# Quantum LDPC Codes Based on Cocyclic Block Matrices
## 3 Quantum Codes Based on Cocyclic Quasi-Jacket Block Matrices
### Definitions and Terms  
Based on the general definition of a square matrix, in this section, we presented a construction method of quantum codes over binary matrices. As the following, for obtaining the stabilizer generators of quantum codes over the complex number field with a bigger size, we consider a class of binaryocyclic Jacket matrices based on a circular permutation matrix to construct its concatenated generator matrix, i.e.,  
\[G_{N}=[G^{x}|G^{z}]_{N\times 2N}, \tag{11}\]  
here, \(G^{x}\) and \(G^{z}\) are generator matrices of the encoding bit and phase, respectively.  
Since a cocyclic approach was introduced in [24], some interesting codes come from useful cocyclic matrices, such as the Hadamard matrix that is the matrix representation of a cocycle [25].  
**Definition 1**.: _If set \(G\) is a finite group of order \(v\) and \(\mathcal{A}\) is an Abelian group of order \(w\), respectively, a mapping: \(\varphi\): \(G\times G\rightarrow\mathcal{A}\) is called a cocycle while it satisfies the following particular cocycle equation_  
\[\varphi(g,h)\varphi(gh,k)=\varphi(g,hk)\varphi(h,k),\ \ \ \forall g,h,k\in G. \tag{12}\]  
From the definition, it is easy to show that \(\varphi(g,1)=\varphi(1,h)=\varphi(1,1),\forall g,h\in G\), so \(\varphi(1,1)=1\), as we call it, is a normalized cycle.  
According to the cocycle mapping, if the rows and columns of a square matrix can be indexed by the elements of \(G\) in terms of some distribution, the matrix is called a cocyclic matrix, which is described as  
\[M_{\varphi}=[\varphi(g,h)]_{g,h\in G}, \tag{13}\]  
where the map \(\varphi(g,h)\) denotes a position \((g,h)\) of the entry in matrix \(M_{\varphi}\). If the cocycle \(\varphi\) is symmetric, \(M_{\varphi}\) will be a symmetric matrix. Furthermore, on the basis of the cocylic matrices, some cocyclic codes are primitively derived from Hadamard matrices or from related block designs [33]. In this case, a \(v\times v\) Hadamard matrix \(H\) with entries \(\{\pm 1\}\) such that \(HH^{T}=vI_{v}\) is used to construct codes with length \(v=4\)\(s\) for integer \(s\geq 1\). If \(1\)s and \(-1\)s are replaced by \(1\) s and \(0\) s, respectively, the Hadamard matrix becomes a binary one. A main superiority of cocyclic codes is that the long-length codes may be easily gained with a circulant manner; hence, it may be applicable to the generalized classical case and further quantum codes.  
**Definition 2**.: _A square matrix \(J_{N}=[a_{ij}]_{N}\), if its inverse is obtained simply by element-wise inverse, i.e., it satisfies_  
\[a_{ij}^{-1}=\left\{\begin{array}{ll}a_{ij}/a&\text{if }a_{ij}\
eq 0,\\ 0&a_{ij}=0,\end{array}\right. \tag{14}\]  
_where a is the normalized constant, we call matrix \(J_{N}\) a block Jacket matrix. Obviously, the Hadamard matrix belongs to the defined Jacket matrix. Furthermore, if there is a permutation matrix \(\mathcal{T}\), and its inverse matrix can be described as \(J_{N}^{-1}=\mathcal{T}\cdot[a_{ij}]_{N}/a\), matrix \(J_{N}\) is a quasi-Jacket block matrix [34]. This class of matrices may be generalized by a simple binary transformation. Matrix decomposition with a Kronecker product of identity matrices can successively design a large length of codes that resorts to lower-order coefficient matrices. Meanwhile, this factorization decomposition generates a family of block circular sparse Jacket matrices which have good performance of LDPC codes. In the applications, the derived circulant permutation matrices with a jacket pattern could lead to a very simple encoding algorithm._