# Adaptive Learned Belief Propagation for Decoding Error-Correcting Codes
## 3 Channel Models
### Optical Fiber Channel  
In this application, we consider a multi-user fiber-optic transmission system using WDM with \(N_{c}\) users, each of bandwidth \(B_{0}\) Hz, as shown in Figure 1. \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**AWGN Channel**} \\ \hline Low rate & High rate \\ \hline BCH \(C_{1}(63,36)\), \(r=0.57\) & QC-LDPC \(\mathcal{C}_{6}(1050,875)\), \(0.83\) \\ \hline QC-LDPC \(C_{2}(3224,1612)\), \(0.5\) & QC-LDPC \(\mathcal{C}_{7}(1050,850)\), \(0.81\) \\ \hline QC-LDPC \(C_{3}(4016,2761)\), \(0.69\) & QC-LDPC \(\mathcal{C}_{8}(4260,3834)\), \(0.9\) \\ \hline Irregular LDPC \(C_{4}(420,180)\), \(0.43\) & Polar \(\mathcal{C}_{9}(1024,854)\), \(0.83\) \\ \hline Irregular LDPC \(C_{5}(128,64)\), \(0.5\) & \\ \hline \multicolumn{2}{c}{**Optical Fiber Channel**} \\ \hline Inner code & Outer code \\ \hline Single-edge QC-LDPC \(C_{10}(4000,3680)\), \(0.92\) & Multi-edge QC-LDPC \(C_{11}(3680,3520)\), \(0.96\) \\ \hline Non-binary multi-edge \(\mathcal{C}_{12}(800,32)\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Codes in this paper. _Transmitter (TX)_: A binary source generates a pseudo-random bit sequence of \(n_{b}\) information bits \(\mathbf{b}_{u}=(b_{u,1},b_{u,2},\ldots,b_{u,n_{b}})\), \(b_{u,j}\in\{0,1\}\), for the WDM channel \(u\in[u_{1}:u_{2}]\), \(u_{1}=-\lfloor N_{c}/2\rfloor\), \(u_{2}=\lceil N_{c}/2\rceil-1\), \(j\in[1:n_{b}]\). Bit-interleaved coded modulation (BICM) with concatenated coding is applied in WDM channels independently. The BICM comprises an outer encoder for the code \(C_{0}(n_{\text{o}},k_{\text{o}})\) with rate \(r_{\text{o}}\), an inner encoder for \(\mathcal{C}_{i}(n_{\text{i}},k_{\text{i}})\) with rate \(r_{\text{i}}\), a permuter \(\pi\), and a mapper \(\mu\), where \(n_{\text{o}}/k_{\text{i}}\) is assumed to be an integer. The concatenated code \(\mathcal{C}=\mathcal{C}_{i}\circ\mathcal{C}_{o}\) has parameters \(k=k_{\text{o}}\) and \(n=n_{\text{o}}/r_{\text{i}}\) and \(r_{\text{total}}=r_{\text{i}}r_{\text{o}}\). Each consecutive subsequence of \(\mathbf{b}_{u}\) of length \(k_{\text{o}}\) is mapped to \(\xi_{u}\in\mathcal{C}_{o}\subset\{0,1\}^{n_{\text{o}}}\) by the outer encoder and subsequently to \(\xi_{u}\in\mathcal{C}\subset\{0,1\}^{n}\) by the inner encoder. Next, \(\mathbf{c}_{u}\) is mapped to \(\xi_{u}=\pi(\mathbf{c}_{u})\) by a random uniform permuter \(\pi:\mathbb{F}_{2}^{n}\to\mathbb{F}_{2}^{n}\). The mapper \(\mu:\mathbb{F}_{2}^{m}\mapsto\mathcal{A}\) maps consecutive sub-sequences of \(\xi_{u}\) of length \(m\) to a symbol in a constellation \(\mathcal{A}\) of size \(\mathfrak{M}=2^{m}\). Thus, the BICM maps \(\mathbf{b}_{u}\) to a sequence of symbols \(\mathbf{s}_{u}=(s_{u,\ell_{1}},s_{u,\ell_{1}+1},\ldots,s_{u,\ell_{2}})\), where \(s_{u,\ell}\in\mathcal{A}\), \(\ell\in[\ell_{1}:\ell_{2}]\), \(\ell_{1}=-\lfloor n_{s}/2\rfloor\), \(\ell_{2}=\lceil n_{s}/2\rceil-1\), \(n_{s}=n_{b}/m\). The symbols \(s_{u,\ell}\) are modulated with a root raised cosine (RRC) pulse shape \(p(t)\) at symbol rate \(R_{s}\), where \(t\) is the time. The resulting electrical signal of each channel \(x_{u}(t)\) is converted to an optical signal and subsequently multiplexed by a WDM multiplexer. The baseband representation of the transmitted signal is  
\[x(t)=\sum_{u=u_{1}}^{u_{2}}\sum_{\ell=\ell_{1}}^{\ell_{2}}s_{u,\ell}\,p(t- \ell T_{s})e^{\beta\pi uB_{\text{o}}t}, \tag{4}\]  
where \(T_{s}=1/R_{s}\) and \(s_{u,\ell}\) are i.i.d. random variables. The average power of the transmitted signal is \(\mathcal{P}\); thus, \(\mathbb{E}|s_{u,\ell}|^{2}=\mathcal{P},\forall u,\ell\). _Encoder_: SC LDPC codes are attractive options for optical communications [59]. These codes approach the capacity of the canonical communication channels [60; 61] and have a flexible performance-complexity trade-off. They are decoded with the BP and the sliding window decoder (SWD).