# Polar Code BP Decoding Optimization for Green 6G Satellite Communication: A Geometry Perspective
## 2 BP Decoding Algorithm for Polar Codes
Figure 8 compares the block error rates of several decoding algorithms under three channel conditions. Figure 8 confirms that the two information geometry-based BP decoding algorithms for polar codes do not suffer from any loss in decoding performance. The average number of iterations is shown in Figure 9. Table 2 presents the ratios of iteration reductions achieved by several early convergence algorithms. Table 3 demonstrates that compared to the average number of iterations of the three existing early convergence algorithms designed based on LLR, ESBP-R reduces the number of iterations by 2-11%, and ESBP-D reduces it by 4-15%. Notably, at \(E_{b}/N_{0}\) values ranging from 1.5 dB to 2.5 dB, ESBP-R and ESBP-D exhibit significant improvements in convergence speed. Simulation results indicate that under various channel conditions, the two information geometry-based BP decoding algorithms for polar codes proposed in this paper, namely ESBP-R and ESBP-D, can terminate the decoding algorithm earlier than other methods, leading to a lower average number of iterations. Figure 8: BLER performance for five polar codes with N = 1024 and K = 512. \(E_{b}/N_{0}\) (energy-per-bit-to noise power spectral density ratio) measures the normalized signal quality. Figure 9: The average number of iterations corresponding to the simulation results in Figure 8.

## 6 Conclusions

This paper addresses the issue of high energy consumption in the Belief Propagation (BP) decoding algorithm for polar codes in Low-Earth Orbit (LEO) satellite communication systems. A geometric framework for polar code BP decoding is established, and two early convergence algorithms for polar code BP decoding based on geometric distance are proposed. Simulation results demonstrate that both algorithms improve the convergence speed of polar code BP decoding without compromising the bit error rate. Our research provides a new theoretical interpretation and method for the BP decoding of polar codes, offering valuable insights for further enhancing the decoding performance of polar codes. It also presents fresh perspectives for achieving sustainable 6G network architectures and communications in all resource-constrained node scenarios. Previous scholars have attempted to utilize geometry to study channel coding problems in Turbo codes and LDPC codes [14; 18; 19; 20; 21]. The geometric framework for polar code BP decoding proposed in this paper can be seen as a continuation of these studies. Through research on early convergence algorithms for polar code BP decoding, this paper validates the effectiveness of the geometric framework. Future work can leverage this geometric framework to improve optimization algorithms for polar code BP decoding and optimize the error correction performance of polar code BP decoding. This research was conducted by all the authors: [PERSON] and [PERSON] conceived the idea, performed the analysis, and designed the scheme; [PERSON] and [PERSON] conducted the numerical simulations; [PERSON], [PERSON] and [PERSON] co-wrote the manuscript. All authors have read and agreed to the published version of this manuscript. This research received no external funding. The data used to support the findings of the current study are available from the authors upon request. The authors declare no conflicts of interest. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\mathbf{E_{b}/N_{0}}\) & **minLLR** & **LMS** & **PLA** & **ESBP-R** & **ESBP-D** \\ \hline
1 & 2.01\% & 3.81\% & 4.01\% & 6.01\% & 7.39\% \\
1.5 & 15.42\% & 19.03\% & 19.78\% & 27.15\% & 25.27\% \\
2 & 42.14\% & 39.59\% & 39.81\% & 43.97\% & 46.38\% \\
2.5 & 55.14\% & 52.87\% & 53.51\% & 56.01\% & 61.17\% \\
3 & 60.74\% & 60.43\% & 60.63\% & 61.40\% & 63.59\% \\
3.5 & 64.55\% & 65.28\% & 65.31\% & 65.71\% & 67.12\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The ratio of the number of iterations reduced by several early convergence algorithms under different channel conditions.