# High Efficiency Continuous-Variable Quantum Key Distribution Based on ATSC 3.0 LDPC Codes
## 4 Performance Analysis  
In this section, we analyze the performance of proposed LDPC codes in terms of reconciliation efficiency \(\beta\), FER, final secret key rate and transmission distance. In order to balance the loss of code rate and the increase in construction complexity, we construct SC-LDPC codes of length 6,480,000 based on ATSC LDPC codes by setting \(n_{b}=100\). Moreover, we construct QC-LDPC codes of length 648,000 based on ATSC LDPC codes with lifting factor \(L=10\). For decoding of LDPC codes, a sum-product algorithm with maximum iteration number 300 is adopted. Furthermore, \(n\) is set to 8.  
Figure 7: The construction of QC-LDPC code based on [\(B\)\(C\)\(I\)] part.  
As can be seen from Figure 8, the ATSC LDPC codes can achieve \(\beta\) values of more than 0.9. Furthermore, it can be clearly seen that the \(\beta\) values of our proposed LDPC codes are much higher than those of the ATSC LDPC codes under the same code rates. When \(R=2/15\), the SC-LDPC codes enable \(\beta\) values to increase by 0.0102, 0.0207, 0.0254, 0.0391 and 0.0455 for FERs of 0.5, 0.2, 0.1, 0.01 and 0.001, respectively. When \(R=5/15\), the SC-LDPC codes enable \(\beta\) values to increase by 0.0026, 0.0074, 0.0111, 0.0172 and 0.022 for FERs of 0.5, 0.2, 0.1, 0.01 and 0.001, respectively. Furthermore, the QC-LDPC codes raise \(\beta\) values by 0.0082, 0.0186, 0.023, 0.0357 and 0.0374 under the code rate of 2/15 for FERs of 0.5, 0.2, 0.1, 0.01 and 0.001, respectively. Moreover, the QC-LDPC codes increase \(\beta\) values by 0.006, 0.0106, 0.0141, 0.0206 and 0.0257 under the code rate of 5/15 for FERs of 0.5, 0.2, 0.1, 0.01 and 0.001, respectively. The reconciliation efficiency of our proposed codes can reach up to 0.96 for FER of 0.2.  
According to Equation (2), it is obvious that for a given FER, the reconciliation efficiency should be as large as possible in order to obtain a high final secret key rate. Our proposed LDPC codes can achieve much higher \(\beta\) values than original ATSC LDPC codes under the same FERs, which indicates that our proposed LDPC codes can lead to higher final secret key rate and longer transmission distance.  
As shown in Figure 9, we also compared the FER curves of ATSC LDPC codes and proposed LDPC codes under code rates of 2/15 and 5/15, respectively. It is easy to see that for a given FER, our proposed LDPC codes require lower SNRs than original ATSC LDPC codes under the same code rate, which results in smaller \(C(s)\) and higher \(\beta\). Therefore, our proposed SC-LDPC codes and QC-LDPC codes will lead to higher reconciliation efficiency.  
Figure 10 depicts the final secret key rate with respect to the transmission distance for the CVQKD system with different LDPC codes under FER of 0.001. The multidimensional reconciliation protocol in Figure 1 was used to obtain a high final secret key rate. Furthermore, the related parameters are as follows: excess noise \(\xi=0.012\), attenuation factor of the quantum channel \(\alpha=0.2\) dB/km, efficiency of the homodyne detection \(\eta=0.6\) and the electronic noise \(\
u_{el}=0.055\). It is obvious that we can effectively increase the transmission distance by improving reconciliation efficiency. The performance of multidimensional reconciliation protocol combined with our proposed LDPC codes is significantly  
Figure 8: Reconciliation efficiency under different frame error rates (FERs).  
better compared to the original ATSC LDPC codes. This comes from the fact that original ATSC LDPC codes achieve lower reconciliation efficiency values, which are less than 0.91 when the FER is 0.001, while our proposed LDPC codes can achieve reconciliation efficiency values of more than 0.95 when the FER is 0.001.