In this paper, we define partial orders (POs) of polar codes based on the Bhattacharyya parameter and the bit-error probability, respectively. These POs are applicable to arbitrary binary memoryless symmetric channel (BMSC). Leveraging the extremal inequalities of polarization transformation, we derive new POs for BMSC based on the corresponding POs observed in the Binary Erasure Channel (BEC). %Additionally, we discover more special POs in the Binary Symmetric Channel (BSC). We provide examples that demonstrate the inability of existing POs to deduce these novel POs. Furthermore, we establish upper bounds for the expansion parameter $\beta$ if the polar codes constructed by $\beta$-expansion method obey these POs.

翻译：本文分别基于Bhattacharyya参数和比特错误概率定义了极化码的偏序关系。这些偏序关系适用于任意二元无记忆对称信道。利用极化变换的极值不等式，我们基于在二元删除信道中观测到的对应偏序关系，推导出面向BMSC的新偏序关系。我们提供的实例表明，现有偏序关系无法推导出这些新偏序关系。此外，若采用β展开法构建的极化码满足这些偏序关系，我们还建立了展开参数β的上界。