# Polar Code BP Decoding Optimization for Green 6G Satellite Communication: A Geometry Perspective
## 2 BP Decoding Algorithm for Polar Codes
(**b**) A schematic diagram of the information iteration process on typical node A from a geometric perspective. ### Geometric Representation of Factor Graph Information

From the perspective of the factor graph, for a polar code with code length \((N,K)\), at time \(t\), the information propagated leftwards contained in all nodes of the factor graph can be represented as an \(N(n+1)\) vector \(\xi_{t}\) as follows:

\[\xi^{t}=(\lambda_{L}^{t})_{N\times(n+1)}. \tag{13}\]

All possible probability distributions of the information transmitted to the left can form an \(N(n+1)\)-dimensional manifold \(M_{L}\), where \(\xi^{t}\) can be seen as a point on \(M_{L}\). Among them, \(\eta_{L}\) represents the soft information transmitted to the left of the leftmost bit node in the factor graph, and the manifold where \(\eta_{L}\) is located is a \(K\)-dimensional submanifold of \(M_{L}\), represented by Equation (8). \(\eta_{L}\) is the projection of \(\xi\) on this submanifold. Similarly, at time \(t\), the information propagated rightwards contained in all nodes of the factor graph can be represented as an \(N(n+1)\) vector \(\zeta_{t}\) as follows:

\[\zeta^{t}=(\lambda_{R}^{t})_{N\times(n+1)}. \tag{14}\]

All possible probability distributions of the information transmitted to the right can form an \(N(n+1)\)-dimensional manifold \(M_{R}\), where \(\zeta^{t}\) can be seen as a point on \(M_{R}\). Among them, \(\theta_{R}\) represents the soft information transmitted to the left of the leftmost bit node in the factor graph, and the manifold where \(\theta_{R}\) is located is a \(N\)-dimensional submanifold of \(M_{R}\). \(\eta_{L}\) is the projection of \(\zeta\) on this submanifold. The parameter space of the codeword after hard judgment is \(K\)-dimensional Hamming space. The iterative process from a geometric perspective is shown in Figure 7:

Among them

\[\xi^{t+1}=\Phi_{L}(\xi^{t},\zeta^{t})\] Figure 4:0

Figure 4:1 Polarization code BP decoding process from a geometric perspective. The process of hard judgment is

\[ \begin{array}{l}\hat{a}^{t}=\text{sgn}\eta_{L}^{t}\\ \hat{a}_{x}^{t}=\hat{x}^{t}G^{-1}\\ \hat{x}^{t}=\text{sgn}\theta_{R}^{t}\end{array} . \tag{16}\]

The iterative process of polar code BP decoding can be seen as the movement of \(\xi\) and \(\zeta\) on their respective manifolds. When decoding is complete, there is \(\xi^{t}=\xi^{t+1}\), \(\xi^{t}=\zeta^{t+1},\theta_{R}^{t}=\theta_{R}^{t+1},\eta_{L}^{t}=\eta_{L}^{t+1 },\hat{a}^{t}=\hat{a}^{t+1},\hat{a}^{t}=\hat{a}_{x}^{t}\). The process of polar code BP decoding from a geometric perspective is represented as Algorithm 2:
```
0:\(y\): received signal vector (length \(N\)), maximum iteration number \(I_{\text{max}}\), frozen bits
0: decoded result \(\hat{u}\)
1:for\(i=1\) to \(N\)do
2: Initialize the coordinates of each node based on the received signal vector \(y\) and frozen bit positions
3:endfor
4:for\(iter=1\) to \(I_{\text{max}}\)do
5: Left-to-right iteration: Update the coordinates of \(\zeta\) based on Equation (15)
6: Right-to-left iteration: Update the coordinates of \(\zeta\) based on Equation (15)
7:if\(\zeta^{t}=\zeta^{t+1}\) and \(\zeta^{t}=\zeta^{t+1}\)then
8: break
9:endif
10:endfor
11: Output: decoded result \(\hat{u}\)
```
**Algorithm 2** Polar Code BP Decoding

## 4 Early Stopping Based on Information Geometry

### Algorithm Flow

In the process of polar code Belief Propagation (BP) decoding, the high number of iterations and significant computational load are crucial factors limiting decoding efficiency. Based on the geometric model for polar code BP decoding established in the previous section, this paper proposes an early convergence algorithm utilizing the geometric model. The objective is to prematurely terminate the decoding process by calculating the distance of soft information in differential geometric space, thereby enhancing decoding efficiency. As mentioned in the previous section, when decoding concludes, we have \(\eta_{L}^{t}=\eta_{L}^{t+1}\).