# Research on Polarization Coding Cooperative Communication Scheme for FSO System
## 3 Construction Principle of System Polarization Code in Cooperative Communication  
Because the system polarization code has better bit-error rate performance than the non-system polarization code, the system polarization code is used in this system. Assuming that the set \(A\) is a reliable channel set, it is a subset of the set \(\{1,\cdots,N\}\), and the set transmitting frozen bits can be represented by a complement \(A^{c}\) of \(A\). The encoding method of polarization code with code length \(N\) is:  
\[x_{1}^{N}=u_{1}^{N}G_{N} \tag{8}\]  
where \(u_{1}^{N}\) is the information bit of source information, \(x_{1}^{N}\) is the encoded information bit, and \(G_{N}\) is the generation matrix of \(N\) order. According to the reliability of the channel, the source information and the generation matrix can be split, \(u_{1}^{N}=(u_{A},u_{A^{c}})\), \(G_{N}=(G_{A},G_{A^{c}})\). Equation (8) can be written as follows:  
\[x_{1}^{N}=u_{A}G_{A}\oplus u_{A^{c}}G_{A^{c}} \tag{9}\]  
In Equation (9), the encoded codeword \(x_{1}^{N}\) can also be divided into \(x_{1}^{N}=(x_{B},x_{B^{c}})\), the set \(B\) is any subset of \(\{1,\cdots,N\}\). Equation (9) can be expressed as Equations (10) and (11):  
\[x_{B}=u_{A}G_{AB}+u_{A^{c}}G_{A^{c}B} \tag{10}\]  
\[x_{B^{c}}=u_{A}G_{AB^{c}}+u_{A^{c}}G_{A^{c}B^{c}} \tag{11}\]  
\(G_{AB}\) can be represented by a submatrix of \(G_{N}\). In the system polarization code, \(x_{B}\) can also be considered part of \(x_{1}^{N}\) containing only information bits. \(x_{B^{c}}\) is the part of \(x_{1}^{N}\) containing frozen bits. For unsystematic polarization code, there is an unsystematic decoder with a given parameter. Then, for the system polarization code, if there is one-toone correspondence between the value of \(u_{A}\) and \(x_{B}\), there is a system decoder with the given parameter \((B,u_{A^{c}})\)[14]. In this system, the relay node uses the DF protocol. The source node \(S\) concatenates the check code and polarization code for the source bits and transmits them to the relay node \(R\) through the \(S-R\) link. The concatenation of the two codewords is for error-checking at the relay node \(R\), which can effectively improve the error correction performance of medium and shortcode long polarization code, to improve the bit-error rate performance of the whole cooperative communication system. The encoding and decoding process of cyclic redundancy check (CRC) cascaded polarization code is shown in Figure 2.  
In the cascade coding of the CRC and system polarization code, all information bits of the source bits participate in CRC code. The information bit length of the input information sequence is \(K\), which can be represented by \(v=\{v_{1},v_{2},\cdots,v_{K}\}\). After CRC coding \(v\), a sequence \(x_{i}\) with code length \(K+M\) is obtained, where \(i\in\{1,2,\cdots,K+M\}\). Then, \(x_{i}\) is sent to the system polarization code encoder to obtain the information sequence with code length of \(N\) and the length of the frozen bit is \(C_{1}^{N}\). After passing through the atmospheric turbulence channel, it is transmitted to the relay node \(R\), and the received signal sequence at \(R\) is \(y_{1}^{N}\). Next, \(y_{1}^{N}\) is decoded at the relay node \(R\), and the decoded sequence is \(\phi_{1}^{N}\). The use of CRC-assisted polarization code decoding can effectively detect the burst errors in the decoding process to improve the decoding performance. In the symmetric decoding and forwarding relay system of polarization code, for a random degenerate relay channel with the orthogonal receiver, as long as the code rate of polarization code \(R<R_{S}^{DF}\) (\(R_{S}^{DF}\) is the symmetric capacity of the system), there is always a polarization code sequence indexed by block length N, and the polarization code has a bit-error rate \(P_{e}\leq O\Big{(}2^{-N^{B}}\Big{)}\) for any \(0<\beta<\frac{1}{2}\), where \(O(\cdot)\) represents the asymptotic property of the function and \(N\) is the code length of polarization code [15].