# 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
as the indices of ones in \(g_{i}\). If \(j\in 1^{i}\), there is \(1^{j}\subseteq 1^{i}\)._  
**Proof. ** Appendix A. **Theorem 2**. : _Denote \(\mathcal{A}^{\prime}_{i}\) as the \(i\)-th element of \(\mathcal{A}^{\prime}\), \(g_{i}\) as the \(i\)-th row of \(\mathbf{G}_{\mathcal{A}\mathcal{T}}\) and \(1^{i}\) as the indices of ones in \(g_{i}\). Assume \(\mathcal{A}^{\prime}_{i}\in\mathcal{P}_{c}\) and \(j\in 1^{i}\), if and only if \(\mathcal{A}^{\prime}_{j}\in\mathcal{P}_{c}\), polar code is systematic._  
**Proof. ** According to Theorem 1, if \(j\in 1^{i}\), the relation \(1^{j}\subseteq 1^{i}\) holds. Obviously, there is \(1^{j}\cap 1^{i}=1^{j}\). If \(\mathcal{A}^{\prime}_{j}\
otin\mathcal{P}_{c}\), \(\mathcal{A}^{\prime}_{j}\) is an information index. By the transformation \(\mathbf{x}_{\mathcal{A}\mathcal{T}}=\mathbf{a}_{\mathcal{A}\mathcal{T}}\mathbf{G}_{\mathcal{ A}\mathcal{T}\mathcal{A}^{\prime}}\), the information bits \(\left[ \begin{subarray}{c}x^{k}_{\mathcal{A}\mathcal{T}},k\in 1^{j}\\ \end{subarray} \right]\) must be superimposed with parity bit \(a^{i}_{\mathcal{A}\mathcal{T}}\). Thus, the systematic code could not be obtained. For example, let \(N_{m}=16\), \(\mathcal{A}^{\prime}=\{8,10,11,12,13,14,15,16\}\), and the corresponding submatrix \(\mathbf{G}_{\mathcal{A}\mathcal{T}}\) is shown in Figure 4. If \(\mathcal{A}^{\prime}_{i}=\mathcal{A}^{\prime}_{4}\) is chosen as an element of \(\mathcal{P}_{c}\), the corresponding position set of ones can be expressed as \(1^{4}=\{2,3,4\}\). According to Theorem 1, the relationship  
Figure 3: Systematic parity check and CRC-aided (PCCA)-polar encoder. \(1^{2}=\{2\}\subset 1^{4}\) and \(1^{3}=\{3\}\subset 1^{4}\) can be deduced. Following Theorem 2, \(\mathcal{A}^{\prime}_{2}\) and \(\mathcal{A}^{\prime}_{3}\) must be the elements of \(\mathcal{P}_{c}\). Thus, when \(\mathcal{P}_{c,1}=\{\mathcal{A}^{\prime}_{2},\mathcal{A}^{\prime}_{3},\mathcal{ A}^{\prime}_{4}\}\) is chosen as the parity check indices, the sequence \(\boldsymbol{a}_{\mathcal{A}^{\prime}}=\{b^{\prime}_{1},p_{1},p_{2},p_{3},b^{ \prime}_{2},b^{\prime}_{3},b^{\prime}_{4},b^{\prime}\}\). From Figure 4, the parity check bits \(\boldsymbol{p}=\left\{a_{\mathcal{A}^{\prime}}^{2},a_{\mathcal{A}^{\prime}}^{3 },a_{\mathcal{A}^{\prime}}^{4}\right\}\) do not affect the information bits \(\left\{x_{\mathcal{A}^{\prime}}^{1},x_{\mathcal{A}^{\prime}}^{5},x_{\mathcal{ A}^{\prime}}^{6},x_{\mathcal{A}^{\prime}}^{7},x_{\mathcal{A}^{\prime}}^{8}\right\}\) by transformation \(x_{\mathcal{A}^{\prime}}=a_{\mathcal{A}^{\prime}}\boldsymbol{G}_{\mathcal{A}^ {\prime}\mathcal{A}^{\prime}}\), in other words, \(x_{\mathcal{A}}=\left\{x_{\mathcal{A}^{\prime}}^{1},x_{\mathcal{A}^{\prime}}^ {5},x_{\mathcal{A}^{\prime}}^{6},x_{\mathcal{A}^{\prime}}^{7},x_{\mathcal{A}^ {\prime}}^{8}\right\}=\{b_{1},b_{2},b_{3},b_{4},b_{5}\}\) and \(\boldsymbol{x}\) is systematic code. However, for \(\mathcal{P}_{c,2}=\{\mathcal{A}^{\prime}_{3},\mathcal{A}^{\prime}_{4},\mathcal{ A}^{\prime}_{5}\}\) and \(\boldsymbol{a}_{\mathcal{A}^{\prime}}=\{b^{\prime}_{1},b^{\prime}_{2},p_{1},p_ {2},p_{3},b^{\prime}_{3},b^{\prime}_{4},b^{\prime}_{5}\}\), the information bit \(x_{\mathcal{A}^{\prime}}^{2}\) is superimposed with parity bit \(p_{2}=a_{\mathcal{A}^{\prime}}^{4}\), so \(x_{\mathcal{A}}=\left\{x_{\mathcal{A}^{\prime}}^{1},x_{\mathcal{A}^{\prime}}^ {2},x_{\mathcal{A}^{\prime}}^{2},x_{\mathcal{A}^{\prime}}^{5},x_{\mathcal{A}^ {\prime}}^{7},x_{\mathcal{A}^{\prime}}^{8}\right\}=\{b_{1},b_{2}\oplus p_{2}, b_{3},b_{4},b_{5}\}\) and \(\boldsymbol{x}\) is not the systematic code. According to Theorems 1 and 2 and 3 GPP TS 38.212, given \(\mathcal{A}^{\prime}\) and \(n_{pc}\), Algorithm 1 is proposed to choose \(\mathcal{P}_{c}\). ```
Input:\(\mathcal{A}^{\prime}\) and \(n_{pc}\). Output:\(\mathcal{P}_{c}\). 1: Initialize:\(\mathcal{P}_{c}=O\).