# Quantum LDPC Codes Based on Cocyclic Block Matrices
## 3 Quantum Codes Based on Cocyclic Quasi-Jacket Block Matrices
### Cocyclic Matrix Derived from Jacket Block Matrices
#### 3.2.1 Quasi-Jacket Block Matrices with Size \(4^{m}\) by \(4^{m}\)  
Resorting to the binary matrixes, we consider building the generator matrix of quantum stabilizer codes. On the one hand, we define a class of fundamental matrices \(\alpha\) and \(\beta\) derived from binary Hadamard matrix \(H\) as follows, respectively:  
\[\alpha\triangleq\bigg{[} \begin{array}{cc}1&0\\ 1&1\end{array} \bigg{]},\ \ \beta\triangleq\bigg{[} \begin{array}{cc}1&1\\ 0&1\end{array} \bigg{]}. \tag{27}\]  
It is easy to check that the introduced matrices have the following properties  
\[\alpha\beta+\beta\alpha=I_{2},\ \ \ \alpha\beta\times\beta\alpha=I_{2},\ \ \alpha^{2}=\beta^{2}=I_{2}. \tag{28}\]Furthermore, define a permutation matrix with \(2q\) order as  
\[\mathcal{T}_{2q}=\left[ \begin{array}{cc}0&1\\ 1&0\end{array} \right]\times I_{q}. \tag{29}\]  
Based on the above fact, with simple computation, we can obtain a 4-order matrix and its inverse matrix  
\[\mathcal{J}_{4}\triangleq\left[ \begin{array}{cc}\alpha&\beta\\ \beta&\alpha\end{array} \right]\text{ and }\mathcal{J}_{4}^{-1}=\left[ \begin{array}{cc} \beta&\alpha\\ \alpha&\beta\end{array} \right]=\mathcal{T}_{4}\times\mathcal{J}_{4}, \tag{30}\]  
which is a quasi-Jacket block matrix in terms of the definition. Similarly, we may also obtain the 16-order quasi-Jacket block matrix as  
\[\mathcal{J}_{16}=\mathcal{J}_{4^{2}}=\mathcal{J}_{4}\otimes \mathcal{J}_{4}=\left[ \begin{array}{cc}\alpha&\beta\\ \beta&\alpha\end{array} \right]^{\otimes 2}\] \[=\left[ \begin{array}{cc}1&0\\ 1&1&0\\ ine1&1&0\\ 0&1\end{array} \right]\otimes\mathcal{J}_{4}=\left[ \begin{array}{cc} \mathcal{J}_{4}&O_{4}\\ \mathcal{J}_{4}&\mathcal{J}_{4}&O_{4}\\ ine\mathcal{J}_{4}&\mathcal{J}_{4}&\mathcal{J}_{4}\end{array} \right], \tag{31}\]  
and its inverse matrix  
\[\mathcal{J}_{16}^{-1}=\mathcal{T}_{16}\times\left[ \begin{array}{cccc} \mathcal{J}_{4}^{-1}&O_{4}&\mathcal{J}_{4}^{-1}&\mathcal{J}_{4}^{-1}\\ \mathcal{J}_{4}^{-1}&\mathcal{J}_{4}^{-1}&O_{4}&\mathcal{J}_{4}^{-1}\\ \mathcal{J}_{4}^{-1}&\mathcal{J}_{4}^{-1}&\mathcal{J}_{4}^{-1}&\mathcal{O}_{4} \\ O_{4}&\mathcal{J}_{4}^{-1}&\mathcal{J}_{4}^{-1}&\mathcal{J}_{4}^{-1}\end{array}  \right]. \tag{32}\]  
According to its definition, the matrix obviously is a quasi-Jacket block matrix. Generally, by resorting to the recursive relationships based on \(\mathcal{J}_{4}\) defined by (30), we can obtain the quasi-Jacket block matrices with larger size \(4^{m}\) by using the Kronecker product, i.e.,  
\[\mathcal{J}_{4^{m}}=\mathcal{J}_{4}\otimes\mathcal{J}_{4^{m-1}}, \quad\text{for }\ m\geq 2, \tag{33}\]  
which may be used to generate the generator matrices \(G^{x}\) (or \(G^{z}\)). On the other hand, to obtain the matrix \(G^{z}\) (or \(G^{x}\)), we consider another class of matrix called the permutation matrix \(P^{h}=(p_{ij})_{p\times p}\),  
\[p_{ij}=\left\{ \begin{array}{cc}1&\text{if }i=(j+h)\text{ mod }p,1\leq h<p,\\ 0&\text{otherwise},\end{array} \right. \tag{34}\]  
for an even number \(p\geq 4\). According to the definition, it is obviously a finite cyclic group \(\mathcal{P}=\{I,P,\ldots,P^{p-1}\}\) formed by a family of matrices \(P^{i},0\leq i\leq p-1\) is an Abelian group based on the matrix-multiplication operation, and \(I=P^{0}\) is its unit element. Here, given that the number \(p\) is an even number and \(h\) is an odd number, respectively, a sequence of matrices whose exponents are odd can form a subset \(\mathcal{P}_{o}=\{P,P^{3},\ldots,P^{p-1}\}\) of Abelian group \(\mathcal{P}\). Based on the generated two classes of matrices, i.e., the cocyclic Jacket matrix and permutation matrix, we consider obtaining the stabilizer for quantum codes. Let \(G^{x}\) with a size of \(N=4^{m}\) be obtained by the cocyclic Jacket matrix described before. In the following, we consider how to construct another matrix \(G^{z}\). Firstly, we take two 4 order basic permutation matrices \(P_{4}\) in Equation (34) with parameters \(h=1,3\) respectively, i.e., \(\{\omega^{2},\omega^{4}\}\), such that  
\[\omega^{2}=P=\left[