# High Efficiency Continuous-Variable Quantum Key Distribution Based on ATSC 3.0 LDPC Codes
## 3 Proposed LDPC Codes for Information Reconciliation
### Construction Method of SC-LDPC Codes for Information Reconciliation  
The structure of the SC-LDPC code can be regarded as a special case of the protograph-based LDPC code. It links multiple unrelated subprotographs into a coupling link through spatial coupling [26], which results in a flexible construction structure for the SC-LDPC code. The detailed steps of our proposed SC-LDPC code based on ATSC LDPC code are described as follows:  
* For a given parity-check matrix of ATSC LDPC code, we first calculate \(\alpha=gcd(M_{b},N_{b})\), where \(gcd(\cdot)\) represents the greatest common divisor function; \(M_{b}\) and \(N_{b}\) denote the number of rows and columns in the parity-check matrix, respectively. * The cutting line is determined by alternately moving \(N_{b}/\alpha\) entries to the right and then \(M_{b}/\alpha\) entries down. Subsequently, we cut the parity-check matrix into upper and lower parts along the cutting line. After obtaining the upper and lower parts, we can construct a new parity-check matrix by re-splicing two parts as shown in Figure 4. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Code Rate** & **Original Structures** & **Proposed Structures of 648,000** & **Proposed Structures of 6,480,000** \\ \hline
2/15 & MET & QC & SC \\
5/15 & MET & QC & SC \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The coding structures corresponding to different code lengths and code rates. Figure 3: The parity-check matrix \(H\) of multi-edge type (MET)-structured LDPC code. * We can repeat the step 2 until the parity-check matrix of desired code length is generated as shown in Figure 5.  
Note that the \(M_{b}(n_{b}+1)\times N_{b}n_{b}\) parity-check matrix of new SC-LDPC code is constructed from a smaller base matrix of ATSC LDPC code of size \(M_{b}\times N_{b}\) via repeatedly copying and pasting as shown in Figure 5. For a given positive integer of repetitions \(n_{b}\), the code rate of proposed SC-LDPC code is given by  
\[ \begin{split} R_{\text{SC}}&=\frac{N_{b}n_{b}-M_{b} (n_{b}+1)}{N_{b}n_{b}}\\ &=\frac{N_{b}n_{b}-M_{b}n_{b}}{N_{b}n_{b}}-\frac{M_{b}}{N_{b}n_{b}} \\ &=R_{ATSC}-\frac{M_{b}}{N_{b}n_{b}},\end{split}  \tag{4}\]  
which faces an unavoidable rate loss of \(M_{b}/N_{b}n_{b}\) compared with the code rate of base matrix. Only when the number of repetitions is large enough can the effect of rate loss be approximately ignored. For a simple example, let \(M_{b}=3\), \(N_{b}=6\) and repetitions \(n_{b}=2\). Then, the base matrix \(H_{b}\) of size \(M_{b}\times N_{b}\) is given by  
Figure 4: The construction of SC-LDPC code based on ATSC LDPC code. Figure 5: New SC-LDPC code with repetitions \(n_{b}\). \[H_{b}= \begin{bmatrix}1&0&1&0&0&1\\ 1&0&0&1&1&0\\ 0&1&0&1&0&1\end{bmatrix} . \tag{5}\]  
According to the construction steps of SC-LDPC codes, we can draw a cutting line in the \(H_{b}\) by alternately moving \(N_{b}/\alpha=6/3=2\) entries to the right and \(M_{b}/\alpha=3/3=1\) entry down as shown in Equation (6):  
\[\widetilde{H_{b}}= \begin{bmatrix}1&0&1&0&0&1\\ 1&0&0&1&1&0\\ 0&1&0&1&0&1\end{bmatrix} . \tag{6}\]  
Subsequently, we can cut \(\widetilde{H_{b}}\) into upper and lower parts along the cutting line, and fill the missing entries of the two parts with 0s, which are given by  
\[H_{upper}= \begin{bmatrix}0&0&1&0&0&1\\ 0&0&0&0&1&0\\ 0&0&0&0&0&0\end{bmatrix} , \tag{7}\]  
\[H_{lower}= \begin{bmatrix}1&0&0&0&0&0\\ 1&0&0&1&0&0\\ 0&1&0&1&0&1\end{bmatrix} . \tag{8}\]  
Accordingly, the SC-LDPC code based on base matrix \(H_{b}\) will be generated by re-splicing the upper part and the lower part:  
\[H_{SC}= \begin{bmatrix}1&0&0&0&0&0\\ 1&0&0&1&0&0\\ 0&1&0&1&0&1\\ 0&0&1&0&0&1\\ 0&0&0&0&1&0\\ 0&0&0&0&0\end{bmatrix} . \tag{9}\]  
Due to the \(n_{b}=2\), we can get the new SC-LDPC code with code length of 12 by repeatedly copying and pasting Equation (9) as follows:  
\[H_{SC}(n_{b})=