## 4 Hybrid Check Matrix Construction (HC) Algorithm
### Hybrid Construction (HC) Algorithm  
The PEG algorithm belongs to the construction algorithm of the randomly constructed non-binary LDPC code check matrix. Although its error correction performance is superior to the structure construction algorithm because of the randomness of the constructed matrix, the hardware implementation complexity is higher than the structure construction algorithm. Although the proposed irPEG algorithm combined with the superposition coding algorithm can reduce the encodingcomplexity of non-binary LDPC codes, it is more suitable for hardware implementation of short codes in non-binary LDPC codes, as the hardware implementation complexity for long codes is still high. At this time, through the loss of certain error correction performance in non-binary LDPC codes, the random construction algorithm and structure construction algorithm of non-binary LDPC codes check matrix are combined and the complexity of the hardware implementation is reduced based on the superposition coding algorithm and a hybrid construction algorithm is proposed. Based on the ir-qc construction algorithm, the hybrid construction algorithm has the structural characteristics of the lower triangles. Simultaneously, the irPEG algorithm is used to construct the binary base matrix \(\mathbf{W}_{J\times L}\), which improves the randomness of non-binary LDPC codes and reduces the loss of error performance due to structured construction. The specific steps of the HC algorithm are as follows:  
* The irPEG algorithm constructs the binary base matrix \(\mathbf{W}_{J\times L}\). For a given degree distribution of a non-binary LDPC code, a binary matrix with a lower triangular structure is constructed according to the irPEG algorithm and the size is \(J\times L\). * Determine the finite field element coefficient matrix \(\mathbf{Gc}_{J\times L}\) and randomly select the value of \(gc_{j,l}\) between \((0,q-1)\) according to the position of the non-zero elements in the base matrix, where \(q=2^{b}\). * From the binary base matrix \(\mathbf{W}_{J\times L}\), determine the cyclic shift coefficient matrix \(S_{J\times L}\). Although the binary base matrix \(\mathbf{W}_{J\times L}\) has a lower triangular structure, after cyclic shift, the check matrix does not necessarily have a lower triangular structure. To ensure the structure always has a lower triangle, the coefficients on the diagonal line of the cyclic shift coefficient matrix were set to 0 and thus, the check matrix constructed has a lower triangular structure. The principle of randomly selecting shift coefficients \(s_{j,l}\) in the shift coefficient matrix \(S_{J\times L}\) was to avoid the short loops in the Tanner graph as much as possible. The necessary and sufficient conditions for avoiding the short loop of length \(2^{i}\) through the binary base matrix \(\mathbf{W}_{J\times L}\) combination are shown in Equation (5). A continuous heuristic was used to determine the shift coefficients \(s_{j,l}\) in the shift coefficient matrix \(\mathbf{S}_{J\times L}\) at the non-zero elements of the binary base matrix \(\mathbf{W}_{J\times L}\). \[\sum_{k=1}^{m}\left(s_{j_{k},l_{k}}-s_{j_{k+1},l_{k}}\right)\
eq 0\text{mod}p \tag{5}\]  
1. First, determine the target square \(\mathbf{P}\) of the binary \(p\times p\) dimension, as shown in Equation (6). If the value of the element \(gc_{j,l}\) in the \(j\)th row and the l-column of the coefficient matrix \(\mathbf{Gc}_{J\times L}\) is not \"0,\" that is, \(gc_{j\times l}\
eq 0\), \(1\leq j\leq J\) and \(1\leq l\leq L\). Randomly select a value of the shift coefficient \(s_{j,l}\) from the set \(\{0,1,\cdots,p-1\}\). For a given degree distribution, according to Equation (5), determine whether the constraint condition is satisfied. If so, the value of the shift coefficient \(s_{j,l}\) can be determined and added to the cyclic shift coefficient matrix \(\mathbf{S}_{J\times L}\), if the constraint condition is not satisfied, the n return to step 1. \[P_{p\times p}=\left[