# Shared Graph Neural Network for Channel Decoding
## 2 Preliminary
As shown in Figure 7, the same weights can be used to provide parameters for the longer codeword, and flexibility is also one of the advantages of GNN. We can adapt to the training LDPC code of simple codewords, but we can test different codewords. At the same time, please note that different code rates appear in training and testing, and the weights need to be adjusted at the same time. Both algorithms \begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**Codes**} & **Number of Parameters in** & **Number of Parameters in** & **Number of Parameters in** \\  & **GNN** & **SGNN1** & **SGNN2** \\ \hline BCH (63,45) & 9640 & 2440 & 4840 \\ BCH (7,4) & 9640 & 2440 & 4840 \\ Regular LDPC (3,6) & 12,657 & 3201 & 6353 \\
5G NR LDPC (60,140) & 18,929 & 4769 & 9489 \\ \hline \end{tabular}
\end{table} 
Table 2: Comparison of parameters in GNN [22] and SGNN-based channel decoding algorithms. Figure 6: BER performance of GNN [22] and SGNN for regular LDPC (3,6) code of BPSK modulation. use \(flooding\) schedules. The updates of its VN nodes and FN nodes, as well as the required number of iterations, will change with different schedules. The difference between the BER performance of SGNN1 and the GNN-based channel decoding algorithm when testing the same code of \(N=140\), \(K=60\) is about 0.5 dB, and the difference between the BER performance of the SGNN2 and GNN-based channel decoding algorithms is about 0.1 dB when testing different codes of \(N=280\), \(K=150\). A small gain can be observed between SGNN and GNN-based channel decoding algorithms. There is little room for improvement in the deep learning GNN-based channel decoding algorithm and the BP algorithm for optimal settings for 5G LDPC codes, but there is room for further improvement in the SGNN algorithm. ### Complexity Analysis

The number of weights in the SGNN1 and SGNN2-based channel decoding algorithms are greatly reduced, but the BER performance is nearly the same or slightly reduced, as listed in Table 2. We can find than the SGNN1 and SGNN2-based channel decoding algorithms have low complexity compared with that of the GNN-based channel decoding algorithm. The dense network is used to initialize graph nodes and return variable node values, and the number of multiplications in this process is small and is related to the output of the dense network. In each iteration of decoding, the MLPs network is used to update factor nodes and update variable nodes, so the number of layers of the MLPs network and the output of each dense network determine the number of multiplications here. As shown in Figure 2, a total of four MLPs are 208w in each iteration. The reduction in the number of parameters therefore requires fewer memory storage resources, which is more beneficial for implementation on resource-constrained communication hardware. But at the same time, investigating how to reduce the number of multiplications is a potential direction for future research. ## 5 Open Issues

As mentioned above, the complexity of GNN is much higher than that of the NBP algorithm due to the multiple use of MLPs. Although the SGN-based channel decoding algorithm reduces the number of parameters to even less than that of the NBP algorithm, the number of multiplications has not been reduced. Such a large amount of calculation limits its implementation in hardware such as FPGA, let alone enabling it to replace traditional decoding algorithms in communication chips. The trained codeword weights cannot reflect the same BER performance in other codewords. Therefore, possible future research directions are as follows:

Figure 7: BER performance of GNN [22] and SGNN for 5G LDPC (60,140) code of BPSK modulation. A possible way to implement a universal GNN-based channel decoder is to train GNN-based channel decoder weights for multiple forward error correction (FEC) parallel codes. * The decoding complexity of the proposed SGNN is higher compared with that of BP, and other complexity reduction methods are put together to further reduce complexity, such as pruning, quantification, etc.