# Quantum LDPC Codes Based on Cocyclic Block Matrices
## 3 Quantum Codes Based on Cocyclic Quasi-Jacket Block Matrices
### Quantum LDPC Codes Based on Cocyclic Block Matrix
Assuming the binary phase shift keying (BPSK) modulation is over an additive white Gaussian noise (AWGN) channel, the bit error performance of the selected quantum codes is shown in this figure. As a result, the gained codes have the good properties of a sparse matrix. In decoding the performance of LDPC, short cycles will degrade the result under an iterative algorithm. The fewer short cycles of code makes its performance better; hence, the prime numbers \(p=5,7\) more than 4 are taken to avoid 4\(-\)cycles in the Tanner graph. It can be seen from the result that the constructed quantum LDPC codes have no apparent performance in the low signal-to-noise ratio (SNR) region, but the cocyclic codes are better than quasi-cyclic codes at high SNR. Furthermore, we also can see that the difference becomes more pronounced when the code length is larger. Therefore, the designed cocyclic codes may be better applied to construct large-length quantum codes. Generally, three types of independent Pauli errors occur in a realistic depolarizing communication channel, i.e., bit-flip errors \(X\), phase-flip errors \(Z\) and \(Y\), and errors that are the combination of \(X\) and \(Z\). Assuming that the total probability denotes as \(f\), the probabilities of the three classes of errors are equally \(f/3\) owing to their independence. Namely, the distributed marginal flip probability can be depicted as \(f_{m}=2f/3\). As a result, the errors of types \(X\) and \(Z\) in the obtained generator matrices \(G^{x}\) and \(G^{z}\) are separately corrected by reason of their isomorphism. Furthermore, in the construction of quantum LDPC codes, the binary generator matrix of a quantum code is constructed by matrices \(H^{x}\) and \(H^{z}\) which can be seen as the parity check matrices of two two classical codes \(C_{x}\) and \(C_{z}\), respectively. The spaces of degenerate \(X\)-codewords \(C_{x}^{\perp}\) and \(Z\)-codewords \(C_{z}^{\perp}\) are derived from the rows of matrices \(H^{x}\) and \(H^{z}\). The minimum distance \(d=\min\{d_{x},d_{z}\}\) of the quantum code is defined as the smaller of two-types distances \(d_{x}\) and \(d_{z}\). According to the structure of the generator matrices, at least the minimum distances are bounded above by \(O(N^{1/2}\log N)\) as the code length \(N\to\infty\). In the following, we compare the proposed quantum cocyclic codes to the quantum Reed-Solomon coding method with the previous work in ref. [38] under the \(0.5\) coding rate of constructed codes in Figure 2. The designed quantum codes with lengths \(N=500,1372\), respectively, and similar length of CSS-type quantum Reed-Solomon (RS) codes are taken for comparison. Here, we take the generator in terms of the similar method from Equation (31) with size \(N=1024,4096\) for parameter \(m=5,6\) respectively. Furthermore, similar lengths of quantum Reed-Solomon codes are constructed according to the circulant permutation of two classical codes in its construction method. Here, we take the prime number \(p=31,61\), so that the codes of size \(N=930,3660\), respectively, are gained for comparison. Figure 1: BER performance comparison of classical cocyclic code with \(N=500,1372\) and classical quasi-cyclic codes with code rate \(R=0.5\). To achieve the bigger length of cycle, the prime numbers \(p=5,7\) more than \(4\) (for avoiding \(4\)-cycles) are taken in the construction method. If the input is taken in the model of the Gaussian white noise channel (GWNC), we apply the quasi-Jacket block matrix to construct generator matrices. On the other hand, the quasi-cyclic codes with similar length are also constructed. Figure 2: Comparison of decoding error rate of the proposed quantum cocyclic codes (red) and conventional quantum Reed–Solomon LDPC codes (blue). Here, the proposed quantum codes with lengths \(N=930,3660\) are generated by \(m=5,6\) on a binary symmetric channel, and the conventional quantum LDPC codes are obtained with similar lengths with rates \(R_{q}=0.50\), respectively.