The Construction Of Several Classes Of Linear Codes With Few Weights

As a special class of linear codes,linear codes with few weights have important applications in authentication codes,association schemes,secret sharing schemes,etc.In this paper,the theoretical knowledge of character and exponential sums is applied to define sets,and bivariate functions are selected to construct linear codes with few weights.Three-,four-,five-and six-weight linear codes are obtained over F3,and the parameters and weight distributions of these linear codes are also determined.Compared with previous work,the results show that most of the codes we constructed are new.The content of this paper is organized as follows:Chapter 1 is the introduction.This chapter first briefly describes the research background and significance of linear codes,secondly focuses on the current status of research on linear codes with few weights,and finally outlines the main content of this paper.Chapter 2 is the preparation.This chapter introduces the algebraic knowledge of linear codes,including trace functions,congruence,character,etc.,and further states some basic coding theories.This knowledge provides theoretical support for the subsequent construction of linear codes with few weights.Chapter 3 defines the set:D1={(x,y)∈F3m2\{(0,0)}:Tr1m(R(x,y))=u,u∈F3},by selecting the bivariate function R(x,y)=x3+y3+xy to construct linear codes with few weights over F3.Finally,the three-weight ternary linear codes with good properties are obtained,and their parameters and weight distributions are further determined using the knowledge of the exponential sums.Chapter 4 defines the sets:D1={(x,y)∈F3m2\{(0,0)}:Tr1m(R(x,y))=u,u ∈ F3},D2={(x,y)∈F3m2\{(0,0)}:(Tr1m(R(x,y)),Tr1m(R(y,x)))=v,v ∈ F32｝,by selecting the bivariate function R(x,y)=x3+xy to construct linear codes with few weights over F3.Finally,the three-,four-,five-and six-weight ternary linear codes with good properties are obtained,and their parameters and weight distributions are further determined using the knowledge of the exponential sums.Chapter 5 is the summary and outlook.This chapter summarizes the main contents of this paper and provides an outlook on the research of linear codes with few weights.