Trace Functions And Their Application To The Construction Of Linear Codes

Linear codes are a very important class of error-correcting codes,which have always been the focus of coding theory and the basis of coding theory.Linear codes with small weights have been applied in the fields of consumer electronics,communication,data storage system,secret sharing,authentication codes and so on,among which two or three weights codes have important applications in strongly regular graphs and association schemes.In this thesis,we consider the construction of linear codes with two or three weighs by defining sets.Let Fq be a finite field of q elements,where q=pm and p is an odd prime.Ding et al.presented the construction of linear codes by using a defining set.Let D ={d1,d2,...,dn} be a subset of Fq.Then a linear code of length n defined over Fq is defined by where D is called the defining set of CD and Tr1m(x)=?i=0m-1xpi is the trace function from F to Fp.Let m1,m2,…,mt be positive integers and T=Fq1×Fq2×…×Fqt,where qi=pmi for 1?i?t in this thesis.A linear code can be constructed by a defining set D,where D=(X??i=1tTr1mi(xi2)=c},x?Fp is a subset of T and c(?)=(?i=1tTr1mi(aixi).We also determine the weight distributions of these linear codes with two or three weights.These linear codes constructed in this thesis can be applied to strongly regular graphs,association schemes and secret sharing schemes.Through verification and calculation,these punctured codes are minimal codes.