# Shared Graph Neural Network for Channel Decoding
## 1 Introduction  
Deep learning (DL) technology has become increasingly crucial in many disciplines, especially in image processing [1; 2; 3], wireless communication [4; 5; 6], speech recognition [7], and other fields. In particular, DL was applied in multiple-input multiple-output (MIMO) channel state information (CSI) feedback [8], intelligent reflecting surfaces [9], and the channel decoding algorithm [10; 11; 12; 13; 14]. Although the bit error ratio (BER) performance has been improved, it brings higher storage complexity and computational complexity compared with traditional decoder algorithm. Tanner graphs and factor graphs are commonly used in decoding algorithms for low-density parity check (LDPC), Hamming, and BCH codes. Graph theory is a very broad branch of mathematics that is applicable to real world problems. Compared with the dimensional disaster [15] caused by the increase of the block code length of the DL-based decoding algorithm, GNN [16; 17; 18] is an extension of existing neural network methods that take graph-passing messages and add aggregation networks. GNN can find a suitable model by training the graph structure data, including edge data and node data. Model complexity is independent of graph size, and the trained model is easily applicable to arbitrarily large graphs. GNN has effectively, stability, and strong generalization ability. A special heterogeneous graph with different node types and edge types is proposed in ref. [19]. A GNN-based scalable polar code is proposed in ref. [20], which is a unique graph called a reconstructed code message passing (PCCMP) diagram for polar code, which changes the successive cancellation list decoding algorithm into a heterogeneous graph, while an iterative message passing algorithm is added to the graph to find the best graph with a minimum frame ratio (FER) or BER for a suitable polar code. A GNN-based channel decoding algorithm that adds weights to Tanner graph edges forshort block codes is proposed in ref. [21]. This algorithm is a graph-theoretic extension of the neural belief propagation (NBP) algorithm. A GNN structure for BCH codes and LDPC codes is proposed in ref. [22]. This network structure is suitable for codewords of any length, and the training will not be limited by the curse of dimensionality. The BER performance of the network for eight iterations exceeds the BER performance of the weighted BP for 20 iterations. However, it uses a large number of multilayer perceptron (MLP) network aggregation messages in each iteration, which requires a lot of computing costs and storage resources. Excessive complexity is very challenging to implement on resource-constrained communication hardware, which is also the problem this paper attempts to solve. In this work, model compression technologies are used in GNN-based channel decoding algorithms. Methods to reduce the number of correction factors of GNN-based decoder are proposed, which involve changing the directed-edge to the non-directed edge, sharing weights in factor nodes, variable nodes, factor-to-variable node messages and variable-to-factor nodes messages, and applying it to the BCH and LDPC decoders. The main contribution we proposed is a low-complexity GNN-based channel decoding algorithm. The novelty of this paper can be summarized as follows:  
* We share the same weights in GNN-based channel decoding algorithm especially in the update of factor nodes, variable nodes, factor-to-variable node messages, and variable-to-factor node messages. * Based on different sharing schemes, we propose two shard GNN (SGNN)-based channel decoding algorithms to balance BER performance and storage complexity. * Furthermore, we apply the SGNN-based channel decoding algorithm to BCH and LDPC decoders, which reduces the storage resources required by GNN-based channel decoders with a slight decrease in BER performance. The rest of this paper can be organized as follows: GNN-based channel decoding is described briefly in Section 2. In Section 3, the SGNN-based channel decoding algorithms is proposed.