## 1 Introduction
Then, using Euclidean distance as the path selection criterion, one decoding result is selected from the \(L\) results as the decoding output. Currently, researchers mainly focus on optimizing polar code BPL decoding in the following aspects:  
1. Optimizing the selection of factor graphs. The study in [8] points out that for polarized codes with code length \(N\), there are \((log2N)!\) species factor diagrams. How to select \(L\) factor graphs for decoding has become a research hotspot in BPL decoding. Early factor graph selection schemes were mainly empirical [7; 9]. As research progresses, researchers have analyzed factor graphs through various methods and selected them based on their reliability. Reference [10] effectively ranks factor graphs based on the number of cycles of length 12 in the permuted factor graphs. Reference [11] uses EXIT charts to analyze the performance of permutation factor graphs. Reference [12] ranks factor graphs based on the upper bound of the error probability of different factor graphs. Reference [13] analyzes the performance of factor graphs under different sorting modes based on frozen bits. Reference [14] treats a polar code of length \(N\) as the concatenation of two subcodes of length \(N/2\), yielding more than \((log2N)!\) factor graphs. 2. Architecture design. Regarding the architecture design for polar code BPL decoding, parallel BPL decoders are currently the main research focus, with some research on serial BPL decoders [9; 10; 15]. Each architecture design for two BPL decoders has its pros and cons. Parallel BPL decoders consume more hardware resources but offer slightly better performance and lower latency. Serial BPL decoders share the same BP decoder, leading to higher latency but are more conducive to hardware design. 3. Using noise perturbation [16] or other post-processing methods [17] to process failed decoding results and improve overall decoding performance. 4. Combining with other decoding methods. Reference [18] proposes an efficient BPL decoding algorithm by combining BPL with renew min-sum (RMS) decoding. Reference [19] proposes a BP-based sparse graph list (BP-SGL) decoding algorithm by combining BPL with low-density parity-check (LDPC) decoding. However, research on polar code BPL decoding algorithms has neglected the study of path selection criteria. Existing path selection schemes generally have the following issues:  
1. Existing path selection schemes need to determine whether the \(\hat{x}=\hat{u}G\) is met in each iteration and discard decoding results that still do not meet \(\hat{x}=\hat{u}G\) after reaching the maximum number of iterations, leading to two adverse consequences: on the one hand, the decision condition \(\hat{x}=\hat{u}G\) is stringent. Simulation results in reference [12] show that, taking a polar code with a code length of (1024, 512) as an example, even after 200 iterations, few decoding results meet the decision condition, resulting in the need for a sufficiently large number of BP decoders; on the other hand, since the decision \(\hat{x}=\hat{u}G\) involves matrix operations in a finite field, combined with a large number of iterations and BP decoders, this leads to excessive hardware resource usage. Even with improved factor graph selection methods, the issue of excessive hardware resource usage is improved but remains unsatisfactory. Due to the limited payload of UAVs, it is necessary to design solutions to reduce hardware resource usage. 2. Existing path selection schemes first perform hard decisions on the decoding results of each decoder and then select one from the \(L\) decoding results as the output codeword. This does not utilize the advantage of BP decoding's soft-input soft-output capability. 3. Existing path selection schemes mainly use Euclidean distance for decision-making, lacking mathematical analysis of the soft information of the decoding output and failing to consider the non-Euclidean nature of the Bernoulli distribution manifold. 4.