# Low Complexity List Decoding for Polar Codes with Multiple CRC Codes
## 4 Simulation Results and Analysis
Thus, the experimental complexity is measured at the high SNR, e.g., \(E_{S}/N_{0}=-1.0\), \(-1.5\) dB for \(N=1024,4096\), respectively, by averaging the number of addition-equivalent operations. The complexity ratio from the experiment is in the parentheses of Tables 2 and 3. It is observed that the experimental values are very close to the estimated ones. Note that the optimization of \(J\)-CRC scheme is performed based on the estimated complexity and the estimation turned out to be pretty accurate. By the way, a large complexity reduction of the ID-SCL decoding in Table 3 is obtained due to the skipping of sorting operations. Figures 5-7 give the performance and complexity comparison for the optimized 2-CRC and 3-CRC schemes in a wide range of SNR. The complexity ratio includes the reduction from the early stopping in the low SNR region, and it is calculated based on the complexity of SCRC (under the basic SCL decoding). Only the parallel bitonic sorting is assumed in the scheme optimization and complexity evaluation. The average complexity is obtained via simulation. Figures 5 and 6 show the performance and complexity ratio of the proposed and conventional 2-CRC schemes. In Figures 5a and 6a, the negligible performance gap between the proposed and conventional schemes is observed, and the error performance loss is within 0.05 dB. However, Figures 5b and 6b display obvious visible difference in complexity. Especially, when the ID-SCL decoding is applied to the SCRC scheme, about 40% of the complexity is reduced. For \(N=1024\), the complexity reduction of O-2 CRC under the basic SCL decoding is about 7% _points_ compared to U-2 CRC at \(E_{S}/N_{0}=-1.0\) dB, but when the ID-SCL decoding is used, about 4% _points_ are reduced compared to \"U-2 CRC with ID\". In addition, at the low SNR, more complexity reduction is acquired thanks to the early stopping. For \(N=4096\), the complexity reduction of O-2 CRC under the basic SCL decoding is about 22% _points_ compared to U-2 CRC at \(E_{S}/N_{0}=-3.0\) dB, and about 18% _points_ are reduced under the ID-SCL decoding compared to \"U-2 CRC with ID\". \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**U-2 CRC**} & \multicolumn{2}{c}{**O-2 CRC**} & \multicolumn{2}{c}{**U-3 CRC**} & \multicolumn{2}{c}{**O-3 CRC**} \\ \hline
**Sorter Type** & **Serial** & **Parallel** & **Serial** & **Parallel** & **Serial** & **Parallel** & **Serial** & **Parallel** \\ \hline \(N=1024\) & 99.6 (99.5) & 99.5 (99.4) & 90.4 (90.6) & 92.6 (92.6) & 99.3 (99.3) & 99.2 (99.1) & 85.6 (85.2) & 88.9 (88.4) \\ \hline \(N=4096\) & 99.7 (99.5) & 99.7 (99.8) & 91.8 (91.7) & 93.5 (93.3) & 99.8 (99.7) & 99.7 (99.7) & 87.2 (87.3) & 90.0 (89.9) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The ratio (%) of complexity of the \(J\)-CRC scheme (_basic SCL decoding_) to the SCRC scheme (_basic SCL decoding_) obtained by analysis and experiment. The experimental values are in parentheses and obtained at \(E_{S}/N_{0}=-1.0\), \(-1.5\) dB for \(N=1024,4096\), respectively. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**U-2 CRC**} & \multicolumn{2}{c}{**O-2 CRC**} & \multicolumn{2}{c}{**U-3 CRC**} & \multicolumn{2}{c}{**O-3 CRC**} \\ \hline
**Sorter Type** & **Serial** & **Parallel** & **Serial** & **Parallel** & **Serial** & **Parallel** & **Serial** & **Parallel** \\ \hline \(N=1024\) & 62.7 (62.8) & 53.3 (53.4) & 57.3 (57.4) & 49.2 (49.3) & 62.9 (62.7) & 53.5 (53.2) & 52.0 (51.7) & 45.1 (44.9) \\ \hline \(N=4096\) & 65.6 (65.7) & 55.6 (55.7) & 60.7 (60.6) & 51.8 (51.8) & 67.3 (67.3) & 56.9 (57.0) & 56.2 (55.9) & 48.2 (48.1) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The ratio (%) of complexity of the \(J\)-CRC scheme (_ID-SCL decoding_) to the SCRC scheme (_basic SCL decoding_) obtained by analysis and experiment. The experimental values are in parentheses and obtained at \(E_{S}/N_{0}=-1.0\), \(-1.5\) dB for \(N=1024,4096\), respectively.