# Polar Code BP Decoding Optimization for Green 6G Satellite Communication: A Geometry Perspective
## 2 BP Decoding Algorithm for Polar Codes  
In the BP decoding algorithm, for a polar code with a code length of \(N=2^{n}\), the factor graph of the polar code contains \(n\) stages and \(N\) levels, totaling \(N(n+1)\) nodes, represented by \((i,j)\) for the \(i\)th node at the \(j\)th stage. Each stage contains \(N/2\) butterfly decoding units, and information propagates through the computing units, as shown in Figure 1. \(L_{ij}\) and \(R_{ij}\) denote the soft information for left-going and right-going messages, respectively, \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Meaning** \\ \hline System Parameters & \\ \(N\) & Code length \\ \(K\) & Information bits length \\ \(E_{b}/N_{0}\) & Energy-per-bit-to-noise power spectral density ratio \\ \(I_{\max}\) & Maximum number of iterations \\ \hline Algorithmic Variables & \\ \(L_{i,j}\),\(R_{ij}\) & Leftward LLR message at node \((i,j)\) \\ \(R_{ij}\) & Rightward LLR message at node \((i,j)\) \\ \(\xi^{t}\) & Leftward propagation vector at iteration \(t\) \\ \(\xi^{t}\) & Rightward propagation vector at iteration \(t\) \\ \hline Geometric Framework & \\ \(\lambda\) & Natural parameter of Bernoulli distribution \\ \(G(\lambda)\) & Metric matrix \\ \(\eta_{L}\) & Soft information at leftmost bit nodes \\ \(\theta_{R}\) & Soft information at rightmost bit nodes \\ \(D(\cdot)\) & Riemannian distance or divergence measure \\ \(I(\lambda)\) & Fisher information matrix \\ \hline \hline \end{tabular} Note: LLR = Log-Likelihood Ratio.  
\end{table} 
Table 1: Summary of key symbols. at node \((i,j)\) on the factor graph, represented by Log-Likelihood Ratios (LLRs). According to reference [3], the information update process for a computing unit can be expressed as follows:  
\[\left\{ \begin{aligned} L_{i,j}^{t+1}&=f(L_{i+1,j}^{t},L_ {i+1,j+2^{n-1}}^{t}R_{i,j+2^{n-1}}^{t})\\ L_{i,j+2^{n-1}}^{t+1}&=L_{i+1,j+2^{n-1}}^{t}f(L_{i+1,j}^{t },R_{ij}^{t})\\ R_{i+1,j}^{t+1}&=f(R_{i,j^{\prime}}^{t},L_{i+1,j+2^{n-1 }}^{t}R_{i+1,j+2^{n-1}}^{t})\\ R_{i+1,j+2^{n-1}}^{t+1}&=R_{i,j+2^{n-1}}^{t}f(R_{i,j}^{t },L_{i+1,j}^{t})\end{aligned} \right. \tag{1}\]  
where  
\[f(\alpha,\beta)=\ln(\frac{e^{\alpha+\beta}+1}{e^{\alpha}+e^{\beta}})\approx sign (\alpha,\beta)\min(|\alpha|,|\beta|) \tag{2}\]  
Equation (2) follows the min-sum approximation commonly used in BP decoding to reduce computational complexity [4]. Figure 2 shows an example of a polar code factor graph with N = 8. From Equation (1), it can be seen that the iterative information of each intermediate node is determined by the iterative information of the remaining three nodes of the left and right butterfly decoders, where the node was located the previous time. As shown in Figure 3, \(u\) represents the original codeword, \(G\) represents the generator matrix, \(x\) represents the codeword after encoding and transmission through the channel, and \(y\) represents the codeword sequence received at the receiver. \(\eta_{L}\) denotes the soft information of the bit nodes at the leftmost side of the factor graph, which, after hard  
Figure 1: Minimal computational unit for BP decoding [3]. Figure 2: An example of a factor graph for a polar code with N = 8 [3]. decision, becomes the decoded result. \(\theta_{R}\) represents the soft information of the bits at the rightmost side of the factor graph, which, after hard decision, becomes \(\hat{x}\) (the predicted value of \(x\)). * Note:
* When [PERSON] introduced the BP decoding for polar codes, he did not specify a concrete stopping criterion. In practical applications, a maximum number of iterations is often set or one of the three approaches mentioned in the introduction is adopted to prematurely terminate the iterations. * In the literature and practical applications, attention is often solely focused on \(\hat{u}\), while \(\hat{x}\) is overlooked. Ref. [4] points out that when the iterations are stopped, \(\hat{x}=\hat{u}G\).