Research On Hamming Weight Distributions And Symbol-pair Distances Of Linear Codes

Coding theory originated in the late 1940s.It is an intersectional research field of mathematics and information science.Linear codes over finite fields are a class of error correcting codes with algebraic structure,which are widely used in practical communication.The study of Hamming weight distributions of linear codes is an important topic in coding theory.The Hamming weight distributions of linear codes can not only determine the error correction ability of codes,but also determine the probability of error detection and error correction of some decoding algorithms.In recent years,linear codes with few Hamming weights have been widely used in secret sharing schemes,authentication codes and association schemes.The research on Hamming weight distributions of linear codes has attracted scholars’ interest and achieved a large number of research results.On the other hand,Cassuto and Blaum proposed a new measure of symbol-pair distance different from Hamming distance based on the symbol-pair error in symbol pair read channel in 2010.This new metric and the symbol-pair code are introduced for the first time.The construction and decoding method of symbol-pair codes are also given.Subsequently,many scholars have been committed to studying the symbol-pair distance of classical linear codes and constacyclic codes,as well as the structure and construction of optimal symbolpair codes based on related symbol-pair distance bounds.Let p be an odd prime,m be a positive integer and Fpmbe a finite field with pm elements.This thesis is carried out on the finite field research.Based on the known research work,this thesis further carries out two aspects of research work.One aspect of this research is to make a profound study of the structure and weight distributions of linear codes with few Hamming weights over finite fields.The other aspect is make a further study on the symbol-pair distance and MDS symbol-pair codes of constacyclic codes over finite fields.The specific research contents are as follows:In Chapter 3,by applying the theory of quadratic form and Gaussian sum over finite fields,we construct several classes of linear codes with few Hamming weights over the finite field Fp,where p is an odd prime number.We also determine the Hamming weight distributions of the constructed linear codes.It is found that the linear codes obtained for special p and m are optimal or almost optimal according to the code table[49]through some examples.The parameters of these linear codes are new in most cases.Moreover,two classes of MDS codes are obtained.In Chapter 4,we study the general linear code CE={(Tr(αx))x∈E|α∈Fpm｝by choosing the defining set E={x∈ Fpm*| Tr(ax2+bx)=0},where a ∈ Fpm*and b ∈ Fpm*.Several classes of linear codes with calculation formula of Hamming weight distributions are obtained.In particular,when the parameters a and b take special values,we obtain the determined Hamming weight distributions of several types of linear codes,and these types of linear codes are new.According to some examples and codes table in[49],it is found that the linear codes obtained for special a and b are optimal or almost optimal.Our results generalize some results in[31,71].In Chapter 5,we study symbol-pair distance of λ-constacyclic codes of length 2ηps over Fpm,where η,s are positive integers and λ∈ Fpm*.Specifically,we consider the following two situations:when λ is not a square element,we study the symbolpair distance of λ-constacyclic codes Ci(η)=<(x2η+ε)i),where-εps=λ,ε∈IFpm*,0≤i≤ps.When λ is a square element in the finite field Fpm,there existsγ∈ Fpm*with γ2ps=λ such that x2ηps-λ=((xη-γ)(xη+γ))ps.We also study the symbol-pair distance of A-constacyclic codes Ci,j(η)=<(xη+γ)i(xη=γ)j),where xη+γ,xη-γ are irreducible and 0 ≤i,j≤ps.Furthermore,all MDS symbol-pair of these λ-constacyclic codes of length 2ηps over the finite field Fpm are obtained.