# Quantum Coding via Quasi-Cyclic Block Matrix
## 4 Long-Length Quantum Coding via Quasi-Cyclic Jacket Matrix
### The Construction Method of Jacket Matrix  
Firstly, the construction manner of the low-parity jacket matrix \(J\) in Equation (18) is investigated. According to their definition, it is obvious that the Hadamard matrix and the Pauli matrices are special jacket matrices. Consider the 2-order Hadamard matrix  
\[H_{2}=\left( \begin{array}{cc}1&1\\ 1&-1\end{array} \right), \tag{19}\]  
as the smallest basic matrix. A binary \(n_{1}\)-size Hadamard matrix \(H=(h_{ij})_{n_{1}\times n_{1}}\) for \(n_{1}=2^{m}\), \(m\geq 3\), under mapping: \(1\to 1,-1\to 0\), may be obtained with the following recursive relation  
\[H_{n_{1}}=H_{n_{1}/2}\otimes\mathrm{H}_{2}. \tag{20}\]  
Generally, on the basis of the tensor product, the \(n_{1}=2^{m}\)-order jacket matrix can be shown as  
\[J_{n_{1}}=\prod_{i=1}^{m}I_{2^{m-i}}\otimes J_{2}\otimes I_{2^{ i-1}}, \tag{21}\]  
where \(J_{2}\) may also be chosen as Pauli matrices except for the Hadamard matrix,  
\[\sigma_{0}=\left( \begin{array}{cc}1&0\\ 0&1\end{array} \right)\ \ \text{or}\ \ \sigma_{1}=\left( \begin{array}{cc}0&1\\ 1&0\end{array} \right). \tag{22}\]  
Correspondingly, one obtains  
\[J_{n_{1}}J_{n_{1}}^{T}=(\prod_{i=3}^{m}I_{2^{m-i}}\otimes(J_{2}J_ {2}^{T})\otimes I_{2^{i-1}})\otimes(J_{2^{3}}^{T}I_{2^{3}}). \tag{23}\]  
Similarly, we also consider a jacket matrix with size \(n_{2}=3^{m}\) based on the fundamental matrix \(J_{3}\) as follows: Denoting a \(1\times 1\) Jacket matrix as \(J_{1}=1\), the direct sum \(J_{3}\) may be written as  
\[J_{3}=J_{1}\oplus J_{2}=\left( \begin{array}{cc}1&0\\ 0&J_{2}\end{array} \right). \tag{24}\]It is easy to check that  
\[J_{3}J_{3}^{-1} = \left( \begin{array}{cc}1&O\\ O&J_{2}\end{array} \right)\left( \begin{array}{cc}1&O\\ O^{T}&J_{2}^{-1}\end{array} \right)=I_{3}, \tag{25}\]  
Then, one may obtain  
\[J_{n_{2}}=J_{n_{2}/3}\otimes J_{3}, \tag{26}\]  
for \(n_{2}=3^{m},m\in\{1,2,\cdots\}\). In terms of number theory, any finite prime number \(n\) may be decomposed into a Kronecker product and the direct sum of 2 and 3 resort to the following form  
\[n=2^{n^{\prime}_{1}}+2^{n^{\prime}_{2}}+\cdots+2^{n^{\prime}_{r}}+3^{n^{\prime }_{1}}+\cdots+3^{m^{\prime}_{s}}, \tag{27}\]  
where \(n^{\prime}_{\mu^{\prime}},m^{\prime}_{v}\) are integers for \(1\leq\mu\leq r,1\leq\
u\leq s\). Therefore, to obtain any prime size of quantum code, we first construct generator matrices with size \(2^{m}\) and \(3^{m}\). In view of the recursive relation, the jacket matrix \(J_{n}\) of large-scale order and its transpose matrix \(J_{n}^{T}\) satisfy  
\[J_{n}J_{n}^{T} = \left(J_{2}^{\otimes n^{\prime}_{1}}J_{2}^{\otimes n^{\prime}_{1} T}\right)\oplus\cdots\left(J_{2}^{\otimes n^{\prime}_{r}}j_{2}^{\otimes n^{ \prime}_{1}T}\right)\oplus\left(J_{3}^{\otimes m^{\prime}_{1}}j_{3}^{\otimes m ^{\prime}_{1}T}\right) \tag{28}\] \[\oplus\cdots\left(J_{3}^{\otimes m^{\prime}_{s}}j_{3}^{\otimes m^ {\prime}_{s}T}\right).\]  
It is obvious that the weight of the gained jacket matrix \(J_{n_{2}}\) is equal to 1 when it involves \(J_{2}\) as one of the Pauli matrices. For clarity, some decomposition methods of the jacket matrix are presented based on 2-order and 3-order _fundamental jacket matrices_\(J_{2}\) (Pauli matrices or the Hadamard matrix) and \(J_{3}\) in Table 1.  
As can be seen from the above table, any large-scale jacket matrix of prime order can be decomposed by these two kinds of matrices. **Example 1**. : _Take parameter \(n=59\) that is decomposed with a Kronecker product and the direct sum of the fundamental Jacket matrices, then the concatenated matrix may be expressed as different methods_  
\[J_{59} = J_{2}^{\otimes 5}\oplus J_{3}^{\otimes 3}=J_{32}\oplus J_{27}= \left( \begin{array}{cc}J_{32}&O\\ \hline O&J_{27}\end{array} \right) \tag{29}\] \[= J_{7}^{\otimes 2}\oplus J_{10}=J_{49}\oplus J_{10}=\left( \begin{array}{cc}J_{49}&O\\ \hline O&J_{10}\end{array} \right)\]  
_Therefore, the same jacket matrix may be described with different forms in the light of its decomposition methods.