# Adaptive Learned Belief Propagation for Decoding Error-Correcting Codes
## 7 Conclusions  
Adaptive decoders are proposed for codes on graphs that can be decoded with message-passing algorithms. Two variants, the parallel WBP and the two-stage decoder, are studied. The parallel decoders search for the best sequence of weights in real time using multiple instances of the WBP decoder running concurrently, while the two-stage neural decoder employs an NN to dynamically determine the weights of WBP for each received word. The performance and complexity of the adaptive and several static decoders are compared for a number of codes over an AWGN and optical fiber channel. The simulations show that significant improvements in BER can be obtained using adaptive decoders, depending on the channel, SNR, the code and its parameters. Future work could explore further reducing the computational complexity of the online learning, and applying adaptive decoders to other types of codes and wireless channels. Conceptualization, A.T. and M.Y. ; Methodology, A.T. and M.Y. ; Software, A.T. and M.Y. ; Formal analysis, A.T. and M.Y. ; Investigation, A.T. and M.Y. ; Writing--original draft, A.T. and M.Y. ; Project administration, M.Y. All authors have read and agreed to the published version of the manuscript. This work has received funding from the European Research Council (ERC) research and innovation program, under the COMNFT project, Grant Agreement no. 805195. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\text{BER}_{i}\)} & **Inner-SD** & **BER\({}_{a}\)** & **BER\({}_{a}\)** & **NCG** & **NCG\({}_{f}\)** & **NCG\({}_{f}\)** & **Cap to NCG\({}_{f}\)** & **Gap to NCG\({}_{f}\)** \\  & **Decoder** & **Inner** & **Total** & **Inner** & **Total** & **Inner** & **Total** & **Inner** & **Total** \\ \hline \multirow{4}{*}{0.012} & AWEMS & \(3.21\times 10^{-8}\) & \(2.74\times 10^{-8}\) & 7.23 & 7.69 & 9.38 & 9.44 & 2.15 & 1.75 \\ \cline{2-10}  & NNNE & \(4.61\times 10^{-8}\) & \(2.11\times 10^{-8}\) & 7.12 & 7.15 & 9.38 & 9.44 & 2.26 & 2.29 \\ \cline{2-10}  & EMS & \(2.44\times 10^{-7}\) & \(8.20\times 10^{-8}\) & 6.60 & 6.75 & 9.38 & 9.44 & 2.78 & 2.69 \\ \hline \multirow{4}{*}{0.025} & AWEMS & \(0.0063\) & \(0.0051\) & 1.73 & 1.80 & 10.20 & 10.28 & 8.47 & 8.48 \\ \cline{2-10}  & NNNE & \(0.0087\) & \(0.0075\) & 1.32 & 1.32 & 10.20 & 10.28 & 8.88 & 8.96 \\ \cline{1-1} \cline{2-10}  & EMS & \(0.025\) & \(0.022\) & \(-0.36\) & \(-0.32\) & 10.20 & 10.28 & 10.56 & 10.60 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Concatenated inner non-binary QC-LDPC code \(\mathcal{C}_{11}\) and outer SC-QC-LDPC code \(\mathcal{C}_{12}\) with \(r_{\text{total}}=0.88\) for the optical fiber channel. The sections for \(\text{BER}_{i}\)\(0.012\) and \(0.025\) correspond to average powers \(-10\) and \(-11\) dBm, respectively. NCGs are in dB. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\text{BER}_{i}\)} & **Inner-SD** & **BER\({}_{a}\)** & **BER\({}_{a}\)** & **NCG** & **NCG\({}_{f}\)** & **NCG\({}_{f}\)** & **Gap to NCG\({}_{f}\)** \\  & **Decoder** & **Inner** & **Total** & **Inner** & **Total** & **Inner** & **Total** & **Inner** & **Total** \\ \hline \multirow{4}{*}{0.012} & AWMS & \(3.29\times 10^{-6}\) & \(4.52\times 10^{-8}\) & 5.64 & 6.93 & 9.38 & 9.44 & 3.74 & 2.51 \\ \cline{2-10}  & NNNE & \(4.02\times 10^{-6}\) & \(5.43\times 10^{-7}\) & 5.56 & 6.13 & 9.38 & 9.44 & 3.82 & 3.31 \\ \cline{1-1}  & MS & \(4.77\times 10^{-6}\) & \(7.75\times 10^{-7}\) & 5.49 & 6.00 & 9.38 & 9.44 & 3.89 & 3.44 \\ \cline{1-1} \cline{2-10}  & AWEMS & \(0.019\) & \(0.017\) & 0.13 & 0.12 & 10.20 & 10.28 & 10.07 & 10.16 \\ \cline{1-1} \cline{2-10}  & NNNE & \(0.02\) & \(0.018\) & 0.04 & 0.03 & 10.20 & 10.28 & 10.16 & 10.25 \\ \cline{1-1} \cline{2-10}  & MS & \(0.023\) & \(0.02\) & \(-0.20\) & \(-0.15\) & 10.20 & 10.28 & 10.40 & 10.43 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Concatenated inner binary QC-LDPC code \(\mathcal{C}_{10}\) and outer SC-QC-LDPC code \(\mathcal{C}_{12}\) with an \(r_{\text{total}}=0.88\) for the optical fiber channel.