Generalized partial orders for polar code bit-channels

We study partial orders (POs) for the synthesized bit-channels of polar codes. First, we consider complementary bit-channel pairs whose Bhattacharyya parameters over the binary erasure channel (BEC) exhibit a symmetry property and provide insight into the alignment of polarized sets of bit-channels for the BEC and general binary-input memo-ryless symmetric (BMS) channels. Next, we give an alternative proof of a recently presented PO for bit-channels and use the underlying idea to extend the bit-channel ordering to some additional cases. In particular, the bit-channel ordering for a given code block-length is used to generate additional bit-channel ordering relationships for larger block-lengths, generalizing previously known POs. Examples are provided to illustrate the new POs.