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.
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.
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.
This paper uses the reconstruction-computation-quantization (RCQ)paradigm to decode low-density parity-check (LDPC) codes. RCQ facilitates dynamic non-uniform quantization to achieve good frame error rate (FER) performance with very low message precision. For message-passing according to a flooding schedule, the RCQ parameters are designed by discrete density evolution. Simulation results on an IEEE 802.11 LDPC code show that for 4-bit messages, a flooding Min Sum RCQ decoder outperforms table-lookup approaches such as information bottleneck (IB) or Min-IB decoding, with significantly fewer parameters to be stored. Additionally, this paper introduces layer-specific RCQ, an extension of RCQ decoding for layered architectures. Layer-specific RCQ uses layer-specific message representations to achieve the best possible FER performance. For layer-specific RCQ, this paper proposes using layered discrete density evolution featuring hierarchical dynamic quantization (HDQ) to design parameters efficiently. Finally, this paper studies field-programmable gate array (FPGA) implementations of RCQ decoders. Simulation results for a (9472, 8192) quasi-cyclic (QC) LDPC code show that a layered Min Sum RCQ decoder with 3-bit messages achieves more than a 10% reduction in LUTs and routed nets and more than a 6% decrease in register usage while maintaining comparable decoding performance, compared to a 5-bit offset Min Sum decoder.
This paper proposes a finite-precision decoding method for low-density parity-check (LDPC) codes that features the three steps of Reconstruction, Computation, and Quantization (RCQ). Unlike Mutual-Information-Maximization Quantized Belief Propagation (MIM-QBP), RCQ can approximate either belief propagation or Min-Sum decoding. MIM-QBP decoders do not work well when the fraction of degree-2 variable nodes is large. However, sometimes a large fraction of degree-2 variable nodes is used to facilitate a fast encoding structure, as seen in the IEEE 802.11 standard and the DVB-S2 standard. In contrast to MIM-QBP, the proposed RCQ decoder may be applied to any off-the-shelf LDPC code, including those with a large fraction of degree-2 variable nodes. Simulations show that a 4-bit Min-Sum RCQ decoder delivers frame error rate (FER) performance within 0.1 dB of floating point belief propagation (BP) for the IEEE 802.11 standard LDPC code in the low SNR region. The RCQ decoder actually outperforms floating point BP and Min-Sum in the high SNR region were FER less than 10-5. This paper also introduces Hierarchical Dynamic Quantization (HDQ) to design the time-varying non-uniform quantizers required by RCQ decoders. HDQ is a low-complexity design technique that is slightly sub-optimal. Simulation results comparing HDQ and optimal quantization on the symmetric binary-input memoryless additive white Gaussian noise channel show a mutual information loss of less than 10-6bits, which is negligible in practice.
In this paper, we discuss the perspectives of utilizing deep neural networks (DNN) to decode Low-Density Parity Check (LDPC) codes. The main idea is to build a neural network to learn and optimize a conventional iterative decoder of LDPC codes. A DNN is based on Tanner graph, and the activation functions emulate message update functions in variable and check nodes. We impose a symmetry on weight matrices which makes it possible to train the DNN on a single codeword and noise realizations only. Based on the trained weights and the bias, we further quantize messages in such DNN-based decoder with 3-bit precision while maintaining no loss in error performance compared to the min-sum algorithm. We use examples to present that the DNN framework can be applied to various code lengths. The simulation results show that, the trained weights and bias make the iterative DNN decoder converge faster and thus achieve higher throughput at the cost of trivial additional decoding complexity.
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.
The min-sum (MS) algorithm can decode Low-density parity-check (LDPC) codes with low computational complexity at the cost of slight performance loss. It is an effective way to realize hardware implementation of the min-sum decoder by quantizing the floating belief messages (i.e., check-to-variable messages and variable-to-check messages) into low-resolution (i.e., 2–4 bits) versions. However, such a way can lead to severe performance degradation due to the finite precision effect. In this paper, we propose a neural-network optimized low-resolution decoding (NOLD) algorithm for LDPC codes to deal with the problem. Specifically, the optimization of decoding parameters (i.e., scaling factors and quantization step) is achieved in a hybrid way, in which we concatenate a NOLD decoder with a customized neural network. All learnable parameters associated with the decoding parameters are assigned to each neuron in the proposed method. What's more, we design a new activation function whose outputs are close to the employed quantizer ones when network parameters are finally optimized off-line. Finally, the performance of the proposed method is verified by numerous experiments. For the case of 2-bit decoding, the proposed approach significantly outperforms several conventional decoders at the expense of slightly increased off-line training time. Besides, the proposed method with 4-bit quantization incurs only 0.1 dB performance loss compared with the floating min-sum decoder at the coded bit-error-rate of 10−5. Moreover, we show that the proposed NOLD decoder works over a wide range of channel conditions for regular and irregular LDPC codes. Simulation code for reproductive results is publicly available1.