# Quantum Coding via Quasi-Cyclic Block Matrix
## 4 Long-Length Quantum Coding via Quasi-Cyclic Jacket Matrix
### The Construction Method of the Exponent Matrix
According to the designed approach, two matrices \(G_{e_{1}}\) and \(G_{e_{2}}\) that generated similarly to \(G_{e}\) will be obtained by operating a cycle-shift operation \(\gamma\) on two groups of vectors \(\vec{e}_{a_{i}}\) and \(\vec{e}_{ij}\) (\(i,j=1,2\)). Finally, we take the obtained matrix \(G_{i}\) (similar matrix \(G\) in Equation (18)) as the parity-check matrix \(H_{i}\) of a classical QC code by resort to the algorithm. However, two conditions of coding construction should be satisfied for the generated matrix \(G\). On the one hand, any two rows in \(G\) should be orthogonal, i.e., it is self-orthogonal. In fact, according to the proposed construction algorithm, take a cyclic-shift operator \(\sigma\) in the no-unit operator group \(\mathcal{K}/\{l\}\), where the group \(\mathcal{K}\) is formed by \(K\) introduced before. Without losing generality, denote \(\vec{v}_{\sigma}=(\sigma^{i_{1}},\sigma^{i_{2}},\ldots,\sigma^{i_{l}})\) and \(\vec{v}^{\prime}_{\sigma}=(\sigma^{i_{l}^{\prime}},\sigma^{i_{2}^{\prime}}, \ldots,\sigma^{i_{l}^{\prime}})\) two row operator vectors in the circulant permutation matrix \(\mathcal{Q}\), \(1\leq i,i^{\prime}\leq l=(n-1)/2\). Namely, exponent vectors \(\vec{e}_{i}=(i_{1},i_{2},\ldots,i_{l})\) and \(\vec{\sigma}_{i}=(i^{\prime}_{1},i^{\prime}_{2},\ldots,i^{\prime}_{l})\) are both in the same matrix \(E_{1}\) or \(E_{2}\). It is obvious that, if any binary row vector \(\vec{c}\) in the jacket matrix \(j\) is both operated by \(\vec{v}_{\sigma}\) and \(\vec{v}^{\prime}_{\sigma}\) with operation \(\hat{\otimes}\), their inner product is equal to \(0\) or \(1\), i.e., \((\vec{c}\hat{\otimes}\vec{v}_{\sigma})\cdot(\vec{c}\hat{\otimes}\vec{v}^{ \prime}_{\sigma})^{T}=0\) or \(1\). Therefore, if any two row vectors \(\vec{x}\) and \(\vec{y}\) in matrix \(J\) are operated by the concatenated vector \(\vec{v}=(\vec{v}_{\sigma}|\vec{v}^{\prime}_{\sigma})\) with size \(2l\) and another vector \(\gamma\vec{v}\), their inner product will always be \(0\) for mode \(2\) with one-to-one mapping, i.e., \((\vec{x}\hat{\otimes}\vec{v})\cdot(\vec{x}\hat{\otimes}\vec{v})^{T}=0\); here, \(\gamma\) is a cyclic-shift operator. This means that \(G\) is a self-orthogonal matrix. On the other hand, it should also be proved that any codewords \(c_{1}\) from \(C_{2}^{\perp}\) are orthogonal to all codewords \(c_{2}\) from \(C_{1}^{\perp}\), i.e., \(G_{1}\cdot G_{2}^{\perp}=0\). In fact, for the basic matrix \(K\), assume its cyclic-shift time of operating \(\gamma\) on it in \(Z_{I_{1}}^{\gamma}\) and \(Z_{I_{2}}^{\gamma}\), respectively. According to the described algorithm, the cyclic-shift steps in the two sets are both \(2\), i.e., with the same operation \(\gamma^{2}\). Therefore, any two matrices \(K_{1}\) and \(K_{2}\) of which exponents from \(Z_{I_{1}}^{\ast}\) and \(Z_{I_{2}}^{\ast}\), can be described as forms like Equation (12), i.e.,  
\[K_{1}=a_{1}K+a_{3}K^{3}+\ldots+a_{n-1}K^{n-1},\] \[K_{2}=a_{2}K^{2}+a_{4}K^{4}+\ldots+a_{n}K^{n}. \tag{32}\]  
It is obvious that \(K_{1}\) and \(K_{2}\) are linearly independent, i.e., the two generated Abelian spaces are orthogonal. As a result, if we take \(G_{e_{1}}\) and \(G_{e_{2}}\) as the exponent matrices constructed by Equation (18), we can obtain two \(N\)-length matrices \(G_{1}\) and \(G_{2}\). Then, the \(N-k_{i}\) rows of the two matrices are randomly chosen as the parity-check matrix \(H_{i}\) of classical code \(C_{i}[N,k_{i},d_{i}]\), \((i=1,2)\), so a generator matrix \(\mathcal{G}_{c}\) of a QC quantum code \(\mathcal{C}[[N,k_{1}+k_{2}-N,\min\{d_{1},d_{2}\}]]\) of the form of Equation (5) is obtained. **Example 2**. : _Let prime number \(n=11,Z_{1}^{\ast}=\{1,2,\cdots,10\}\), so \(l=(n-1)/2=5\), and sets \(Z_{I_{1}}^{\ast}=\{1,3,5,7,9\}\), \(Z_{I_{2}}^{\ast}=\{2,4,6,8,10\}\). Take \(\vec{e}_{11}=(1,3,5,7,9)\) and \(\vec{e}_{21}=(3,5,7,9,1)\) as the  
Figure 2: The circuit of the constructing generator matrix \(G\).