# LDPC Decoder Design Using Compensation Scheme of Group Comparison for 5G Communication Systems
## 4 Proposed Compensation Scheme for NPMSA
### Simulation Results  
The proposed methods implemented the floating-point bit-error rate (BER) performance on NMSA and NPMSA using improved methods based on layer decoding under the rate-5/6 (1944, 324) QC-LDPC code; with early termination and different improving algorithms, row weight \(w_{r}\) = 20, scaling factor \(\gamma\) = 0.75, and maximum iteration = 20 for the IEEE 802.11n, as shown in Figure 2. The NPMSA-2 and NPMSA-4 represent NPMSAs with 2 Group and 4 Group searches, respectively. The BER performance of INPMSA-4 with \(\alpha\) = 0.5 and \(\beta\) = 0.5 is worse than that of the NPMSA-2. However, the INPMSA-4 with \(\alpha\) = 0.125 and \(\beta\) = 0.875 significantly improves the BER performance when \(\alpha\) is selected using a smaller value of error probability listed in Table 2. The BER performance between the INPMSA-4 with \(\alpha\) = 0.125 and \(\beta\) = 0.875, and the ExMin algorithm using the 4 Group search with \(\theta\) = 0.5, are approximately the same. There is a similarity in the decoding performance between SNR = 4.0 dB and SNR = 4.5 dB, showing that the INPMSA-4 with \(\alpha\) = 0.125 and \(\beta\) = 0.875 can achieve better decoding performance than the NPMSA-4, without improvement in the SNR. In comparison to the SMA-MSA, the INPMSA-4 can achieve a coding gain of about 0.15 dB when the BER = \(10^{-5}\). This is because the SMA-MSA does not consider the reversal problem, which causes the degradation of the decoding performance due to the use of the inappropriate new second minimum. In the NAND flash channel model, the BER performance is based on the rate-0.89 (9728, 1024) LDPC code, which uses early termination and different improvement algorithms, a scaling factor \(\gamma\) = 0.75, row weight \(w_{r}\) = 36, and maximum iteration = 8. The dashed lines in Figure 3 denote the results of the lower page. The solid lines denote the results of the upper page. The proposed decoder [9] suggests that dividing \(Q_{i,j}^{k}\) into four groups can achieve the best decoding performance and hardware implementation. Therefore, we adopted four groups of minima finders. There is a reversal problem in the performance of the SMA-MSA, which caused the decoding performance to decline. The performance of the INPMSA is similar to that of the rExMin algorithm when \(Q_{i,j}^{k}\) is divided into four groups. Compared to the NPMSA, the INPMSA can reach approximately 0.028 and 0.02 dB in the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Group \(G\)** & & **2** & **4** & **8** & **16** \\ \hline \multirow{4}{*}{
\begin{tabular}{} \end{tabular} } & 8 & 42.9\% & 14.3\% & 0\% & 0\% \\  & 16 & 46.7\% & 20\% & 6.7\% & 0\% \\ \cline{1-1}  & 19 & 47.2\% & 20.8\% & 7.6\% & 1\% \\ \cline{1-1}  & 32 & 48.3\% & 22.6\% & 9.7\% & 3.2\% \\ \cline{1-1}  & 64 & 49.2\% & 23.8\% & 11.1\% & 4.8\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Error probability of probabilistic second minimum for different groups and inputs. upper and lower pages, respectively, when the BER = \(10^{-5}\). When the BER performance drops below \(10^{-6}\), the decoding capability starts to converge, with the exception of the NMSA. From the above simulation results, the INPMSA is capable of avoiding the reversal problem and efficiently achieves the correction effect to improve the decoding capability. In addition, we applied the INPMSA to models implemented in related studies using the NPMSA to observe the practicability of the NAND flash channel model. We referred to the partially stopped probabilistic min-sum algorithm (PS-PMSA) [22], which discards the unnecessary operations in check node updating using error-cor  
Figure 3: Simulation results of different algorithms in the NAND flash channel model. Figure 2: Simulation results of different algorithms in IEEE 802.11n. shown in Figure 4. Comparing the NPMSA and the PS-PMSA shows a 0.05 dB coding gain when the BER = \(10^{-5}\).