Some Algebraic Properties Of Polar Codes Spectrum And Cycles Distribution

E Arikan discovered the phenomenon of channel combination and splitting in 2009,and proposed the Polar codes based on this polarization phenomenon.Both the strengths and limitations of Polar codes lie in their rigid structure.The special and stable structure is convenient for people to study and modify it,and then propose the encoding and decoding algorithm with lower complexity.The algebraic structure and properties contained in the structure have very important research significance.The commonly used selection method of information bits for Polar codes is to select according to the channel reliability in the serial cancellation(SC)decoding algorithm,but this selection method is not suitable for all situations.In the improved list(SCL)decoding algorithm based on the SC decoding algorithm,the selection of information bits needs to consider the code weight spectrum and reliability,so that better performance can be obtained.Moreover,the Polar codes structure of encoding and decoding contains many short cycles,which limits its application range.In this paper,we firstly propose a new representation method of rows,and verify the number range of the information set to obtain a unique code spectrum on its basic.Then we introduce the principle of inclusion and exclusion to derive a general formula for calculating code weight in binary field.Aiming at the existing short cycles,with the help of the structure of QC LDPC codes and QC Polar codes,the distribution rules of 4-cycles in Polar codes are analyzed,and a conjecture about the selection method of information set based on 4-cycles distribution is proposed.The details are as follows:First of all,the research status of Polar codes is summarized.Secondly,the basic principles of Polar codes is introduced.Then,the spectrum of Polar codes obtained based on the new row representation method is studied.The basic structure of the Polar codes is introduced firstly.Then based on the basic structure of the Polar codes and the original row representation method,a new row representation method is proposed.On this basis,the linear combination of code weights is analyzed.Introducing the principle of tolerance and exclusion,which is related to the linear combination of rows,we can obtain the general formula for analyzing the code weight,and proposes the conjecture about the selection of the information set based on the code spectrum.In addition,the distribution of cycles in Polar codes is analyzed based on quasi-cyclic construction.The definitions and properties of QC LDPC codes and QC Polar codes are introduced.On this basis,the characteristics of quasi-cyclic structures are explained by analyzing the correctness of the Z-fold simplification algorithm.Based on the strict structure of Polar codes,the distribution of 4-cycles is analyzed and some conjectures about the larger cycles are put forward.Then we can have the simulation results,and propose a remark about the selection method of the information set based on the distribution of the 4-cycles is inferred.Finally,the work of this paper is summarized,and future research directions are proposed for the problems arising in the analysis process.