# Improving Polar-Coded SCMA System by Information Coupling and Parity Check
## 3 Design of Partially Information-Coupled Polar Code based on Parity Check
### Systematic PCCA-Polar Code  
According to 3 GPP TS 38.212, based on the idea of \"using parity check bits for polar decoding,\" PCCA-polar code has been adopted for the uplink control channel coding since it prevents premature deletion of the correct candidate path in list decoding. For short-length packet, PCCA-polar code  
Figure 2: SCMA factor graph with six users and four resources. exhibits superior performance compared to CA-polar code. The performance of PIC polar code should be further enhanced by using PCCA-polar code as the base code. However, PCCA-polar code, in its standard form, is a non-systematic code. In other words, the information bits do not appear as part of the codeword transparently. Therefore, the traditional PCCA-polar code cannot be directly used to construct the PIC polar code. Since any linear code can be turned into a systematic code, [28] gives encoding and decoding methods for the systematic polar code that preserves the same frame error rate. However, this method does not work for all types of polar codes, such as PCCA-polar code and DCA-polar code. In this subsection, we give the encoding method of systematic PCCA-polar code as follows. For systematic PCCA-polar code, the encoding is performed as Figure 3. The information block (including CRC bits) is defined as \(\mathbf{b}=\{b_{1},b_{2},\ldots,b_{K^{\prime}}\}\), and \(\mathcal{A}\) denotes the information indices, where \(|\mathcal{A}|=K^{\prime}\). As presented in [8], the recursive construction of the generator matrix can be defined as \(\mathbf{G}=\left[ \begin{array}{cc}1&0\\ 1&1\end{array} \right]^{\otimes\log_{2}\left(N_{m}\right)}\), where \(N_{m}\) is the mother code length and \(\otimes\) denotes the Kronecker product. \(\mathbf{G}_{\mathcal{A}\mathcal{A}}\) is the submatrix of \(\mathbf{G}\) consisting of the array of elements \(\left(G_{i,j}\right)\) with \(i,j\in\mathcal{A}\), and similarly for the other submatrices. The module with parameter \(\mathbf{G}_{\mathcal{A}\mathcal{A}}\) can implement the mapping \(\mathbf{b}\mapsto\mathbf{b}^{{}^{\prime}}\) by computing \(\mathbf{b}^{{}^{\prime}}=\mathbf{b}\mathbf{G}_{\mathcal{A}\mathcal{A}}\). Then, by inserting parity check bits \(\mathbf{p}\) and frozen bits \(0\) into \(\mathbf{b}^{{}^{\prime}}\), \(\mathbf{a}=\left\{a_{1},\ldots,a_{N_{m}}\right\}\) can be obtained. The indices of parity check bits are denoted by \(\mathcal{P}_{c}\) and \(|\mathcal{P}_{c}|=n_{\mathcal{P}c}\). Thus, the indices of information with assistant bits (including CRC bits and parity check bits) can be represent by \(\mathcal{A}^{\prime}=\mathcal{A}\cup\mathcal{P}_{c}\), where \(\mathcal{A}^{\prime}\subset\{1,\ldots,N_{m}\}\) corresponds to the \(K^{\prime}+n_{\mathcal{P}c}\) most reliable indices (skipping shortened or punctured positions) and \(\mathbf{a}_{\mathcal{A}\mathcal{T}}=\left\{\mathbf{b}^{{}^{\prime}},\mathbf{p}\right\}\). Then, the basic polar encoding is performed by \(\mathbf{x}=\mathbf{a}\mathbf{G}\). Since \(\mathbf{G}_{\mathcal{A}\mathcal{A}}\) and \(\mathbf{G}_{\mathcal{A}\mathcal{T}}\) are lower-triangular with ones on the diagonal, they are invertible (in fact, the inverse of \(\mathbf{G}_{\mathcal{A}\mathcal{A}}\) and \(\mathbf{G}_{\mathcal{A}\mathcal{T}}\) are themselves). Thus, \(\mathbf{x}_{\mathcal{A}}=\mathbf{b}^{{}^{\prime}}\mathbf{G}_{\mathcal{A}\mathcal{A}}=\mathbf{b}\) and \(\mathbf{x}_{\mathcal{A}^{\prime}}=\mathbf{a}_{\mathcal{A}\mathcal{T}}\mathbf{G}_{\mathcal{ A}\mathcal{T}}\) can be get. It can be seen, the mother code \(\mathbf{x}\) is systematic. Finally, \(\mathbf{c}^{{}^{\prime}}\) is obtained by rate matching. Since \(\mathcal{A}^{\prime}\) is chosen by skipping shortened or punctured positions, \(\mathbf{c}^{{}^{\prime}}\) is also systematic code. According to the preceding description, to ensure that \(\mathbf{x}_{\mathcal{A}}=\mathbf{b}\), the choice of \(\mathcal{P}_{c}\) is the key issue. First, we give two theorems as follow. **Theorem 1**. : _Denote \(g_{i}\) as the \(i\)-th row of \(\mathbf{G}_{\mathcal{A}\mathcal{T}}\) and \(1^{i}\).