## 3 ALF Decoder
### Design of the Simplified Flip Metric
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**T = 15**} \\ \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 & 19.710 & 0.00\% & 7.355 & 0.00\% & 4.360 & 0.00\% \\ \hline z = 1 & 19.915 & \(-1.04\%\) & 7.586 & \(-3.15\%\) & 4.355 & 0.11\% \\ \hline z = 2 & 21.315 & \(-8.14\%\) & 7.524 & \(-2.30\%\) & 4.353 & 0.17\% \\ \hline z = 3 & 19.798 & \(-0.45\%\) & 7.468 & \(-1.53\%\) & 4.350 & 0.22\% \\ \hline z = 4 & 20.265 & \(-2.81\%\) & 7.436 & \(-1.11\%\) & 4.350 & 0.22\% \\ \hline z = 5 & 19.011 & **3.55\%** & 7.293 & 0.83\% & 4.337 & **0.53\%** \\ \hline z = 6 & 21.127 & \(-7.19\%\) & 7.424 & \(-0.94\%\) & 4.350 & 0.23\% \\ \hline z = 7 & 20.153 & \(-2.25\%\) & 7.418 & \(-0.87\%\) & 4.346 & 0.32\% \\ \hline z = 8 & 20.785 & \(-5.45\%\) & 7.304 & **0.70\%** & 4.347 & 0.30\% \\ \hline z = 9 & 19.985 & \(-1.39\%\) & 7.598 & \(-3.31\%\) & 4.356 & 0.09\% \\ \hline z = 10 & 21.960 & \(-11.41\%\) & 7.658 & \(-4.12\%\) & 4.362 & \(-0.05\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the average complexity performance of D-SCLF2 using \(f_{\beta=0.4}\) and D-SCLF2 using \(f_{\beta=0.4}^{line}\), with \(L=4\), \(T=15\), and different \(z\) values, for \(PC(512,256+24)\). Based on the above discussion, a fixed and appropriate \(z\) value should satisfy two points: (i) The FER performance of the DSCLF-2 using \(f_{\beta=0.4}^{line}\) cannot be significantly worse than that of the D-SCLF2 using \(f_{\beta=0.4}^{line}\); (ii) The \(L_{av}\) value of the DSCLF-2 using \(f_{\beta=0.4}^{line}\) cannot be significantly greater than that of the D-SCLF2 using \(f_{\beta=0.4}\). Therefore, we choose a fixed \(z\) value based on the above two points and think of the first point as having a higher priority. Finally, we choose a \(z\) value of 5. Then, we can obtain the following equation:  
\[f_{\beta}^{line}(x)= \begin{cases}-0.28x+1.72,&\text{if $x\geq 0$ and $x\leq 5$;}\\ -0.05x+0.59,&\text{if $x\geq 5$ and $x\leq 10$;}\\ 0,&others.\end{cases}  \tag{8}\]  
Thus, our simplified flip metric satisfies the equation below:  
\[M_{\beta}^{line}(S_{t}\bigcup\{i\})=E_{1}^{(i)}+\sum_{k\in S_{t}}E_{1}^{(k)} +\sum_{k<-i,k\in\mathcal{A}\setminus\mathcal{A}^{\prime}}f_{\beta=0.4}^{line} (E_{1}^{(k)}), \tag{9}\]  
where \(i>i_{t}\), and \(i_{t}\) is the last element in \(S_{t}\). Moreover, we use the more understandable \(M_{\beta}^{line}(S_{t}\bigcup\{i\})\) to represent the flip metric of high-order flip operations. In particular,  
\[M_{\beta}^{line}(S_{t})=\sum_{k\in S_{t}}E_{1}^{(k)}+\sum_{k<-i_{t},k\in \mathcal{A}\setminus\mathcal{A}^{\prime}}f_{\beta=0.4}^{line}(E_{1}^{(k)}), \tag{10}\]  
and  
\[M_{\beta}^{line}(\{i\})=E_{1}^{(i)}+\sum_{k<-i,k\in\mathcal{A}\setminus \mathcal{A}^{\prime}}f_{\beta=0.4}^{line}(E_{1}^{(k)}). \tag{11}\]  
It is worth mentioning that the default value of \(z\) for our new flip metric is 5, which means that the simplified function remains consistent with Equation (8). Moreover, we will prove in Section 4 that the new flip metric with a \(z\) value of 5 is valid under different operating conditions.