The Study Of Applications Of Exponential Sums In Cyclotomic Number Problem And Coding Theory

Exponential sums are one of the most important and fundamental objects in numbertheory, which have been studied extensively and deeply. Gauss and Jacobi sums are twotypes of exponential sums with a series of systematical and in-depth results.In this thesis, we apply Gauss and Jacobi sums into the following problems:Problem1. Using the formulas of Gauss sums in the index2case, present two familiesof cyclotomic numbers of order l and2l overFq, for prime l such that3l≡3(mod4).Problem2. Using quadratic reduced Jacobi sums and Gaussian periods, determine a newclass of the weight distribution of the cyclic codes constructed in[66]with arbitrarynumber of zeros.To be specific, for Problem1, we use the main results of[33]and[65], i.e., the formulasof Gauss sums in the index2case with order N=l,2l, and show the explicit formulasof corresponding cyclotomic numbers. Our main results are related with Nl(λ)(2λl-1), the number of rational points of certain elliptic curve, called “Legendre curve”.And the properties and value distribution of such number are also presented.For Problem2, recently,[66]constructed a class of cyclic codes (a1,···,at)with arbi-trary number of zeros. We determine the weight distribution for a new family of suchcyclic codes. This is achieved by certain new methods, such as theory of reduced Jacobisums over finite fields and subtle treatment of complicated combinatorial identities.