# Polar Code Parameter Recognition Algorithm Based on Dual Space
## 3 Principle and Method of Polar Code Parameter Recognition
### Model System  
Assuming that \(N\) polar codewords \(c\)= \([c_{1,1},c_{1,2},\cdots,c_{1,n},\cdots,c_{N,1},c_{N,2},\cdots,c_{N,n}]\) with code length \(n\) are transmitted, after binary phase shift keying (BPSK) modulation and utilizing the AWGN channel, its soft decision sequence \(r\)= \([r_{1,1},r_{1,2},\cdots,r_{1,n^{\prime}},\cdots,r_{N,1},r_{N,2},\cdots,r_{N,n}]\) is intercepted, and then the constructed code word matrix is  
\[\mathbf{R}=\begin{bmatrix}r_{1,1}&r_{1,2}&\cdots&r_{1,n}\\ r_{2,1}&r_{2,2}&\cdots&r_{2,n}\\ \vdots&\vdots&\ddots&\vdots\\ r_{N,1}&r_{N,2}&\cdots&r_{N,n}\end{bmatrix}, \tag{13}\]  
where  
\[r_{i,j}=2c_{i,j}-1+n_{i,j^{\prime}}^{\eta},n_{i,j}^{\eta}\sim\mathcal{N}(0, \sigma^{2}). \tag{14}\]  
The model system is shown in Figure 3.  
The matrix formed by all the column vectors in \(\mathbf{H}_{n}\) is  
\[\mathbf{H}=\begin{bmatrix}h_{1,i}&h_{1,i+a}&\cdots&h_{1,i+c}\\ h_{2,i}&h_{2,i+a}&\cdots&h_{2,i+c}\\ \vdots&\vdots&\ddots&\vdots\\ h_{n,i}&h_{n,i+a}&\cdots&h_{n,i+c}\end{bmatrix}. \tag{15}\]  
Let \(\mathbf{T}=\mathbf{R}\cdot\mathbf{H}\), so that  
\[\mathbf{T}=\begin{bmatrix}\sum\limits_{t=1}^{n}r_{1,t}h_{t,i}&\sum\limits_{t=1}^{ n}r_{1,t}h_{t,i+a}&\cdots&\sum\limits_{t=1}^{n}r_{1,t}h_{t,i+c}\\ \sum\limits_{t=1}^{n}r_{2,t}h_{t,i}&\sum\limits_{t=1}^{n}r_{2,t}h_{t,i+a}& \cdots&\sum\limits_{t=1}^{n}r_{2,t}h_{t,i+c}\\ \vdots&\vdots&\ddots&\vdots\\ \sum\limits_{t=1}^{n}r_{n,t}h_{t,i}&\sum\limits_{t=1}^{n}r_{n,t}h_{t,i+a}& \cdots&\sum\limits_{t=1}^{n}r_{n,t}h_{t,i+c}\end{bmatrix}. \tag{16}\]  
In the ideal state, \(\mathbf{T}\) is an all-zero matrix--that is, the value of each element in the matrix \(\mathbf{T}\) is 0--but in an actual communication system, the codeword will be interfered with by noise, so even if \(\mathbf{H}_{n}\) is orthogonal to the original polar codeword, \(\mathbf{T}\) will not be an all-zero matrix. To measure whether \(\mathbf{H}_{n}\) is orthogonal to the original polar codeword under the AWGN channel, we consider the probability that the \(K^{\text{th}}\) column vector \(\mathbf{h}_{K}=\left[h_{1,K},h_{2,K},\cdots,h_{n,K}\right]^{T}\) in \(\mathbf{H}_{n}\) is orthogonal to the \(i^{\text{th}}\) codeword vector \(\mathbf{r}_{i}=\left[r_{i,1},r_{i,2},\cdots,r_{i,n}\right]\):  
\[P_{i,K  
\(\mathbf{H_{0}}\)**: \(\mathbf{H_{n}}\) is not orthogonal to the codeword matrix \(\mathbf{R}\). \(\mathbf{H_{1}}\)**: \(\mathbf{H_{n}}\) is orthogonal to the codeword matrix \(\mathbf{R}\). All possible dual matrices for the intercepted codeword will be constructed and tested to ascertain whether any one of them satisfies the relationship between the decision statistic and decision threshold in the hypothesis test to find the dual matrix that is orthogonal to the codeword. When the correct dual matrix is found, the constructing code length and rate are considered the real parameters of the intercepted codeword, and the information-bit positions \(A\) can be calculated by the GA algorithm. Then, the generator matrix can be uniquely determined, and the parameter recognition is finally realized. Finally, the polar code parameter recognition problem is transformed into a hypothesis-testing problem, and the key is to find the decision threshold.