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

9406614 Reed该奖项支持南加州大学的Irving S. Reed教授和P. Vijay Kumar教授，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教授在应用数学研究方面进行合作。一般来说，他们的工作领域是代数几何和编码理论之间的相互作用。他们合作的一个主要焦点是研究伽罗瓦环上的指数和，并将其应用于设计和研究纠错码。另一个重要的组成部分涉及格罗布纳基技术，包括它们在有限域上的计算和它们在循环码和代数几何码解码中的应用。为了补充美国小组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贡献了必要的专业知识，以及在伽罗瓦环和格罗伯纳基方面的经验。他还提供了挪威的大型并行计算机的访问权限，这些计算机将用于一些联合研究。斯蒂特诺斯教授是代数几何方面的专家，也是代数几何密码领域的世界领军人物。他们的才能的结合创造了一个非常强大的协作团队。所提出的研究是离散数学、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和新的和改进的解码算法的纠错码。这样的结果将在数学、计算机科学和工程的几个领域有有用的应用。＊＊＊

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. ***