Algebraic Function Fields and Codes
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This book has two objectives. The first is to fill a void in
the existing mathematical literature by providing a modern,
self-contained and in-depth exposition of the theory of
algebraic function fields. Topics include the Riemann-Roch
theorem, algebraic extensions of function fields,
ramifications theory and differentials. Particular emphasis
is placed on function fields over a finite constant field,
leading into zeta functins and the Hasse-Weil theorem.
Numerous examples illustrate the general theory.
Error-correcting codes are in widespread use for the
reliable transmission of information. Perhaps the most
fascinating of all the ties that link the theory of these
codes to mathematics is the construction by V.D. Goppa, of
powerful codes using techniques borrowed fromalgebraic
geometry. Algebraic function fields provide the most
elementary approach to Goppa's ideas, and the second
objective of this book is to provide an introduction to
Goppa's algebraic-geometric codesalong these lines. The
codes, their parameters and links with traditional codes
such as classical Goppa, Peed-Solomon and BCH codes are
treated atan early stage of the book. Subsequent chapters
include a decoding algorithmfor these codes as well as a
discussion of their subfield subcodes and tracecodes.
Stichtenoth's book will be very useful to students and
researchers in algebraic geometry and coding theory and to
computer scientists and engineers interested in information
transmission.
the existing mathematical literature by providing a modern,
self-contained and in-depth exposition of the theory of
algebraic function fields. Topics include the Riemann-Roch
theorem, algebraic extensions of function fields,
ramifications theory and differentials. Particular emphasis
is placed on function fields over a finite constant field,
leading into zeta functins and the Hasse-Weil theorem.
Numerous examples illustrate the general theory.
Error-correcting codes are in widespread use for the
reliable transmission of information. Perhaps the most
fascinating of all the ties that link the theory of these
codes to mathematics is the construction by V.D. Goppa, of
powerful codes using techniques borrowed fromalgebraic
geometry. Algebraic function fields provide the most
elementary approach to Goppa's ideas, and the second
objective of this book is to provide an introduction to
Goppa's algebraic-geometric codesalong these lines. The
codes, their parameters and links with traditional codes
such as classical Goppa, Peed-Solomon and BCH codes are
treated atan early stage of the book. Subsequent chapters
include a decoding algorithmfor these codes as well as a
discussion of their subfield subcodes and tracecodes.
Stichtenoth's book will be very useful to students and
researchers in algebraic geometry and coding theory and to
computer scientists and engineers interested in information
transmission.
This book has two objectives. The first is to fill a void in
the existing mathematical literature by providing a modern,
self-contained and in-depth exposition of the theory of
algebraic function fields. Topics include the Riemann-Roch
theorem, algebraic extensions of function fields,
ramifications theory and differentials. Particular emphasis
is placed on function fields over a finite constant field,
leading into zeta functins and the Hasse-Weil theorem.
Numerous examples illustrate the general theory.
Error-correcting codes are in widespread use for the
reliable transmission of information. Perhaps the most
fascinating of all the ties that link the theory of these
codes to mathematics is the construction by V.D. Goppa, of
powerful codes using techniques borrowed fromalgebraic
geometry. Algebraic function fields provide the most
elementary approach to Goppa's ideas, and the second
objective of this book is to provide an introduction to
Goppa's algebraic-geometric codesalong these lines. The
codes, their parameters and links with traditional codes
such as classical Goppa, Peed-Solomon and BCH codes are
treated atan early stage of the book. Subsequent chapters
include a decoding algorithmfor these codes as well as a
discussion of their subfield subcodes and tracecodes.
Stichtenoth's book will be very useful to students and
researchers in algebraic geometry and coding theory and to
computer scientists and engineers interested in information
transmission.
Table of contents:
1. Foundations of the Theory of Algebraic Function Fiels.- 2. Geometric Goppa Codes.- 3. Extensions of Algebraic Function Fields.- 4. Differentials of Algebraic Function Fields.- 5. Algebraic Function Fields over Finite Constant Fields.- 6. Examples of Algebraic Function Fields.- 7. More about Geometric Goppy Codes.- 8. Subfield Subcodes and Trace Codes.- Appendix A. Field Theory.- Appendix B. Algebraic Curves and Algebraic Function Fields.- Bibliography.- List of Notations.- Index
the existing mathematical literature by providing a modern,
self-contained and in-depth exposition of the theory of
algebraic function fields. Topics include the Riemann-Roch
theorem, algebraic extensions of function fields,
ramifications theory and differentials. Particular emphasis
is placed on function fields over a finite constant field,
leading into zeta functins and the Hasse-Weil theorem.
Numerous examples illustrate the general theory.
Error-correcting codes are in widespread use for the
reliable transmission of information. Perhaps the most
fascinating of all the ties that link the theory of these
codes to mathematics is the construction by V.D. Goppa, of
powerful codes using techniques borrowed fromalgebraic
geometry. Algebraic function fields provide the most
elementary approach to Goppa's ideas, and the second
objective of this book is to provide an introduction to
Goppa's algebraic-geometric codesalong these lines. The
codes, their parameters and links with traditional codes
such as classical Goppa, Peed-Solomon and BCH codes are
treated atan early stage of the book. Subsequent chapters
include a decoding algorithmfor these codes as well as a
discussion of their subfield subcodes and tracecodes.
Stichtenoth's book will be very useful to students and
researchers in algebraic geometry and coding theory and to
computer scientists and engineers interested in information
transmission.
Table of contents:
1. Foundations of the Theory of Algebraic Function Fiels.- 2. Geometric Goppa Codes.- 3. Extensions of Algebraic Function Fields.- 4. Differentials of Algebraic Function Fields.- 5. Algebraic Function Fields over Finite Constant Fields.- 6. Examples of Algebraic Function Fields.- 7. More about Geometric Goppy Codes.- 8. Subfield Subcodes and Trace Codes.- Appendix A. Field Theory.- Appendix B. Algebraic Curves and Algebraic Function Fields.- Bibliography.- List of Notations.- Index