## 4 Design of MIM-QSMS Decoder for RC-QC-LDPC Codes
### Proposed SMIM-DE
#### 4.1.2 CN Update  
In [15], we assume that the V2C messages sent to the CNs located in the same layer have equal contributions so that we consider their expected pmfs for the CN update. However, this is not the case in designing the RC-MIM-QSMS decoder because the CNs at the same layer are supposed to receive the V2C messages not only from different layers but also from different target LDPC codes when using the joint edge-degree distributions. Since the number of layers may be different for each target LDPC code, we consider the weighted expectation of the V2C message pmfs in the proposed SMIM-DE by introducing a parameter called the weight coefficient. Define \(\mathbf{I}_{k}\) of size \(N_{b,\max}\) as the layered indicator of the \(k\)th target LDPC code. Each entry \(i_{k,j}\in\mathbf{I}_{k}\) for \(j=1,2,\ldots,N_{b,\max}\) is of binary values such that  
\[i_{k,j}=\begin{cases}1,&j\leq N_{b,k},\\ 0,&\text{otherwise},\end{cases} \tag{10}\]where the value 1 means that the corresponding layer exists for the \(k\)th target LDPC code. At the \(j\)th layer, we define the weight coefficient of the V2C message pmfs as  
\[w_{j}=\frac{\sum_{k=1}^{K}i_{k,j}}{\sum_{k=1}^{K}\sum_{j=1}^{N_{b,\max}}i_{k,j}}. \tag{11}\]  
At the \(t\)th iteration, a CN of degree \(d\) (\(d\in\mathcal{D}_{c}\)) in layer \(j\) receives the V2C messages that were updated at iteration \(t-1\) from layers \(j^{\prime}>j\) and have been updated at iteration \(t\) from layers \(j^{\prime}<j\). Therefore, the expected pmf of the V2C message received at layer \(j\) can be expressed as  
\[\tilde{P}_{R|X}^{(t,j)}(r|x)=\frac{\sum_{j^{\prime}\
eq j,i_{j^{\prime}}=1}^{N _{b,\max}}w_{j^{\prime}}\cdot P_{R|X}^{(t-1(j^{\prime}>j),f^{\prime})}(r|x)}{ \sum_{j^{\prime}\
eq j,i^{\prime}=1}^{N_{b,\max}}w_{j^{\prime}}}. \tag{12}\]  
Note that we have \(P_{R|X}^{(0,j)}=P_{L|X}\) for \(j=1,2,\ldots,N_{b,\max}\). Define the vector of V2C messages received at a degree-\(d\) (\(d\in\mathcal{D}_{c}\)) CN by \(\mathbf{R}\in\mathcal{R}^{d-1}\), where \(\mathbf{r}=(r_{1},r_{2},\ldots,r_{d-1})\in\mathcal{R}^{d-1}\) is a realization of \(\mathbf{R}\). Denote \(\mathbf{x}=(x_{1},x_{2},\ldots,x_{d-1})\) as the vector of coded bits associated to its neighboring VNs. With the independent and identical distribution (i.i.d.) assumption [4], the joint pmf at layer \(j\) and iteration \(t\) is given by  
\[\tilde{P}_{\mathbf{R}|X}^{(t,j)}(\mathbf{r}|x)=\left(\frac{1}{2}\right)^{d-2} \sum_{\mathbf{x}\in\mathbf{x}=x}\prod_{k=1}^{d-1}\tilde{P}_{R|X}^{(t,j)}(r_{k }|x_{k}), \tag{13}\]  
where \(x\in\mathcal{X}\) is a realization of \(X\), and \(\oplus\mathbf{x}=x\) means the checksum of the CN is satisfied. For each received realization \(\mathbf{r}\), the CN update function \(\Phi_{c}^{\text{MS}}\) computes the corresponding output \(s\) as in (5). By considering the fraction \(\rho_{d}\) in (8), the conditional pmf of the C2V message at layer \(j\) and iteration \(t\) can be obtained by  
\[P_{S|X}^{(t,j)}(s|x)=\sum_{d\in\mathcal{D}_{c}}\rho_{d}\cdot\sum_{\begin{subarray} {c}\mathbf{r}\in\mathcal{R}^{d-1}\\ \phi_{c}^{\text{MS}}(\mathbf{r})-s\end{subarray}}\tilde{P}_{\mathbf{R}|X}^{(t,j)}(\mathbf{r}|x). \tag{14}\]  
Note that the computation of (14) can be conducted recursively such as in [4; 16] to achieve low complexity, which is not a big issue for an off-line process.