# Polar Code BP Decoding Optimization for Green 6G Satellite Communication: A Geometry Perspective
## 2 BP Decoding Algorithm for Polar Codes
* Practicality: Analyzing and optimizing characteristics such as metrics, curvature, and divergence on the manifold with geometric tools can effectively improve the convergence performance of decoding algorithms, providing powerful tool support for subsequent performance optimization. ### Geometric Representation of Node Information

From a microscopic perspective, at time \(t\), the left information of any node can be represented by the coordinate \(\lambda_{L}^{t}\), and the manifold it resides on can be denoted as \(M_{1}\); similarly, the right information can be represented by the coordinate \(\lambda_{R}^{t}\), and the manifold it resides on can be denoted as \(M_{2}\). Therefore, at time t, the coordinates of the node's information on the manifold can be expressed as \(\lambda^{t}(\lambda_{L}^{t},\lambda_{R}^{t})\). By taking the Cartesian product of the manifold where the left information coordinate \(\lambda_{L}^{t}\) resides and the manifold where the right information coordinate \(\lambda_{R}^{t}\) resides, we obtain

\[M=M_{1}\times M_{2}, \tag{9}\]

which is a two-dimensional manifold with coordinates representing left information and right information, respectively. For any node, its iterative information at any time can be represented by coordinates on \(M\). In this way, the iterative process of the node's information can be represented as the trajectory of a point moving on the manifold. As shown in Figure 4, if four information nodes are represented as \((A,B,C,D)\), then Equation (1) can be expressed as

\[ \begin{array}{l}\lambda_{AL}^{t+1}=f(\lambda_{BL}^{t},\lambda_{DR}^{t} \lambda_{CL}^{t})\\ \lambda_{DL}^{t+1}=\lambda_{CL}^{t}(\lambda_{BL}^{t},\lambda_{AR}^{t})\\ \lambda_{BR}^{t+1}=f(\lambda_{AR}^{t},\lambda_{CL}^{t}\lambda_{DR}^{t})\\ \lambda_{CR}^{t+1}=\lambda_{DR}^{t}f(\lambda_{AR}^{t},\lambda_{BL}^{t})\\ \end{array}  \tag{10}\]

We correlate the four information nodes \((A,B,C,D)\) with the four statistical manifolds \((M_{A},M_{B},M_{C},M_{D})\). The probabilistic information embodied within these information nodes is equivalent to a point on the statistical manifold where the probability density function resides, thereby yielding four information points: \(\lambda_{A}^{t}\), \(\lambda_{B}^{t}\), \(\lambda_{C}^{t}\), \(\lambda_{D}^{t}\). The process of information transmission on the butterfly decoding unit can be represented by Figure 5.

For any intermediate node \(A\), as shown in Figure 6a, its left information is determined by the information from the other three nodes \((B,C,D)\) in the butterfly decoder at the previous time step, and similarly, its right information is determined by the information from the other three nodes \((G,F,E)\) in another butterfly decoder at the previous time step. This information iteration is independent of the information of node \(A\) itself at the previous time step, and such information iteration can be termed as extrinsic information. Figure 4: Another representation of the decoding unit. From the perspective of information geometry, seven information nodes \((A,B,C,D,\)\(E,F,G)\) correspond to seven statistical manifolds \((M_{A},M_{B},M_{C},M_{D},M_{E},M_{F},M_{G})\). The iterative process can be expressed as

\[\lambda_{A}^{t+1}=g(\lambda_{BL}^{t},\lambda_{DR}^{t},\lambda_{CL}^{t},\lambda_ {EL}^{t},\lambda_{FR}^{t},\lambda_{GR}^{t}), \tag{11}\]

or as

\[\lambda_{A}^{t+1}=h(\lambda_{B}^{t},\lambda_{D}^{t},\lambda_{C}^{t},\lambda_{ E}^{t},\lambda_{F}^{t},\lambda_{G}^{t}). \tag{12}\]

The information iteration process on node A is shown in Figure 6b, where the green arrows represent right information and the orange arrows represent left information. Figure 5: A schematic diagram of the iterative information process of a unit decoder from a geometric perspective. (**a**) A description of the schematic diagram of the left information iterative process. (**b**) A description of the schematic diagram of the right information iterative process. Figure 6: (**a**) The information carried by typical node A is related to six adjacent information nodes.