Construction Of Some Three-weight Linear Codes And Their Complete Weight Enumerators

Linear codes are a class of error-correcting codes,which have a good algebraic structure and important applications in data storage,communications,and so on.Determining the complete weight enumerators of linear codes is an important research direction for coding theory.The complete weight enumerators of linear codes contain some important information:the error detection and correction capability for codes,the probability of detecting and correcting errors,etc.Especially,linear codes with few weights have been widely used in the research of strong regular graphs,secret sharing schemes,authentication codes,and combination schemes,etc.In this thesis,for an odd prime p,some three-weight linear codes over the finite field Fp are constructed from the defining set,and their complete weight enumerators are determined by using Weil sums,and then a class of these codes with parameters[p2-1,3,p2-p-1]are optimal with respect to the Griesmer bound.Especially,these codes are minimal in some cases,and then they can be suitable for applications in secret sharing schemes.The results of this thesis are improvements of some results given by Jian et al(Finite Fields and Their Applications,2019,57:92-107).