## 5 Simulation Results and Discussion
### FER Performance  
Figure 2 shows the FER performance of different decoders for the length-1296 IEEE 802.11n LDPC codes with code rates \(2/3\), \(3/4\), and \(5/6\). We can see that the proposed RC-MIM-QSMS decoder achieves almost the same FER compared to its rate-dependent counterparts for the same bit width settings and the same LDPC codes. Moreover, the \((4,8)\)-RC-MIM-QSMS decoder can outperform the \((4,8)\)-QSMS decoder by at least \(0.1\) dB and approaches the performance of the \((\infty)\)-SBP decoder within \(0.15\) dB. In addition, the \((4,8)\)-RC-MIM-QSMS decoder also surpasses both the RC-MIM-QMS decoder\({}^{\text{R2}}\) and the RC-FAID decoder with the same bit width settings at most \(0.25\) dB for all simulated codes. With the \((3,7)\) bit width settings, the RC-MIM-QSMS decoder has an FER performance close to the \((4,8)\)-QSMS decoder for code rate \(5/6\) and even outperforms the \((4,8)\)-QSMS decoder for code rates \(3/4\) and \(2/3\). Compared to the \((3,7)\)-RC-FAID decoder, the \((3,7)\)-RC-MIM-QSMS decoder achieves a performance gain of at least \(0.2\) dB for all code rates. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Code Types** & \(R_{\text{c}}\) & \multicolumn{2}{c}{**RD-MIM-QSMS**} & \multicolumn{2}{c}{**RC-MIM-QSMS**} \\ \cline{3-5}  & & **(3,7)** & **(4,8)** & **(3,7)** & **(4,8)** \\ \hline \multirow{4}{*}{\(802.11\text{n}\)} & \(2/3\) & \(0.6877\) & \(0.7002\) & \multirow{4}{*}{\(0.6058\)} & \multirow{4}{*}{\(0.6219\)} \\ \cline{2-2} \cline{5-5}  & \(3/4\) & \(0.6148\) & & & \\ \cline{2-2} \cline{5-5}  & \(5/6\) & \(0.5392\) & & & \\ \hline \multirow{4}{*}{\(5\)G} & \(3/4\) & \multirow{4}{*}{-} & \(0.6705\) & \multirow{4}{*}{-} & \multirow{4}{*}{\(0.6050\)} \\ \cline{2-2} \cline{5-5}  & \(5/6\) & & & & \\ \cline{1-1} \cline{2-2} \cline{5-5}  & \(5/6\) & & & & \\ \cline{1-1} \cline{2-2} \cline{5-5}  & \(8/9\) & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Design Noise Standard Deviations for the RC-MIM-QSMS and RD-MIM-QSMS Decoders. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Code Types** & \(R_{\text{c}}\) & **Degree Distributions (\(\rho(\xi),\theta(\xi)\))** \\ \hline \multirow{4}{*}{\(802.11\text{n}\)} & \(2/3\) & \(\rho(\xi)=\xi^{10}\) \\  & \(2/3\) & \(\theta(\xi)=0.1591\xi+0.4091\xi^{2}+0.1591\xi^{6}+\xi 0.2727\xi^{7}\) \\ \cline{2-2} \cline{5-5}  & \(3/4\) & \(\rho(\xi)=0.3182\xi^{13}+0.6818\xi^{14}\) \\  & \(\theta(\xi)=0.1136\xi+0.4091\xi^{2}+0.4773\xi^{5}\) \\ \cline{2-2} \cline{5-5}  & \(5/6\) & \(\rho(\xi)=0.7412\xi^{20}+0.2586\xi^{21}\) \\  & \(\theta(\xi)=0.0706\xi+0.1765\xi^{2}+0.7529\xi^{3}\) \\ \hline \multirow{4}{*}{\(5\)G} & \(\rho(\xi)=0.0246\xi^{2}+0.0574\xi^{6}+0.0656\xi^{7}+0.1475\xi^{8}+0.0820\xi^{9 }+0.6230\xi^{18}\) \\  & \(3/4\) & \(\theta(\xi)=0.0492+0.0328\xi^{8}+0.1230\xi^{2}+0.1967\xi^{3}\) \\  & \(+0.4508\xi^{4}+0.0656\xi^{7}+0.0820\xi^{9}\) \\ \cline{2-2} \cline{5-5}  & \(\rho(\xi)=0.0313\xi^{2}+0.0833\xi^{7}+0.0938\xi^{8}+0.7917\xi^{18}\) \\ \cline{2-2} \cline{5-5}  & \(5/6\) & \(\rho(\xi)=0.0313+0.0625\xi+0.2813\xi^{2}+0.5000\xi^{3}+0.0521\xi^{4}+0.0729\xi^{6}\) \\ \cline{2-2} \cline{5-5}  & \(8/9\) & \(\rho(\xi)=0.0380\xi^{2}+0.9620\xi^{18}\) \\ \cline{2-2} \cline{5-5}  & \(\theta(\xi)=0.0127+0.0759\xi^{5}+0.7975\xi^{2}+0.0506\xi^{3}+0.0633\xi^{4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Degree Distributions of the IEEE 802.11n LDPC Codes and the 5G LDPC Codes. Figure 3 depicts the FER performance of different decoders for the 5G LDPC codes with lifting size 26 and code rates 2/3, 3/4, and 5/6. As shown by the figure, the proposed \((4,8)\)-RC-MIM-QSMS decoder achieves almost the same FER compared to its rate-dependent counterparts for the same simulated code rates. Furthermore, the \((4,8)\)-RC-MIM-QSMS decoder performs better than the \((4,8)\)-QSMS decoder by up to 0.4 dB and approaches the performance of the \((\infty)\)-SBP decoder within 0.2 dB.