Universitext: Algebraic Function Fields and Codes

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This book aims to address a gap in mathematical literature by offering a modern, self-contained exposition of algebraic function fields. It covers essential topics such as the Riemann-Roch theorem, algebraic extensions, ramifications theory, and differentials, with a focus on function fields over finite constant fields, leading to zeta functions and the Hasse-Weil theorem. Numerous examples are provided to illustrate the general theory. Additionally, the book explores the fascinating connections between algebraic geometry and error-correcting codes, particularly through V.D. Goppa's construction of powerful codes. It introduces Goppa's algebraic-geometric codes using the elementary approach of algebraic function fields. Early chapters discuss the codes, their parameters, and their relationships with traditional codes like classical Goppa, Reed-Solomon, and BCH codes. Later sections include a decoding algorithm for these codes and an examination of subfield subcodes and trace codes. This work will be valuable for students and researchers in algebraic geometry and coding theory, as well as computer scientists and engineers focused on information transmission.