ERROR-CORRECTING CODES AND FINITE FIELDS

We investigate the properties of modern error-correcting codes from an algebraic perspective. First, using techniques of linear algebra over finite fields, we develop the basic concepts of linear codes such as minimum distance, dimension, and error-correcting capabilities. We then use the structure of polynomial rings to define an example of cyclic codes, the Reed-Solomon code, and derive some of its properties. Finally, we introduce algebraic function fields and reinterpret Reed-Solomon codes from that perspective, then introduce the BCH code from the perspective of both cyclic codes and function fields.