# Quantum Coding via Quasi-Cyclic Block Matrix
## 4 Long-Length Quantum Coding via Quasi-Cyclic Jacket Matrix
### The Construction Method of the Exponent Matrix
Therefore, the low complexity of the coding method should obey the following rules:  
1. the length of the taken matrices between the Kronecker-product operator should be larger;
2. to obtain the long QC quantum code, the number in the decomposition method should include more direct-sum operators than Kronecker-product operators. As a result, the constructed quantum codes show good performance for low density code. Figure 4: Comparison of decoding error rate of the proposed QC quantum codes (bule) and conventional QC LDPC CSS codes (red). Here, the proposed QC quantum codes with lengths \(N=7832\) and \(N=5256\) are generated by \(n=89\) and \(n=73\) for \(n-1\) times of iterations on a binary symmetric channel, and the conventional QC LDPC codes are obtained with similar lengths with rates \(R_{q}=0.50\), \(0.48\), respectively. Based on the equal error rate of one of the classical pair codes \(C_{1}\) and \(C_{2}\), the bit error rate of the marginal flip probability \(f_{m}\) of \(X\) and \(Z\) errors is shown. In addition, we consider the low-density properties of the constructed matrix. After first being proposed by [PERSON], LDPC codes were rediscovered several decades later [31; 32]. Classical LDPC codes have good performance, which approach the channel capacity in the limit of large block size. There has been considerable interest in finding quantum versions of these codes [33; 34; 35]. However, quantum LDPC codes are far less studied than their classical counterparts. In spite of its good performance, one main obstacle is how to obtain a highly efficient algorithm for iterative coding. In our paper, we have tentatively presented a method of constructing quantum LDPC codes and the results further enrich the theoretical achievements in this field. To ensure that the Tanner graph of the LDPC codes is free of 4-cycles, the girth may be at least 6 because the cycle of short length will reduce the performance of the LDPC codes. To meet the requirement, we just take the parameter \(n\) of the basic jacket matrix to be no less than 7, i.e., \(l\geq 3\), in our construction algorithm. In classical codes, a bipartite Tanner graph consists of check nodes and variable nodes [40]. The variable and check node degrees are denoted as \(\eta\) and \(\rho\), respectively, which correspond to the row and column minimum distances of the sparse parity-check matrix \(H\). Assume the length of the shortest cycle on the graph, and that the number of independent iterations are referred to as the girth \(g\) and \(T\), respectively [41; 42]. Correspondingly, a useful iteration gain is generally bounded by  
\[T<g/4\leq T+1. \tag{39}\]  
Obviously, the gained block length should be sufficient while the girth is given, so that the bound satisfies [42]  
\[N\geq 1+\sum_{i=2}^{x+1}\eta(\eta-1)^{i-2}(\rho-1)^{i-1} \tag{40}\]  
for a specific girth \(g=4x+2\), where \(x\) is an integer. According to the bound (40), the relation of the code length \(N\) and the weights \(\eta\) and \(\rho\) is shown in Figure 5. Here, as \(x\) is taken as 2, 3 and 4 with hypothesis \(\eta=\rho\), then the cycle-10, cycle-14 and cycle-18 are gained, respectively. In this figure, it is shown that the length increases exponentially with the cycle length and its minimum weight \(\eta\). Provided that the required cycle is determined, i.e., the cycle is bigger than a 4-cycle, for balancing the weights \(\eta,\rho\) and the code length \(N\), it is necessary to consider the fundamental construction of the jacket matrix. In terms of the characters of the proposed method, the minimum distance of the obtained codes will grow linearly with increase in the parameter \(n\). Thereby, we can apply the taken prime number to generate jacket matrices to adjust the relation between \(\eta\) and \(N\). The main rule for reducing the density and increasing the cycle of the parity check matrix \(H\) is that \(n\) in the decomposition method should be larger.