# Adaptive Learned Belief Propagation for Decoding Error-Correcting Codes
## 3 Channel Models
### Optical Fiber Channel
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter Name** & **Value** \\ \hline  & Transmitter parameters \\ \hline WDM channels & 5 \\ \hline Symbol rate \(R_{s}\) & 32 Gbaud \\ \hline RRC roll-off & 0.01 \\ \hline Channel frequency spacing & 33 GHz \\ \hline Fiber channel parameters \\ \hline Attenuation (\(\alpha\)) & 0.2 dB/km \\ \hline Dispersion parameter (D) & 17 ps/nm/km \\ \hline Nonlinearity parameter (\(\gamma\)) & 1.21/(W\(\cdot\)km) \\ \hline Span configuration & 8 \(\times\) 80 km \\ \hline EDFA gain & 16 dB \\ \hline EDFA noise figure & 5 dB \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The parameters of the fiber-optic link. symmetric split-step Fourier method is applied to compensate for most of the linear and nonlinear fiber impairments [76]. After DSP, the symbols are still subject to signal-dependent noise, which is mitigated by the bit-interleaved coded demodulator (BICD). Let \(\mathbf{y}\in\mathbb{C}^{n_{s}}\) denote the equalized signal samples for the transmitted symbols \(\mathbf{s}\in\mathcal{A}^{n_{s}}\) in the WDM channel of interest. Given that the deterministic effects were equalized, we assume that the channel \(\mathbf{s}\mapsto\mathbf{y}\) is memoryless so that \(\Pr(\mathbf{y}|\mathbf{s})=\prod_{\ell=1}^{n_{s}}\Pr(y_{\ell}|s_{\ell})\). For \(s\in\mathcal{A}\), let \(\mu^{-1}(s)=(b_{1}(s),\cdots,b_{m}(s))\). From the symbol-to-symbol channel \(\Pr(y|s)\), \(s\in\mathcal{A}\), \(y\in\mathbb{C}\), we obtain \(m\) bit-to-symbol channels  
\[\Pr_{j}(y|b)=\sum_{s\in\mathcal{A},b_{j}(s)=b}\Pr(y|s), \tag{6}\]  
where \(b\in\mathbb{F}_{2}\), and \(j\in[m]\). Let \(\bar{\mathbf{c}}=(b_{1}(s_{1}),\cdots,b_{m}(s_{1}),\cdots,b_{1}(s_{n_{s}}), \cdots,b_{m}(s_{n_{s}}))\), \(n=mn_{s}\). The LLR function \(L:\mathbb{C}^{n_{s}}\mapsto\mathbb{R}^{n}\) of \(\mathbf{c}\) conditioned on \(\mathbf{y}\) is, for each \(i\in[n]\),  
\[[\mathrm{L}(\mathbf{y})]_{i} \stackrel{{\Delta}}{{=}}\log\biggl{(}\frac{\Pr(c_{i }=0\mid\mathbf{y})}{\Pr(c_{i}=1\mid\mathbf{y})}\biggr{)}\] \[=\log\biggl{(}\frac{\Pr(\bar{c_{i^{\prime}}=0\mid\mathbf{y})}}{ \Pr(\bar{c_{i^{\prime}}=1\mid\mathbf{y})}}\biggr{)}\] \[=\log\biggl{(}\frac{\Pr_{j}(y\mid b=0)}{\Pr_{j}(y\mid b=1)} \biggr{)}, \tag{7}\]  
where \(i^{\prime}\) is obtained from \(i\) according to \(\pi\), \(j=i^{\prime}\mod m\), and \(\Pr_{j}(y|b)\) is defined in (6). DecoderThe decoding of \(\mathcal{C}_{i}\circ\mathcal{C}_{o}\) consists of two steps. First, \(\mathcal{C}_{i}\) is decoded using an adaptive WBP in Appendix B, which takes the soft information \(\mathrm{L}(\mathbf{y})\in\mathbb{R}^{n}\) and corrects some errors. Second, \(\mathcal{C}_{o}\) is decoded using the min-sum (MS) decoder with SWD in Appendix B, which further lowers the BER, and outputs the decoded information bits \(\hat{\mathbf{b}}\). The LLRs in the inner decoder are represented with 32 bits, and in the outer decoder are quantized at 4 bits with per-window configuration. In optical communication, the forward error correction (FEC) overhead 6-25% is common [77]. Thus, the inner code typically has a high rate of \(\geq\)0.9 and a block length of several thousands, achieving a BER of \(10^{-6}\)-\(10^{-2}\). The outer code has a length of up to tens of thousands, lowering the BER to an error floor to \(\sim\)\(10^{-15}\).