# Quantum Coding via Quasi-Cyclic Block Matrix
## 4 Long-Length Quantum Coding via Quasi-Cyclic Jacket Matrix
### The Construction Method of the Exponent Matrix
We first consider the coding performance of one of the pair of classical codes \(C_{1}\) and \(C_{2}\) in Figure 3. Here, if different prime numbers are taken \(n=29,23,19\) in the construction method, their corresponding code lengths are \(N=n(n-1)=812,506,342\), respectively. With a 0.5 coding rate, we compare their corresponding QC codes. In the signal-to-noise ratio (SNR) range, it is shown that the two shorter lengths of the taken QC quantum codes show better performance from 0.2 to 0.4 dB than the longer one. However, owing to the close cycles and their bit-flip error correction capability, a greater length of code shows its superiority with increasing length. Furthermore, we consider the probability of three classes of Pauli errors in the communication channel. Assuming the channel is viewed as a possibly more realistic depolarizing channel, it can independently generate a list of \(X\) bit-flip errors, \(Z\) phase-flip errors and \(Y\) errors which are a combination of bit and phase flip with equal probability \(f/3\) for a total probability \(f\). Hence, the marginal flip probability distributed identically is \(f_{m}=2f/3\). Owing to the construction methods of all CSS-type quantum codes obtained by the two classical codes \(C_{1}\) and \(C_{2}\), the \(X\)-and-\(Z\) errors may be separately decoded and corrected with a standard classical correction algorithm [39]. In Figure 4, we compare the proposed QC quantum codes with the conventional quantum error-correction codes proposed in [22]. Here, we take, respectively, the quantum coding rate of these codes as 0.50 and 0.48. The proposed codes of lengths \(N=7832\) and \(N=5256\) are generated by jacket matrices with parameters \(n=89\) and \(n=73\), respectively. According to the construction method in [22], two codes with similar lengths \(N=6806\) and \(N=4970\) are compared. Because the pair of classical codes \(C_{1}\) and \(C_{2}\) are isomorphic, the average decoding performances are similar in simulation. Therefore, we just focus on analysis of the single classical code in Figure 4. According to the presented coding method, the obtained generator matrix  
Figure 3: BER performance comparison of code rate \(R=0.5\) classical QC code with \(N=812,N=506\) and \(N=342\). To achieve a greater length of cycle, the decomposition numbers of prime \(n\) more than 4 (to avoid 4\(-\)cycles) are used in our construction method. Assuming the input is a Gaussian white noise channel model, we take the prime numbers \(n=29,23,19\) and the maximum iterations of operator \(\gamma\) as \(n-1\) to construct the jacket matrixes; then the code lengths are \(812,506,342\), respectively. has sparse properties in terms of the character of the jacket block matrix; hence, a larger minimum distance for the generator matrix can be obtained. As a result, the bit error rate of our gained codes is higher than for the conventional codes, although the floor of no error for both codes is down to about \(10^{-6}\). However, with increasing \(f_{m}\), their BERs tend to the same level. Therefore, the proposed QC quantum codes show better performance for the coding of large-scale data. Assume \(J_{i}\) is a \(t\times t\) jacket in Equation (23) constructed with Pauli matrices \(\sigma_{0}\) and \(\sigma_{1}\) (not including the Hadamard matrix) in the described algorithm. To analysis the computation complexity, we first consider the \(0\) and \(1\) densities in the binary matrix denoted by  
\[d_{0}=\frac{\iota_{1}}{\iota_{0}+\iota_{1}}, \tag{38}\]  
where \(\iota_{0}\) and \(\iota_{1}\) are the \(0\) and \(1\) numbers respectively. Therefore, the density \(d_{0}\) of the jacket matrix equals \(1/\iota\) and becomes lower with increasing length \(\iota\). According to the described algorithm, there are two operators, i.e., the Kronecker-product operator '\(\otimes\)' and the direct-sum operator '\(\oplus\)', involved. It is obvious that the calculation complexity of the direct sum is \(O(1)\) that is less than the Kronecker-product operator's \(O(\iota)\).