# Higher-Order Belief Propagation Correction Decoder for Polar Codes
## 4 Numerical Results  
In this section, the simulation results are verified on a binary-input additive white Gaussian noise (BI-AWGN) channel in terms of the BLER performance and decoding latency for polar codes based on the proposed schemes. Information set \(\mathcal{I}\) is selected based on the Monte Carlo approach [1] at 2.0 dB. First, the BLER performance of the proposed MBPC-\(\Omega\) scheme is compared with that of the BPC decoder in [22] and the EBPF decoder in [16]. MBPC-\(\Omega\) with order \(\Omega=1\) is a one-bit correction scheme, the same as MBPC. The parameters of the proposed metric are set to \(\alpha=1.0\), \(\beta=0.75\) for code length \(N=512\) and rate \(R=1/2\). To ensure a fair comparison, the sizes of the correction set are set to \(T=20\) in the MBPC-1 and the BPC, while the size of the flipping set in the EBPF is set to \(T=40\). The parameters of correction sets in the MBPC-2 and the MBPC-3 decoders are set to \(T_{1}=20\), \(T_{2,1}=T_{3,2}=20\) and \(T_{2,2}=T_{3,3}=20\), respectively. As seen in Figure 3, the MBPC-1 decoder has a better BLER performance than the EBPF decoder and achieves a slight performance gain than that of the BPC decoder in high SNR regions. The proposed higher-order algorithm can evidently effectively improve the BLER performance, e.g., at \(\text{BLER}=10^{-3}\), the MBPC-2 decoder performs 0.9 dB better than the original BP 0.45 dB and better than the MBPC-1. As \(\Omega\) increases, the maximum number of decoding attempts increases dramatically, bringing challenges to the complexity, but the performance gain gradually decreases. Accordingly, in the further comparisons with the existing multiple flipping scheme, only MBPC-1 and MBPC-2 are studied. Figure 3: The BLER performance comparisons in polar code with length \(N=512\), \(R=1/2\), and \(I_{max}=60\). A 16-bit CRC is used. The BLER performance and average number of iterations of the proposed schemes for \(N=2048,R=1/2\) are simulated for a comparison with the original BP, BPC, higher-order GBPF proposed in [16], and multiple BPF decoder designed based on the critical set (CS) in [15] as shown in Figures 4 and 5. The 24-bit CRC is used. The parameters in Equation (6) are set to \(\alpha=0.65\) and \(\beta=3.5\). The size of the correction set in the MBPC-1 is \(T=20\). In the higher-order scenario, the number of maximum flipping attempts in the GBPF-2 decoder and the maximum correction operations in the MBPC-2 decoder are set to \(T=80+20\times 40\) and \(T=40+4\times 10\times 20\), respectively, to keep the maximum decoding attempts the same as those for the BPF-2, i.e., \(T=4\times|CS|,|CS|=210\). As seen in Figure 4, the performance of the proposed MBPC-2 decoder is significantly improved compared with that of the conventional BP decoder. When compared with the BPF-2 and GBPF-2 decoders, the MBPC-2 decoder achieves a similar performance in low to medium SNR regions but has a 0.1 dB performance gain when \(\text{BLER}=10^{-5}\). Compared with the CA-SCL decoder with \(L=4\), the gain is around 0.2 dB in almost all simulated SNR regions. In addition, the BLER of the MBPC-2 decoder can compete with the CA-SCL decoder with \(L=8\) when \(\text{SNR}\leq 2.2\) dB, but the performance degrades at high SNR regions. The BP-based decoding complexity and latency can be reflected as the number of iterations. Figure 5 presents a comparison of the average number of iterations. As expected, the number of average iterations of the MBPC-1 decoder is approximately half that of the BPC and EBPF decoders, while the MBPC-2 decoder also reduces the complexity and latency by half compared with the BPF-2 and GBPF-2 decoders at a similar BLER performance. Specifically, the proposed MBPC-2 decoder saves around 45% of the average number of iterations at 2.0 dB than the BPF-2 decoder; also observed is that, with the increase in SNR, the average number of iterations of the proposed MBPC decoder can approach that of the conventional BP decoder.