U.S.-Norway-Germany Cooperative Research: Some Problems in Galois Rings, Grobner Bases and their Applications (Applied Mathematics)

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9406614 Reed This award supports Professors Irving S. Reed and P. Vijay Kumar of the University of Southern California, together with a postdoctoral associate, to collaborate in applied mathematics research with Professors Tor Helleseth of the University of Bergen, Norway, and Henning Stichtenoth of the University of Essen, Germany. In general, they are working in the area of the interaction between algebraic geometry and coding theory. A major focus of their collaboration is the study of exponential sums over Galois rings, and the application of this to the design and study of error correcting codes. Another significant component involves Grobner basis techniques, both their computation over finite fields and their application for the decoding of cyclic and algebraic-geometric codes. To complement the U.S. groups' strengths in error correcting codes, Professor Helleseth contributes necessary expertise in the theory of algebraic functions and their applications to coding theory as well as experience with Galois rings and Grobner bases. He also offers access to massively parallel computing machines in Norway which will be used for some of the joint research. Professor Stichtenoth is an expert on algebraic geometry and a world leader in the area of algebraic- geometric codes. The combination of their talents creates an extremely powerful collaborative team. The proposed research is an interdisciplinary combination of discrete mathematics, symbolic computation and algebraic computation theory. Results will contribute to advances in the theory of exponential sums and new and improved decoding algorithms for error-correcting codes. Such results will have useful applications in several fields of mathematics, computer science and engineering. ***