# Syndrome-Flow Consistency Model Achieves One-step Denoising Error Correction Codes

arXiv: https://arxiv.org/abs/2512.01389 · HTML: https://mlsanity.com/paper/2512.01389/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Discrete and text diffusion, one of 14 directions in Diffusion models.
- Builds on 32 of its 42 references that are in the map.
- First posted on arXiv 1 Dec 2025.

## Placed in

- Diffusion models / [Discrete and text diffusion](https://mlsanity.com/diffusion/discrete-and-text-diffusion/)

## Builds on (most cited first, 30 of 32)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [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)
- [SDXL: Improving Latent Diffusion Models for High-Resolution Image Synthesis](https://mlsanity.com/paper/2307.01952/) (2023)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [Align your Latents: High-Resolution Video Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2304.08818/) (2023)
- [Large Language Diffusion Models](https://mlsanity.com/paper/2502.09992/) (2025)
- [Discrete Diffusion Modeling by Estimating the Ratios of the Data Distribution](https://mlsanity.com/paper/2310.16834/) (2023)
- [Improved Techniques for Training Consistency Models](https://mlsanity.com/paper/2310.14189/) (2023)
- [Latent Video Diffusion Models for High-Fidelity Long Video Generation](https://mlsanity.com/paper/2211.13221/) (2022)
- [DIFUSCO: Graph-based Diffusion Solvers for Combinatorial Optimization](https://mlsanity.com/paper/2302.08224/) (2023)
- [Consistency Models Made Easy](https://mlsanity.com/paper/2406.14548/) (2024)
- [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)
- [Neural Offset Min-Sum Decoding](https://mlsanity.com/paper/1701.05931/) (2017)
- [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)
- [Autoregressive Belief Propagation for Decoding Block Codes](https://mlsanity.com/paper/2103.11780/) (2021)
- [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)
- [Recent Advances in Deep Learning for Channel Coding: A Survey](https://mlsanity.com/paper/2406.19664/) (2024)

## Built on by (most cited first, 1 of 1)

- [Scalable Mamba-Based Message-Passing Neural Decoder for Error-Correcting Codes](https://mlsanity.com/paper/2605.10681/) (2026)

## Abstract (arXiv)

Error Correction Codes (ECC) are fundamental to reliable digital communication, yet designing neural decoders that are both accurate and computationally efficient remains challenging. Recent denoising diffusion decoders achieve state-of-the-art performance, but their iterative sampling limits practicality in low-latency settings. To bridge this gap, consistency models (CMs) offer a potential path to high-fidelity one-step decoding. However, applying CMs to ECC presents a significant challenge: the discrete nature of error correction means the decoding trajectory is highly non-smooth, making it incompatible with a simple continuous timestep parameterization. To address this, we re-parameterize the reverse Probability Flow Ordinary Differential Equation (PF-ODE) by soft-syndrome condition, providing a smooth trajectory of signal corruption. Building on this, we propose the Error Correction Syndrome-Flow Consistency Model (ECCFM), a model-agnostic framework designed specifically for ECC task, ensuring the model learns a smooth trajectory from any noisy signal directly to the original codeword in a single step. Across multiple benchmarks, ECCFM attains lower bit-error-rate (BER) and frame-error-rate (FER) than transformer-based decoders, while delivering inference speeds 30x to 100x faster than iterative denoising diffusion decoders.