# 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 \(4^{m}\)
\begin{array}{cccc}0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ 1&0&0&0\end{array} \right]\text{ or }\omega^{4}=P=\left[ \begin{array}{cccc}0&0&0&1 \\ 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\end{array} \right]. \tag{35}\]  
Then, it provides that \(P_{N}=P_{4}\otimes P_{N-1}\), and it takes \(G_{z}=P_{4}\otimes P_{N/4}\). As a result, the concatenated generator matrix \(G\) is built. In the following, we use the mathematical induction to deduce that two matrices \(G^{x}\) and \(G^{z}\) in the obtained concatenated generator matrix \(G\) satisfy self-orthogonal and commuting conditions, i.e., the obtained generator matrix should meet the conditions of self-orthogonal and commuting, so that it may be used to construct a stabilizer of quantum codes. In fact, it is obvious that matrix \(G^{z}\) satisfies the self-orthogonal, so we firstly consider that the case of \(m=1\) for \(G^{x}\) meets the self-orthogonal constrain. Namely, in the following matrix  
\[[G^{x}|G^{z}]_{4\times 8}=[\mathcal{J}_{4}|P_{4}]=\left[ \begin{array}{cccc }1&0&1&1\\ 1&1&0&1\\ ine1&1&1&0\\ 0&1&1&1\end{array}\begin{array}{cccc}0&0&0&1\\ 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\end{array} \right], \tag{36}\]  
any two rows in \(G^{x}\) and \(G^{z}\) are self-orthogonal and commuting. Assuming it meets the case of \(m-1\), we consider the result for \(m\). In fact, if any two rows in concatenated block submatrix \([\alpha\ \beta]\) meet the conditions, it can also do so in \([\beta\alpha]\). Without a loss of generality, here, we only consider the rows in submatrix \([\alpha\ \beta]\). By making use of the Kronecker product of any two row vectors in \(G^{x}_{4}\), it is easy to check that they meet the orthogonal condition. And then, the symplectic inner product in Equation (6) is proved as following. Define complement operators \(\vec{1}=0\) and \(\vec{0}=1\), according to number theory; then, it is possible to easily check that matrices \(\vec{P}=(\bar{p}_{ij})\) with parameters \(h=1,3\) commute. Namely, any two row vectors in concatenated matrices \((\vec{P}|P)\) satisfy the commuting condition in Equation (7). In fact, it can be checked that the elements in the Abelian group \(\mathcal{P}\) commute when their cycle indexes both are odd or even numbers. The matrices satisfy isomorphic \(\mathcal{J}\cong P\) when its corresponding orthogonal matrix also satisfies isomorphic \(\mathcal{J}\cong\vec{P}\). The complement of \(G^{x}\) is an element in the circulant permutation group \(\mathcal{P}_{0}\); hence, two pairs of vectors \(\vec{x}_{1}\), \(\vec{x}_{2}\) and \(\vec{z}_{1}\), \(\vec{z}_{2}\) all belong to this group. The product of any two vectors in \(G^{x}\) and \(G^{z}\) is equal to \(0\) or \(1\) simultaneously; i.e., it means that two pairs of vectors of the inner product with module \(2\) is equal to \(0\). Similarly, we consider the generator matrix based on the block matrix \(\mathcal{J}_{16}\) and \(16\)-order permutation matrix stemmed from \(P_{4}\). \[[G^{x}|G^{z}]_{16\times 32}=[\mathcal{J}_{16}|P_{4}^{\otimes 2}]\] \[=\left[ \begin{array}{ccccc|cccc}\mathcal{J}_{4}&\mathcal{O}_{4}& \mathcal{J}_{4}&\mathcal{J}_{4}&\mathcal{O}_{4}&\mathcal{O}_{4}&\mathcal{O}_{4 }&\mathcal{P}_{4}\\ \mathcal{J}_{4}&\mathcal{J}_{4}&\mathcal{O}_{4}&\mathcal{J}_{4}&\mathcal{P}_{4 }&\mathcal{O}_{4}&\mathcal{O}_{4}&\mathcal{O}_{4}\\ \mathcal{J}_{4}&\mathcal{J}_{4}&\mathcal{J}_{4}&\mathcal{O}_{4}&\mathcal{O}_{4 }&\mathcal{O}_{4}&\mathcal{O}_{4}&\mathcal{O}_{4}\\ \mathcal{O}_{4}&\mathcal{J}_{4}&\mathcal{J}_{4}&\mathcal{J}_{4}&\mathcal{O}_{4 }&\mathcal{O}_{4}&\mathcal{O}_{4}&\mathcal{O}_{4}\end{array} \right]. \tag{37}\]  
Generally, the \(N\)-order generator matrix for \(N=4^{m},m\geq 2\) can be generated by  
\[[G^{x}|G^{z}]_{N\times 2N}=[\mathcal{J}_{N}|P_{4}\otimes P_{N/4}]. \tag{38}\]  
On the other hand, from the factorization of a large number, \(4^{m}\), in Equation (22), we can obtain a group \(\mathcal{A}\) with Formula (23) whose element \(A^{i}_{pm}=I_{4^{i-1}}\otimes\mathcal{J}_{4}\otimes I_{4^{m-i}}\). Any element \(A^{i}_{4^{m}}\) in the group can be taken as the generator \(G^{x}\).