## 3 ALF Decoder
### Design of the Simplified Flip Metric
However, at some \(z\) values, the D-SCLF2 using \(f_{\beta=0.4}^{line}\) may have greater \(L_{av}\) than the D-SCLF2 using \(f_{\beta=0.4}\). The reason is that the new flip metrics corresponding to some z values are not effective enough, which makes the D-SCLF2 using \(f_{\beta=0.4}^{line}\) need more decoding attempts to find the correct path. Thus, a fixed \(z\) value is required to ensure that the D-SCLF2 using \(f_{\beta=0.4}^{line}\) has similar or better performance to the D-SCLF2 using \(f_{\beta=0.4}\) under different operating conditions. It is worth mentioning that \(L_{av}\) represents the complexity because it is proportional to the number of recursions of the LLR of the polar code, but \(L_{av}\) does not count the calculations in the flip metric. Therefore, as long as the FER and \(L_{av}\) of the D-SCLF2 using \(f_{\beta=0.4}^{line}\) and the D-SCLF2 using \(f_{\beta=0.4}^{line}\) are similar, our new flip metric is valid. To better identify the fixed \(z\) value, we created Tables 1-3 to provide more details on the complexity comparison found in Figures 3-5. In these tables, \(R_{c}\) represents the reduction ratio of the current \(L_{av}\) as compared to the \(L_{av}\) corresponding to a \"no z\" situation; a \"no z\" situation refers to the D-SCLF2 using \(f_{\beta=0.4}\), while a \"z = x\" situation refers to the D-SCLF2 using \(f_{\beta=0.4}^{line}\) with the \(z\) value of \(x\). For easier understanding, we use the data in Table 1 as an example. In Table 1, the \(L_{av}\) value of the D-SCLF2 using \(f_{\beta=0.4}\) is 8.495 when \(E_{b}/N_{0}=1.5\) dB, and the corresponding \(R_{c}\) satisfies \(R_{c}=(8.495-8.495)/8.495=0\%\). Similarly, the \(L_{av}\) value of the D-SCLF2 using \(f_{\beta=0.4}^{line}\) with \(z=1\) is 7.991 when \(E_{b}/N_{0}=1.5\) dB, and the corresponding \(R_{c}\) satisfies \(R_{c}=(8.495-7.991)/8.495=5.94\%\). Figure 5: Performance comparison of D-SCLF2 using \(f_{\beta=0.4}\) and D-SCLF2 using \(f_{\beta=0.4}^{line}\), with \(L=4\), \(T=50\), and different \(z\) values, for \(PC(512,256+24)\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**T = 3**} \\ \hline  & \multicolumn{2}{c}{**1.5 dB**} & \multicolumn{2}{c}{**2.0 dB**} & \multicolumn{2}{c}{**2.5 dB**} \\ \cline{2-7}  & **Lav** & **Rc** & **Lav** & **Rc** & **Lav** & **Rc** \\ \hline no z & 8.495 & 0.00\% & 5.117 & 0.00\% & 4.154 & 0.00\% \\ \hline z = 1 & 7.991 & 5.94\% & 5.110 & 0.13\% & 4.150 & 0.11\% \\ \hline z = 2 & 8.004 & 5.77\% & 5.059 & 1.12\% & 4.159 & \(-0.12\%\) \\ \hline z = 3 & 8.224 & 3.20\% & 5.154 & \(-0.74\%\) & 4.151 & 0.08\% \\ \hline z = 4 & 7.979 & 6.08\% & 5.099 & 0.34\% & 4.143 & **0.27\%** \\ \hline z = 5 & 7.889 & 7.13\% & 5.102 & 0.29\% & 4.151 & 0.08\% \\ \hline z = 6 & 7.878 & **7.27\%** & 5.096 & 0.40\% & 4.151 & 0.09\% \\ \hline z = 7 & 8.111 & 4.53\% & 5.043 & **1.44\%** & 4.160 & \(-0.12\%\) \\ \hline z = 8 & 7.909 & 6.90\% & 5.107 & 0.19\% & 4.150 & 0.10\% \\ \hline z = 9 & 7.974 & 6.13\% & 5.108 & 0.18\% & 4.150 & 0.10\% \\ \hline z = 10 & 7.998 & 5.85\% & 5.096 & 0.39\% & 4.159 & \(-0.12\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of the average complexity performance of D-SCLF2 using \(f_{\beta=0.4}\) and D-SCLF2 using \(f_{\beta=0.4}^{line}\), with \(L=4\), \(T=3\), and different \(z\) values, for \(PC(512,256+24)\).