## 4 Hybrid Check Matrix Construction (HC) Algorithm
### Hybrid Construction (HC) Algorithm
\begin{array}{cccc}1&0&0&\cdots&0\\ 0&1&0&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\cdots&0\\ 0&0&0&\cdots&1\end{array} \right] \tag{6}\]If the value of the elements of the \(j\)th row and the l-column of the coefficient matrix \(\mathbf{Gc}_{l\times L}\) is \"0,\" that is, \(gc_{j\times l}=0\), \(1\leq j\leq J\) and \(1\leq l\leq L\), the n \(s_{j,l}=\infty\). To summarize, the form of the cyclic shift coefficient matrix \(\mathsf{S}_{J\times L}\) is as shown in Equation (7). \[S_{J\times L}=\left[ \begin{array}{cccccc}s_{1,1}&\cdots&s_{1,L-l}&0&0&\cdots &0\\ s_{2,1}&\cdots&s_{2,L-l}&s_{2,L-l+1}&0&\cdots&0\\ \vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots\\ s_{l,1}&\cdots&s_{l,L-l}&s_{l,L-l+1}&s_{l,L-l+2}&\cdots&0\end{array} \right]\] (7)
3. The check matrix \(\mathbf{H}\) is constructed using the binary base matrix \(\mathbf{W}_{J\times L}\), the cyclic shift coefficient matrix \(\mathsf{S}_{J\times L}\) and the coefficient matrix \(\mathbf{Gc}_{l\times L}\). If \(w_{j,l}=0\), the sub-matrix \(\mathbf{H}_{j,l}=0_{p\times p}\), that is, \(\mathbf{H}_{((j-1)p+1;jp),((l-1)p+1;lp)}=0_{p\times p}\). If \(w_{j,l}=1\), the n \(\mathbf{H}_{j,l}=gc_{j,l}\mathbf{P}_{p\times p}\big{(}s_{j,l}\big{)}\), that is, \(\mathbf{H}_{((j-1)p+1;jp),((l-1)p+1;lp)}=gc_{j,l}\mathbf{P}_{p\times p}\big{(}s_{j,l} \big{)}\). The check matrix \(\mathbf{H}\) is in the form of Equation (8). \[\mathbf{H}=\left[\begin{array}{ccccc}\mathbf{H}_{1,1}&\  
non-binary LDPC-CPM model is constructed [32], as shown in Figure 4. At the transmitter side, the dimension of the check matrix of the non-binary LDPC code is \(m\times n\) and the length of the non-binary LDPC code is \(n\), the bit of information is \(k=n-m\) and the code rate is \(r=k/n=1-m/n\). The information sequence \(\mathbf{U}^{0}\) gets the code word sequence \(\mathbf{C}^{\circ}\) after mapping and non-binary LDPC coding and then through interleaving and symbol mapping into the M-ary CPM, where CPM is decomposed into CPE and memoryless modulation (MM) [33], CPE has continuous phase encoding of the input information as input of memory modulation MM and MM selects a suitable waveform to send to the channel. The dimension of the non-binary LDPC encoder and the higher-order CPM modulator is equal, that is, \(q=M\). The demodulation and decoding process is accomplished by the two subsystems M-ary Continuous Phase Modulation Soft-Input Soft-Output (CPM-SISO) iteration (called \"external iteration\"). The M-ary CPM-SISO subsystem uses the Log-MAP algorithm, the q-ary LDPC-SISO subsystem uses the Log-FFT-BP iterative decoding algorithm and the iteration is called the internal iteration. \(\boldsymbol{\pi}^{i}(\mathbf{c};I)\) and \(\boldsymbol{\pi}^{i}(\mathbf{u};I)\) are input probability sequences of the M-ary CPM-SISO subsystem, respectively and \(\boldsymbol{\pi}^{i}(\mathbf{c};O)\) and \(\boldsymbol{\pi}^{i}(\mathbf{u};O)\) are output probability sequences of code words and information words of M-ary CPM-SISO subsystem; \(\boldsymbol{\pi}^{o}(\mathbf{c};I)\) and \(\boldsymbol{\pi}^{o}(\mathbf{u};I)\) are input probability sequences of code words and information words of the q-ary LDPC-SISO subsystems and \(\boldsymbol{\pi}^{o}(\mathbf{c};O)\) and \(\boldsymbol{\pi}^{o}(\mathbf{u};O)\) are output probability sequences of q-ary LDPC-SISO subsystem code words and information words, respectively. After the information words probability sequence \(\boldsymbol{\pi}^{i}(\mathbf{u};O)\) output by the M-ary CPM-SISO subsystem is deinterleaved, it is used as the input probability sequence \(\boldsymbol{\pi}^{o}(\mathbf{c};I)\) of the q-ary LDPC-SISO subsystems code words and the output code words probability sequence of the q-ary LDPC-SISO subsystem. After being interleaved, \(\boldsymbol{\pi}^{o}(\mathbf{c};O)\) is input into M-ary CPM-SISO subsystem as the input word probability sequence \(\boldsymbol{\pi}^{i}(\mathbf{u};I)\) and the process is iterated repeatedly several times. The final iteration result is output by the q-ary LDPC-SISO subsystem as a hard decision.