CAREER: Computationally Efficient Error-Correcting Codes and Their Applications

本研究的目标是提高纠错码的计算效率，并将此技术应用于计算和信息理论的其他问题。纠错码是所有电子通信系统的基本组成部分。新的纠错码结构可以非常有效地编码和解码，最近已经开发出来。本研究的目标是通过以下方式扩展这项初步工作：提高基于先前开发的代码的纠错代码的显式构建的容错性和速度。开发可以证明随机结构质量的算法。利用相关技术提高信息理论和电子通信中其他重要任务的计算效率。开发代码的高质量实现，这将使工程师更容易测试这些代码在他们自己的应用领域中的执行情况。拟议调查的教育目标范围更广。它们涉及到课程的设计，这些课程向研究生讲授本研究所依赖的更大的数学框架。这些课程应该吸引数学和计算机科学的学生来学习新的有趣的方法，这些方法可以应用数学中的思想来解决计算机科学中的重要问题。这些课程旨在教授如何在研究中建立联系，并为他们提供许多成熟的联系领域。最后，这些课程将教授学生另一个主要研究者研究方法的基本要素：在数学研究中使用计算实验。他们将学习如何使用计算机实验来制作和测试组合猜想。当他们开始将计算机视为一种可以帮助他们进行自己研究的工具时，他们将不可避免地关注计算算法的效率，并获得直觉，数学思想可能会导致实用的算法

The goals of this research are to improve the computational efficiency of error-correcting codes and to apply techniques from this effort to other problems of computation and information theory. Error-correcting codes are essential components of all electronic communication systems. New constructions for error-correcting codes that could be encoded and decoded very efficiently have recently been developed. The objectives of this research are to extend this initial work by: Improving the error-tolerance and speed of the explicit constructions of error-correcting codes based upon previously developed codes. Developing algorithms that can certify the quality of the randomized constructions. Using related techniques to improve the computational efficiency of other tasks vital to information theory and electronic communication. Developing quality implementations of the codes that will make it easier for engineers to test how these codes would perform in their own application areas. The educational goals of the proposed investigation are broader in scope. They involve the design of classes that teach graduate students about the larger mathematical framework on which this research rests. These classes should attract students from mathematics as well as computer science to learn about the new fascinating ways in which ideas from mathematics can be applied to solve important problems in computer science. These classes are designed to teach how connections in research are made as well as provide them with many areas ripe for connection. Eventually, these classes will teach the students another essential element of the principal investigator's research methodology: that of using computational experiments in mathematical research. They will learn how one can use computer experiments to make and test combinatorial conjectures. As they begin to see the computer as a tool that can aid them in their own research, they will inevitably become concerned with the efficie ncy of computational algorithms and gain intuition for which mathematical ideas might result in practical algorithms.***