## 3 ALF Decoder
### Design of the Simplified Flip Metric
To illustrate our simplified function more clearly, we created the graph in Figure 2. In Figure 2, the black dashed line indicates the \(f_{\beta}(\cdot)\) function, and the red line represents our simplified function. Evidently, our simplified function is mainly composed of two straight line segments, and these two straight line segments are generated based on three points on the black dashed line. Since the \(E_{\alpha=1}\) value in Equation (4) satisfies \(E_{\alpha=1}>0\), we selected (\(0\),\(f_{\beta}(0)\)) as the first point. Based on the experience provided by the existing simplification, we selected (\(10\),\(f_{\beta}(10)\)) as the third point. Finally, we selected (\(z\),\(f_{\beta}(z)\)) as the second point,where \(z\) is a positive integer and \(1\leq z\leq 10\). Therefore, the simplified function \(f_{\beta}^{line}(\cdot)\) satisfies the following equation:  
\[f_{\beta}^{line}(x)=\left\{ \begin{matrix}c_{1}x+f_{\beta}(0),&\text{ if }x\geq 0 \text{ and }x\leq z;\\ c_{2}x+c_{3},&\text{ if }x>z\text{ and }x\leq 10;\\ 0,&others.\end{matrix} \right. \tag{6}\]  
where \(c_{1}=\frac{f_{\beta}(z)-f_{\beta}(0)}{z}\), \(c_{2}=\frac{f_{\beta}(z)-f_{\beta}(10)}{z-10}\), and \(c_{3}=f_{\beta}(z)-z\times\frac{f_{\beta}(z)-f_{\beta}(10)}{z-10}\). Additionally, \(c_{1}\), \(c_{2}\), and \(c_{3}\) in this paper generally retain two decimal places. In particular, when \(z=10\),  
\[f_{\beta}^{line}(x)=\left\{ \begin{matrix}c_{1}x+f_{\beta}(0),&\text{if }x\geq 0 \text{ and }x\leq 10;\\ 0,&others.\end{matrix} \right. \tag{7}\]  
It is clear that we used a combination of one multiplication and one addition to replace the multiplication, addition, exponential, and logarithmic operations included by the original \(f_{\beta}(x)=\frac{1}{\beta}ln(1+e^{-\beta x})\). Furthermore, Ref. [24] demonstrated through simulation that outstanding performance can be reached with a fixed \(\beta\) of 0.4, so this paper uses a default \(\beta\) value of 0.4. To explore the impact of different \(z\) values on performance, we created Figures 3-5 to show a comparison of the performance of D-SCLF2 using \(f_{\beta=0.4}\) and that of D-SCLF2 using \(f_{\beta=0.4}^{line}\), with different \(z\) values for \(PC(512,256+24)\). Note that the D-SCLF2 using \(f_{\beta=0.4}\) refers to the D-SCLF2 decoder using the original flip metric, and the D-SCLF2 using \(f_{\beta=0.4}^{line}\) refers to the D-SCLF2 decoder using the new simplified flip metric. The list size of these decoders is 4. In these figures, the left sub-graphs represent the FER performance, and the right sub-graphs represent the corresponding average complexity performance. Since the average complexity of the D-SCLF ( or CA-SCL) decoder is \(\mathcal{O}((T_{av}+1)\times LNlogN)\) (or \(\mathcal{O}(LNlogN)\)), we can use \(L_{av}\) to represent the average complexity, where \(L_{av}=(T_{av}+1)\times L\) for the list flip decoder (or \(L_{av}=L\) for the CA-SCL decoder ). \(T_{av}\) represents the average number of extra decoding attempts. The dashed lines in these figures represent the performance of DSCLF-2 with the original \(f_{\beta=0,4}(\cdot)\) function, and the solid line represents the performance of D-SCLF2 with the new \(f_{\beta=0,4}^{line}(\cdot)\) function. The AWGN channel, BPSK modulation, and the Gaussian Approximation (GA) construction algorithm [26] with a fixed \(E_{b}/N_{0}\) of 4 dB are used. The generator polynomial of the 24 CRC bits is \(g(x)=x^{24}+x^{23}+x^{6}+x^{5}+x^{1}+1\). Figure 4: Performance comparison 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 \(PC(512,256+24)\). Figures 3-5 show that under the same \(E_{b}/N_{0}\), the D-SCLF2 using \(f_{\beta=0.4}^{line}\) has a similar FER performance to that of the D-SCLF2 using \(f_{\beta=0.4}\). Therefore, our simplified flip metric does not reduce the high-order error correction capability brought by the original flip metric in the D-SCLF decoder, regardless of the \(z\) value.