# Improved Adaptive Successive Cancellation List Decoding of Polar Codes
## 3 Improved Segmented CRC-Based Adaptive SCL Algorithm
### Improved Algorithm Based on Segmented Adaptive SCL  
Since SC decoding has error propagation characteristics, the number of decoded error bits will be very large when errors occur. SCL also shares a similar phenomenon. In [24], there are two types of errors in SCL, which can be divided into disappearing errors and selection errors. Disappearing errors are when the correct paths are not included in the \(L\) paths of SCL. Selection errors are when error paths are selected from the \(L\) paths which include correct paths, because the value PM of correct paths is not enough to be selected. When the decoding has disappearing errors, the correct paths will never be selected after the decoding of this bit. In the SCAD-SCL algorithm, if the first part using SCL decoding with the parameter \(L=L_{\text{max}}\) fail to pass CRC, it means that the correct decoding result is not included in the \(L\) paths in the first part. The decoding error at this time is a disappearing error. The decision of SCAD-SCL is to take the optimal path as an output, and then the algorithm begins the next part of adaptive SCL decoding. However, the decoding of the latter part is based on the previous part, no matter how large \(L\) is, SCL cannot decode the correct codes of the latter part. In addition, due to the failure of the CRC, the adaptive SCL decoding of this segment will increase from \(L_{\text{min}}\) to \(L_{\text{max}}\), which greatly increases delay and complexity. Thus, the decoding of the latter part under this situation is completely unnecessary. The SAD-SCL algorithm proposed in this paper makes full use of the above analysis and improves the existing SCAD-SCL algorithm in two aspects. Figure 3: SCL decoding structure diagram starting from \(S\). Improvement 1: The SAD-SCL algorithm uses the method described in 3.1 to find the value \(S\) before each update value \(L\) is re-decoded and takes \(S\) as the starting bit for decoding. Improvement 2: When the SAD-SCL algorithm cannot get the correct decoding result of the first part, the SAD-SCL algorithm directly uses SC decoding to complete the decoding of the second part. The flow chart of the SAD-SCL decoding algorithm is shown in Figure 4. Let \(\hat{u}\) denote the decoding output value of SAD-SCL. The decoding algorithm can be divided into the following decoding steps:  
Step 1: \(L\) is set as \(L_{\min}=1\), then \(\hat{u}_{1}^{1}\) and \(R_{1}^{1}\) are obtained after the SCL decoding starts in the first part, where \(\hat{u}_{1}^{1}\) represents the SCL decoding result of the first part, and \(R_{1}^{1}\) is the value LLR corresponding to \(\hat{u}_{1}^{1}\). Step 2: If the CRC of \(\hat{u}_{1}^{1}\) fails, then the decoding process of the first part begins to re-decode by the improved adaptive SCL algorithm. The decoder makes a comparison between \(L\) and \(L_{\max}\). If \(L\) is less than \(L_{\max}\), then sets \(L=2L\), and calculates \(S\) according to \(R_{1}^{1}\), where \(S\) indicates the starting bit of re-decoding. Then \(\hat{u}_{1}^{2}\) and \(R_{1}^{2}\) are obtained by re-decoding, if the CRC of \(\hat{u}_{1}^{2}\) fails, continue to increase \(L\) and update the value \(S\), then \(\hat{u}_{1}^{3}\) and \(R_{1}^{3}\) are obtained by re-decoding and so on, the algorithm goes to step 3 until the CRC of \(\hat{u}_{1}^{1}\) passes (\(0<t_{1}\leq log_{2}L_{max}\), where \(t_{1}\) denotes the number of re-decoding of the first part). In addition, when \(L=L_{\max}\), if the decoding result still fails to pass CRC, the first part of the codes is judged to include error bits, and the decoder goes directly to step 4. Step 3: SAD-SCL begins the decoding of the second part, the decoding result and its corresponding LLR values of second part are \(\hat{u}_{2}^{t_{2}}\) and \(R_{2}^{t_{2}}\) (\(0<t_{2}\leq log_{2}L_{max}\), where \(t_{2}\) denotes the number of re-decoding of the second part), respectively. The decoding process of the second part is the same as that of the improved adaptive SCL in step 2.