# Polar Code BP Decoding Optimization for Green 6G Satellite Communication: A Geometry Perspective
## 2 BP Decoding Algorithm for Polar Codes
Firstly, we calculate the first-order derivative:

\[\frac{\partial}{\partial\lambda}\ln p(x;\lambda)=\frac{\partial}{\partial \lambda}[\lambda x-\ln(1+e^{\lambda})]=x-\frac{e^{\lambda}}{(1+e^{\lambda})}. \tag{18}\]

The second-order derivative is

\[\frac{\partial^{2}}{\partial\lambda^{2}}\ln p(x;\lambda)=\frac{\partial^{2}}{ \partial\lambda^{2}}[\lambda x-\ln(1+e^{\lambda})]=-\frac{e^{\lambda}}{(1+e^ {\lambda})^{2}}. \tag{19}\]

The Fisher information matrix of a statistical manifold composed of Bernoulli distributions is

\[I(\lambda)=-E[\frac{e^{\lambda}}{\left(1+e^{\lambda}\right)^{2}}]=\frac{e^{ \lambda}}{\left(1+e^{\lambda}\right)^{2}}. \tag{20}\]

In this article, the inverse matrix of the Fisher information matrix is used as a metric for statistical manifolds, and the metric for statistical manifolds composed of Bernoulli distributions is defined as

\[G(\lambda)=I(\lambda)^{-1}=\frac{\left(1+e^{\lambda}\right)^{2}}{e^{\lambda}}. \tag{21}\]In a one-dimensional Bernoulli distribution manifold, the Riemann distance can be approximated as follows:

\[D(\lambda_{1},\lambda_{2})=\int_{\lambda_{1}}^{\lambda_{2}}\sqrt{G(\lambda)}d \lambda=\int_{\lambda_{1}}^{\lambda_{2}}\sqrt{\frac{\left(1+e^{\lambda}\right)^ {2}}{e^{\lambda}}}d\lambda. \tag{22}\]

First, calculate the indefinite integral as

\[\int\sqrt{\frac{\left(1+e^{\lambda}\right)^{2}}{e^{\lambda}}}d\lambda=-2e^{- \frac{1}{2}\lambda}+2e^{\frac{1}{2}\lambda}\enspace. \tag{23}\]

So, the Riemann distance of a one-dimensional Bernoulli distribution manifold can be expressed as

\[D(\lambda_{1},\lambda_{2})=2\bigg{|}e^{\frac{\lambda_{2}}{2}}-e^{-\frac{ \lambda_{2}}{2}}-e^{\frac{\lambda_{1}}{2}}+e^{-\frac{\lambda_{1}}{2}}\bigg{|}. \tag{24}\]

#### 4.2.2 Divergence

In applications, divergence is also an important concept for describing the difference between two points on a manifold. The most widely used in statistical manifolds is Kullback-Leibler divergence (KL divergence). Using Riemann distance or divergence to calculate the difference between two probability distributions on a statistical manifold is a common method in information geometry. The divergence formula on the manifold mentioned above is (see Appendix A for details) as follows:

\[ \begin{array}{l}D_{KL}(\lambda_{1}\|\lambda_{2})=\sum\limits_{x}p(x; \lambda_{1})\log\frac{p(x;\lambda_{1})}{p(x;\lambda_{2})}\\ =\sum\limits_{x}\exp[\lambda_{1}x-\ln(1+e^{\lambda_{1}})]\log\frac{\exp[ \lambda_{1}x-\ln(1+e^{\lambda_{1}})]}{\exp[\lambda_{2}x-\ln(1+e^{\lambda_{2}}) ]}\\ =(\lambda_{1}-\lambda_{2})\frac{e^{\lambda_{1}}}{1+e^{\lambda_{1}}}-\ln(\frac {1+e^{\lambda_{1}}}{1+e^{\lambda_{2}}})\end{array} \enspace. \tag{25}\]

The above two distance formulas are incorporated into the decoding algorithm framework, resulting in two polar code BP decoding algorithms based on information geometry: the BP polar decoding algorithm with early stopping based on Riemann distance (ESBP-R) and the BP polar decoding algorithm with early stopping based on divergence (ESBP-D). ## 5 Simulation Results and Comparisons

In this section, we compare the performances of the proposed early convergence algorithms, namely ESBP-R and ESBP-D, with several early convergence algorithms designed using LLR (Log-Likelihood Ratio) through simulations. These include the min Log-Likelihood Ratio (minLLR) algorithm proposed in reference [4], an LLR magnitude-aided (LMA) early convergence algorithm based on the convergence properties of LLR messages proposed in reference [7], and a partial LLR-aided (PLA) early convergence algorithm proposed in reference [12]. The performance metrics include the block error rate (BER) and the average number of iterations. The polar codes studied in the simulations have a code length of \(N=1024\), an information bit length of \(K=512\), a code rate of \(r=0.5\), and a maximum number of iterations of \(30\). The channel is an Additive White Gaussian Noise (AWGN) channel, the polar code construction follows the Bhattacharyya parameter method as described in reference [1], and the modulation scheme is BPSK.