# Polar Code BP Decoding Optimization for Green 6G Satellite Communication: A Geometry Perspective
## 2 BP Decoding Algorithm for Polar Codes
The process of polar code BP decoding is represented as Algorithm 1:  
```
0: Received signal vector \(y\) (length \(N\)), maximum iterations \(I_{\max}\), frozen bit set \(\mathcal{A}\)
0: Decoded result
1:for\(i=1\) to \(N\)do
2: Initialize \(L_{i,n+1}=y_{i}\)
3:for all \(j\)do
4: Set \(L_{i,j}=0\) and \(R_{i,j}=0\)
5:endfor
6:if\(i\in\mathcal{A}\)then
7: Set \(R_{i,1}=+\infty\) {Indicate frozen bit position}
8:endif
9:endfor
10:for\(\text{iter}=1\) to \(I_{\max}\)do
11:for\(j=1\) to \(N\)do
12: Compute right-to-left messages based on Equation (1)
13:endfor
14:for\(j=1\) to \(N\)do
15: Compute left-to-right messages based on Equation (1)
16:endfor
17:if convergence criterion met (e.g., \(L_{i,1}^{\text{iter}}=L_{i,1}^{\text{iter}+1}\) for all \(i\)) then
18: Break
19:endif
20:endfor
21: Decision: Output decoded result based on final \(L_{i,1}\) values

Figure 3: A schematic diagram of the BP decoding process for polar codes *. ## 3 Information Geometry of BP Polar Decoding

### Geometric Representation of Soft Information

The probability distribution of each codeword is a Bernoulli distribution, which we usually use as follows:

\[ \begin{pmatrix}0&1\\ p&1-p\end{pmatrix} . \tag{3}\]

The exponential deformation of Bernoulli distribution is defined as follows:

\[p(x)=p^{x}(1-p)^{1-x}=\exp[x\ln x+(1-x)\ln(1-p)]. \tag{4}\]

Let \(\lambda=LLR=\ln\frac{p}{1-p}\), and then \(p=\frac{e^{\lambda}}{1+e^{\lambda}}\); it follows that

\[ \begin{array}{l}p(x;\lambda)=\exp[x\ln\left(\frac{e^{\lambda}}{1+e^{\lambda }}\right)+(1-x)\ln\left(1-\frac{e^{\lambda}}{1+e^{\lambda}}\right)]\\ =\exp[\lambda x-\ln(1+e^{\lambda})].\end{array}  \tag{5}\]

The statistical manifold on which the \(LLR\) (Log-Likelihood Ratio) lies is

\[M=\{p(x;\lambda)=\exp[\lambda x-\ln(1+e^{\lambda})]|x\in\{0,1\},\lambda\in R\}. \tag{6}\]

This is a one-dimensional manifold. For a polar code with code length \((N,K)\), the output contains soft information for \(K\) information bits. As mentioned earlier, the soft information for one information bit can be represented as a point on a statistical manifold consisting of one-dimensional Bernoulli distributions. Therefore, the soft information for \(K\) information bits can be represented as a point on a \(K\)-dimensional statistical manifold, which is composed of the joint probability distribution of \(K\) one-dimensional Bernoulli distributions. The joint probability distribution can be expressed as

\[p(x;\lambda)=\prod_{i}\exp[\lambda_{i}x-\ln(1+e^{\lambda_{i}})]=\exp[\sum_{i} \lambda_{i}x-\sum_{i}\ln(1+e^{\lambda_{i}})]\enspace. \tag{7}\]

In this formula, \(\lambda=(\lambda_{1},\lambda_{2},\cdots,\lambda_{K}),i\in[1,K]\). This \(K\)-dimensional statistical manifold can be represented as

\[M=\{p(x;\lambda)=\exp[\sum_{i}\lambda_{i}x-\sum_{i}\ln(1+e^{\lambda_{i}})]\}. \tag{8}\]

Soft information is mapped to a point on a statistical manifold, whose value can be represented by the coordinates on the manifold, while the iterative process of soft information can be regarded as the movement of points on the statistical manifold. This geometric treatment brings the following three advantages:

* Intuitiveness: The geometric representation dynamically demonstrates the process of polar code BP decoding, providing a clear perspective for a deep understanding of the nature of decoding. * Flexibility: The construction of the manifold can be flexibly adjusted according to research objectives. For example, when studying a particular information node, a two-dimensional manifold can be constructed through the Cartesian product, with left information and right information corresponding to different manifold coordinates, respectively. When studying the left information of N nodes, an N-dimensional manifold can be constructed, with each coordinate component representing the left information of one node.