## 2 Code Length Estimation and Blind Synchronization
### Principles of the Proposed Recognition Algorithm in Soft Decision Situations
#### 2.2.2 Adaptive Processing of MPCM  
Note that the matrix \(Hb_{\min}(\alpha^{i})\) is not sparse, so a fault decision symbol has negative influences on many syndromes. Considering that the unreliable decision bits have higher probabilities of being error decisions than the reliable decision bits, the authors of [28] proposed an adaptive processing algorithm for the binary MPCM \(Hb_{\min}(\alpha^{i})\) to reduce the influences of the unreliability decision bits when decoding the RS codes by utilizing the belief propagation (BP) algorithm. In this paper, we adopt that idea on the calculation of \(P_{r}(\textbf{S}_{j,i}=0)\). The detailed adaptive processing steps for a given binary MPCM \(Hb_{\min}(\alpha^{i})\) and a codeword \(C_{j}\) with the code length \(l\) are listed below:  
**Step 1**: Combine \(Hb_{\min}(\alpha^{i})\) and \(C_{j}^{T}\) to form a new matrix \(H^{*}(\alpha^{i})\) as follows:  
\[H^{*}(\alpha^{i})=\left[ \begin{array}{cccc}r_{1}&r_{2}^{2}&\cdots&r_{l}\\ \hline h_{11}&h_{12}&\cdots&h_{l}\\ h_{21}&h_{22}&\cdots&h_{2l}\\ \vdots&\vdots&\ddots&\vdots\\ h_{m1}&h_{m2}&\cdots&h_{ml}\end{array} \right] \tag{15}\]where \(r_{1}\), \(r_{2}\), \(\ldots\), \(r_{i}\) are the soft decision bits of the codeword \(C_{j}\), \(\{h_{i_{j}}\,|\,1\leq i\leq m,1\leq j\leq l\}\) are the elements of \(Hb_{\min}(\alpha^{i})\) in GF(2). **Step 2**: Replace each \(r_{u}\) (\(1\leq u\leq l\)) in \(H^{*}(\alpha^{i})\) with their absolute values to form a matrix \(H_{r}^{{}^{*}}(\alpha^{i})\), adjust the positions of the columns in \(H_{r}^{{}^{*}}(\alpha^{i})\) to make the first row in \(H_{r}^{{}^{*}}(\alpha^{i})\) is ranked from the lowest to the highest and record the indexes. The absolute values of \(\{r_{u}\,|\,1\leq u\leq l\}\) denote the reliability of the received soft decision bits. As shown in Equation (16), \(\left|r_{i}\right|\leq\left|r_{i_{1}}\right|\leq\cdots\leq\left|r_{i_{r}}\right| \text{and}i_{1},i_{2},\cdots,i_{l}\) are the column indexes of \(r_{i_{1}},r_{i_{2}},\cdots,r_{i_{l}}\) in \(H^{*}(\alpha^{i})\). \[H_{r}^{{}^{*}}(\alpha^{i})=\left[ \begin{array}{c|ccc}\left|r_{i_{1}}\right| &\left|r_{i_{2}}\right|&\cdots&\left|r_{i_{l}}\right|\\ \hline h_{i_{1}}&h_{i_{2}}&\cdots&h_{i_{l}}\\ h_{i_{2}}&h_{i_{2}}&\cdots&h_{i_{l}}\\ \vdots&\vdots&\ddots&\vdots\\ h_{m_{i_{1}}}&h_{m_{i_{2}}}&\cdots&h_{m_{i_{l}}}\end{array} \right] \tag{16}\]  
**Step 3**: Transform \(H_{r}^{{}^{*}}(\alpha^{i})\) by elementary operations to make the last \(m\) elements of the first column in \(H_{r}^{{}^{*}}(\alpha^{i})\) has only one \"1\" at the top, as shown in Equation (17). The first row does not join the elementary transformation. \[H_{r}^{{}^{*}}(\alpha^{i})=\left[ \begin{array}{c|ccc}\left|r_{i_{1}}\right| &\left|r_{i_{2}}\right|&\cdots&\left|r_{i_{l}}\right|\\ \hline 1&x&\cdots&x\\ 0&x&\cdots&x\\ \vdots&\vdots&\ddots&\vdots\\ 0&x&\cdots&x\end{array} \right] \tag{17}\]  
This transformation limits the influences of the most unreliability decision bit to only one syndrome element, which is \(S_{j,1}\) in Equation (10). Furthermore, we continue the elementary transformation on \(H_{r}^{{}^{*}}(\alpha^{i})\)to limit the numbers of \"1\" in the following some columns to one (except the first row), as shown in Equation (18):  
\[H_{r}^{{}^{*}}(\alpha^{i})=\left[ \begin{array}{c|ccc}\left|r_{i_{1}}\right| &\left|r_{i_{2}}\right|&\left|r_{i_{1}}\right|&\cdots&\left|r_{i_{l}}\right|& \left|r_{i_{l}}\right|\\ \hline 1&0&0&\cdots&x&x\\ 0&1&0&\cdots&x&x\\ 0&0&1&\cdots&x&x\\ \vdots&\vdots&\vdots&\ddots&x&x\\ 0&0&0&\cdots&x&x\end{array} \right] \tag{18}\]  
When the elementary transformation is unavailable, stop the operation. The number available operation times equal to the rank of \(Hb_{\min}(\alpha^{i})\). Then the last \(m\) rows in \(H_{r}^{{}^{*}}(\alpha^{i})\) form a new matrix. We recovery its original column orders and call it \(Hb_{\min\_a}(\alpha^{i})\).