# Polar Code BP Decoding Optimization for Green 6G Satellite Communication: A Geometry Perspective
## 1 Introduction
The second approach analyzes each information bit or frozen bit based on the characteristics of polarized channels and designs early termination algorithms according to their reliability. Ref. [PERSON] (1958) points out that observing a group of information bits, known as the Worst of Information Bits (WIBs), can detect successful decoding. WIBs are bits transmitted through channels with higher error probabilities and thus require more decoding iterations for successful decoding compared to other bits. Additionally, Ref. [PERSON] (1958) proposes an early termination criterion based on the bit error rate (BER) for frozen bits of polar codes. In Ref. [PERSON] (1958), researchers suggest using optimal frozen bits to terminate decoding, as these partially frozen bits have lower average error probabilities. The advantage of this approach is that it leverages channel polarization theory and combines the characteristics of polar codes, while the disadvantage is that it does not incorporate the transmission of soft information for information nodes and does not utilize the advantages of BP decoding. 3. The third approach leverages the SISO characteristics of polar code BP decoding to design early termination algorithms based on the Log-Likelihood Ratio (LLR). Since the BP decoding algorithm outputs soft information, the LLR of each information node represents the reliability of the information. Designing early termination algorithms by calculating the LLR exploits the advantages of polar code BP decoding algorithms. Ref. [4] proposes the min Log-Likelihood Ratio (minLLR) criterion, which was later simplified in subsequent research [11] with the decision condition that all LLR values are greater than a set threshold indicating successful decoding. Furthermore, in Ref. [7], an LLR magnitude-aided (LMA) early termination scheme based on the convergence properties of LLR messages is proposed, which terminates iterations early when the LLR of an information node remains unchanged for two consecutive times. Ref. [12] introduces a partial LLR-aided (PLA) early termination scheme, which terminates iterations early when the difference in LLR values of an information node over two consecutive times is less than a threshold. This approach fully exploits the characteristics of soft information transmission in polar code BP decoding, but existing algorithms directly use Euclidean distance to calculate LLR differences, lacking a mathematical analysis of LLR. Moreover, some researchers have combined two of the above three approaches, as seen in [4,7,13]. Existing methods are typically designed based on empirical rules and lack support from a systematic theoretical framework, with the potential in the direction of early termination yet to be fully explored. In recent years, information geometry theory has gradually emerged as a novel research tool in the field of channel coding and wireless communication [14,15]. This theory transforms complex communication problems into geometric optimization problems by mapping probability distribution functions to points on geometric manifolds, utilizing geometric tools such as metrics, curvature, and divergence for performance optimization [16]. This characteristic of information geometry provides a new perspective for optimizing the iterative process of polar code BP decoding, aiding in significantly reducing the number of iterations without sacrificing the bit error rate (BER), thereby achieving the goal of reducing energy consumption. To better leverage the SISO characteristics of polar code BP decoding, this paper conducts a mathematical analysis of the changes in LLR values during the iterative process of polar code BP decoding and proposes a geometric framework for polar code BP decoding based on information geometry theory. This framework not only aids in intuitive understanding of iterative decoding but also facilitates the use of various geometric tools such as metrics, Riemannian distance, and divergence to address the issue of high iteration counts.