Error-correcting Codes and Finite Fields

Starting with the elementary ideas of parity check codes, this work takes the reader via BCH and Reed-Solomon codes all the way to the geometric Goppa codes. The necessary mathematics is developed in parallel with the applications.

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This book provides the reader with all the tools necessary to implement modern error-processing schemes. It assumes only a basic knowledge of linear algebra and develops the mathematical theory in parallel with the codes. Central to the text are worked examples which motivate and explain the theory. The book is in four parts. The first introduces the basic ideas of coding theory. The second and third parts cover the theory of finite fields and give a detailed treatment of BCH and Reed Solomon codes. These parts are linked by their use of Euclid's algorithm as a central technique. The fourth part is devoted to Goppa codes, both classical and geometric, concluding with the Skorobogatov-Vladut error processor. A special feature of this part is a simplified treatment of the geometry of curves.