# Low Complexity List Decoding for Polar Codes with Multiple CRC Codes
## 4 Simulation Results and Analysis  
This section provides performance and complexity of the proposed and conventional schemes. In this paper, a Monte Carlo simulation has been carried out to evaluate the decoding performance and complexity. The block lengths for simulation are 1024, 4096 and only rate one-half is considered. We use a binary-input additive white Gaussian noise channel. The number of total CRC parities \(P=16\) and the list size \(L=32\) for all schemes. For \(J\)-tuple CRC schemes, \(J=2,3\) are considered, and \(\mathbf{p}=(8,8),(4,4,8)\) are used, respectively. All codes are optimized at \(E_{s}/N_{0}=-1\) dB by the density evolution [28]. As the conventional schemes, the SCRC scheme and the \(J\)-tuple CRC scheme that has \(J\) uniform message subblocks [22] are evaluated for comparison. The former is abbreviated as \"SCRC\" and the latter as \"U-JCRC\". The \(J\)-tuple CRC scheme is optimized in terms of the basic SCL decoding or the ID-SCL decoding. They are all abbreviated as \"O-JCRC\". For all the schemes, simulation is performed under the basic and the ID-SCL decoding. Table 1 shows the message vector \(\mathbf{k}\) of O-JCRC and U-JCRC. In general, the optimal \(\mathbf{k}\), \(\mathbf{k}^{*}\), differs depending on whether the ID-SCL decoding is applied or not. The threshold \(\sigma_{\text{Th}}\) for the ID-SCL decoding also differs according to the block length and \(J\) because it is determined empirically to minimize the performance loss, and maximize the complexity reduction for the SCRC scheme. The ratio \(|\mathcal{A}_{g}|/|\mathcal{A}|\) is about 0.746 and 0.806 for \(N=1024,4096\), respectively. The comparison of the average temporary list sizes between the proposed and conventional 3-CRC schemes for \(N=1024\), \(L=32\) and \(P=16\) is shown in Figure 4. Figure 4a is under the basic SCL decoding, and Figure 4b under the ID-SCL decoding. For SCRC in Figure 4a, the temporary list size slowly increases up to \(L\) until the bit index reaches about 220. In the case of U-3 CRC, the average temporary list size decreases from \(L\) to \(1+31/(2^{4})=2.94\) two times due to use of two 4-bit CRC codes. Its temporary list size, however, quickly increases to \(L\) again. For the proposed scheme, the temporary list size of O-3 CRC is also dropped two times, but the list size is maintained small for a burst of consecutive frozen bits. In Figure 4b, the trend of the temporary list size is similar to that of Figure 4a. However, for O-3 CRC, the list size stays small longer due to instant decisions of highly reliable bits on top of frozen operations. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline  & & **2-CRC** & & & **3-CRC** \\ \hline  & **U-2 CRC** & **O-2 CRC** & **O-2 CRC with ID** & **U-3 CRC** & **O-3 CRC** & **O-3 CRC with ID** \\ \hline \(N=1024\) & (256,256) & (136,376) & (107,405) & (170,170,172) & (18,118,376) & (15,92,405) \\ \hline \(N=4096\) & (1024,1024) & (548,1500) & (485,1563) & (682,682,684) & (79,469,1500) & (64,421,1563) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The message vector \(\mathbf{k}\) of the \(J\)-tuple CRC schemes for both serial and parallel sorters. The threshold \(\sigma_{\text{Th}}\) for the ID-SCL decoding is \(10^{-5}\) and \(5\times 10^{-8}\) for \(N=1024,4096\), respectively. Figure 4: The average temporary list size of the proposed and conventional 3-CRC schemes for \(N=~{}1024\), \(L=32\) and \(P=16\); (**a**) with the basic SCL decoding; (**b**) with the ID-SCL decoding. Tables 2 and 3 show the complexity ratio of the \(J\)-tuple CRC schemes to the SCRC scheme with the basic SCL and the ID-SCL decoding. The complexity ratios are estimated by the analysis in Section 3, and the estimation is validated by simulation results. Here, O-2 CRC and O-3 CRC schemes are optimized for serial and parallel sorters separately. As mentioned, the analysis is done under the assumption that there is no early stopping of decoding that indicates a detected codeword error. This assumption is consistent with a high SNR environment.