## 3 Research on BPL Decoding Algorithm for Polar Codes Based on Information Geometry
### Path Selection Criterion Based on Soft Information Centroid
#### 3.2.3 Bit Error Rate Calculation
Since \(\sum\limits_{j=1}^{L}w_{j}\lambda_{j}\) follows a normal distribution with parameters \((\sum\limits_{j=1}^{L}w_{j}\mu_{j},\sum\limits_{j=1}^{L}{w_{j}}^{2}{\sigma_{j} }^{2})\), the bit error rate of BPL decoding using the path selection criterion proposed in this paper is:  
\[p(\frac{\sum\limits_{j=1}^{L}w_{j}\lambda_{j}}{\sum\limits_{j=1}^{L}w_{j}}\leq 0 )=p(\sum\limits_{j=1}^{L}w_{j}\lambda_{j}\leq 0)=\Phi(\frac{\sum\limits_{j=1}^{L} w_{j}\mu_{j}}{\sqrt{\sum\limits_{j=1}^{L}w_{j}\sigma_{j}^{2}}})=\Phi(-\frac{ \sum\limits_{j=1}^{L}w_{j}\mu_{j}}{\sqrt{\sum\limits_{j=1}^{L}w_{j}\sigma_{j} ^{2}}}). \tag{21}\]  
When the BPL bit error rate is lower than the bit error rate of each BP decoder, we have:  
\[p(\frac{\sum\limits_{j=1}^{L}w_{j}\lambda_{j}}{\sum\limits_{j=1}^{L}w_{j}}\leq 0 )<p(\lambda_{1}\leq 0), \tag{22}\]  
i.e.,  
\[\Phi(-\frac{\sum\limits_{j=1}^{L}w_{j}\mu_{j}}{\sqrt{\sum\limits_{j=1}^{L}w_{ j}\sigma_{j}^{2}}})<\Phi(-\frac{\mu_{L}}{\sigma_{L}}). \tag{23}\]  
Since the cumulative function of the normal distribution function is a monotonically increasing function, (3)-(17) hold if and only if:  
\[\frac{\sum\limits_{j=1}^{L}w_{j}\mu_{j}}{\sqrt{\sum\limits_{j=1}^{L}w_{j}^{2} \sigma_{j}^{2}}}>\frac{\mu_{1}}{\sigma_{1}}. \tag{24}\]  
Since both the mean and variance are positive numbers, after deformation, we obtain:  
\[\sum\limits_{j=1}^{L}w_{j}\mu_{j}\sigma_{1}>\sqrt{\sum\limits_{j=1}^{L}w_{j} ^{2}\sigma_{j}^{2}}\mu_{1}. \tag{25}\]  
Expanding the left side of (25), we obtain:  
\[\sum\limits_{j=1}^{L}w_{j}\mu_{j}\sigma_{1}>Lw_{L}\mu_{L}\sigma_{1}. \tag{26}\]Expanding the right side of (25), we obtain:  
\[p(\frac{\lambda_{1}+\lambda_{2}+\cdots+\lambda_{L}}{L}\leq 0)=p(\lambda_{1}+ \lambda_{2}+\cdots+\lambda_{L}\leq 0)=\Phi(\frac{0-L\mu}{\sqrt{L}\sigma})=\Phi(- \frac{\sqrt{L}\mu}{\sigma}). \tag{19}\]0  
When (28) is satisfied, (22) holds:  
\[p(\frac{\lambda_{1}+\lambda_{2}+\cdots+\lambda_{L}}{L}\leq 0)=p(\lambda_{1}+ \lambda_{2}+\cdots+\lambda_{L}\leq 0)=\Phi(\frac{0-L\mu}{\sqrt{L}\sigma})=\Phi(- \frac{\sqrt{L}\mu}{\sigma}). \tag{19}\]1  
After deformation, we obtain:  
\[p(\frac{\lambda_{1}+\lambda_{2}+\cdots+\lambda_{L}}{L}\leq 0)=p(\lambda_{1}+ \lambda_{2}+\cdots+\lambda_{L}\leq 0)=\Phi(\frac{0-L\mu}{\sqrt{L}\sigma})=\Phi(- \frac{\sqrt{L}\mu}{\sigma}). \tag{19}\]2  
Since \(\frac{w_{1}}{w_{1}}<1\), when \(\frac{\Phi^{-1}(Pe_{1})}{\Phi^{-1}(Pe_{L})}<\sqrt{L}\), there always exists a set of weights \(w_{j}\) that satisfies Equation (22). This subsection proves that when the performance gap between the \(L\) factor graphs is not significant (\(\frac{\Phi^{-1}(Pe_{1})}{\Phi^{-1}(Pe_{1})}<\sqrt{L}\)), the bit error rate of BPL decoding using the path selection criterion proposed in this paper is less than the original BP decoding bit error rate. Since the traditional path selection criterion selects one decoding result from the \(L\) factor graph decoding results for output, the bit error rate of BPL decoding using the path selection criterion proposed in this paper is less than that based on the traditional path selection criterion. In Appendix A, the bit error rate in non-Gaussian channels was derived and the same conclusion was reached.