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（POs）。这些偏序关系适用于任意二元无记忆对称信道（BMSC）。利用极化变换的极值不等式，我们基于二元删除信道（BEC）中观测到的相应偏序关系，推导出BMSC下的新偏序关系。此外，我们提供了实例证明现有偏序关系无法推导出这些新型偏序关系。进一步地，我们建立了当采用β-展开法构造的极化码满足这些偏序关系时，展开参数β的上界。