Denoising diffusion error correction code (DDECC) is a recent state-of-the-art neural decoding framework by casting decoding as a deterministic reverse diffusion process conditioned on parity-check errors. However, its heuristic step parameterization and lack of sample diversity limit decoding performance. We propose a time-conditioned diffusion-based decoder that reformulates the decoding task as a discrete-time reverse diffusion process driven by an advanced diffusion sampler. Our method introduces two key components: (i) a Transformerbased denoiser conditioned on diffusion time steps, and (ii) an initialization that maps the received signal to a valid diffusion state. To further enhance error correction, we propose a multiple diffusion sampling (MDS) algorithm that leverages stochasticity to generate L parallel decoding trajectories for enhanced solution exploration. Experiments on $(63,45) \mathbf{B C H}$ and $(64,42)$ polar codes show that our decoder can achieve superior decoding accuracy over DDECC with reduced complexity. As L increases, MDS approaches maximum-likelihood performance.
Denoising diffusion error correction code (DDECC) is a recent state-of-the-art neural decoding framework by casting decoding as a deterministic reverse diffusion process conditioned on parity-check errors. However, its heuristic step parameterization and lack of sample diversity limit decoding performance. We propose a time-conditioned diffusion-based decoder that reformulates the decoding task as a discrete-time reverse diffusion process driven by an advanced diffusion sampler. Our method introduces two key components: (i) a Transformerbased denoiser conditioned on diffusion time steps, and (ii) an initialization that maps the received signal to a valid diffusion state. To further enhance error correction, we propose a multiple diffusion sampling (MDS) algorithm that leverages stochasticity to generate L parallel decoding trajectories for enhanced solution exploration. Experiments on (63,45)BCH and (64,42) polar codes show that our decoder can achieve superior decoding accuracy over DDECC with reduced complexity. As L increases, MDS approaches maximum-likelihood performance.
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.
We introduce a diffusion framework for neural decoding of error-correcting codes (ECCs). We interpret decoding as a conditional diffusion process that progressively predicts codeword bits while treating the physical channel observation as fixed side information. At each stage, a neural model estimates bitwise posteriors conditioned on the physical observation and the current noisy state of the codeword. We focus particularly on masked diffusion, for which the reverse process naturally induces a Maxwell beam search (MBS) procedure inspired by the Maxwell construction of Measson et al. Rather than committing to a single uncertain decision, the decoder maintains multiple candidate trajectories, removes parity-infeasible candidates, and ranks the remaining candidates using accumulated reverse-transition log-probabilities. We evaluate the framework across multiple code families on binary phase-shift keying (BPSK) over additive white Gaussian noise (AWGN). Experiments show competitive decoding performance across Polar, low-density parity-check (LDPC), and Bose-Chaudhuri-Hocquenghem (BCH) codes. The MBS decoder consistently improves upon the corresponding single-trajectory diffusion decoder and, in all evaluated settings, improves upon available neural decoding results.
Quantum computers could solve problems beyond the reach of classical devices, but this potential depends on quantum error correction (QEC) to protect fragile quantum states from noise. A central challenge in QEC is decoding: inferring likely physical errors from syndrome patterns generated by repeated stabilizer measurements. Existing decoders, including graph-based and neural approaches, typically return a single correction hypothesis and therefore discard the richer posterior structure of the error distribution conditioned on the observed syndrome. Here we recast QEC decoding as posterior inference using discrete denoising diffusion, exploiting the analogy between stochastic error accumulation and the forward diffusion process. We introduce DiffQEC, a generative decoder that combines a syndrome processor for multi-round spatial-temporal syndrome histories with syndrome feature modulation to condition denoising on the observed syndrome throughout inference. On experimental data from Google's superconducting quantum processor, DiffQEC reduces logical error rates by up to 10.2% relative to minimum-weight perfect matching and by about 5% relative to tensor-network decoding. These improvements persist for larger code distances up to 17 under depolarizing noise and for logical circuits of increasing depth. Beyond accuracy, the learned posterior provides confidence estimates for post-selection and reveals physically meaningful error structure, establishing posterior generative decoding as a practical framework for QEC.
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.
Error-correcting codes enable reliable communication, yet practical soft decoding remains challenging across code families and block lengths. We propose SB-ECC, a score-based decoder that casts decoding as continuous-time denoising. A neural denoiser defines a probability-flow ordinary differential equation (ODE) that iteratively updates the noisy channel observation toward a valid codeword, guided by parity constraints. The model is trained across noise levels without time/SNR conditioning, enabling inference without SNR estimation and supporting a direct latency accuracy trade off controlled by the ODE solver budget. We use the raw signed channel observation as input for learning a continuous denoising field. Across 42 code/SNR settings, SB-ECC achieves the best BER in 39/42 entries, with an average SNR gain of 0.17dB and a maximum gain of 0.46dB over the strongest competing baseline, we showed that swapping the solver from Euler to DPM preserves -ln(BER) while reducing end-to-end decoding time by 8.86% on average (up to 12.82%).
The realization of fault-tolerant quantum computation relies on Quantum Error-Correcting Codes (QECC). Of particular interest are codes that combine a high encoding rate with sufficient distance. Existing algebraic constructions, such as lifted product (LP) codes, build a large quantum low-density parity-check (qLDPC) code from a classical base matrix. In this work we introduce LiftDiff, a generative approach to the discovery of QECC. A discrete denoising diffusion model is trained to generate base matrices over the group algebra F2[Zℓ], which are then expanded into LP codes. The reverse diffusion process is guided by an energy function that evaluates quantum code properties, including girth, degree, sparsity and decoding performance, steering the sampler toward matrices that yield LP codes with favorable parameters. Our method discovers codes with parameters such as [[169,13,11]], [[143,15,9]], and [[260,24,10]], including a [[247,19,14]] code that achieves the highest kd2/n among the baselines. Moreover, several discovered codes outperform the Gross code across a broad range of the tested physical error rates under both code-capacity and circuit-level noise.