Neural Normalized MinSum (N-NMS) decoding delivers better frame error rate (FER) performance on linear block codes than conventional Normalized MinSum (NMS) by assigning dynamic multiplicative weights to each check-to-variable node message in each iteration. Previous N-NMS efforts primarily investigated short block codes (N < 1000), because the number of N-NMS parameters required to be trained scales proportionately to the number of edges in the parity check matrix and the number of iterations. This imposes an impractical memory requirement for conventional tools such as Pytorch and Tensorflow to create the neural network and store gradients. This paper provides efficient methods of training the parameters of N-NMS decoders that support longer block lengths. Specifically, this paper introduces a family of Neural 2-dimensional Normalized (N-2D-NMS) decoders with various reduced parameter sets and shows how performance varies with the parameter set selected. The N-2D-NMS decoders share weights with respect to check node and/or variable node degree. Simulation results justify a reduced parameter set, showing that the trained weights of N- NMS have a smaller value for the neurons corresponding to larger check/variable node degree. Further simulation results on a (3096,1032) Protograph-Based Raptor-Like (PBRL) code show that the N-2D-NMS decoder can achieve the same FER as N- NMS while also providing at least a 99.7% parameter reduction. Furthermore, the N-2D-NMS decoder for the (16200,7200) DVBS- 2 standard LDPC code shows a lower error floor than belief propagation. Finally, this paper proposes a hybrid decoder training structure that utilizes a neural network which combines a feedforward module with a recurrent module. The decoding performance and parameter reduction of the hybrid training depends on the length of recurrent module of the neural network.
Neural Normalized MinSum (N-NMS) decoding delivers better frame error rate (FER) performance on linear block codes than conventional Normalized MinSum (NMS) by assigning dynamic multiplicative weights to each check-to-variable node message in each iteration. Previous N-NMS efforts primarily investigated short block codes (N < 1000), because the number of N-NMS parameters required to be trained scales proportionately to the number of edges in the parity check matrix and the number of iterations. This imposes an impractical memory requirement for conventional tools such as Pytorch and Tensorflow to create the neural network and store gradients. This paper provides efficient methods of training the parameters of N-NMS decoders that support longer block lengths. Specifically, this paper introduces a family of Neural 2-dimensional Normalized (N-2D-NMS) decoders with various reduced parameter sets and shows how performance varies with the parameter set selected. The N-2D-NMS decoders share weights with respect to check node and/or variable node degree. Simulation results justify a reduced parameter set, showing that the trained weights of N- NMS have a smaller value for the neurons corresponding to larger check/variable node degree. Further simulation results on a (3096,1032) Protograph-Based Raptor-Like (PBRL) code show that the N-2D-NMS decoder can achieve the same FER as N- NMS while also providing at least a 99.7% parameter reduction. Furthermore, the N-2D-NMS decoder for the (16200,7200) DVBS- 2 standard LDPC code shows a lower error floor than belief propagation. Finally, this paper proposes a hybrid decoder training structure that utilizes a neural network which combines a feedforward module with a recurrent module. The decoding performance and parameter reduction of the hybrid training depends on the length of recurrent module of the neural network.
The success of deep learning has encouraged its applications in decoding error-correcting codes, e.g., LDPC decoding. In this paper, we propose a model-driven deep learning method for normalized min-sum (NMS) low-density parity-check (LDPC) decoding, namely neural NMS (NNMS) LDPC decoding network. By unfolding the iterative decoding progress between checking nodes (CNs) and variable nodes (VNs) into a feed-forward propagation network, we can harvest the benefits of both the model-driven deep learning and the conventional normalized min-sum (NMS) LDPC decoding method. In addition, we proposed a shared parameters NNMS with the LeakyReLU and a 12-bit quantizer (SNNMS-LR-Q) which reduces the number of required multipliers and correction factors by sharing parameters, increasing the nonlinear fitting ability by adding LeakyReLU. By utilizing the 12-bit quantizer, we can improve the confrontation ability. Thorough experiments with different code lengths, code rates, channel conditions, and check matrices are implemented to demonstrate the advantages and robustness of our proposed networks. The BER performance of the proposed NNMS is 1.5 dB better than the NMS, using fewer iterations. Meanwhile, The SNNMS-LR-Q outperforms the NNMS regarding the BER performance and efficiency.
