## 5 Simulation Results and Discussion
### Memory Requirement  
We further investigated the overall memory requirement of the proposed RC-MIM-QMS decoder and compare it to that of different quantized LDPC decoders. Here, we considered the decoders that are implemented based on software-defined radios or digital signal processors so that the LUTs are stored in memories [11]. We divided the memory into two types according to their usage, i.e., the memories for arithmetic calculation and those for storing the LUTs. For the conventional QSMS decoders, all \(q_{m}\)-bit V2C messages need to be stored for the arithmetic calculation of the node updates at each iteration [27], while the RC-FAID decoder with flooding schedule uses the memories for arithmetic calculation to store two \(q_{m}\)-bit C2V messages for each CN and one a posteriori message of \(q_{v}\) bit width for each VN per iteration. Note that we considered the parity-check matrix of the rate-2/3 802.11n LDPC code to evaluate the maximum memory requirement for arithmetic calculation since it had the largest size among all simulated codes. Moreover, we assumed that the memories for arithmetic calculation could be reused between two consecutive iterations for improving efficiency. On the other hand, the memory requirement for storing one LUT could be computed by \((E\cdot q_{t})/8\) in bytes, where \(E\) is the number of entries in one LUT, and \(q_{t}\) refers to the maximum bit width of an entry. According to [11], at each iteration, the RC-FAID decoder requires \(d_{v,\max}-1\) cascaded LUTs for updating the VNs of different degrees and one LUT for message alignment process. Since the RC-FAID decoder exchanges \(q_{m}\)-bit messages within the decoder and adopts two-input LUTs, we had \(E=2^{2q_{m}}\) for each LUT and \(q_{t}=q_{m}\). In addition, there was one extra LUT of \(q_{t}=1\) at each iteration for making the hard decision. For the two MIM-QMS decoders with \(q_{m}=4\), all LUTs had a single input so that there were two reconstruction LUTs of size \(E=2^{q_{m}}\) and one quantization LUT of size \(E=2^{q_{m}}-1\). The bit width of each entry in each LUT could be obtained by \(q_{t}=\lfloor\log_{2}(|z_{\max}|)\rfloor\), where \(|z_{\max}|\) is the maximum magnitude of the entry. Table 5 summarizes the overall memory requirements of different quantized LDPC decoders with \(q_{m}=4\) and \(T_{\max}=15\) for the 802.11n LDPC codes. As shown in the table, our proposed RC-MIM-QMS decoder requires almost the same memory as the RC-MIM-QMS decoder and it only has a slight increase in the memory demand of 14.45% compared to that of the conventional QSMS decoder. More importantly, the RC-MIM-QMSMS decoder can reduce the memory demand by 28.26% compared to its rate-dependent counterparts, and significantly saves 87.54% of memory demand when compared with the RC-FAID decoder. Table 6 presents the overall memory requirements of different quantized LDPC decoders with \(q_{m}=4\) and \(T_{\max}=15\) for the 5G LDPC codes. It shows that our proposed RC-MIM-QMSMS decoder only increases the memory demand by 16.81% compared to the conventional QSMS decoder and also requires less memory demand when compared with the RC-MIM-QMS decoder. More significantly, the RC-MIM-QMSMS decoder requires \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Decoders**} & \multirow{2}{*}{**Arithmetic**} & \multicolumn{2}{c}{**LUTs**} & \multirow{2}{*}{**Total**} \\ \cline{3-4}  & & \(R_{\epsilon}=\) **2/3** & \(R_{\epsilon}=\) **3/4** & \(R_{\epsilon}=\) **5/6** \\ \hline RC-FAID [11] & 0.42 kB & & 15.47 kB & 15.89 kB \\ \hline RC-MIM-QMS [14] & & & 0.26 kB & 1.99 kB \\ \hline Conventional QSMS [27] & \multirow{2}{*}{1.73 kB} & \multirow{2}{*}{-} & \multirow{2}{*}{1.73 kB} \\ \cline{3-4} RD-MIM-QMS [15] & & & 0.32 kB & 0.34 kB & 0.37 kB \\ \hline RC-MIM-QMSMS & & & 0.25 kB & 1.98 kB \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The Overall Memory Requirement for Different Quantized Decoders with \(q_{m}=4\) and \(T_{\max}=15\) for the 802.11n LDPC codes.