# Improving Polar-Coded SCMA System by Information Coupling and Parity Check
## 3 Design of Partially Information-Coupled Polar Code based on Parity Check
### Partially Information-Coupled PCCA-Polar Code  
In [20], the authors have constructed a PIC polar code based on systematic CA-polar code. Multiple systematic polar codes blocks are concatenated to form a large code frame by sharing a few information bits between every two consecutive CBs. More specifically, dummy bits are inserted into the first CB and the last CB to terminate the frame. To improve the PIC polar code, in this subsection, we introduce a new type of PIC polar code based on systematic PCCA-polar code. However, following the encoding method presented in [20], the dummy bits are punctured in the end, and the length of the last CB is different from other CBs. More importantly, the windowed decoding of TB will be terminated when a CB is error decoded. Thus, the un-decoded CBs could not supply the extrinsic message for iterative system, such as SCMA. Therefore, we propose a new construction and decoding method for our PIC PCCA-polar code as follows. Let \(\mathbf{u}=\left\{\mathbf{u}_{1},\mathbf{u}_{2},\ldots,\mathbf{u}_{L_{c}}\right\}\) denote the information sequence of a TB, where \(L_{c}\) is the number of CBs, \(\mathbf{u}_{k}=\left\{u_{k}^{1},u_{k}^{2},\ldots,u_{k}^{K}\right\}\) corresponding to the \(k\)-th CB. Let \(\mathcal{A}\) denote the information indices with length \(K+K_{c}\) for the coupled information block \(\mathbf{u}^{{}^{\prime}}_{k}\). \(\mathcal{B}=\left\{\mathcal{B}_{o},\mathcal{B}_{e}\right\}\) expresses the coupling indices, where \(\mathcal{B}_{o}\) and \(\mathcal{B}_{e}\) correspond to the subset with odd and even indices of \(\mathcal{B}\), respectively. The indices of \(\mathcal{B}_{o}\) and \(\mathcal{B}_{e}\) in \(\mathcal{A}\) are respectively denoted as \(\mathcal{B}^{{}^{\prime}}_{o}\) and \(\mathcal{B}^{{}^{\prime}}_{e}\). Figure 5 shows the coupling relation between every two consecutive CBs. For the first CB (\(k=1\)), an all-zero dummy block with length \(K_{c}\) is inserted into indices \(\mathcal{B}^{\prime}{}_{o}\) of \(\mathbf{u}^{{}^{\prime}}_{1}\), and get \(\mathbf{u}^{{}^{\prime}}_{1}=\left\{0,\mathbf{u}_{1}\right\}\). For \(1<k\leq L_{c}\), the information bits that are on the indices \(\mathcal{B}^{\prime}{}_{e}\) of the (\(k\)-1)-th CB are inserted into the position \(\mathcal{B}^{{}^{\prime}}_{o}\) of the \(k\)-th CB to obtain \(\mathbf{u}^{{}^{\prime}}_{k}=\left\{\mathbf{u}^{{}^{\prime}}_{k-1,\mathcal{B}^{\prime }{}_{e}},\mathbf{u}_{k}\right\}\), where \(\mathbf{u}^{{}^{\prime}}_{k,\mathcal{B}^{\prime}{}_{o}}=\mathbf{u}^{{}^{\prime}}_{k-1, \mathcal{B}^{\prime}{}_{e}}\). Thus, the CBs have been coupled by sharing a few information bits between every two consecutive CBs. Then, for every CB, the coupled information block \(\mathbf{u}^{{}^{\prime}}_{k}\) is attached to the CRC bits and encoded by systematic PCCA-polar encoder as shown in Figure 6. For \(1\leq k\leq L_{c}\), the sequence \(\mathbf{x}_{k}\) satisfies \(\mathbf{x}_{1,\mathcal{B}_{o}}=0\) and \(\mathbf{x}_{k,\mathcal{B}_{o}}=\mathbf{x}_{k-1,\mathcal{B}_{e}}\). Finally, \(\mathbf{x}_{k,\mathcal{B}_{o}}\) is punctured to obtain \(\mathbf{c}_{k}\). Figure 5: The coupling relation between every two consecutive code blocks (CBs). In this scheme, dummy block is not transmitted, and coupled information blocks are only transmitted once even though each of them is shared by two consecutive CBs. Hence, the effective code rate of the PIC PCCA-polar code is \(R=K/N\), where \(N\) is the code length of CB. In addition, we define \(\delta=K_{c}/K\) as coupling ratio of the proposed PIC PCCA-polar code, and the mother code rate is \(r=(K+K_{c})/N_{m}\). In [20], the authors proposed a windowed decoding based on the look-back and go-back scheme for PIC polar code. The algorithm is divided into feed-forward decoding and feed-back decoding. When the \(k\)-th CB is decoded correctly in the feed-forward decoding, the (\(k\)-1)-th CB can be corrected by the perfect extrinsic messages of the \(k\)-th CB in feed-back decoding. This decoding scheme is also suitable for our PIC PCCA-polar code.