# Adaptive Learned Belief Propagation for Decoding Error-Correcting Codes
## 6 Performance and Complexity Comparison
### AWGN Channel
The acquisition function retains \((\mathbf{c},\mathbf{y})\) if \(\epsilon_{1}=0\) (no error) or if \((\epsilon_{1},\epsilon_{2})\) falls in the rectangle in Figure 5 (with error). The rectangle is defined such that \(\Pr((\epsilon_{1},\epsilon_{2})\in\mathcal{S})\geq 0.95\). It is ensured that 70% of examples satisfy \(\epsilon_{1}=0\) and 30% with \((\epsilon_{1},\epsilon_{2})\) in the rectangle \(\mathcal{S}\). In this example, \(\epsilon_{1}\in\{80,81,\cdots,100\}\), and \(\epsilon_{2}\in[\mu-2\sigma,\mu+2\sigma]\), \(\mu=149.97\), \(\sigma=12.2\). We use \(\mathfrak{b}_{1}=2000\), \(\mathfrak{b}_{2}=2000\), \(\mathfrak{b}_{s}=500\) and the learning rate \(0.001\). For the adaptive decoder, we consider five parallel decoders with \(T_{1}=4\) iterations. The decoder for the binary codes \(\mathcal{C}_{6}\), \(\mathcal{C}_{7}\) and \(\mathcal{C}_{8}\) is WMS with Type \(T_{a}VC\) sharing. The output of the decoder with the smallest syndrome is continued with an MS decoder with \(T_{2}=4\) iterations. The polar code, however, is decoded with either a cyclic redundancy check (CRC) and successive cancellation list (SCL) with list size \(L\)[88] or the optimized successive cancellation (OSC) [89]. Figure 6 shows the performance of the adaptive and static MS decoders for \(\mathcal{C}_{6}\), \(\mathcal{C}_{7}\) and \(\mathcal{C}_{9}\). The polar code \(\mathcal{C}_{9}\) with 24 CRC bits is simulated using AFF3 CT software toolbox ([90] v3.0.2). It can be seen that at high SNRs, \(E_{b}/N_{0}\geq 4.6\), \(\mathcal{C}_{6}\) and \(\mathcal{C}_{7}\) decoded with adaptive parallel decoders outperform \(\mathcal{C}_{9}\). Given this, and the higher complexity of decoding the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multirow{2}{*}{\(\boldsymbol{\gamma^{*(t)}}\)} & \multirow{2}{*}{\(\boldsymbol{\alpha^{*(t)}}\)} & \multicolumn{3}{c}{**Average RM per Iteration**} \\ \cline{3-6}  & & & \(\mathcal{C}_{1}\) & \(\mathcal{C}_{2}\) & \(\mathcal{C}_{3}\) \\ \hline \multicolumn{6}{c}{No weight sharing} \\ \hline WMS [16] & \(\gamma_{\varphi,\varepsilon}^{(t)}\) & 1 & 768 & 25,792 & 40,160 \\ \hline \multicolumn{6}{c}{Weight sharing} \\ \hline WMS, Type \(T_{a}\) & \(\gamma_{\varphi,\varepsilon}\) & 1 & 768 & 25,792 & 40,160 \\ \hline WMS, Type \(T_{a}VC\) & \(\gamma\) & 1 & 63 & 3226 & 4016 \\ \hline Parallel WMS & \(\gamma^{(t)}\) & 1 & 1440 & 77,376 & 84,336 \\ Type \(T_{b}VC,\
u=16\) & \(\gamma^{(t)}\) & 1 & – & \(\simeq\)3.09 \(\times\) 10\({}^{5}\) & \(\simeq\)3.37 \(\times\) 10\({}^{5}\) \\ \hline Parallel WMS & \(\gamma^{(t)}\) & 1 & 92,340 & – & – \\ Type \(T_{b}VC,\
u=1024\) & \(\gamma^{(t)}\) & 1 & 92,340 & – & – \\ \hline Two-stage decoder & \(\gamma^{(t)}\) & 1 & \(\simeq\)300 & \(\simeq\)14,093 & \(\simeq\)17,558 \\ Type \(T_{b}VC,\tau_{\mathrm{prun}}=0.001\) & \(\gamma^{(t)}\) & 1 & 153 & 8060 & 9287 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Computational complexity of decoders, for the AWGN channel. polar code with either SCL or OSC [88], the choice of QC-LDPC codes for the inner code for the optical fiber channel in Section 6.2 is justified. Figure 7 shows the performance of \(\mathcal{C}_{8}(4260,3834)\) with rate 0.9. The adaptive WMS decoder with \(T_{1}+T_{2}=8\) iterations outperforms the static MS decoder with \(T=8\) iterations at \(E_{b}/N_{0}=5\) by an order of magnitude in BER. Figure 5: The scatter plot of \((\epsilon_{1},\epsilon_{2})\) for \(\mathcal{C}_{9}\) at \(E_{b}/N_{0}=4.25\) dB, for the AWGN channel. The scaled Gaussian approximation curve is fitted per axis. Figure 6: Performance of the polar code \(\mathcal{C}_{9}(1024,854)\) versus QC-LDPC codes \(\mathcal{C}_{6}(1050,875)\) and \(\mathcal{C}_{7}(1050,850)\), for the AWGN channel in the high-rate regime. The curve for OSC decoder is from [89]. The gains of WBP depend on parameters such as the block length or SNR [91] (Section IV. d [44]). In general, the gain is decreased when the block length is increased, with other parameters remaining fixed.