Correlation Distribution Of P - Element Sequences Over Finite Fields And Weight Distribution Of Some Cyclic Codes

Exponential sums over finite fields not only are fundamental and important objects in number theory, but also have wide applications in signal communication. For several decades, a series of profound results on the lower and upper bounds of exponential sums have been found. Exponential sums over finite fields have important applications in coding and cryptography. For example, in coding theory exponential sums are used for estimating the minimum distances of linear codes. In CDMA and OFDM communication systems, we need sequences with low auto- and cross-correlations. It is equivalent to find a series of exponential sums which have small absolute values.In this dissertation, by the theory of quadratic forms on finite fields, we will study some exponential sums and determine the value distributions. As applica-tions, we determine the weight distribution of the corresponding cyclic code and the cross-correlation distribution of p-ary m-sequences. The thesis is organized as follows.Firstly, we mainly introduce some background and previous work on our re-search topic.Secondly, a class of exponential sums from half quadratic binomials are studied. Combining previous work, by the theory of quadratic forms and linearized polynomi-al over finite fields, we determine the value distribution of the sum where m/k. is odd. As applications, firstly by using the m-sequence and its decimat-ed sequence, we construct a p-ary sequence family with large size and low correla-tion, where the magnitude of correlation values is upper bounded by p+1/2pm+1 (when k = 1). Then we determine the weight distributions of several classes of linear codes. Some of the dual codes have minimum distance four, which is optimal with respect to the Hamming bound.Thirdly, Luo et al. investigated the cross-correlation of a p-ary m-sequence {st} of period pn-1 and its decimated sequence {sdt}, where p = 3 mod 4, d=(Pm+1)2/2(pk+1), k|m, and m is an odd integer. We extend the decimation into a general case. Let d be a positive integer satisfy d(pk+1)= pm+1 mod p2m - 1, where 2m/gcd(k,2m) is odd. By discussing the value distribution of we consider the cross-correlation between a p-ary m-sequence{st} of period pn-1 and its decimated sequence{sdt}. Our results shows that the cross-correlation is six-valued and can be completely determined.Finally, we give the distribution of the third class of exponential sum where d satisfy d(pk+1)= pm+1 mod q - 1, and 2m/gcd(2m, k)is odd. As ap-plications, we determine the weight distributions of several classes of linear codes. Some of the dual codes have optimal parameters. On the other hand, we give the cross-correlation of a p-ary m-sequence{st} of period pn - 1 and its decimated sequence{sdt+1}(0≤l<pm+1/2).