# Quantum Coding via Quasi-Cyclic Block Matrix
## 4 Long-Length Quantum Coding via Quasi-Cyclic Jacket Matrix
### The Construction Method of the Exponent Matrix
After taking the proper prime number \(n\), two exponent matrices \(G_{e_{1}}\) and \(G_{e_{2}}\) may be obtained by resort to the operator ‘\(\odot\)’ on matrices \(E_{A_{l}}\) and \(E_{B_{j}}\) using a cyclic shift to vectors \(\vec{e}_{a_{i}}\) and \(\vec{e}_{b_{j}}\), respectively. Correspondingly, the quasi-cyclic matrix \(H\) of the quantum codes can be obtained based on the two classical quasi-cyclic matrices \(G_{1}\) and \(G_{2}\), which are yielded by the same jacket matrix and two circulant permutation matrices by resort to the operation ‘\(\odot\)’. first vectors to construct the sub-matrices \(E_{A1}\) and \(E_{A2}\), respectively, so that \(G_{e1}\) is obtained by combining the two sub-matrices. Similarly, also take \(\vec{e}_{22}=(2,4,6,8,10)\) and \(\vec{e}_{22}=(4,6,8,10,2)\) as the first vectors, then \(G_{e2}\) will be constructed by coupling \(E_{B1}\) and \(E_{B2}\). Correspondingly, in terms of the basic matrix \(K\) with parameter \(h=1\) in Equation (13), the exponent matrices \(G_{e_{1}}\) and \(G_{e_{2}}\) may be generated as_  
\[G_{e_{1}}=\left( \begin{array}{cccccccc}E_{A1}\odot E_{A2}\end{array} \right)= \left(\begin{array}{cccccccc}E_{A1}&\left| \begin{array}{cccccccc}E_{A2} \end{array} \right.\right)\] \[=\left( \begin{array}{cccccccc}1&3&5&7&9&3&5&7&9&1\\ 9&1&3&5&7&1&3&5&7&9\\ 7&9&1&3&5&9&1&3&5&7\\ 5&7&9&1&3&7&9&1&3&5\\ 3&5&7&9&1&5&7&9&1&3\end{array} \right). \tag{33}\]  
_Similarly, by making use of a similar method, matrix \(G_{e_{2}}\) may be obtained as follows:_  
\[G_{e_{2}}=E_{B1}\odot E_{B2}=\left(\begin{array}{cccccccc}E_{B1}&\left| \begin{array}{cccccccc}E_{B2}\end{array} \right.\right)\] \[=\left( \begin{array}{cccccccc}2&4&6&8&10&4&6&8&10&2\\ 10&2&4&6&8&2&4&6&8&10\\ 8&10&2&4&6&10&2&4&6&8\\ 6&8&10&2&4&8&10&2&4&6\\ 4&6&8&10&2&6&8&10&2&4\end{array} \right). \tag{34}\]  
_We also take the 11 by 11 basic-matrix \(J_{11}=J_{6}\oplus J_{5}\). Then, the matrices \(G_{1}\) and \(G_{2}\) with size \(55\times 110\) are obtained by combining the two exponent matrices and the basic-matrix. It is obvious that the obtained matrix can encode 55 information bits at most. To increase the coding rate \(k/N\), we can also add the rows \(E^{\prime}_{A1}\) and \(E^{\prime}_{A2}\) (\(E^{\prime}_{B1}\) and \(E^{\prime}_{B2}\)) of the exponent matrix in terms of the algorithm described before, such as_  
\[(E^{\prime}_{A1}|E^{\prime}_{A2})=\left( \begin{array}{cccccccc}5&7&9&1&3&3&5 &7&9&1\\ 7&9&1&3&5&5&7&9&1&3\\ 9&1&3&5&7&7&9&1&3&5\\ 1&3&5&7&9&9&1&3&5&7\\ 3&5&7&9&1&1&3&5&7&9\end{array} \right) \tag{35}\]  
_and_  
\[(E^{\prime}_{B1}|E^{\prime}_{B2})\] \[=\left( \begin{array}{cccccccc}6&8&10&2&4&4&6&8&10&2\\ 8&10&2&4&6&6&8&10&2&4\\ 10&2&4&6&8&8&10&2&4&6\\ 2&4&6&8&10&10&2&4&6&8\\ 4&6&8&10&2&2&4&6&8&10\end{array} \right). \tag{36}\]  
_As the result, the exponent matrices \(G_{e_{1}}\) and \(G_{e_{2}}\) can also be shown as the following \(110\times 110\) matrix_  
\[G_{e_{1}}=\left(\begin{array}{c|c}E_{A1}&\left| \begin{array}{c}E_{A2}\\ E^{\prime}_{A1}\end{array} \right.\right)\text{and }G_{e_{2}}=\left( \begin{array}{c|c}E_{B1} \\ E_{B1}^{\prime}\end{array} \right.\left| \begin{array}{c}E_{B2}\\ E^{\prime}_{B2}\end{array} \right.\right). \tag{37}\]  
_Therefore, the two matrices \(G_{1}\) and \(G_{2}\) are used to form the generator matrix \(\mathcal{G}_{c}\) in Equation (5)._  
For example, if we wish to obtain a quantum code with a coding rate of 0.4, by applying the proposed approach, the two classical codes \(C_{1}[110,80,12]\) and \(C_{2}[110,74,14]\) can be constructed, respectively. Therefore, according to the construction method of CSS-type, a quantum code \(\mathcal{C}[110,44,12]\) is obtained. With greatly increasing code length of QC codes, some methods of computing the minimum distance are required [37; 38]. In the following, we use different parameters to analysis the bit error ratio (BER) of the presented QC codes with a jacket matrix of 0.5 code rate.