## 4 Simulation and Analysis of the MRAI-LDPC Decoding Algorithm
### Complexity Analysis  
Here, we analyze the computational complexities of the presented MRAI-LDPC algorithm per iteration. The hard decision \(\mathbf{z}^{(k)}\) requires \(nr\) real comparisons (RCs). To compute the syndrome vector \(\mathbf{s}^{(k)}\), \(n\gamma\) Galois-field multiplications (GMs) and \(n(\gamma-1)\) Galois-field additions (GAs) are required for computing. Then, \(n\gamma\) GAs and \(n\gamma\) GMs are required to obtain the extrinsic information \(\sigma_{i,j}^{(k)}\). Counting the extrinsic information frequency \(n(\sigma_{i,j}^{(k)})\) requires \(n\gamma\) real additions (RAs). Additionally, the mapped processing information \(w_{j}\) is computed by \(n\gamma\) RCs. It takes \(n\gamma(r-1)\) RAs to compute the correlated reliability \(\rho(\sigma_{i,j}^{(k)},w_{j})\) between \(\sigma_{i,j}^{(k)}\) and \(w_{j}\). We require \(3n\gamma\) GAs for computing the total reliability \(M_{j}(\sigma_{i,j}^{(k)})\). Finally, updating the new decision vector involves \(n(\gamma-1)\) RCs in the MRAI-LDPC algorithm.  
Let \(\delta\) be the total edges of the nodes; note that \(\delta=n\gamma=m\rho\). When an iteration is completed, 0 real multiplications (RMs), \(5\delta-n\) GAs, \(2\delta\) GMs, \(\delta r\) RAs, and \(\delta+n(2r-1)\) RCs are required for the MRAI-LDPC algorithm. The total complexities are shown in Table 2. The MRAI-LDPC algorithm increases \(2\delta\) GA, \(\delta(r-1)\) RA, and \(n(2r-1)\) RC operations compared to the wtd-AlgB algorithm. However, lower energy consumption can be obtained in hardware implementation since the MRAI-LDPC algorithm does not involve the real-multiplication operation.  
Under the simulation of the \(F_{16}(225,147)\) QC-LDPC code with finite-field construction, when the decoding is successful, the average number of iterations for the wtd-AlgB algorithm and the MRAI-LDPC algorithm is 9 and 5 at an SNR of 4.5 dB. The intuitive data  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Decoding Algorithm** & **RM** & **GA** & **GM** & **RA** & **RC** \\ \hline wtd-AlgB & \(\delta\) & \(3\delta-n\) & \(2\delta\) & \(\delta\) & \(\delta\) \\ Presented MRAI-LDPC & 0 & \(5\delta-n\) & \(2\delta\) & \(\delta r\) & \(\delta+n(2r-1)\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Computational complexities per iteration with various decoding algorithms.  
Figure 7: The ANI performance comparison of the \(F_{16}(255,175)\) QC-LDPC code.  
are given here, and the numerical results of the total computational complexities are shown in Table 3. Specifically, the total computational complexities of the MRAI-LDPC algorithm are 29,025 fewer than those of the wtd-AlgB algorithm.  
Similarly, under the \(F_{16}(255,175)\) regular NB-LDPC code and an SNR of 4.6 dB, the average number of iterations for the wtd-AlgB algorithm and the MRAI-LDPC algorithm is 10 and 4, respectively. According to the values in Table 4, compared with the wtd-AlgB algorithm, the total computational complexities of the MRAI-LDPC algorithm are reduced by 121,890, which greatly reduces the difficulty of hardware implementation.  
To summarize the results in Tables 3 and 4, it is clear that for a certain SNR, the presented MRAI-LDPC algorithm has the lowest complexity compared to the wtd-AlgB algorithm, which is an advantage that can reduce energy consumption and cost.