## 6 Encoding Complexity Analysis  
Under the same error correction performance, the coding complexity was analyzed. According to the irPEG algorithm, the check matrix constructed by the algorithm had a lower triangular structure. Therefore, when the generated matrix was obtained, the computational complexity was lower, so the hardware was easier to implement. Based on improving the QC-LDPC algorithm, the irPEG algorithm was introduced to construct the base matrix. The obtained HC algorithm had no change in encoding complexity. However, the HC algorithm has a simpler matrix storage structure in terms of storage complexity than the irPEG algorithm. The encoding complexities of the PEG algorithm, irPEG algorithm and HC algorithm are shown in Table 4, where \(w\) is the average column weight of the generator matrix, \(n\) is the code length and \(k\) is the information bit length. In terms of storage complexity, the HC algorithm is a structured construction method. The constructed non-binary LDPC code has a quasi-cyclic structure. When storing the matrix, only one \(p\times p\) dimensional target matrix \(\mathbf{P}\), one \(J\times L\) dimensional non-binary coefficient matrix \(\mathbf{G}c_{I\times L}\) and one \(J\times L\) cyclic shift coefficient matrix \(\mathbf{S}_{I\times L}\) are needed. However, a check matrix of the same size that was randomly constructed by the irPEG algorithm needs to store a check matrix of size \(p\times J\times p\times L\) \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Matrix** & & & & \\
**Construction Algorithm** & **Pretreatment** & **Additions Operations** & **Multiplication Operations** & **Encoding Complexity** \\ \hline PEG algorithm & \(\sigma(n^{3})\) & \((w-1)n\) & \(wn\) & \(o(n^{2})\) \\ IrPEG algorithm & **0** & \((w-1)(n-k)\) & \(w(n-k)\) & \(o(n)\) \\ HC algorithm & **0** & \((w-1)(n-k)\) & \(w(n-k)\) & \(o(n)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Encoding complexity analysis. Figure 14: The relationship between signal-to-noise ratio (SNR) ratio and energy efficiency. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Check Matrix** & & & **Non-Binary** \\
**Construction Algorithm** & **HC Algorithm** & **Coding Algorithm** & **Iterative Algorithm** \\ \hline Decoding algorithm & Log-FFT-BP algorithm & Modulation & CPM \\ Inner iterations & 4 & Outer iterations & 8 \\ Degree distribution & \(\lambda(x)=0.38354x+0.04237x^{2}+0.57409x^{3}\) & Information bit & 1536 bit \\ Code rate & \(2/3\) & q & 8 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The parameters of the non-binary LDPC-CPM system. The check matrix constructed by the HC algorithm has a simpler matrix storage structure than the irPEG algorithm. In addition, the HC algorithm has a certain code structure that overcomes the shortcomings of the long search time when the PEG algorithm and the irPEG algorithm construct the long code. In terms of encoding complexity, Gaussian elimination coding algorithm was used for PEG algorithm and the non-binary superposition coding algorithm was used for the irPEG and HC algorithms. The encoding complexity calculation based on Gaussian elimination includes two parts: one to eliminate the check matrix into the structure of the lower triangular matrix, meaning the check matrix is preprocessed and the computational complexity is \(o\big{(}n^{3}\big{)}\); and the other part is the encoding complexity \(o\big{(}n^{2}\big{)}\). The complexity of the encoding depends on the sparsity of the generator matrix. The entire encoding process requires \(wn\) multiplication operations and \((w-1)n\) addition operations. Even if the check matrix of the non-binary LDPC code is very sparse, the generator matrix is not sparse, the ratio of \(w\) and \(n\) cannot be ignored, so that the encoding complexity is proportional to the square of the code length. The complexity of the non-binary superposition coding algorithm depends entirely on the sparsity of the non-binary LDPC code check matrix.