# Polar Code BP Decoding Optimization for Green 6G Satellite Communication: A Geometry Perspective
## 1 Introduction  
Low-Earth Orbit (LEO) satellite communication systems are emerging as a vital component of future 6G networks, offering the potential for high-speed communication across the globe thanks to their low latency, high coverage, and robustness. However, with the expansion of LEO satellite communication systems and the development of mega-constellations, energy consumption has become a core challenge limiting their sustainable development. Satellites rely on limited solar power, and efficient energy utilization in communication links not only determines the lifespan and operational efficiency of satellites but also profoundly impacts the green development and long-term reliability of the entire constellation. Therefore, reducing energy consumption while ensuring communication performance has become a critical issue driving advancements in LEO satellite communication technology. This poses higher requirements for technologies such as signal processing, communication protocols, and channel coding within the 6G network architecture, while also presenting new research opportunities for achieving a sustainable 6G network architecture. Among them, channel coding technology serves as the core of wireless communication systems, directly impacting the reliability and energy consumption performance of communication systems. In recent years, polar codes, as the first channel coding scheme theoretically proven to achieve [PERSON]'s limit [PERSON] (1958), have not only been incorporated into the 5G standard but are also regarded as one of the important candidate coding technologies for 6G networks. Leveraging their soft-input soft-output (SISO) characteristics and high parallel decoding capability, the Belief Propagation (BP) decoding algorithm for polar codes excels in high-throughput communication scenarios [PERSON] (1958); [PERSON] (1958); [PERSON] (1958). However, a major drawback of BP decoding is its high number of iterations and slow convergence speed, requiring numerous iterations to complete decoding in low signal-to-noise ratio (SNR) communication environments. This not only consumes considerable decoding time but also occupies excessive hardware resources and incurs high energy consumption costs, contradicting the sustainability requirements of 6G networks. Therefore, developing efficient early termination algorithms to reduce BP decoding energy consumption and enhance energy efficiency and convergence speed has become one of the key focuses in polar code research. To address the issues of high iteration counts and slow convergence speed in BP decoding algorithms, various early termination algorithms have been proposed. In summary, current early termination algorithms can be categorized into the following three approaches:  
1. The first approach involves designing a verification condition to judge the result of each iteration. If the condition is met, the iteration stops, and the current decoding result is output. For instance, the early termination criterion based on the G-matrix proposed in [PERSON] [PERSON] (1958) determines whether to terminate early by comparing the estimated messages and codewords at the end of each BP decoding iteration. [PERSON]. [PERSON] (1958) improves upon the method in [PERSON] (1958) by proposing the Bit Difference Ratio (BDR) criterion, which focuses on the number of differing bits between the estimated codeword and the codeword calculated based on the estimated messages. [PERSON] [PERSON] (1958) introduces an early termination criterion based on the H-matrix, which has lower complexity compared to the G-matrix criterion in [PERSON] (1958). [PERSON] [PERSON] (1958) incorporates Cyclic Redundancy Check (CRC) to more accurately perceive convergence properties. This approach draws inspiration from early termination algorithms for LDPC codes and belongs to the general idea of linear block code decoding. Its advantage lies in its simplicity and directness, while the disadvantage is that it does not incorporate the characteristics of polar codes. 2.