  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.
