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Consistency Flow Model Achieves One-step Denoising Error Correction Codes

Haoyu Lei, Chin Wa Lau, Kaiwen Zhou, Nian Guo, Farzan Farnia

TL;DR

ECCFM introduces a one-step, consistency-based decoding framework for error correction codes by formulating the reverse denoising process as a PF-ODE and enforcing a differentiable time condition via soft-syndrome. The method is architecture-agnostic and trains a single function to map any noisy observation directly to the original codeword, achieving competitive BER with substantial latency reductions. Empirical results across BCH, Polar, LDPC, and other codes demonstrate strong BER performance, especially for long codes, while delivering 30x–100x faster inference than diffusion-based decoders. The work advances practical neural ECC decoders by combining diffusion-inspired consistency ideas with differentiable syndrome conditioning, enabling high-throughput, low-latency decoding suitable for real-world systems.

Abstract

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 with transformer backbones achieve state-of-the-art performance, but their iterative sampling limits practicality in low-latency settings. We introduce the Error Correction Consistency Flow Model (ECCFM), an architecture-agnostic training framework for high-fidelity one-step decoding. By casting the reverse denoising process as a Probability Flow Ordinary Differential Equation (PF-ODE) and enforcing smoothness through a differential time regularization, ECCFM learns to map noisy signals along the decoding trajectory directly to the original codeword in a single inference step. Across multiple decoding benchmarks, ECCFM attains lower bit-error rates (BER) than autoregressive and diffusion-based baselines, with notable improvements on longer codes, while delivering inference speeds up from 30x to 100x faster than denoising diffusion decoders.

Consistency Flow Model Achieves One-step Denoising Error Correction Codes

TL;DR

ECCFM introduces a one-step, consistency-based decoding framework for error correction codes by formulating the reverse denoising process as a PF-ODE and enforcing a differentiable time condition via soft-syndrome. The method is architecture-agnostic and trains a single function to map any noisy observation directly to the original codeword, achieving competitive BER with substantial latency reductions. Empirical results across BCH, Polar, LDPC, and other codes demonstrate strong BER performance, especially for long codes, while delivering 30x–100x faster inference than diffusion-based decoders. The work advances practical neural ECC decoders by combining diffusion-inspired consistency ideas with differentiable syndrome conditioning, enabling high-throughput, low-latency decoding suitable for real-world systems.

Abstract

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 with transformer backbones achieve state-of-the-art performance, but their iterative sampling limits practicality in low-latency settings. We introduce the Error Correction Consistency Flow Model (ECCFM), an architecture-agnostic training framework for high-fidelity one-step decoding. By casting the reverse denoising process as a Probability Flow Ordinary Differential Equation (PF-ODE) and enforcing smoothness through a differential time regularization, ECCFM learns to map noisy signals along the decoding trajectory directly to the original codeword in a single inference step. Across multiple decoding benchmarks, ECCFM attains lower bit-error rates (BER) than autoregressive and diffusion-based baselines, with notable improvements on longer codes, while delivering inference speeds up from 30x to 100x faster than denoising diffusion decoders.

Paper Structure

This paper contains 24 sections, 1 theorem, 21 equations, 11 figures, 9 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Preliminaries
  4. Method: Consistency Flow Model for One-step decoding
  5. Error Correction Consistency Flow Property
  6. Differential Consistency Condition for Smooth Trajectory
  7. Error Correction Consistency Flow Training Dynamics
  8. Numerical Results
  9. Conclusions and Limitations
  10. Appendix
  11. Preliminary on Consistency Models
  12. Consistency Sampling Algorithm
  13. Proof of Proposition \ref{['prop']}
  14. Detailed Experimental Setting
  15. Additional experimental results
  16. ...and 9 more sections

Key Result

Proposition 4.1

Let $\mathcal{L}_{\text{Standard-CM}}$ be defined by the Total Variation distance, $\text{TV}(\cdot,\cdot)$, and $\mathcal{L}_{\text{EC-CM}}$ be defined by the Binary Cross Entropy, $\text{BCE}(\cdot,\cdot)$. For any timesteps $t, r$ and ground truth $\mathbf{x}_0$, the following semi–triangle inequ

Figures (11)

  • Figure 1: Illustration of our proposed Error Correction Consistency Flow Model (ECCFM). ECCFM learns to map received signals from the trajectories to a single, consistent codeword prediction $\mathbf{x}_0$, represented by a $\delta$ function.
  • Figure 2: Training Dynamics from iterative denoising to 1-step consistency decoding. DDECC's iterative diffusion denoising learns a noise predictor, $\epsilon_\theta(\cdot, e_t)$, requiring a multi-step iterative process to reverse the noise and decode the codeword. Our ECCFM, $f_\theta(\cdot, e^\dagger)$, directly learns the mapping from any noisy signal to the original clean codeword. By using the smooth soft-syndrome condition ($e^\dagger$), it achieves successful decoding in a single step.
  • Figure 3: Decoding trajectories for models conditioned on Hard Syndrome versus Soft Syndrome on a POLAR(64,48) code. The soft-syndrome conditioning results in a smoother path to a valid codeword. Additional Results under low SNR are available in Appendix \ref{['abl_snr']}, Figure \ref{['fig:syn_images']}.
  • Figure 4: Performance comparison of various decoding baselines on medium-to-long block codes. The plot shows the Bit Error Rate (BER) at different Signal-to-Noise Ratios (SNRs), from 2 dB to 6 dB and divided by 0.5 dB.
  • Figure 5: Comparison of Inference Time (top) and Throughput (bottom) across various decoding baselines and code types.
  • ...and 6 more figures

Theorems & Definitions (2)

  • Proposition 4.1
  • proof