# Adaptive Learned Belief Propagation for Decoding Error-Correcting Codes
## 6 Performance and Complexity Comparison
### AWGN Channel
Table 3 presents the mean and variance of this distribution, and \(x_{k}^{(t)}\), for the three codes considered. For the two-stage decoder, we use a CNN with \(F_{1}=5\), \(S_{1}=3\), \(F_{2}=8\), and \(S_{2}=2\), determined by cross-validation. The CNN is trained with a dataset of size 80,000, batch set size 300, and the quantile loss function with \(\xi=0.75\). The number of iterations of the WMS decoder for each code is the same as above. Figure 4 illustrates the BER vs. SNR for different codes, and different decoders for the same code. In each of Figure 4a-d, one can compare different decoders at about the same complexity (except for the parallel decoder with the largest \(\
u\) that shows the smallest achievable BER). For instance, it can be seen in Figure 4a that the two-stage decoder achieves half the BER of the WMS with SS-PAN decoder at SNR 6.4 dB for the short length code \(\mathcal{C}_{1}\), or approximately 0.32 dB gain in SNR at a BER of \(10^{-4}\). For this code, the parallel WMS decoders with 3 iterations and \(\
u=9\) outperforms the tanh-based BP with nearly the same complexity. Figure 4b,c show that the two-stage decoder offers about an order-of-magnitude improvement in the BER compared to the Type \(T_{a}\) WMS decoder at 4.2 dB for moderate-length codes \(\mathcal{C}_{2}\) and \(\mathcal{C}_{3}\), or over 0.1 dB gain at a BER of \(10^{-6}\). The performance gains vary with the code, parameters, and SNR. Figure 4d-f compare decoders with different complexities at about the same performance. The proposed adaptive model-based decoders achieve the same performance of the model-agnostic static decoders, with far fewer parameters. The computational complexity of the decoders are presented in Table 4. For the CNN, from (17), \(\text{RM}=n(8T+95)-24T-310\), which is further reduced by a factor of 4 upon pruning at the threshold \(\tau_{\text{prun}}=0.001\), with minimal impact on BER. Thus, the two-stage decoder requires less than half of the RM of the WMS decoder with no or Type \(T_{a}\) parameter sharing. Moreover, the two-stage decoder requires approximately one-fifth of the RM of the parallel decoders with \(\
u=16\). Compared to the WMS decoder with SS-PAN [18], the two-stage decoder has nearly double the complexity, albeit with much lower BER, as seen in Figure 4a. _High-rate regime:_ To further investigate the error correction performance of the decoders at high rates, we consider three single-edge QC-LDPC codes, \(\mathcal{C}_{6}(1050,875)\), \(\mathcal{C}_{7}(1050,850)\), and \(\mathcal{C}_{8}(4260,3834)\), associated, respectively, with the PCMs \(\mathrm{H}_{6}(\lambda=7,\omega=42,M=25,\mathrm{P}_{6})\), \(\mathrm{H}_{7}(8,42,25,\mathrm{P}_{7})\), and \(\mathrm{H}_{8}(6,60,71,\mathrm{P}_{8})\). These codes have rates \(r=0.84\), \(0.81\), and \(0.9\), respectively, and are constructed using the PEG algorithm. The PEG algorithm requires the degree distributions of the Tanner graph, which are optimized using the stochastic extrinsic information transfer (EXIT) chart described in Appendix C. Additionally, we include the polar code \(\mathcal{C}_{9}(1024,854)\) with \(r=0.84\) from the 5G-NR standard as a state-of-the-art benchmark. The code parameters, including degree distribution polynomials and the exponent matrices, are given in Appendix C.  
The acquisition function with active learning in the parallel decoders is based on Figure 5. The figure shows the scatter plot of the pre-FEC error \(\epsilon_{1}=\|\mathbf{c}-\hat{\mathbf{y}}\|\) versus post-FEC error \(\epsilon_{2}=\|\mathbf{c}-\hat{\mathbf{c}}\|_{H}\), for 340 examples \((\mathbf{c},\mathbf{y})\) for \(\mathcal{C}_{6}\) at \(E_{b}/N_{0}=4.25\) dB. Here, \(\hat{\mathbf{y}}\) is the hard decision of the LLRs at the channel output defined in (11), and \(\hat{\mathbf{c}}\) is decoded with the best decoder at epoch \(e\), i.e., the WMS with weights from epoch \(e-1\).