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

9406614里德这个奖项支持教授欧文S。Reed和南加州大学的P. Vijay Kumar，以及一名博士后，与挪威卑尔根大学的Tor Helleseth教授和德国埃森大学的Henning Stichtenoth教授合作进行应用数学研究。 在一般情况下，他们的工作领域之间的相互作用代数几何和编码理论。 他们合作的一个主要重点是研究伽罗瓦环上的指数和，以及将其应用于纠错码的设计和研究。 另一个重要的组成部分涉及Grobner基础技术，包括它们在有限域上的计算以及它们在循环和代数几何码解码中的应用。 为了补充美国团体在纠错码方面的优势，Helleseth教授在代数函数理论及其在编码理论中的应用方面提供了必要的专业知识，以及Galois环和Grobner基地的经验。 他还提供访问挪威的大规模并行计算机器，将用于一些联合研究。 Stichtenoth教授是代数几何学专家，也是代数-几何编码领域的世界领导者。 他们的才能结合在一起，创造了一个非常强大的协作团队。 该研究是离散数学、符号计算和代数计算理论的跨学科结合。 结果将有助于指数和理论的进步以及新的和改进的纠错码解码算法。 这些结果将在数学、计算机科学和工程等多个领域中有有用的应用。 ***

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. ***