Exponential Sums Over Galois Rings And Their Applications In Communications

Gauss sums and Jacobi sums over finite fields, as two special cases of general exponential sums, not only are important objects in number theory but also have many remarkable applications in communications. Exponential sums over Galois rings is a generalization of exponential sums over finite fields. With the successive research on error-correcting codes over finite fields, Galois rings and exponential sums over Galois rings have become very important tools to construct good error-correcting codes, sequences with good correlation and so on.This dissertation studies the Gauss sums and Jacobi sums over the Galois ring GR(p2,r) and their applications in communication. First, we present the additive characters and multiplicative characters of GR(p2,r) and define Gauss sums and Jacobi sums over GR(p2,r). We give explicit computations on Gauss sums and Jacobi sums over GR(p2,r). It is shown that Gauss sums and Jacobi sums over GR(p2,r) can be reduced to Gauss sums and Jacobi sums over the finite field Fr in all nontrivial cases. A connection between Gauss sums and Jacobi sums over GR(p2,r) is established. Further, an important relationship between Gauss sums and Jacobi sums over GR(p2,r) and Gauss sums and Jacobi sums over R(l)=GR(p2,rl)(l≥1) is obtained.Then, we mainly consider two applications of Gauss sums and Jacobi sums over GR(p2,r). On the one hand, the weight distributions of some classes of linear codes over Zp2are determined by employing Gauss sums over GR(p2,r). Based on this, we use Davenport-Hasse lifting of Gauss sum over GR(p2,r) to obtain the weight distributions of linear codes C(l)={cβ(l)=(Tr(l)(βx))x∈H(l):β∈R(l)} over Zp2, where H(l) is a subgroup of the units of R(l), and Tr(l) is the trace map from R(l) to Zp2. Under a Grary map, many p-ary linear and nonlinear codes with good parameters are obtained. On the other hand, Several new series of approximately mutually unbiased bases are constructed by using Gauss sums and Jacobi sums over GR(p2,r), and tensor method.