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.
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.
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.
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.
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.
Maximum-likelihood (ML) decoding of linear block codes is NP-hard, so neural decoders are evaluated by how closely they approach it. We observe that proving a candidate optimal can be far easier than finding it, and build a decoder that does both. We cast decoding as a search over an absorbing-state discrete diffusion model, whose sequential commitment of coordinates turns generation into a tree, and run a probability beam search over it. A dual bound from a linear-programming relaxation and a distance-based test, both derived from the code, then certify per frame that the current candidate is the global ML solution; because both are theorems, certified frames incur no accuracy-computation trade-off. Across 15 codes and five operating points we certify up to 99.9% of frames, the two tests dividing the space as their derivations predict, one strong for sparse parity-check matrices and the other for codes with a large minimum distance. Certification also behaves as a per-frame reliability signal, with 96.7-100% of all frame errors falling on uncertified frames. Against an exact ML reference at 5 dB the decoder is within one standard deviation on three of the four codes where the reference is uncensored, and against four neural baselines it attains the best bit-error rate in 71 of 75 cells.
Low-density parity-check (LDPC) codes have been one of the most popular error correction candidates in modern communication and data storage systems for the past three decades due to their capacity-approaching error correction capability under the Belief Propagation (BP) decoding algorithms. In this dissertation, we investigate three topics related to efficient and effective decoding algorithms and the code design of LDPC codes. In the first part of this dissertation, we propose a methodology with the help of artificial intelligence to design Finite Alphabet Iterative Decoders (FAID) for LDPC codes, which are not only of great theoretical but also practical importance. This model-driven neural network (NN) framework unfolds the decoding iterations, introduces quantized neural networks (QNN) to model the FAIDs, and proposes a new cost function, namely, bit error rate (BER), to address the inefficiency in discrete-output channel scenarios. The low precision activations in the QNN and quantization in the cost functions cause a critical issue that their gradients vanish almost everywhere, making it difficult to use classical backward propagation. We design and leverage straight-through estimators as surrogate gradients to tackle this issue and provide a joint training scheme. These QNN-aided FAIDs, are capable of surpassing the floating-point BP algorithms, with much lower complexity and faster convergence. The QNN-aided approach is general, as it can be applied to both continuous and discrete-output channels. In the second part of this dissertation, we propose a new code construction of LDPC codes based on the conventional parity-check matrices of Reed-Solomon (RS) codes. We investigate the girth and cycle structure and introduce an efficient searching algorithm to reduce short cycles in their Tanner graphs and to construct LDPC codes with a girth of at least eight. We use masking to further reduce the number of short cycles and to adjust the column and row weights of the parity-check matrices of the constructed codes for performance improvement. Our designs are systematic and flexible, which balance the requirements of good waterfall performance and low error-floor while maintaining a structure that facilitates implementation. In the third part of this dissertation, we investigate binary quasi-cyclic (QC) LDPC codes and their capability to correct phased bursts of erasures. We prove that the peeling algorithm when applied to such parity-check matrices of column weight two can correct all erasures that can be corrected by a maximum likelihood (ML) decoder. We show that by modifying the parity-check matrices of column weight two and globally coupling them, the erasure correcting capability can be further enhanced. For the Additive White Gaussian Noise (AWGN) channel, we design QC LDPC codes with parity-check matrices of column weight three or more that can correct phased bursts of erasures and perform well over the AWGN channel. Our analysis of QC-LDPC codes with parity-check matrices of column weight two is rather comprehensive as we determine, for all such codes, their dimensions, minimum Hamming distances, and their capabilities to correct phased bursts.
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%).