## 3 Proposed Scheme
### Error-Detecting and -Correcting Scheme
#### 3.2.2 Error Correcting  
Since the error-correcting process occurs after an error symptom is detected, analyzing the error pattern underlying various symptoms should be the first step of error correction. According to Table 4, the first parity check is failed while the second parity check is passed, indicating 2 possible error patterns. If both of the parity checks are failed, there are 5 possible error patterns. The basic idea of error correcting is as follows: based on the error symptom, try to find the highest possible error pattern, then flip the error bits. To decide which one or two bits to flip, the Error Occur Probability (EOP) of these error patterns needs to be calculated and sorted, so that the bit-flipping can applied according to the EOP. The basic idea is to flip the bit or combination of bits with the lowest reliability, which is transmitted through the channel with the lowest capacity, and perform the parity check again; if it passes the check, the error is corrected; otherwise, the correction is failed. **(a) Bit Reliable Probability**  
Suppose there are two consecutive rows as Figure 3 displays: \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Row-1**} & \multicolumn{2}{c}{**Row-2**} & \multicolumn{2}{c}{**Parity Check**} \\ \hline  & **UBA** & **RBA** & **UBA** & **RBA** & **Row-1** & **Row-2** \\ \hline \multirow{2}{*}{Single Error} & \(\surd\) & \(-\) & \(-\) & \(-\) & \(\times\) & \(\surd\) \\ \cline{2-7}  & \(-\) & \(\surd\) & \(-\) & \(\surd\) & \(\times\) & \(\times\) \\ \hline \multirow{4}{*}{Double Errors} & \(\surd\surd\) & \(-\) & \(-\) & \(-\) & \(\times\) & \(\surd\) \\ \cline{2-7}  & \(-\) & \(\surd\) & \(-\) & \(-\) & \(\times\) & \(\times\) \\ \cline{2-7}  & \(\surd\) & \(\surd\) & \(-\) & \(\surd\) & \(\times\) & \(\times\) \\ \cline{2-7}  & \(-\) & \(\surd\) & \(\surd\) & \(\surd\) & \(\times\) & \(\times\) \\ \cline{2-7}  & \(\surd\) & \(-\) & \(\surd\) & \(-\) & \(\times\) & \(\times\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Error symptom classification. The Bit Reliable Probability (BRP) is used to illustrate the probability that a certain bit could cause an error, whose value is equal to 1 minus the polarized channel capacity that transfers the bit:  
\[P_{br(i,j)}=1-C_{ij} \tag{10}\]  
\(P_{br(i,j)}\) and \(C_{i,j}\) represent the BRP and the polarized channel capacity of the bit located in the \(i_{th}\) row and \(j_{th}\) column in the shifted matrix. Then, according to the section X column shift, the BRP relationship of these 1s in Figure 3 is  
\[P_{br(i+1,9)}>P_{br(i+1,10)}>P_{br(i,A)}>P_{br(i,5)}>P_{br(i,6)}>P_{br(i,7)}>P_{ br(i,8)} \tag{11}\]  
For an error pattern including only one bit, the EOP equals the BRP. However, for other error patterns, the EOP equals the multiplication of the bits that relate to it. An error is detected in the two consecutive rows in Figure 3. Table 5 shows the EOP of distinct error patterns based on Table 4.  
**(b) Error-Correcting Principle**  
Since trying to manipulate the bit-flipping by assuming the possible error patterns is resource-consuming, only part of the error patterns should be tested. Consequently, there are two suggested ways to decide which part. One is setting a certain number of error patterns that would be tested under certain error symptoms; the other is setting an EOP number under which the error patterns would be tested. Suppose the split channel capacity of these columns in Figure 3 is  
\[C_{i,A}=0.42,C_{i,5}=0.48,C_{i,6}=0.52 \tag{12}\] \[C_{i,7}=0.60,C_{i,8}=0.64,C_{i,9}=0.35,C_{i,10}=0.38\]  
**(c) Error correction process:**  
The error correction process includes the following steps:  
Step 1: Determine the error mode: After the error is detected at the first error detection and correction mark position, continue decoding to the first marker position. If the first marker position check result is 0 (the first error mode), perform Step 2; if the first marker  
Figure 3: Two consecutive rows in shifted matrix.