Study On The Constructions And Properties Of Several Classes Of Linear Codes With A Few Weights

Linear codes have important applications in information security,data storage and communication due to their efficient encoding algorithms and good algebraic structure.Moreover,the weight distribution of linear code not only indicates the error correcting ability of the code,but also can be used to calculate the proba-bility that the information will produce an error in the transmission process.It is extremely difficult to determine the weight distribution of linear codes,so it has always been an important research direction of coding theory,and linear codes with a few weights can be applied in association schemes,authentication codes,combi-natorial designs,secret sharing schemes,data storage systems,etc.Choosing the appropriate definition set to construct linear codes with a few weights is one of the hot spots of linear code theory research in recent years.If the definition set is properly selected,some optimal codes with a few weights can be obtained.In this dissertation,we construct several classes of linear codes with a few weights from new defining sets,and obtain their explicit weight distributions using exponential sums.At the same time,we illustrate the validity of our results using Magma program and discuss their applications in secret sharing schemes.The main work as follows:(1).We construct a class of four-weight and six-weight linear codes with the preimage of linear function and quadratic form function as the defining set,deter-mine weight distributions of the linear codes,and the application of the linear codes in secret sharing is also discussed.In addition,there are some almost optimal codes with respect to the Singleton bound in our constructed codes.(2).We obtain a class of linear codes with four weights and six weights with the the preimage of univariate function and bivariate function as the defining set And we also determine their weight distributions.(3).When the defining set is vectors.we construct a class of two-weight linear codes and investigate their weight distributions.Finally,we discuss the application of the linear codes in secret sharing.