Exponential Sums Over Finite Fields And Their Applications In Linear Codes

Let p be an odd prime,q=pn(n ? Z+),f(x)?Fq[x].As early as the 19th century,Gauss defined exponential sums Sq(f(x)),and then obtained the computing formula of quadratic exponential sums over Fp.Based on the definition of exponential sums and the properties of finite fields,the value of Sq(f(x))is studied.Daqing Wan studied the algebraic degrees and the periods of the exponential sums Sq(f(x))over the finite field Fq.In 2020,Chao Chen,et al.,determined the values of Sq(xd)for the case both p?-1(mod d),p?1(mod 4)and q=p2.In this thesis,we generalize their results and study the computing formula of Sq(xd)for the case p ?-1(mod d)and q=p2r(r?Z+)by using elementary methods and techniques,and then obtain the all possible values of Sq(xd)in this case.The exponential sums also play an important role in coding theory and cryptography.Especially,in 2007,Cunshen Ding proposed the general ways to construct linear codes by using definition sets,and obtained the weight distribution of these codes by using exponential sums.In 2020,Yansheng Wu,et al.,constructed t-weight binary codes(t=2,3,4,5,6)over F2+uF2(u2=0)by employing simplicial complexes,and obtained the weight distribution of codes by using exponential sums.This thesis is based on the above researches,we present two new constructions of t-weight binary codes(t=2,3,4)over F2 by employing simplicial complexes.In addition,we determine the weight distributions of two classes of codes over F2 by using exponential sums.Finally,two concrete examples are given.