# Adaptive Learned Belief Propagation for Decoding Error-Correcting Codes
## 6 Performance and Complexity Comparison
### Optical Fiber Channel  
We simulate a 16-QAM WDM optical communication system described in Section 3.2, with parameters described in Table 2. The continuous-time model (5) is simulated with the split-step Fourier method with a spatial step size of 100 m and a simulation bandwidth of 200 GHz. DBP with two samples/symbol is applied to compensate for the physical effects and to obtain the per-symbol channel law \(\Pr(y|s)\), \(s\in\mathcal{A}\), \(y\in\mathbb{C}\). For the inner code in the concatenated code, we consider two QC-LDPC codes of rate \(r_{i}=0.92\): binary single-edge code \(\mathcal{C}_{10}(4000,3680)\) and non-binary multi-edge code \(\mathcal{C}_{11}(800,32)\) over \(\mathbb{F}_{32}\), respectively, with PCMs \(\mathrm{H}_{10}(4,50,80,\mathrm{P}_{10})\) and \(\mathrm{H}_{11}(2,25,\,32,\,\mathrm{P}_{11})\), given in Appendix C. For the component code used in the outer spatially coupled code, we consider multi-edge QC-LDPC code \(\mathcal{C}_{12}(3680,3520)\) with the PCM \(\mathrm{H}_{12}(1,23,160,\mathrm{P}_{12})\). For \(m_{s}=2\) and \(L=100\), the resulting SC-QC-LDPC code has the PCM \(\mathrm{H}_{\mathrm{GC}}(1,23,160,\mathrm{P}_{12},2,100,\mathrm{\bar{B}}_{12})\), where \(\mathrm{\bar{B}}_{12}\) is the spreading matrix. The outer SC-QC-LDPC code is encoded with the sequential encoder [92]. This requires that the top-left \(\lambda M\times\omega M\) block \(\mathrm{H}_{0}(0)\) of \(\mathrm{H}_{\mathrm{SC}}\) in Equation (A2) is of full rank. Thus, \(\mathrm{\bar{B}}_{12}\) is designed to fulfill this condition. In Equation (A2), we have \(\mathrm{H}_{t}(\lambda=1,\omega=23,M=160)\stackrel{{\Delta}}{{=} }\mathrm{H}\in\mathbb{F}_{2}^{160\times 3680}\), \(t=0,1,2\), \(\mathrm{H}_{0}\) is of full rank, and \(\mathrm{H}_{\mathrm{SC}}\in\mathbb{F}_{2}^{16320\times 368000}\). The rate of the component code is \(r_{\mathrm{QC}}=1-1/23\approx 0.956\), and the rate of outer SC code is \(r_{\mathrm{o}}=r_{\mathrm{QC}}-\frac{m_{s}\lambda}{L\omega}\approx 0.955\), so \(r_{\mathrm{total}}=r_{\mathrm{i}}\cdot r_{\mathrm{o}}\approx 0.88\). The \(\mathrm{P}_{12}\) and \(\mathrm{\bar{B}}_{12}\) matrices are constructed heuristically and are given in Appendix C.  
The inner code is decoded with the parallel decoder, with five decoders with four iterations each. The decoder for the binary code \(\mathcal{C}_{9}\) is WMS with Type \(T_{a}VC\) parameter sharing, while for the non-binary code, \(\mathcal{C}_{11}\) is WEMS with Type \(T_{a}VCQ\) sharing and \(\beta_{c,o,q^{\prime}}^{(t)}=1\). The static EMS algorithm [93] is parameterized as in Section 4, initialized with the LLRs computed from Equation (1) [81]. The outer code is decoded with the SWD, with the static MS decoding of a maximum of 26 iterations per window.  
Tables 5 and 6 contain a summary of the numerical results. BER\({}_{i}\) is pre-FEC BER, and the reference BER for the coding gain is BER\({}_{o}=10^{-12}\). At \(\mathcal{P}=-10\) dBm, the total gap to  
Figure 7: Performance of the static and adaptive MS decoder for \(\mathcal{C}_{8}(4260,3834)\) at \(E_{b}/N_{0}=4\) dB, for the AWGN channel in the high-rate regime.  
NCG\({}_{f}\) for the adaptive weighted min-sum AWSM (resp., WEMS) decoder is 2.51 (resp., 1.75), while this value is 3.31 (resp., 2.29) and 3.44 (resp., 2.69), respectively, for the NNMS (resp., NNEMS) and MS (resp., EMS) decoders. Thus, the adaptive WBP provides a coding gain of 0.8 dB compared to the static NNMS decoder with about the same computational complexity and decoding latency.