## 3 Candidate Generation in Parallel Decoding of Rate-1 Nodes
### Candidate Generation Based on Partially Ordered Candidate Sets
Accordingly, the curves plotted in red and green correspond to insufficient and sufficient values for \(S_{\text{Rate-1}}\) and \(k_{c}\) in ExPOS-Fast-SSCL decoding. Additional FER curves are plotted in blue for \(L=8\) as corner cases with an error-correction performance loss equal to \(\Delta\) at the selected target FER. For a code rate \(R=1/6\), the FER plots for \(L\in\{2,4,8,16\}\) are presented in Figure 9. It can be seen that \(S_{\text{Rate-1}}\in\{0,0,1,1\}\) is sufficient to avoid an error-correction performance loss \(>\Delta\) for the presented list sizes, respectively. Note that \(S_{\text{Rate-1}}=0\) implies that no splitting is carried out in Rate-1 nodes, and, therefore, ExPOS-Fast-SSCL decoding is not feasible. POS-Fast-SSCL decoding with \(S_{\text{Rate-1}}=0\) for \(L=8\) exhibits a loss equal to \(\Delta\) at \(\text{FER}=10^{-5}\), which increases for decreasing FER. For ExPOS-Fast-SSCL decoding with \(L\in\{8,16\}\), \(S_{\text{Rate-1}}=1\) and \(k_{c}=L\) are sufficient. This results in considering only the candidates with \(x=0\) and the one most probable candidate with \(x=1\) originating from input path \(l=0\) (see also Figure 6d). Figure 10 shows the FER plot of code rate \(R=1/2\). The values of \(S_{\text{Rate-1}}\in\{1,1,2\}\) for \(L\in\{2,4,8\}\) correspond to the results of [20]. For \(L=16\), we find \(S_{\text{Rate-1}}=2\) for POS-Fast-SSCL decoding to be sufficient to preserve error-correction performance at \(\text{FER}=10^{-5}\) with respect to the chosen \(\Delta=0.02\,\text{dB}\). Setting \(S_{\text{Rate-1}}=1\) results in a FER performance degr

Figure 8: Error-correction performance of Polar codes with \(N=1024\) and \(R\in\{1/6,1/2,5/6\}\), under SCL and PO-Fast-SSCL decoding with \(L=16\). \[P=\min(N_{v},S_{\text{Rate-1}}). \tag{31}\]0 Error-correction performance of Polar code \(N=1024\), \(R=1/6\). dation of \(0.09\,\)dB. For ExPOS-Fast-SSCL decoding with \(L\in\{4,8,16\}\) and the appropriate values of \(S_{\text{Rate-1}}\), we observe an adequate error correction with \(k_{c}\in\{3,4,4\}\), respectively. Excluding more candidates by setting \(k_{c}\in\{4,6,5\}\) results in an error-correction performance degradation \(>\Delta\). As a corner case, we point out the combination \(L=8\), \(S_{\text{Rate-1}}=2\) and \(k_{c}=5\), where the performance loss is equal to \(\Delta\) at \(\text{FER}=10^{-5}\). This can be considered negligible, especially when a smaller FER, e.g., \(10^{-3}\) is targeted. The observations for code rate \(R=5/6\), presented in Figure 11, correspond to those for code rate \(R=1/2\). Based on the sufficient \(S_{\text{Rate-1}}\in\{1,2,2,3\}\) in POS-Fast-SSCL decoding with \(L\in\{2,4,8,16\}\), it can be seen that ExPOS-Fast-SSCL decoding with \(k_{c}\in\{3,3,6\}\), respectively, preserves the error-correction performance with respect to \(\Delta\). An error-correction performance degradation \(>\Delta\) is observed for \(k_{c}\in\{4,5,7\}\). As a corner case, \(k_{c}=4\) for \(L=8\) results in an FER performance loss equal to \(\Delta\). However, the error-correction degradation for the presented ExPOS cases with \(L\in\{8,16\}\) is close to \(\Delta\). Therefore, \(k_{c}\in\{4,5\}\) with \(L=8\) and \(k_{c}=7\) with \(L=16\) should also be considered for a small target FER, e.g., \(10^{-3}\), since these values provide an adequate error-correction performance in that FER region. \[P=\min(N_{v},S_{\text{Rate-1}}). \tag{31}\]1 Error-correction performance of Polar code \(N=1024\), \(R=5/6\). \[P=\min(N_{v},S_{\text{Rate-1}}). \tag{31}\]2 Error-correction performance of Polar code \(N=1024\), \(R=1/2\). ### Complexity Estimations

In this section, we analyze the complexity reduction gained by ExPOS-based candidate generation compared to the serial approach of Fast-SSCL with speed optimization, i.e., with threshold \(S_{\text{Rate-1}}\), as presented in [20], and to the parallel approach of MC set-based Fast-SSCL in [15].