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Non-uniform message quantization techniques such as reconstruction-computation-quantization (RCQ) improve error-correction performance and decrease hardware complexity of low-density parity-check (LDPC) decoders that use a flooding schedule. Layered MinSum RCQ (L-msRCQ) enables message quantization to be utilized for layered decoders and irregular LDPC codes. We investigate field-programmable gate array (FPGA) implementations of L-msRCQ decoders. Three design methods for message quantization are presented, which we name the Lookup, Broadcast, and Dribble methods. The decoding performance and hardware complexity of these schemes are compared to a layered offset MinSum (OMS) decoder. Simulation results on a (16384, 8192) protograph-based raptor-like (PBRL) LDPC code show that a 4-bit L-msRCQ decoder using the Broadcast method can achieve a 0.03 dB improvement in error-correction performance while using 12% fewer registers than the OMS decoder. A Broadcast-based 3-bit L-msRCQ decoder uses 15% fewer lookup tables, 18% fewer registers, and 13% fewer routed nets than the OMS decoder, but results in a 0.09 dB loss in performance. 

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.