混合多核极化码的极化性

The Polarization of Hybrid Multi-kernel Polar Codes

Arikan于2009年提出的极化码是纠错编码理论领域的一大突破,也是近年来的研究热点,已广泛应用于5G通信等领域。本文主要研究作为极化码的推广的混合多核极化码的极化性。首先,利用随机切换信道概念,将以对称二元输入离散无记忆信道(BIDMC)为子信道构成的并行广播信道(PBC)的信道容量的一个重要下界推广到子信道中包含非对称BIDMC的情形;然后,放宽极化码构造中通用的信道组合与分裂策略(CAST)对基础信道对称性及等价性要求,并在基础信道是非对称BIDMC时,利用PBC信道容量的下界对CAST生成的各虚拟信道的最大对称容量进行估计;最后,通过分析混合多核极化码的编码矩阵与递归构造中使用的各CAST的关系,并利用CAST虚拟信道对称容量的估计,首次在基础信道为一般BIDMC的条件下,对混合多核极化码的极化性给出了严格证明。

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.