# Quantum LDPC Codes Based on Cocyclic Block Matrices
## 3 Quantum Codes Based on Cocyclic Quasi-Jacket Block Matrices
### Quantum LDPC Codes Based on Cocyclic Block Matrix  
To encode \(k\) information qubits into a QC quantum code \(\mathcal{C}\) with parameter \([[N,k]]\), one should firstly gain two \(N\times N\) cocyclic circulant permutation matrices \(G\) based on the previously constructed matrices \(G^{x}\) and \(G^{z}\) of satisfying the orthogonal condition. According to the recursive relationship, we take any \(N-k\) rows of the two generated matrices as the generators \(\mathcal{G}^{x}\) and \(\mathcal{G}^{z}\). It means that the concatenated matrix \(\mathcal{G}_{N}\) in Equation (3) meets the construction condition of quantum code. In classic coding theory, a class of very important linear codes called LDPC code is widely explored and used in practice. Its theoretical importance mainly embodies that its good performance of linear-time decoding can achieve Shannon capacity [14; 16]. Its quantum version, known as quantum LDPC (QLDPC) codes, plays a very important role in quantum information. However, compared to its classical counterparts, the achieved fruits of quantum LDPC codes are still far less. The main problem is the lack of efficient methods for obtaining iterative coding. Generally, the proportion of \(1\)s in a binary matrix iscalled the matrix's destiny. Given that \(\iota_{0}\) and \(\iota_{1}\) are the numbers of 0s and 1s in a matrix, respectively, the destiny is  
\[d_{0}=\frac{\iota_{1}}{\iota_{0}+\iota_{1}}. \tag{44}\]  
According to the construction method described in Equation (41), it obvious that the total number of 0s and 1s in a \(4p\times 4p\) matrix is \(16p^{2}\). Furthermore, each of the 12 matrices \(I\), \(P^{-h}\) and \(P^{h}\) in \(\mathcal{G}\) contain \(p\) 1s; hence, there are \(12p\) 1s in this matrix: i.e, the 1s density in \(\mathcal{G}\) is  
\[d_{4p}=\frac{12p}{16p^{2}}=\frac{3}{4p}. \tag{45}\]  
From the ratio, it means that the constructed cocylic matrix has a good performance of low-density property. On the other hand, there are \(N(\log 2N-1)\) additions and \(2N(\log 2N-1)\) multiplications for the computation complexity in processing the obtained matrix with size \(N=4^{m}\). The check matrix \(H\) of LDPC codes is generally represented by a simpler and more intuitive Tanner graph with a one-to-one map. If there are equal degrees of all variable nodes in the Tanner graph, its corresponding LDPC code is called a regular code. Otherwise, it is called an irregular code. Denote by \(\eta\) and \(\rho\) the variable and check the nodes' degrees corresponding respectively to the minimum weights of the row and column of the check matrix. Assume that the girth \(g\) in the graph refers to the length of the shortest cycle [36; 37]. Then, the following bounds should be satisfied, i.e.,  
\[N\geq 1+\sum_{i=2}^{x+1}\eta(\eta-1)^{i-2}(\rho-1)^{i-1} \tag{46}\]  
for girth \(g=4x+2\), and  
\[N\geq 1+\sum_{i=1}^{x+1}\eta[(\eta-1)(\rho-1)]^{i-1} \tag{47}\]  
for girth \(g=4x\), where \(x\) is an integer, it can be seen from the above bounds that the code length will increase exponentially, while the cycle length and weights also are rising. To obtain the good performance, having no 4-cycles in the Tanner graph of the quantum LDPC codes is required. A cycle is composed of a group of interconnected vertices in a Tanner graph, with one of these vertices serving as both the starting and ending points, and passing through each vertex only once. If the length of a cycle is called the number of lines it contains, the girth of a graph is defined as the minimum cycle length in the graph. The traditional cycle detection methods are generally based on block matrix sequences [15]. When using this algebraic description to detect cycles in a given check matrix, there is redundancy due to the possibility that different block matrix sequences may form the same girth. To avoid redundancy caused by repeated traversal, we propose the following method for detection. Based on the cocyclic block matrices in Equations (38) and (43), the constructed quantum codes have no \(4-\)cycles in the Tanner graph.