# Shared Graph Neural Network for Channel Decoding
## 2 Preliminary
The number of trainable parameters in ref. [12] is 2-3 orders of magnitude larger than our decoder. However, multiplication in our decoder is much larger than that in traditional communication decoding algorithms. As shown in Figures 4 and 5, the BER performances of the Uncoded, BP, Weighted BP, GNN, SGNN1, and SGNN2-based channel decoding algorithms are consistently improving for the BCH (7,4) code of BPSK modulation and QPSK modulation. The GNN-based channel decoding algorithm with only four iterations achieves an almost equivalent BER performance compared to the traditional BP decoder with twenty iterations. Overall, the BER performance of the SGNN1-based channel decoding algorithm degrades by 0.15 dB compared to the GNN-based channel decoding algorithm with the same number of iterations, and the BER performance of the SGNN2-based channel decoding algorithm degrades by 0.05 dB. However, SGNN1 and SGNN2 only use 2440 and 4840 parameters compared to the GNN, which uses 9640 parameters. BCH (7,4) has the same number of parameters as BCH (63,45), and the parameters are reduced to a quarter and a half, respectively, as listed in Table 2. According to our experience, the training weights of BCH (63,45) cannot achieve the desired BER performance when used in BCH (7,4). \begin{table}
\begin{tabular}{c c c c} \hline
**Paramter** & **BCH Codes** & **5G LDPC Code** & **Regular LDPC Code** \\ \hline activation & tanh & ReLU & tanh \\ T & 8 & 10 & 10 \\ \(f^{F_{n}}\) & 20 & 16 & 16 \\ \(f^{F_{n}}\) & 20 & 16 & 16 \\ hidden units MLP & 40 & 48 & 64 \\ MLP layers & 2 & 3 & 2 \\ aggregation function & mean & sum & mean \\ LLR clipping & \(\infty\) & 20 & \(\infty\) \\ learning rate & \(1\times 10^{-3},1\times 10^{-4},1\times 10^{-5}\) & \(1\times 50^{-4},1\times 10^{-4},1\times 10^{-5}\) & \(1\times 20^{-4},1\times 10^{-4},1\times 10^{-5}\) \\ batch size & 256 & 128 & 150 \\ train iteration & \(635,000\) & \(635,000\) & \(550,000\) \\ \hline \end{tabular}
\end{table} 
Table 1: Parameter of simulation. Figure 4: BER performance of GNN [22] and SGNN for BCH (_7.4_) code of BPSK modulation. Figure 5: BER performance of GNN [22] and SGNN for BCH (_7.4_) code of QPSK modulation. Figure 3: BER performance of GNN [22] and SGNN for BCH (63,45) code of BPSK modulation. ### Regular LDPC Code

In order to increase the applicability of the GNN, we compare the GNN and the SGNN-based channel decoding algorithms for the regular LDPC code (v = 3, c = 6), when the training parameter T is 10 and the codeword length \(N\) is 100. In order to achieve fast convergence of training, MLP adds multiple neural network layers, the result is shown in Figure 6. The blue vertical line is the shannon limit of the code rate corresponding to reg LDPC codes. The SGNN1-based channel decoding algorithm performs slightly better than the twenty iterations of the weighted BP algorithm. In addition, when \(N=100\), the BER performances of the SGNN1 and SGNN2-based channel decoding algorithms are basically the same as that of the GNN-based channel decoding algorithm. As the length of the code increases, the BER performance gap between the SGNN algorithm and the GNN-based channel decoding algorithm gradually becomes larger, with a difference of about 0.2 dB; however, the BER performance gains between the BP algorithms and the GNN-based channel decoding algorithm tend to diminish with increasing n, but the weight is reduced from 28,700 to 32,016,352, as listed in Table 2.

### 5g NR LDPC Code

In order to increase the applicability of the GNN network, we compare the 5G NR LDPC code [27] GNN network with the SGNN network, when the training parameter T is 10 and the codeword length \(N=140\), \(K=60\). The blue vertical line is the shannon limit of the code rate corresponding to 5G NR LDPC codes. As the 5G NR LDPC code requires rate-matching, the codewords tested all contain the same code rate, such as \(N=280\), \(K=120\), \(N=420\), \(K=180\).