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Zusatztext '... the text is clear and well-written with many helpful examples... this book should prove useful to students trying to learn about geometric codes.' Klappentext A fascinating branch of mathematics since antiquity, the geometry of curves has been extensively developed and become highly abstract. Recently links have been made with the subject of error correction, leading to the creation of geometric Goppa codes, a new and important area of coding theory. This book is an expanded and updated version of one part of the author's successful book Error-Correcting Codes and Finite Fields. Here he gives an elementary introduction to Goppa codes and includes many examples, calculations, and applications. The first part of the book emphasizes motivations, giving precedence to applications over proofs. The second part then provides the formal theory, with some concepts and proofs simplified without sacrificing rigor. Zusammenfassung The geometry of curves has fascinated mathematicians for 2500 years, and the theory has become highly abstract. Recently links have been made with the subject of error correction, leading to the creation of geometric Goppa codes, a new and important area of coding theory. This book is an updated and extended version of the last part of the successful book Error-Correcting Codes and Finite Fields. It provides an elementary introduction to Goppa codes, and includes many examples, calculations, and applications. The book is in two parts with an emphasis on motivation, and applications of the theory take precedence over proofs of theorems. The formal theory is, however, provided in the second part of the book, and several of the concepts and proofs have been simplified without sacrificing rigour. Inhaltsverzeichnis Part 1: Codes from algebraic curves 1: Introduction: curves and codes 2: Algebraic curves 3: Functions on algebraic curves 4: A survey of the theory of algebraic curves 5: Geometric Goppa codes 6: Basic error processing 7: Full error processing Part 2: Fields of algebraic functions 8: Introduction: the algebraic approach 9: Function fields and places 10: Valuations 11: Divisors 12: Repartitions and differentials 13: Extensions of function fields 14: Curves and function fields 15: More on Goppa codes ...
