Belief propagation (BP) decoding achieves near-optimal performance with large block lengths but faces performance degradation with short blocks. Recent advances in deep learning have spurred research into neural decoders to overcome the limitations of BP decoding. The error correction code transformer (ECCT) offers state-of-the-art decoding performance, yet it demands substantial computational complexity and memory usage. This paper analyzes the hardware complexity of ECCT compared to the basic neural decoder, neural BP (NBP). We also explore strategies for model simplification, showing that applying quantization techniques can reduce resource overhead while maintaining high performance. This paper highlights the potential for advancing neural decoders to achieve greater efficiency and performance.
Belief propagation (BP) decoding achieves near-optimal performance with large block lengths but faces performance degradation with short blocks. Recent advances in deep learning have spurred research into neural decoders to overcome the limitations of BP decoding. The error correction code transformer (ECCT) offers state-of-the-art decoding performance, yet it demands substantial computational complexity and memory usage. This paper analyzes the hardware complexity of ECCT compared to the basic neural decoder, neural BP (NBP). We also explore strategies for model simplification, showing that applying quantization techniques can reduce resource overhead while maintaining high performance. This paper highlights the potential for advancing neural decoders to achieve greater efficiency and performance.
Error correction codes (ECC) are crucial for ensuring reliable information transmission in communication systems. Choukroun & Wolf (2022b) recently introduced the Error Correction Code Transformer (ECCT), which has demonstrated promising performance across various transmission channels and families of codes. However, its high computational and memory demands limit its practical applications compared to traditional decoding algorithms. Achieving effective quantization of the ECCT presents significant challenges due to its inherently small architecture, since existing, very low-precision quantization techniques often lead to performance degradation in compact neural networks. In this paper, we introduce a novel acceleration method for transformer-based decoders. We first propose a ternary weight quantization method specifically designed for the ECCT, inducing a decoder with multiplication-free linear layers. We present an optimized self-attention mechanism to reduce computational complexity via codeaware multi-heads processing. Finally, we provide positional encoding via the Tanner graph eigendecomposition, enabling a richer representation of the graph connectivity. The approach not only matches or surpasses ECCT's performance but also significantly reduces energy consumption, memory footprint, and computational complexity. Our method brings transformer-based error correction closer to practical implementation in resource-constrained environments, achieving a 90% compression ratio and reducing arithmetic operation energy consumption by at least 224 times on modern hardware.
In communication and storage systems, error correction codes (ECCs) are pivotal in ensuring data reliability. As deep learning's applicability has broadened across diverse domains, there is a growing research focus on neural network-based decoders that outperform traditional decoding algorithms. Among these neural decoders, Error Correction Code Transformer (ECCT) has achieved the state-of-the-art performance, outperforming other methods by large margins. To further enhance the performance of ECCT, we propose two novel methods. First, leveraging the systematic encoding technique of ECCs, we introduce a new masking matrix for ECCT, aiming to improve the performance and reduce the computational complexity. Second, we propose a novel transformer architecture of ECCT called a double-masked ECCT. This architecture employs two different mask matrices in a parallel manner to learn more diverse features of the relationship between codeword bits in the masked self-attention blocks. Extensive simulation results show that the proposed double-masked ECCT outperforms the conventional ECCT, achieving the state-of-the-art decoding performance with significant margins.
With the broadening applications of deep learning, neural decoders have emerged as a key research focus, specifically aimed at improving the decoding performance of conventional decoding algorithms. In particular, error correction code transformer (ECCT), which utilizes the transformer architecture, has achieved state-of-the-art performance among neural network-based decoders. We present three technical contributions to significantly enhance the performance of ECCT. First, we propose a novel transformer architecture of ECCT, termed themultiple-masks ECCT (MM ECCT). We employ multiple masked self-attention blocks with different mask matrices in a parallel manner to learn diverse relationships among the codeword bits. Second, we discover that constructing mask matrices based on systematic parity check matrices (PCMs) can make the attention mapssparse, which not only enhances the decoding performance but also reduces computational complexity. Finally, we propose using complementary mask matrices derived from cyclic permutations of the systematic PCM. These complementary mask matrices are specifically designed to enhance the decoding of cyclic codes. Our extensive simulation results show that the proposed MM ECCT architecture with carefully designed mask matrices outperforms the original ECCT by a large margin, achieving state-of-the-art decoding performance among neural decoders. The source code is available at https://github.com/iil-postech/mm-ecct.
