Error-Correcting Codes

Syndrome-based neural decoding (SBND) has emerged as a promising deep learning approach for soft-decision decoding of high-rate, short-length codes. However, this approach still has substantial room for improvement. In this paper, we show how to leverage code automorphisms to enhance the ability of existing SBND models to learn and generalize through data augmentation during training and inference. As a result, for the short high-rate codes considered, we obtain models that closely approach MLD performance using small datasets and proper training. Our findings also suggest that many prior results for SBND models in the literature underestimate their true correction capability due to undertraining. Code to reproduce all results is available at: https://github.com/lebidan/sbnd.

This paper presents a semantic-enhanced receiver framework for transmitting natural language sentences over noisy wireless channels using multiple short block codes. After ASCII encoding, the sentence is divided into segments, each independently encoded with a short block code and transmitted over an AWGN channel. At the receiver, segments are decoded in parallel, followed by a semantic error correction (SEC) model, which reconstructs corrupted segments using language model context. We further propose the semantic list decoding (SLD), which generates multiple candidate reconstructions and selects the best one via weighted Hamming distance, and a semantic confidence-guided HARQ (SHARQ) mechanism that replaces CRC-based error detection with a confidence score, enabling selective segment retransmission without CRC overhead. All modules are designed and trained using bidirectional and auto-regressive transformers (BART). Simulation results demonstrate that the proposed scheme significantly outperforms conventional capacity-approaching short codes and long codes at the same rate. Specifically, SEC provides approximately 0.4 dB BLER gain over plain short-code transmission, while SLD extends this to 0.8 dB. Compared to transmitting the entire sentence as a single long 5G LDPC codeword, our approach significantly improves semantic fidelity and reduces decoding latency by up to 90%. SHARQ further provides an additional 1.5 dB gain over conventional HARQ.

Concatenating quantum error correction codes scales error correction capability by driving logical error rates down double-exponentially across levels. However, the noise structure shifts under concatenation, making it hard to choose an optimal code sequence. We automate this choice by estimating the effective noise channel after each level and selecting the next code accordingly. In particular, we use learning-based methods to tailor small, non-additive encoders when the noise exhibits sufficient structure, then switch to standard codes once the noise is nearly uniform. In simulations, this level-wise adaptation achieves a target logical error rate with far fewer qubits than concatenating stabilizer codes alone--reducing qubit counts by up to two orders of magnitude for strongly structured noise. Therefore, this hybrid, learning-based strategy offers a promising tool for early fault-tolerant quantum computing.

Quantum error correction (QEC) is essential for scalable quantum computing. However, it requires classical decoders that are fast and accurate enough to keep pace with quantum hardware. While quantum low-density parity-check codes have recently emerged as a promising route to efficient fault tolerance, current decoding algorithms do not allow one to realize the full potential of these codes in practical settings. Here, we introduce a convolutional neural network decoder that exploits the geometric structure of QEC codes, and use it to probe a novel "waterfall" regime of error suppression, demonstrating that the logical error rates required for large-scale fault-tolerant algorithms are attainable with modest code sizes at current physical error rates, and with latencies within the real-time budgets of several leading hardware platforms. For example, for the $[144, 12, 12]$ Gross code, the decoder achieves logical error rates up to $\sim 17$x below existing decoders - reaching logical error rates $\sim 10^{-10}$ at physical error $p=0.1\%$ - with 3-5 orders of magnitude higher throughput. This decoder also produces well-calibrated confidence estimates that can significantly reduce the time overhead of repeat-until-success protocols. Taken together, these results suggest that the space-time costs associated with fault-tolerant quantum computation may be significantly lower than previously anticipated.

Diffusion models generate high-quality images but pose serious risks like copyright violation and disinformation. Watermarking is a key defense for tracing and authenticating AI-generated content. However, existing methods rely on threshold-based detection, which only supports fuzzy matching and cannot recover structured watermark data bit-exactly, making them unsuitable for offline verification or applications requiring lossless metadata (e.g., licensing instructions). To address this problem, in this paper, we propose Gaussian Shannon, a watermarking framework that treats the diffusion process as a noisy communication channel and enables both robust tracing and exact bit recovery. Our method embeds watermarks in the initial Gaussian noise without fine-tuning or quality loss. We identify two types of channel interference, namely local bit flips and global stochastic distortions, and design a cascaded defense combining error-correcting codes and majority voting. This ensures reliable end-to-end transmission of semantic payloads. Experiments across three Stable Diffusion variants and seven perturbation types show that Gaussian Shannon achieves state-of-the-art bit-level accuracy while maintaining a high true positive rate, enabling trustworthy rights attribution in real-world deployment. The source code have been made available at: https://github.com/Rambo-Yi/Gaussian-Shannon

A constant weight binary code consists of $n$-bit binary codewords, each with exactly $w$ bits equal to 1, such that any two codewords are at least Hamming distance $d$ apart. $A(n,d,w)$ is the maximum size of a constant weight binary code with parameters $n,d,w$. We establish improved lower bounds on $A(n,d,w)$ by constructing new larger codes, for 24 values of $(n,d,w)$ with $6 \leq d \leq 18$ and $18 \leq n \leq 35$. The improved lower bounds come from two strategies. The first is a tabu search that operates at the level of bit swaps. The second is a novel greedy heuristic that repeatedly chooses the candidate codeword that maximizes a randomly-scored histogram of distances to previously-added codewords. These strategies were proposed by CPro1, an automated protocol that generates, implements, and tests diverse strategies for combinatorial constructions.

We study neural networks (NNs) operating under hardware-induced memory faults and computational errors, which can degrade reliability in safety-critical applications. To address this challenge, we propose a novel real-valued error correction coding (ECC) framework capable of detecting and correcting both types of errors. Our approach imposes structured real-valued linear constraints on network weights, enabling efficient error detection and recovery without increasing the number of parameters or compromising classification performance. The proposed method leverages linear programming and sparse recovery techniques to achieve low-complexity error correction. To the best of our knowledge, this is the first real-number-based ECC scheme that simultaneously addresses memory and computational errors in neural networks.