Journal of Guangxi Normal University(Natural Science Edition) ›› 2021, Vol. 39 ›› Issue (3): 69-82.doi: 10.16088/j.issn.1001-6600.2020082902

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The Polarization of Hybrid Multi-kernel Polar Codes

SONG Rui, XU Ming, TANG Yuansheng*   

  1. School of Mathematical Sciences, Yangzhou University, Yangzhou Jiangsu 225002, China
  • Received:2020-08-29 Revised:2020-10-28 Published:2021-05-13

Abstract: The proposal of polar codes by Arikan in 2009 is a significant breakthrough in coding theory. It has been one of the hottest topics for researchers in the field of error-correcting codes in recent years, and has been widely applied in 5G communication systems, etc. This paper mainly deals with the polarization of hybrid multi-kernel polar codes which are generalizations of the conventional polar codes. Firstly, by introducing random switch channels, a lower bound for the symmetric capacities of parallel broadcast channels (PBCs) is generalized to the case that the constituent channels contain some asymmetric binary-input discrete memoryless channels (BIDMCs). Secondly, for the common tool of combining-and-splitting tactics (CAST) for the construction of polar codes, the restriction on symmetry and equivalence for the underlying channels are removed. Under the condition that the underlying channels are general BIDMCs, a lower bound for the largest symmetric capacity of the synthetic channels is generated by a CAST. Next, for any hybrid multi-kernel polar code, the exact relation between its coding matrix and the CASTs used in the iterative construction are determined. Finally, a rigorous proof for its polarization is demonstrated by using the lower bound on the symmetric capacity of synthetic channels generated by CASTs, when the underlying channel is a general BIDMC.

Key words: polar codes, symmetric capacity, successive cancellation decoding, hybrid multi-kernel polar code, combining and splitting tactics

CLC Number: 

  • O236.2
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