For short to moderate-length block codes, neural belief propagation (NBP) decoders are developed to ameliorate the performance of the belief propagation algorithm (BPA). The core concept underlying these decoders is to model belief propagation (BP) as a neural network, allowing adaptable decoding through trainable weights. The article proposes a novel Gated neural min-sum (GNMS) decoding technique with learnable parameters as a more hardware-efficient substitute for conventional NBP decoders. By using parameterised updates and gating techniques, our approach reformulates the min-sum (MS) algorithm, which is an approximation of BP, while preserving the flexibility of neural approaches and drastically lowering computing overhead. Additionally, we develop an autoencoder-assisted framework that supports flexible, learning-based optimisation for GNMS and Gated Neural Offset Min-Sum (GNOMS) decoders. The suggested decoders achieve error-correction performance at par with the most advanced NBP decoders by evaluating them on a variety of short to moderate-length block codes. Notably, when compared to traditional BP, the suggested approach lowers the bit error rate (BER) 10-fold for signal-to-noise ratio (SNR) greater than 3 dB. Furthermore, the efficacy of the proposed work is exhibited by comparing the computational complexity of the neural network-based decoders, highlighting the advantages in terms of practical implementation and hardware feasibility. The simulation results demonstrate that GNMS decoding with learnable parameters offers a convincing trade-off between complexity and performance, offering a potential path for future decoder design in contemporary communication systems.
Quantum error correction (QEC) is critical for scalable fault-tolerant quantum computing. Topological codes, such as the toric code, offer hardware-efficient architectures but their Tanner graphs contain many girth-4 cycles that degrade the performance of belief-propagation (BP) decoding. For this reason, BP decoding is typically followed by a more complex second stage decoder such as minimum-weight perfect matching. These combined decoders achieve a remarkable performance, albeit at the cost of increased complexity. In this paper we propose two key improvements for the decoding of toric code. The first one is replacing the BP decoder by a neural BP decoder, giving rise to the neural belief-matching decoder which substantially decreases the average decoding complexity. The main drawback of this approach is the high cost associated with the training of the neural BP decoder. To address this issue, we impose a convolutional architecture on the neural BP decoder, enabling weight sharing across the spatially homogeneous structure of the code's factor graph. This design allows a model trained on a modest-size topological code to be directly transferred to much larger instances, preserving decoding quality while dramatically lowering the training burden. Our numerical experiments on toric-code lattices of various sizes demonstrate that this technique does not result in a noticeable loss in performance.
Neural belief propagation (NBP) decoding can improve the performance of belief propagation (BP) decoding for high-density parity check (HDPC) codes, at the expense of higher memory storage requirement and computational complexity due to the addition of trainable weight coefficients. To reduce the high storage requirement of NBP, the cyclically equivariant neural BP (CENBP) algorithm makes full use of the cyclically invariant property of the cyclic code, optimizes and reuses the weight coefficients of the NBP algorithm, at the expense of further increasing the computational complexity of NBP. In this paper, we propose low-complexity, in terms of both memory storage requirement and computational complexity, NBP and CENBP decoding algorithms based on Tensor Ring (TR) decomposition. First, in order to reduce the memory storage and computational complexity of the NBP algorithm, we propose a TR-based compression algorithm to compress the messages and mathematical calculations in the NBP decoding algorithm, called TR-NBP algorithm. Second, to address the high computational complexity of the CENBP algorithm, we propose to apply TR decomposition-based compression to the odd layers of the CENBP decoding algorithm, called TR-CENBP, to reduce the computational complexity, and further reduce the required memory storage requirement of the CENBP algorithm. Furthermore, we use TR decomposition-based compression to simplify the mathematical computations associated with thetanhfunction in the NBP algorithm to further reduce the complexity of the hardware implementation. Experimental results show that direct compression of BP algorithm using TR decomposition results in significant performance degradation and our proposed low complexity TR-NBP algorithm and TR-CENBP algorithm can greatly reduce both the memory storage requirement and computation complexity, without significant performance degradation for typical BCH and LDPC codes.