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.***