For recent years, deep learning has been studied and utilized among decoders in wireless and optical communication. For low density parity check (LDPC) codes, most works use neural network (NN) to implement the belief propagation (BP) decoder and unfold its iterations into layers activated by different functions. However, the whole decoder composed by neural network performs lots of non-linear operations, which are time consuming and unfriendly to hardware implementation. In this work, a neural network-aided normalized min-sum (NN-aided NMS) decoder for LDPC code is proposed, in which the normalization factors are inferred from input log likelihood ratio (LLR) by apre-trained NN, while maintaining check and variable nodes processing the same as in min-sum (MS) decoder. In order to test the proposed decoder with irregular LDPC code, a rate-7/8 one was constructed using progressive edge growth (PEG) algorithm, showing better error correction capability than the standard of consultative committee for space data systems (CCSDS) LDPC (8176,7154) code by 0.08 dB with the same rate. Simulation results show that, NN-aided NMS decoder outperforms MS decoder by 0.3 dB, and its performance approaches BP decoder with a gap of 0.02 dB, which is 50% closer than the best NMS decoder using 0.7 as the normalization factor. The proposed decoder has good performance for both regular and irregular LDPC codes, and needs much less computation complexity than BP decoder. It introduces more linear computation than NMS decoder but achieves better bit error rate (BER) performance, thus becoming a good complement between BP and NMS decoder when trading off among complexity, error correction performance and hardware implementation for wireless and satellite communication.
In this work, we propose a neural adjusted min-sum (NAMS) decoder for low-density parity-check (LDPC) codes. In particular, we improve the traditional normalized min-sum (NMS) decoder by introducing a selection mechanism to adjust the check-node update step, where either the min-sum rule or the belief propagation (BP) rule is selected. Besides, we unfold the modified decoder into a model-driven neural network, where layer-dependent trainable parameters are introduced as weights in the Tanner graph and optimized by gradient descent-based methods during network training. Simulation results demonstrate that the proposed NAMS decoder is able to provide superior error-correction performance as compared to the neural NMS decoder, with only slightly increased computational complexity. Moreover, in certain circumstances, the proposed NAMS decoder even outperforms the neural BP decoder, with much lower computational complexity.
Low-density parity-check (LDPC) codes have been successfully commercialized in communication systems due to their strong error correction capabilities and simple decoding process. However, the error-floor phenomenon of LDPC codes, in which the error rate stops decreasing rapidly at a certain level, presents challenges for achieving extremely low error rates and deploying LDPC codes in scenarios demanding ultra-high reliability. In this work, we propose training methods for neural min-sum (NMS) decoders to eliminate the error-floor effect. First, by leveraging the boosting learning technique of ensemble networks, we divide the decoding network into two neural decoders and train the post decoder to be specialized for uncorrected words that the first decoder fails to correct. Secondly, to address the vanishing gradient issue in training, we introduce a block-wise training schedule that locally trains a block of weights while retraining the preceding block. Lastly, we show that assigning different weights to unsatisfied check nodes effectively lowers the error-floor with a minimal number of weights. By applying these training methods to standard LDPC codes, we achieve the best error-floor performance compared to other decoding methods. The proposed NMS decoder, optimized solely through novel training methods without additional modules, can be integrated into existing LDPC decoders without incurring extra hardware costs. The source code is available at https://github.com/ghy1228/LDPC_Error_Floor .
Recently, neural networks have improved MinSum message-passing decoders for low-density parity-check (LDPC) codes by multiplying or adding weights to the messages, where the weights are determined by a neural network. The neural network complexity to determine distinct weights for each edge is high, often limiting the application to relatively short LDPC codes. Furthermore, storing separate weights for every edge and every iteration can be a burden for hardware implementations. To reduce neural network complexity and storage requirements, this paper proposes a family of weight-sharing schemes that use the same weight for edges that have the same check node degree and/or variable node degree. Our simulation results show that node-degree-based weight-sharing can deliver the same performance requiring distinct weights for each node. This paper also combines these degree-specific neural weights with a reconstruction-computation-quantization (RCQ) decoder to produce a weighted RCQ (W-RCQ) decoder. The W-RCQ decoder with node-degree-based weight sharing has a reduced hardware requirement compared with the original RCQ decoder. As an additional contribution, this paper identifies and resolves a gradient explosion issue that can arise when training neural LDPC decoders.
Ensuring extremely high reliability in channel coding is essential for 6G networks. The next-generation of ultra-reliable and low-latency communications (xURLLC) scenario within 6G networks requires frame error rate (FER) below 10-9. However, low-density parity-check (LDPC) codes, the standard in 5G new radio (NR), encounter a challenge known as the error floor phenomenon, which hinders to achieve such low frame error rates. To tackle this problem, we introduce an innovative solution: boosted neural min-sum (NMS) decoder. This decoder operates identically to conventional NMS decoders, but is trained by novel training methods including: i) boosting learning with uncorrected vectors, ii) block-wise training schedule to address the vanishing gradient issue, iii) dynamic weight sharing to minimize the number of trainable weights, iv) transfer learning to reduce the required sample count, and v) data augmentation to expedite the sampling process. Leveraging these training strategies, the boosted NMS decoder achieves the state-of-the art performance in reducing the error floor as well as superior waterfall performance. Remarkably, we fulfill the 6G xURLLC requirement for 5G LDPC codes without a severe error floor. Additionally, the boosted NMS decoder, once its weights are trained, can perform decoding without additional modules, making it highly practical for immediate application. The source code is available athttps://github.com/ghy1228/LDPC_Error_Floor.