# Adaptive Learned Belief Propagation for Decoding Error-Correcting Codes
## 3 Channel Models
### Optical Fiber Channel
Another class of codes in optical communication is the SC product-like codes, braided block codes [62], SC turbo product codes [63], staircase codes [64; 65; 66] and their generalizations [67; 68]. These codes are decoded with iterative, algebraic hard decision algorithms and prioritize low-complexity, hardware-friendly decoding over coding gain. In this paper, the encoding in BICM combines an inner (binary or non-binary) QC-LDPC code \(\mathcal{C}_{i}\) with an outer SC QC-LDPC code \(\mathcal{C}_{o}\) whose component code is a multi-edge QC-LDPC code, as outlined in Table 1. The construction and parameters of the codes are given in Appendix B.1 and Appendix C, respectively. The choice of the inner code is due to the decoder complexity. Other options have been considered in the literature, for instance, algebraic codes, e.g., the BCH (Section 3.3 [69]) or Reed-Solomon codes (Section 3.4 [69]), or polar codes [70]. However, the QC-LDPC codes are simpler to decode, especially at high rates. The outer code can be an LDPC code [71; 72], a staircase code [71; 73], or a SC-LDPC code [72]. _Fiber-optic link_: The channel is an optical fiber link with \(N_{sp}\) spans of length \(L_{sp}\) of the standard single-mode fiber, with parameters in Table 2.  
Figure 1: Block diagram of an optical fiber transmission system. Let \(q(t,z):\mathbb{R}\times\mathbb{R}^{+}\mapsto\mathbb{C}\) be the complex envelope of the signal as a function of time \(t\) and distance \(z\) along the fiber. The propagation of the signal in one polarization over one span of optical fiber is modeled by the nonlinear Schrodinger equation [74]  
\[\frac{\partial q(t,z)}{\partial z}=-\frac{\alpha}{2}q(t,z)-\frac{j\beta_{2}}{2} \frac{\partial^{2}q(t,z)}{\partial t^{2}}+j\gamma|q(t,z)|^{2}q(t,z), \tag{5}\]  
where \(\alpha\) is the loss constant, \(\beta_{2}\) is the chromatic dispersion coefficient, \(\gamma\) is the Kerr nonlinearity parameter, and \(j=\sqrt{-1}\). The transmitter is located at \(z=0\) and the receiver at \(z=\mathcal{L}\). The continuous-time model (5) can be discretized to a discrete-time discrete-space model using the split-step Fourier method (Section III.B [75]). The optical fiber channel described by the partial differential Equation (5) differs significantly from the AWGN channel due to the presence of nonlinearity. An erbium doped fiber amplifier (EDFA) is placed at the end of each span, which compensates for the fiber loss, and introduces amplified spontaneous emission noise. The input \(x_{i}(t)\)-output \(x_{o}(t)\) relation of the EDFA is given by \(x_{o}(t)=Gx_{i}(t)+n(t)\), where \(G=e^{a\mathcal{L}_{\pi}}\) is the amplifier's gain, and \(n(t)\) is zero-mean circularly symmetric complex Gaussian noise process with the power spectral density  
\[\sigma^{2}=\frac{1}{2}(G-1)hf_{0}\mathrm{NF},\]  
where NF is the noise figure, \(h\) is a Planck constant, and \(f_{0}\) is the carrier frequency at 1550 nm. _Receiver:_ The advent of the coherent detection paved the way for the compensation of transmission effects in optical fiber using digital signal processing (DSP). As a result, the linear effects in the channel, such as the chromatic dispersion and polarization-induced impairments, and some of the nonlinear effects, can be compensated with DSP. At the receiver, a demultiplexer filters the signal of each WDM channel. The optical signal for each channel is converted to an electrical signal by a coherent receiver. Next, DSP followed by bit-interleaved coded demodulation (BICD) is applied. The continuous-time electrical signal is converted to the discrete-time signals by analogue-to-digital converters, down-sampled, and passed to a digital signal processing unit for the mitigation of the channel impairments. For equalization, digital back-propagation (DBP) based on the