# The Polar Code Construction Method in Free Space Optical Communication
## 4 Construction Method for the Polar Code
\begin{array}{c}\beta^{4}+\beta^{3}>\beta^{2}+\beta+1\\ \beta^{3}+\beta+1>\beta^{4}+\beta^{3}\\ \beta^{3}+\beta^{2}+1>\beta^{4}+\beta^{3}\\ \beta^{3}+\beta^{2}+\beta>\beta^{4}+\beta+1\\ \beta^{3}+\beta^{2}+\beta+1>\beta^{4}+\beta^{3}+\beta^{2}\end{array} \right. \tag{11}\]  
According to (9), the value range of \(\beta\) is shown in Table 3.  
Combining Table 2 with Table 3, the value range of \(\beta\) could be obtained when SNR \(<5.25\) dB. When \(N=32\), \(\beta\in(1.1787,1.2207)\). When STPO constructed the polar  
Figure 5: Error probability of sub-channels (7,11,13,14,15,19,24,28) in strong turbulence. \begin{table}
\begin{tabular}{c c} \hline \hline
**STOP, \(N=32\)** & **Range of \(\beta\) Values** \\ \hline (7,24) & \(\beta>1.1787\) \\ (11,24) & \(1<\beta<1.2207\) \\ (13,24) & \(1<\beta<1.2720\) \\ (14,19) & \(1<\beta<1.7549\) \\ (15,28) & \(1<\beta<1.2207\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Range of \(\beta\) values when \(N=32\). code, any value of \(\beta\) was taken within this range, and the construction of the polar code sequence was completed by combining with (6). When SNR > 5.25 dB, according to Figure 5, the reliability relationship between sub-channel 7 and 24 changed, and the error probability of sub-channel 24 was greater than that of sub-channel 7. At this time, \(PW_{7}>PW_{24}\) was satisfied. According to (6), \(\beta<1.1787\) was obtained, and combined with the reliability relationship of other sub-channel pairs in Table 3, \(\beta\in(1,1.1787)\) was obtained. In order to use the value of \(\beta\) in the construction of polar code with a longer code length, a more accurate value of \(\beta\) was needed. When the code length was extended to 64 bits, 10 groups of uncertain sub-channels needed to be obtained according to UPO. After statistics, the number of uncertain sub-channel pairs is shown in Table 4.  
From Table 4, as the code length increased, the number of sub-channels whose reliability relationship needed to be determined increased exponentially. When the code length was 1024, the number of sub-channels that needed to be determined increased to about 390, and the 1024-bit polar code had met most of the communication requirements, and there it was unnecessary to further expand the code length. For different code lengths, through the MC simulation, the relationship between uncertain sub-channel pairs was obtained, and the SNR was obtained at the point where the relationship between the sub-channels pair error probability changed. Table 5 shows the mean SNR (M-SNR) of the error probability change point at different code lengths. Through M-SNR, the value range of \(\beta\) was divided into two parts: SNR was less than M-SNR, and SNR was larger than M-SNR. Table 6 shows the \(\beta\) value range when SNR was less than M-SNR and the \(\beta\) value range when SNR was greater than M-SNR in different code lengths. From Tables 5 and 6, when the code length was 1024, the range of \(\beta\) is selected with SNR = 5 dB as the dividing point. When SNR < 5 dB, then \(\beta\in(1.1890,1.1893)\). When SNR > 5 dB, then \(\beta\in(1.1725,1.1729)\). The accuracy of the traditional MC method is related to the number of simulations. In order to ensure a high accuracy, it usually requires at least 100,000 simulation times to ensure the accuracy of \(10^{-4}\). Each simulation requires a polar code encoding and decoding operation. The asymptotic complexity of polar code encoding and decoding is \(O(N\log_{2}N)\). So, the complexity of the MC method is \(O(MN\log_{2}N)\), and the \(M\) is the number of simulations. STPO method is used to construct polar code by combining UPO with MC. \begin{table}
\begin{tabular}{c c} \hline \hline \(N\) & **The Number of Uncertain Sub-Channel Pairs** \\ \hline
32 & 9 \\
64 & 19 \\
128 & \(\sim\)50 \\
256 & \(\sim\)100 \\
512 & \(\sim\)190 \\
1024 & \(\sim\)390 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The number of uncertain sub-channel pairs in different code lengths.