# The Polar Code Construction Method in Free Space Optical Communication
## 4 Construction Method for the Polar Code
Figure 4 shows the error probability curves of all sub-channels with the increase in SNR. In the strong turbulence channel, sub-channel 3 was more reliable than sub-channel 4 and sub-channel 8 for all the interested SNR values. For the error probability, sub-channel 6 was higher than that of sub-channel 9. Moreover, sub-channel 7 had a better performance than sub-channel 12 in terms of sub-channel reliability. Therefore, when the code length was 16, the polar code was constructed as a single chain by the STPO method. \begin{table}
\begin{tabular}{c c} \hline \hline \(N\) & Proportion of Reliability Relationship \\ \hline
4 & 100\% \\
8 & 75\% \\
16 & 45\% \\
32 & \(\sim\)31.8\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Reliability ratio of subchannels. Figure 3: Hasse diagram with \(N=16\). Polarization weight (PW) is one of the methods that can be used to evaluate the reliability of the polar sub-channel. In this paper, the polarization weight formula was used to evaluate the reliability of sub-channel in strong turbulence channels. The Formula shows as [34]:  
\[PW_{i}=\sum_{j=0}^{n-1}B_{j}\beta^{j}(\beta>1) \tag{8}\]  
where \(n=\log_{2}N\), \(N\) represents code length, and \(PW_{i}\) represents the reliability of \(i\)-th sub-channel. Higher \(PW_{i}\) represents a more reliable sub-channel. For the \(i\)-th sub-channel, its channel serial number \(i\) can be described as \(B_{n-1}B_{n-2}\cdots B_{0}\) in binary representation. \(\beta\) means base for decimal extension, which can represent any real. Take the code length \(N=16\) as an example, which shows as:  
\[PW_{0}=f(0000)=0,PW_{1}=f(0001)=1,\cdots,PW_{15}=f(1111)=\beta^{3}+\beta^{2}+ \beta+1 \tag{9}\]  
Polarization weight is used to directly represent the reliability of each sub-channel, but the \(\beta\) value is different in different channels. Therefore, a suitable \(\beta\) value was determined to construct polar code in strong turbulence channel in this paper. When \(N=16\), the \(\beta\) value met the reliability relationship of the sub-channel pair (3,4) (3,8) (6,9) (7,12):  
\[\left\{ \begin{array}{c}\beta+1>\beta^{2}\\ \beta+1>\beta^{3}\\ \beta^{2}+\beta+1>\beta^{3}+\beta^{2}\\ \beta^{2}+\beta<\beta^{3}+1\end{array} \right. \tag{10}\]  
According to the set of inequalities of (8), the range of \(\beta\) values was calculated and shown in Table 2: \begin{table}
\begin{tabular}{c c} \hline
**STOP, \(N=16\)** & **Range of \(\beta\) Values** \\ \hline (3,4) & \(1<\beta<1.618\) \\ (3,8) & \(1<\beta<1.3247\) \\ (7,12) & \(1<\beta<1.3247\) \\ (6,9) & \(\beta>1\) \\ \hline \end{tabular}
\end{table} 
Table 2: Range of \(\beta\) values when \(N=16\). Figure 4: Error probability of sub-channels (3,4,6,7,8,9,12) in strong turbulence. According to Table 2, when \(\beta\in(1,1.3247)\), a value within this range was sufficient to construct a polar code sequence of \(N=16\), but a 16-bit code length could not satisfy the FSO system for information transmission. When \(N=32\), the pair of sub-channels to be determined increased by (7, 24) (11, 24) (13, 24) (14, 19) (15, 28). The turbulent channel of \(\sigma_{R}^{2}=3.5\) was still selected, MC was adopted for the simulation, and the error probability curve is shown in Figure 5.  
According to the error probability curve of five groups of sub-channels shown in Figure 5, it can be seen that the relationship among (11, 24) (13, 24) (14, 19) (15, 28) four groups of sub-channels was relatively clear. When SNR was less than 5.25 dB, the error probability of sub-channel 7 was greater than sub-channel 24. With the increase in SNR, the error probability of subchannel 7 was gradually exceeded by subchannel 24. At this time, sub-channel 7 was selected to transmit data to obtain a lower BER. When SNR \(<5.25\) dB, the reliability relationship among sub-channel pairs (7, 24) (11, 24) (13, 24) (14, 19) (15, 28) was obtained from Figure 5, and the \(\beta\) value was satisfied:  
\[\left\{