# An Enhanced Belief Propagation Flipping Decoder for Polar Codes with Stepping Strategy
## 4 Numerical Results
### Analysis of the BLER Performance  
Using the stepping strategy, some bits in FS are skipped during the procedure of flipping with negligible performance loss. The BLER performance of the EBPSF-\(\Omega\) and EGBPSF-\(\Omega\) algorithms is depicted in Figures 11 and 12. It can be observed that the algorithms which apply the proposed step-flipping strategy are better in terms of the BLER performance. Unlike the BPF-\(\Omega\) and GBPF-\(\Omega\) algorithms, which flip the right information of the leftmost nodes in the DFG, we flip the left information of the rightmost nodes by (16). In contrast to the CA-SCL decoding, the EBPSF-\(\Omega\) and EGBPSF-\(\Omega\) algorithms can achieve comparable decoding performance.  
When \(T\) = 39, Figure 11a indicates that the EBPSF-1 (\(\beta\) = 1) algorithm performs similarly to the BLER of GBPF-1 algorithm. Additionally, the EBPSF-1 algorithm has 0.23 dB gain at BLER = \(4\times 10^{-3}\) against the OA-BP decoder when \(T\) = 256. The BPF-\(\Omega\) and GBPF-\(\Omega\) algorithms flip the right information of the leftmost nodes in the DFG. However, the proposed algorithm flips the left information of the rightmost nodes, and the EBPSF algorithm obtains a gain over the OA-BP decoder. With the same parameters, the EBPSF-1 (\(T\) = 256) outperforms the CA-SCL (\(L\) = 4) decoder by 0.13 dB when BLER = \(1\times 10^{-2}\). The one-bit flipping BLER performance for \(N\) = 512 is depicted in Figure 11b. When \(T\) = 60 and \(\beta\) = 0.5, the EBPSF-1 algorithm for BLER compared to the BPF-1 algorithm shows a gain of 0.09 dB at BLER = \(2\times 10^{-3}\). At \(T\) = 60, the performance of the EGBPSF-1 (\(\Gamma\) = 60, \(\beta\) = 0.5) algorithm is approaching the BLER of the CA-SCL (\(L\) = 2) decoder.  
The two-bit flipping performance for \(N\) = 64 is illustrated in Figure 12a. With \(T\) = 32, the EBPSF-2 (\(\beta\) = 1) algorithm has 0.47 dB, 0.53 dB and 0.41 dB gain compared with the BPF-2, GBPF-2 and OA-BP-2 algorithms at BLER = \(5\times 10^{-2}\), respectively. Furthermore, the EBPSF-2 (\(T\) = 32, \(\beta\) = 1) algorithm achieves the BLER performance between CA-SCL (\(L\) = 4) and (\(L\) = 8). Figure 12b presents the two-bit flipping performance for \(N\) = 256. The EBPSF-2 (\(\beta\) = 1) algorithm, when \(T\) = 256, shows an improvement of 0.21 dB compared with the OA-BP-2 algorithm at BLER = \(5\times 10^{-4}\). The EBPSF-2 (\(T\) = 256, \(\beta\) = 1) algorithm outperforms the CA-SCL (\(L\) = 16) decoder by 0.02 dB at BLER = \(1\times 10^{-3}\).  
Figure 10: The average number of iterations for GBPF-2 [(18)] ([PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] 2020), BPF-2 [(15)] ([PERSON]; [PERSON]; [PERSON]; [PERSON] 2019) and EBPSF-2. (**a**) The average number of iterations for different algorithms with \(N\) = 64. (**b**) The average number of iterations for different algorithms with \(N\) = 256.  
Figure 11: The BLER comparison of BP [10] ([PERSON]; [PERSON]; [PERSON] 2009), OA-BP-1 [18] ([PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] 2020), CA-SCL [9] ([PERSON]; [PERSON] 2012), GBPF-1 [17] ([PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] 2020), BPF-1 [15] ([PERSON]; [PERSON]; [PERSON]; [PERSON] 2019), EGBPSF-1 and EBPSF-1. (**a**) The BLER comparison of different algorithms for \(P\)(256, 128). **(b**) The BLER comparison of different algorithms for \(P\)(512, 256).  
Figure 12: The BLER comparison of OA-BP-2 [18] ([PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] 2020), CA-SCL [9] ([PERSON]; [PERSON] 2012), GBPF-2 [18] ([PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] 2020), BPF-2 [15] ([PERSON]; [PERSON]; [PERSON]; [PERSON] 2019) and EBPSF-2. (**a**) The BLER comparison of different algorithms for \(P\)(64, 32). **(b**) The BLER comparison of different algorithms for \(P\)(256, 128).