Generalized Partial Orders for Polar Code Bit-Channels

• Published in: IEEE transactions on information theory Vol. 65; no. 11; pp. 7114 - 7130
• Main Authors: Wu, Wei, Siegel, Paul H.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.11.2019; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Bhattacharyya parameter; Channels; Decoding; Degradation; erasure channel; Error probability; function composition; Hamming weight; Indexes; induction; Parameter identification; partial order; Polar codes; Reliability; Symmetry
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2019.2930292

We study partial orders (POs) for the synthesized bit-channels of polar codes. First, we give an alternative proof of an existing PO for bit-channels with the same Hamming weight 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. Next, we consider POs especially for the binary erasure channel (BEC). We identify a symmetry property of the Bhattacharyya parameters of complementary bit-channel pairs on the BEC and provide a condition for the alignment of polarized sets of bit-channels for the BEC and general binary-input memoryless symmetric (BMS) channels. Numerical examples and further properties about the POs for the bit-channels with different Hamming weights are provided to illustrate the new POs. The bit-channels with universal ordering positions, which are independent of the channel erasure probability, are verified for all of the code block lengths. Finally, we show the threshold behavior of the Bhattacharyya parameters of some bit-channels by approximating the threshold values. The corresponding value for a bit-channel can be used to determine whether it is good or bad when the underlying channel is known.