## 4 Design of MIM-QSMS Decoder for RC-QC-LDPC Codes
### Remarks  
Denote the noise standard deviation of the selected AWGN channel to design the RC-MIM-QSMS decoder as \(\sigma_{d}\). Given the tuple \((q_{m},q_{v})\), the degree distributions \((\rho(\xi),\,\rho(\xi))\), and the maximum number of iterations \(T_{\max}\), different choices of \(\sigma_{d}\) result in different sets of reconstruction and quantization LUTs. Compared to the conventional DE [4] that focuses on the decoding error probability, the design of the RC-MIM-QSMS decoder is similar to that of other MIM-FAIDs, e.g., [12; 15; 16], which aim to achieve a certain MI value approaching \(1\) for the a posteriori messages. Let \(\sigma^{*}\) be the optimal noise standard deviation for designing the RC-MIM-QSMS decoder, which is determined by following the steps below. Denote the alphabet set of a posteriori messages by \(\mathcal{Q}\), where \(Q\in\mathcal{Q}\) is the random variable for a posterior message. With the iteration-specific reconstruction LUTs and the conditional pmf \(\bar{P}_{S|X}^{(t)}\) given by (20), we first obtain \(\mathcal{Q}\) from (6) and compute the conditional pmf \(P_{Q|X}\) based on (18) by considering \((l,\mathbf{s})\in\mathcal{L}\times\mathcal{S}^{d}\). Define \(\hat{X}\) as the hard decision of \(Q\), which takes values from \(\mathcal{X}=\{0,\,1\}\). We select \(\sigma^{*}\) equal to the maximum \(\sigma_{d}\) that achieves a mutual information between \(X\) and \(\hat{X}\) greater than \(1-\epsilon\) after \(T_{\max}\) iterations, i.e.,  
\[\sigma^{*}=\sup\Big{\{}\sigma_{d}:I^{(\sigma_{d},T_{\max})}(X;\hat{X})>1- \epsilon\Big{\}}. \tag{21}\]As shown in the literature [12; 15; 16], we set \(\epsilon=10^{-3}\) in this paper for achieving a desirable error rate performance across a wide range of SNRs. We note that the proposed RC-MIM-QSMS decoder directly determines the coded bits at each iteration according to \(\hat{X}\) rather than using the quantization LUT for bit decision as in [14; 15]. This is more efficient for hardware implementation compared to the RD-MIM-QSMS decoder [15] and the MIM-FAID [14].  
In addition, we also notice that it is possible for the RC-MIM-QSMS decoder to cause a performance degradation compared to the RD-MIM-QSMS decoders since the joint degree distributions considered in the LUT design are mismatched with the specific degree distributions of any target LDPC codes. However, the performance loss can be limited within a certain range by selecting appropriate target LDPC codes for rate-compatible design. Intuitively, for preset bit widths (\(q_{m}\), \(q_{\sigma}\)) and the maximum number of iterations \(T_{\text{max}}\), the target LDPC codes that have the design noise standard deviations close to each other are preferable to be optimized jointly. This is because the MI values of these target LDPC codes are likely to approach 1 after \(T_{\text{max}}\) iterations. To verify this hypothesis, we quantify \(\sigma_{d}\) for a given code rate \(R_{c}\) by the corresponding design SNR, which is denoted as  
\[\tau\stackrel{{\Delta}}{{=}}-10\text{log}_{10}(2R_{c}\cdot\sigma _{d}^{2})\text{ [dB]}. \tag{22}\]  
Based on extensive simulation results, we let any two target LDPC codes have different values of \(\tau\) within 1.2 dB for achieving a desirable error rate performance. As we can see in the later simulations, this criterion leads to the RC-MIM-QSMS decoders operating on three different code rates for the length-1296 IEEE 802.11n LDPC codes [30] and the fifth-generation (5G) LDPC codes [31].