On The Polarization Of Hybrid Multi-Kernel Polar Codes

Since Shannon proposed the modern error-correcting coding theory,searching for the error-correcting code that approximates or even reaches channel capacity has been the goal of this field.From Reed-Solomon codes and Bose-Chaudhuri-Hocquenghem codes,which were first discovered,to Turbo codes and low density parity cheek codes,which are widely used now,these codes are gradually approaching the channel capacity in terms of performance,and the most outstanding performance are the Polar Codes first proposed by Arikan in 2009.Polar Codes have been one of the hottest topics for researchers in the field of error-correcting since they were put forward,and have been widely applied in 5G communication systems,etc,and they have been successfully adopted in the coding scheme of 5G communication.In this paper,we mainly deal with and prove the polarization of hybrid multi-kernel polar codes which are generalizations of the conventional polar codes.At first,by introducing a concept of random switch channels,an important 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 memory less channels(BIDMCs).Then,for the common tool of combing-and-splitting tactics(CAST)for the construction of polar codes,we drop the restriction on symmetry and equivalence for the underlying channels.Under the condition that the underlying channels are general BIDMCs,we show a lower bound for the largest symmetric capacity of the synthetic channels generated by a CAST.In the final,for any hybrid multi-kernel polar code,we determine the exact relation between its coding matrix and the CASTs used in the iterative construction,and then show a rigorous proof for its polarization,by using the lower bound on the symmetric capacity of synthetic channels generated by CASTs,when the underlying channel is a general BIDMC.