## 3 ALF Decoder
### The Adoption of the Adaptive List  
Inspired by [22], we naturally thought of applying an adaptive list to the D-SCLF decoder to reduce its average complexity. Compared with [22], we provide a more in-depth discussion on the reasons for this application as well as more in-depth theoretical analyses of the impact of this application on performance. Furthermore, we verify these theoretical analyses by simulation in Section 4. Since a list flip decoder is converted into a list decoder when \(T=0\), we use the default value of \(T>0\) to better analyze the combination of the list flip decoder and the adaptive list. **The negative impact of applying an adaptive list in list flip decoders is lower than that of applying the adaptive list in list decoders. ** We denote \(L_{av}^{w}\) as the \(L_{av}\) for the worst case, and \(L_{max}\) as the maximum list size. Then, the \(L_{av}^{w}\) of the CA-SCL decoder and the CA-SCL decoder using an adaptive list [8] is \(L_{max}\) and \(2\times L_{max}-1\), respectively. The \(L_{av}^{w}\) of a list flip decoder and the list flip decoder with an adaptive list is \((T+1)\times L_{max}\) and \((T+2)\times L_{max}-1\), respectively. Since \(T>0\), we can obtain \(\frac{2\times L_{max}-1}{L_{max}}>\frac{(T+2)\times L_{max}-1}{(T+1)\times L_{ max}}\). Therefore, the negative impact of applying the adaptive list on the list flip decoder is lower. In particular, \(\frac{(T+2)\times L_{max}-1}{(T+1)\times L_{max}}\) approaches 1 when \(T\) increases. **A list flip decoder with an adaptive list (LFD-w) and the same list flip decoder without the adaptive list (LFD-o) have similar error-correction performance. ** To illustrate this, we divided all possible decoding processes of an LFD-w into three cases:  
* Case 1: The LFD-w can output the path passing CRC when the current list size is \(L_{cur}\) and \(L_{cur}<L_{max}\);
* Case 2: The LFD-w can output the path passing CRC when the current list size is \(L_{cur}\) and \(L_{cur}=L_{max}\);
* Case 3: The LFD-w cannot output the path passing CRC. For case 1, the PM value of the path passing CRC is small enough to always be retained by a small list, which means that the path has a high probability of being retained by a larger list as well. In other words, the LFD-o can find this path with a high probability. For case 2, the path passing CRC can be found by the LFD-w when \(L_{cur}=L_{max}\), which means the correct path can also be found by the LFD-o with a list of list size \(L_{max}\). For case 3, if the path passing CRC cannot be found by the LFD-w, the LFD-o also fails to find the path since the LFD-w includes the LFD-o. According to the above analyses, the error-correction performance of the LFD-w may be consistent with that of the LFD-o. However, there are two special situations included by the above cases. One is that a small size list (\(L<L_{max}\)) can preserve the correct path, but a large size list (\(L=L_{max}\)) may prune the correct path since the \(PM\) value of the correct path may be more than that of other candidate paths in this large size list. The other is that a wrong path passing CRC can be preserved by a small size list (\(L<L_{max}\)) but be removed by a larger size list (\(L=L_{max}\)). Both of these special situations will affect the error-correction performance of the LFD-w and the LFD-o to a certain extent. Considering that these two cases are relatively rare and that CRC is a reliable verification method, the LFD-w and the LFD-o have similar error-correction performance. **The introduction of the adaptive list greatly reduces the complexity of the list flip decoders at a high SNR. ** We denote \(L_{av}^{b}\) as the \(L_{av}\) for the best case. Then, the \(L_{av}^{b}\) of an LFD-w and an LFD-o is 1 and \(L_{max}\), respectively. Therefore, the LFD-w can efficiently reduce the complexity of the LFD-o when \(L_{max}>1\) and the SNR is high. Based on the above analyses, we believe the combination of the adaptive list and the list flip decoder is very suitable.