## 2 Code Length Estimation and Blind Synchronization
### Principles of the Proposed Recognition Algorithm in Soft Decision Situations  
#### 2.2.1 Calculation of \(p_{i}\) in Soft Decision Situations  
To develop the application of this algorithm to be suitable for soft decision cases, we should modify Equation (4) to utilize the soft output of the channel. To calculate \(p_{i}\) in Equation (4) in soft decision situations, we define the minimal parity check matrix (MPCM) \(H_{\min}(\alpha^{i})\) corresponding to the element \(\alpha^{i}\) in GF(\(2^{m}\)) as follows:  
\[H_{\min}(\alpha^{i})=\left((\alpha^{i})^{l-1},(\alpha^{i})^{l-2},\cdots,( \alpha^{i})^{l},(\alpha^{i})^{0}\right) \tag{5}\]  
where \(l\) is the code length. According to the coding theories, if \(\alpha^{i}\) is a root of a codeword \(C_{j}(1\leq j\leq M)\), we have:  
\[H_{\min}(\alpha^{i})\times C_{j}=0 \tag{6}\]Soft outputs of the channel can provide more information about the reliability of each decision symbol. Instead of verifying whether \(\alpha^{i}\) is a root of a codeword polynomial \(c_{j}(x)\), we calculate \(p_{j,i}(1\leq i\leq 2^{m}-1,1\leq j\leq M)\), which is the probability of \(\alpha^{i}\) being a root of \(c_{j}(x)\), and calculate \(p_{i}\) in Equation (4) as follows:  
\[p_{i}=\frac{\sum_{j=1}^{M}p_{j,i}}{\sum_{i=1}^{2^{m}-1}\sum_{j=1}^{M}p_{j,i}} \tag{7}\]  
To calculate \(p_{j,i}\), we transform the MPCM defined in Equation (5) to its binary form by replacing the symbol elements in \(H_{\min}(\alpha^{i})\) with their binary column vector patterns. For example, \(H_{\min}(\alpha^{3})\), a BCH code corresponding to the element \(\alpha^{3}\) in GF(2\({}^{3}\)) with code length \(l=7\), is as follows:  
\[H_{\min}(\alpha^{3})=\left(\alpha^{18}\quad\alpha^{15}\quad\alpha^{12}\quad \alpha^{9}\quad\alpha^{6}\quad\alpha^{3}\quad 1\right) \tag{8}\]  
Based on the primitive polynomial \(p(x)=x^{3}+x+1\), because the symbol \(\alpha\) is a root of \(p(x)\), we have \(\alpha^{3}+\alpha+1=0\), and \(\alpha^{3}=\alpha+1\). Other symbols are processed similarly. Finally, we can calculate all the symbols in \(H_{\min}(\alpha^{3})\) and get their binary vector patterns listed in Table 1. By replacing the symbols in \(H_{\min}(\alpha^{3})\) with the binary patterns, the MPCM can be written in its binary form as follows:  
\[Hb_{\min}(\alpha^{3})=\left( \begin{array}{cccccc}1&1&1&1&1&0&0\\ 1&0&1&0&0&1&0\\ 0&1&1&0&1&1&1\end{array} \right) \tag{9}\]  
Note that the number of rows in \(H_{\min}(\alpha^{i})\) equals to \(m\), which is the degree of GF(2\({}^{m}\)). And then the syndromes corresponding to the MPCM \(H_{\min}(\alpha^{i})\) and codeword \(C_{j}(1\leq j\leq M)\) are calculated in binary forms as follows:  
\[\mathbf{S}_{j,i}=[S_{j,1},\;S_{j,2},\;\cdots,\;S_{j,m}]^{T}=Hb_{\min}(\alpha^{ i})\times C_{j} \tag{10}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbols** & **Polynomial expressions** & **Vector form** \\ \hline
1 & 1 & 001 \\ \(\alpha\) & \(\alpha\) & 010 \\ \(\alpha^{3}\) & \(\alpha+1\) & 011 \\ \(\alpha^{6}\) & \(\alpha^{2}+1\) & 101 \\ \(\alpha^{9}\) & \(\alpha^{3}\) & 100 \\ \(\alpha^{12}\) & \(\alpha^{2}+\alpha+1\) & 111 \\ \(\alpha^{15}\) & \(\alpha\) & 010 \\ \(\alpha^{18}\) & \(\alpha^{2}+\alpha\) & 110 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Symbols in \(H_{\min}(\alpha^{3})\) and their binary vector patterns. Let \(c/(x)\) be the codeword polynomial of \(C_{j}\), if the element \(\alpha^{i}\) corresponding to \(Hb_{\min}(\alpha^{i})\) in Equation (10) is a root of \(c_{j}(x)\), then we have \(\textbf{S}_{j,i}=0\). So the probability of \(\textbf{S}_{j,i}=0\) can be considered as \(p_{j,i}\) in Equation (7) and calculated by the mean probabilities of \(S_{j,k}=0\) (\(1\leq k\leq m\)) :  
\[p_{j,i}=P_{r}(\textbf{S}_{j,i}=0)=\frac{1}{m}\sum_{k=1}^{m}P_{r}(S_{j,k}=0) \tag{11}\]  
In Equation (11) and the remainder of the paper we use \(P_{r}(\Psi)\) to depict the probability of the event \(\Psi\).