Algebraic and geometric coding theory

Abstract

Coding theory is concerned with methods for “packaging” and “unpackaging” messages in order that the most information can be reliably send over a communication channel. In this chapter, a greater emphasis is given to the roles of geometry and group theory in communication problems than is usually the case in presentations of this subject. Geometry and group theory enter in problems of communication in a surprising number of different ways. These include the use of finite groups and sphere packings in highdimensional spaces for the design of error-correcting codes (such as those due to Golay and Hamming). These codes facilitate the efficient and robust transmission of information. Additionally, Lie groups enter in certain decoding problems related to determining the state of various motion sensors.