algorithms via Deep Learning](https://mlsanity.com/paper/1805.09317/) (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)
- [Autoregressive Belief Propagation for Decoding Block Codes](https://mlsanity.com/paper/2103.11780/) (2021)

## Built on by (most cited first, 24 of 24)

- [CDDM: Channel Denoising Diffusion Models for Wireless Semantic Communications](https://mlsanity.com/paper/2309.08895/) (2023)
- [Enhancing Deep Reinforcement Learning: A Tutorial on Generative Diffusion Models in Network Optimization](https://mlsanity.com/paper/2308.05384/) (2023)
- [CrossMPT: Cross-attention Message-Passing Transformer for Error Correcting Codes](https://mlsanity.com/paper/2405.01033/) (2024)
- [CDDM: Channel Denoising Diffusion Models for Wireless Communications](https://mlsanity.com/paper/2305.09161/) (2023)
- [Generative Diffusion Models for Wireless Networks: Fundamental, Architecture, and State-of-the-Art](https://mlsanity.com/paper/2507.16733/) (2025)
- [Latent Diffusion Model-Enabled Low-Latency Semantic Communication in the Presence of Semantic Ambiguities and Wireless Channel Noises](https://mlsanity.com/paper/2406.06644/) (2024)
- [Deep Quantum Error Correction](https://mlsanity.com/paper/2301.11930/) (2023)
- [Learning Linear Block Error Correction Codes](https://mlsanity.com/paper/2405.04050/) (2024)
- [Diffusion Models for Wireless Communications](https://mlsanity.com/paper/2310.07312/) (2023)
- [CRISP: Curriculum based Sequential Neural Decoders for Polar Code Family](https://mlsanity.com/paper/2210.00313/) (2022)
- [Factor Graph Optimization of Error-Correcting Codes for Belief Propagation Decoding](https://mlsanity.com/paper/2406.12900/) (2024)
- [Accelerating Error Correction Code Transformers](https://mlsanity.com/paper/2410.05911/) (2024)
- [DeepPolar: Inventing Nonlinear Large-Kernel Polar Codes via Deep Learning](https://mlsanity.com/paper/2402.08864/) (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)
- [Disturbance-based Discretization, Differentiable IDS Channel, and an IDS-Correcting Code for DNA-based Storage](https://mlsanity.com/paper/2407.18929/) (2024)
- [5G LDPC Linear Transformer for Channel Decoding](https://mlsanity.com/paper/2501.14102/) (2025)
- [CoDiPhy: A General Framework for Applying Denoising Diffusion Models to the Physical Layer of Wireless Communication Systems](https://mlsanity.com/paper/2503.10297/) (2025)
- [On the Fragility of AI-Based Channel Decoders under Small Channel Perturbations](https://mlsanity.com/paper/2602.01582/) (2026)
- [Diffusion Denoiser Achievable Analysis for Finite Blocklength Unsourced Random Access](https://mlsanity.com/paper/2604.09904/) (2026)
- [Scalable Mamba-Based Message-Passing Neural Decoder for Error-Correcting Codes](https://mlsanity.com/paper/2605.10681/) (2026)
- [Variational Diffusion Channel Decoder](https://mlsanity.com/paper/2605.18902/) (2026)
- [Score Based Error Correcting Code Decoder](https://mlsanity.com/paper/2605.28358/) (2026)
- [Distance Prediction for Quantum Codes](https://mlsanity.com/paper/2610.18130/) (2026)

## Abstract (arXiv)

Error correction code (ECC) is an integral part of the physical communication layer, ensuring reliable data transfer over noisy channels. Recently, neural decoders have demonstrated their advantage over classical decoding techniques. However, recent state-of-the-art neural decoders suffer from high complexity and lack the important iterative scheme characteristic of many legacy decoders. In this work, we propose to employ denoising diffusion models for the soft decoding of linear codes at arbitrary block lengths. Our framework models the forward channel corruption as a series of diffusion steps that can be reversed iteratively. Three contributions are made: (i) a diffusion process suitable for the decoding setting is introduced, (ii) the neural diffusion decoder is conditioned on the number of parity errors, which indicates the level of corruption at a given step, (iii) a line search procedure based on the code's syndrome obtains the optimal reverse diffusion step size. The proposed approach demonstrates the power of diffusion models for ECC and is able to achieve state of the art accuracy, outperforming the other neural decoders by sizable margins, even for a single reverse diffusion step.