## 4 Hybrid Check Matrix Construction (HC) Algorithm  
### IrPEG Construction Algorithm  
The check matrix was constructed prior to decoding. Given the disadvantage of the higher encoding complexity of the existing PEG algorithm, this paper proposes an improved non-regular PEG algorithm with a lower triangle structure based on an improved non-binary superposition coding algorithm, namely the irPEG construction algorithm. The algorithm is improved from two aspects: encoding scheme and constructing check matrix, reducing the complexity of the algorithm and improving error correction performance. Since the PEG algorithm is a random construction algorithm, each time the non-zero elements on the Galois field are filled, the position of the fill is random [28; 29]. To reduce the encoding complexity of the check matrix construction algorithm, a base matrix with a lower triangular structure was constructed using specified bit node degree distribution, as shown in Figure 2. Compared with the PEG algorithm, the check matrix constructed by the irPEG algorithm has a certain structural constraint and then the values of all check node positions were determined by adopting the backward superposition method. The specific steps of the irPEG algorithm are as follows. (1) Determine the parameters in the base matrix: the number of rows and columns and the sequence of variable node  
Figure 2: Check matrix with lower triangle structure. degrees and initialize the base matrix information, including the set of check nodes connected to the variable node and its complement. (2) Construct the lower triangular part on the right side of the base matrix. First, the last variable node was constructed using the backward iteration method and the check nodes were connected forward according to the degree sequence of the variable node. The position of the first non-zero element in each column had to be connected to the check node on the diagonal and the remaining non-zero elements were added below the diagonal, which can be used with the linear encoding complexity of the improved non-binary superposition coding algorithm for encoding. The benefit of the irPEG algorithm is that it has a lower encoding complexity than the PEG algorithm. Then, all the check node sets connected with the variable node were searched and the check node set with the smallest degree was filtered out. If the set contained multiple elements, the check nodes that could form four rings and six rings were deleted. Finally, one of the remaining check nodes was randomly connected and if there was only one element, the node was directly connected. If the set contained multiple elements, the check nodes that could form four rings and six rings were deleted Finally, one of the remaining check nodes was randomly connected and if there was only one element, the node was directly connected. Construct the first \(n-m\) columns of the base matrix. The selection is constructed forward from the \(n-m\) variable node and the constructed method is similar to the lower triangle. Select the check node with the smallest degree according to the degree sequence of the variable nodes to ensure that the row weight of each row is not considerably different from the average row weight. After the check nodes from the fourth ring are deleted, the connection is randomly selected from the remaining check nodes. \[\mathbf{H}=\left[ \begin{array}{cccccccccccccccc}0&3&0&0&7&0&4&13&1&0&0&0&0&0&0&0\\ 0&0&3&3&11&1&0&0&14&1&0&0&0&0&0&0\\ 11&0&0&0&0&0&0&0&6&12&1&0&0&0&0&0\\ 14&3&0&0&0&0&3&0&0&7&7&1&0&0&0&0\\ 0&5&8&8&0&0&0&0&14&0&0&1&1&0&0&0\\ 0&0&0&0&0&7&0&3&0&0&0&0&12&1&0&0\\ 13&0&7&7&1&0&10&0&0&0&0&0&0&8&1&0\\ 9&0&0&0&8&7&0&0&0&0&12&0&0&0&4&1\end{array} \right] \tag{3}\]  
Since the irPEG algorithm belongs to a random check matrix construction algorithm and the constructed check matrix has no certain structure, the hexadecimal \(\mathbf{H}_{8\times 16}\) of Equation (3) is used as an example to illustrate the structure of the proposed check matrix constructed and improved irPEG algorithm.