# Improving Polar-Coded SCMA System by Information Coupling and Parity Check
## 2 System Model  
Figure 1 shows the uplink information-coupled PC-SCMA system. In this system, a TB consists of \(L_{c}\) mutually coupled CB. For CB \(k,1\leq k\leq L_{c}\), the information bits of \(V\) users \(\mathbf{U}_{k}=\left\{\mathbf{u}_{1,k},\mathbf{u}_{2,k},\cdots,\mathbf{u}_{V,k}\right\}\) are encoded to \(\mathbf{C}_{k}=\left\{\mathbf{c}_{1,k},\mathbf{c}_{2,k},\cdots,\mathbf{c}_{V,k}\right\}\) by information coupling, CRC, and systematic encoder. For user \(v\), its information bits and coded bits are denoted as \(\mathbf{u}_{v,k}=\left\{u_{v,k}^{1}\,u_{v,k}^{2},\cdots,u_{v,k}^{K}\right\}\) and \(\mathbf{c}_{v,k}=\left\{c_{v,k}^{1}\,c_{v,k}^{2},\cdots,c_{v,k}^{N}\right\}\), respectively. Then, each user's codeword is interleaved by the random interleaver, which is denoted by \(\mathbf{d}_{v,k}=\Pi\left(\mathbf{c}_{v,k}\right)\). Assuming the \(V\) users multiplexed over \(M\left(M<V\right)\) orthogonal resources. Then, every \(Q=\log_{2}(M)\) bits of \(\mathbf{d}_{v,k}\) are mapped to a \(J\)-dimensional complex codeword \(\mathbf{s}_{v,k}=\left\{s_{v,k}^{1},s_{v,k}^{2},\cdots,s_{v,k}^{J}\right\}\) by SCMA mapper, and the number of SCMA codewords is \(L_{k}=N/Q\). Thus, the overloading factor of SCMA is defined as \(\lambda=V/J\) and the structure of an SCMA code can be represented by a \(J\times V\) binary mapping matrix \(\mathbf{F}\). As shown in Figure 2, a SCMA factor graph with six variable nodes (VNs) and four function nodes (FNs) \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Abbreviation** & **Full Name** & **Abbreviation** & **Full Name** \\ \hline
5G & Fifth generation & AWGN & Additive white Gaussian noise \\ BC & Bayes construction & BLER & Block error rate \\ BMC & Binary-input memoryless channel & BP & Belief propagation \\ CA & CRC-aided & CB & Code block \\ CBER & Code block error rate & CRC & cyclic redundancy check \\ DCA & Distributed CRC-aided & eMBB & enhanced Mobile BroadBand \\ EXIT & Extrinsic information transfer & FN & Function node \\ HC & Hybrid construction & IDD & Iterative detection and decoding \\ JIDD & Joint iterative detection and decoding & JIDS & Joint iterative detection and SCL decoding \\ JSC & Joint successive cancellation & LDPC & Low-density parity-check \\ LDPC-SCMA & LDPC coded sparse code multiple access & LDS & Low-density signatures \\ LLR & Log-likelihood rate ratio & log-BP & log-Belief propagation \\ mMTC & massive Machine Type Communications & MPA & Message passing algorithm \\ MUD & Multi-user detection & MUSA & Multi-user shared access \\ NOMA & Non-orthogonal multiple access & NR & New radio \\ OMA & Orthogonal multiple access & PCCA & Joint parity check and CRC aided \\ PC-SCMA & Polar-coded sparse code multiple access & PDMA & Pattern division multiple access \\ PE & Processing element & PIC & Partially information-coupled \\ QoS & Quality of service & SC & Successive cancellation \\ SCAN & Soft cancellation & SCL & Successive cancellation list \\ SCMA & Sparse code multiple access & SISO & Soft-input-soft-output \\ TB & Transport block & TBER & TB error rate \\ TC & Transfer characteristic & URLLC & Ultra-Reliable and Low \\ VN & Variable node & & Latency Communications \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of abbreviations. Figure 1: Uplink information-coupled polar-coded sparse code multiple access (PC-SCMA) system. is illustrated, where VNs and FNs represent the users and resources, respectively. The corresponding mapping matrix is denoted as  
\[\mathbf{F}=\left[ \begin{array}{cccccc}0&1&1&0&1&0\\ 1&0&1&0&0&1\\ 0&1&0&1&0&1\\ 1&0&0&1&1&0\end{array} \right]. \tag{1}\]  
The iterative detection and decoding are performed after the whole TB is obtained by the receiver. Thus, the number of SCMA codeword \(\mathbf{s}_{v}=\left\{\mathbf{s}_{v,1},\ldots,\mathbf{s}_{v,L_{v}}\right\}=\left\{\mathbf{s}_ {v}^{1},\ldots,\mathbf{s}_{v}^{L}\right\}\) is \(L=L_{c}N/Q\). Supposing the channel gains matrix between the base station and user \(v\) at the \(l\)-th block is denoted as  
\[\mathbf{H}_{v}^{l}=\left[