# Improving Polar-Coded SCMA System by Information Coupling and Parity Check
## 4 Joint Iterative Detection and SCL Decoding Receiver
### Updated a Priori Information of MPA Detector  
The prior information of MPA detector is updated by the extrinsic messages of coupled polar decoder. In this subsection, we mainly focus on the SCL decoding algorithm of base polar code. Note that in iterative receiver, in order to iteratively exchange their messages between the MPA detector and channel decoder, they should be soft-input-soft-output. Therefore, the coupled polar decoder must reconstruct the soft output information that can be used for iteration. On the basis of LLR-based SCL algorithm [11], we introduce the construction of extrinsic messages of the SCL decoder and extend it to the coupled polar decoder. For the \(k\)-th CB of user \(v\), after the decoding of the last bit, the SCL decoder outputs the \(L_{\text{o}}\) most likely path \(\hat{\mathbf{a}}_{v,k}^{l^{\prime}}\) and corresponding path metric \(P_{v,k^{\prime}}^{l^{\prime}}\), \(1\leq l^{\prime}\leq L_{o}\). For each candidate path \(l^{\prime}\), the corresponding codeword can be obtained by encoding the estimated sequence \(\hat{\mathbf{a}}_{v,k}^{l^{\prime}}\). \[\hat{\mathbf{c}}_{v,k}^{l^{\prime}}=\hat{\mathbf{a}}_{v,k}^{l^{\prime}}\mathbf{G}. \tag{8}\]  
In order to calculate the extrinsic messages, the path metric \(P_{v,k}^{l^{\prime}}\) is normalized as follows  
\[p_{v,k}^{l^{\prime}}=\exp\big{(}P_{v,k}^{l^{\prime}}\big{)}/\sum \
olimits_{l^{\prime}=1}^{L_{\text{o}}}\exp\Big{(}P_{v,k}^{l^{\prime}}\big{)}. \tag{9}\]  
Then, the probability of the bit \(c_{v,k}^{i}\) can be written as  
\[p\big{(}c_{v,k}^{i}\big{)}=\sum\limits_{1\leq l^{\prime}\leq L_{ \text{o}}}\;\;p_{v,k}^{l^{\prime}}. \tag{10}\] \[\hat{c}_{v,k}^{i,l^{\prime}}=c_{v,k}^{i}\]Thus, for the mother code \(x_{k}\), by the Bayes rule, the extrinsic messages of the SCL decoder \(L_{x,k}^{v}=\left(L_{x,k}^{v,1},\ldots,L_{x,k}^{v,N_{m}}\right)\) can be calculated as  
\[L_{x,k}^{v,i}\left(i\right)=\ln\left(\frac{p\left(c_{v,k}^{i}=0\right)}{p\left( c_{v,k}^{i}=1\right)}\right)=\left(\ln\left( \begin{array}{c}+\infty&p\left(c_{v,k}^{ i}=1\right)=0\\ \sum\limits_{1\leq l^{\prime}\leq L_{\mathrm{o}}}&p_{v,k}^{l^{\prime}}\\ \sum\limits_{v,k}^{i\
u,l^{\prime}}=0&\\ \sum\limits_{1\leq l^{\prime}\leq L_{\mathrm{o}}}&p_{v,k}^{l^{\prime}}\\ \delta_{v,k}^{i\
u,l^{\prime}}=1&\\ -\infty&p\left(c_{v,k}^{i}=0\right)=0\end{array} \right)\right. \tag{11}\]  
With CRC aiding, if the path \(l^{\prime}\) passes the CRC detection, the sign of \(L_{x,k}^{v,i}\) is modified by  
\[L_{x,k}^{v,i}=\left(1-2\delta_{v,k}^{i,\
u,i}\right)\left|L_{x,k}^{v,i}\right| \tag{12}\]  
We call the method described above a Bayes construction (BC) algorithm, which can be directly used in list decoder. But, the accuracy of the extrinsic message depends on the list size and the result of CRC check. When the list size is small and the check fails, the constructed extrinsic messages of SCL decoding by the BC algorithm are not accurate enough. Therefore, the hybrid construction (HC) algorithm can be used to construct the extrinsic messages of the SCL decoder. When the CRC check fails, the SCAN algorithm is used to construct the extrinsic message as follows. Based on the SCAN algorithm, the extrinsic messages are calculated on the factor graph. For a \(N_{m}\)-length polar mother code, the factor graph is composed of \(n+1=\log_{2}(N_{m})+1\) columns, and every two adjacent columns comprise \(N_{m}/2\) processing elements (PEs) shown in Figure 7. The pair \((i,j)\) refers to the \(i\)-th variable node at column \(j\) in the factor graph, where \(1\leq i\leq N_{m}\) and \(1\leq j\leq n+1\). The left and right propagating messages of node \((i,j)\) are denoted as \(L(i,j)\) and \(R(i,j)\), respectively. The left messages and right messages are passed toward the left and right direction of the factor graph, respectively. The \(L(i,n+1)\) and \(R(i,1)\) are respectively initialized by the prior LLR information \(z_{v,k}\) and (13). \[R(i,1)=\left\{ \begin{array}{cc}0&i\in\mathcal{A}^{\prime}\\ +\infty&i\in\mathcal{A}^{\prime c}\end{array}