A Class Of Exponential Sums And Their Applications

Exponential sums have many fundamental applications to coding theory,combi-natorics,and number theory.In number theory,since the time of Gauss,exponential sums have appeared in the study of reciprocity laws.More recently,in the study of coding theory dealing with exponential sums,take Kloosterman sums as an example,they played a central role in several areas.In this thesis,we first consider a class of exponential sums using number theory and finite field.The exponential sums are proven to be four-valued.Furthermore,Cyclic codes,a subclass of linear codes,have attracted a lot of research interest for decades as they have efficient encoding and decoding algorithms.So as an significant application of the exponential sums,we determines the weight distributions of a family of p-ary cyclic codes over Fp whose duals have two zeros.Then obviously the weight distribution can give the minimal distance of the code,and hence the error correcting capability.Moreover,the weight distribution of a code allows the computation of the error probability of error detection and correction,This is of great significance.in ensuring the security of information transmission.On top of these,a class of authentication codes with the generic coding-theory construction are described at last.The authentication codes presented in this paper are better than several known authentication codes with comparable parameters.