Study On Several Classes Of Exponential Sums Over Finite Fields

Exponential sums over finite fields are intimately connected to many research fields in mathematics. For example, number of rational points on an algebraic curve can be expressed as a certain combination of exponential sums. In communication theory, correlation function of a periodic sequence is a class of exponential sums. Moreover, for cyclic codes, Hamming weight of each codeword can also be expressed as a ceratin combination of exponential sums. In this thesis, we mainly investigate some exponential sums, and give the corresponding applications. The details are shown as follows.In Chapter 1, we introduce background and previous results on exponential sums.In Chapter 2, we give some basic definitions and conclusions, which will be used in later chapters.In Chapter 3, firstly, based on the theory of binary quadratic form over finite fields, the possible values of exponential sum are determined, and the cross-correlation between a binary m-sequence (st) of length 2m - 1 and its d-decimated sequences (sdt+u) is studied. It is shown that the maximum magnitude of cross-correlation values is 2m/2+1+1. Secondly, a new sequence family with maximum correlation magnitude 2m/2+1+1 and family size 2m/2 is proposed. Finally, the value distribution of the exponential sum is given, and the weight distribution of a class of binary cyclic codes is determined. Here q= 2m,m= 2k, l, k are two odd integers with 0<l< k and gcd(l,k)=1.In Chapter 4, a new family S of p-ary sequences with period N= pn - 1 is proposed. The sequences in S are constructed by adding a p-ary sequence to its two decimated sequences with different phase shifts. From the theory of nonbinary quadratic form over finite fields. the auto-correlation and cross-correlation distribu-tion among sequences in S is completely determined. It is shown that the maximum magnitude of nontrivial correlations of S is upper bounded by and the family size of S is N2. Compared with the known sequence families, our se-quence family is new and has a larger family size. Here p is an odd prime, n≥ 3 and k are two positive integers with e=gcd(n, k).In Chapter 5, firstly, by the theory of nonbinary quadratic form over finite fields, the value distribution of the exponential sum is determined, the cross-correlation distribution between a p-ary m-sequence (st) of period pm - 1 and its decimated sequence (sdt+i) is given. Furthermore, the sequence family F is constructed and the cross-correlation distribution among sequences in the family F is also determined. Our result shows that the maximal magnitude of the correlation value is low when e= 1. Secondly, the value distribution of the exponential sum is given, and the weight distribution of a class of p-ary cyclic codes is determined. Here p is an odd prime, m and k are two positive integers, e=gcd(m, k), and d=(pk+1)(pm+1)/4,0≤l≤gcd(pm-1,d).