| Linear codes with a few weights can be applied in secret sharing schemes,au-thentication codes,association schemes,data storage systems,etc.Constructing linear codes from defining sets is an important research direction in coding theory.If the defining sets are well chosen.some optimal linear codes with few weights can be obtained.In this paper,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 is as follows:(1).We construct a class of two-weight,three-weight and five-weight linear codes for the defining set be the preimage of general quadratic function,determine weight distributions of the linear codes which include some optimal codes and almost optimal codes.(2).We obtain a class of linear codes with two weights and three weights for the defining set be the preimage of bivariate function and determine their weight distributions.(3).We extend the defining set to the kernel of multivariate function,construct a class of two-weight and three-weight linear codes and investigate their weight distributions.Results show that our construction include some maximum distance separable codes(MDS)and almost optimal codes with respect to the Singleton Bounds.And all of the linear codes we constructed can be used in the secret sharing schemes. |
