Codes and Algebraic Curves

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ISBN: 9780198500391 | 0198500394
Cover: Hardcover
Copyright: 3/5/1998

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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 anupdated 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, andapplications 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.

List of tables

xii

I CODES FROM ALGEBRAIC CURVES

(86)

1 Introduction: curves and codes

(4)

1 The Greeks.

2 The introduction of coordinates.

3 The use of algebraic structures.

4 The origins of codes.

5 Codes found by evaluating functions.

6 Codes and algebraic curves.

7 Chapter layout.

8 Notational conventions.

2 Algebraic curves

(14)

1 Defining a curve.

2 Irreducible polynomials in K[x, y].

3 Unique factorization in K[x, y].

4 Increasing the supply of curves.

5 Absolute irreducibility.

6 Existence of absolutely irreducible polynomials.

7 Projective transformations.

8 Final definition of a curve.

9 Curves over finite fields.

10 An example of a cubic curve.

11 The Klein quartic.

12 A quintic.

13 Exercises.

3 Functions on algebraic curves

(17)

1 Congruence.

2 The coordinate ring.

3 A convention.

4 The function field.

5 Independence of coordinate system.

6 Evaluating functions at points.

7 Shifting to the origin.

8 Divisibility by x - Alpha

9 Simple points.

10 Order functions.

11 Good orders and simple points.

12 A test for singularity.

13 Smooth curves.

14 Conjugate points.

15 The degree of a point.

16 Exercises.

4 A survey of the theory of algebraic curves

(10)

1 Existence of zeros.

2 Counting poles and zeros.

3 The horizon theorem.

4 Applying the horizon theorem.

5 Divisors.

6 A special case.

7 Riemann's theorem and the genus.

8 The Plucker formula for smooth plane curves.

9 The Klein quartic.

10 A Hermitian quintic.

11 Exercises.

5 Geometric Goppa codes

(12)

1 Error-correcting codes.

2 Weight and distance.

3 Dual Goppa codes.

4 Parameters of function codes.

5 Points of the Klein quartic.

6 Primary Goppa codes.

7 Generator and check matrices.

8 Parameters of residue codes.

9 Exercises.

6 Basic error processing

(11)

1 Syndromes.

2 Error locators.

3 Finding an error locator.

4 Conditions for the SV-algorithm.

5 The Skorobogatov-Vladut error processing algorithm.

6 Verification.

7 Another example.

8 A comparison with Reed-Solomon codes.

9 Exercises.

7 Full error processing

(18)

1 Auxiliary divisors.

2 The syndrome table.

3 Existence of error locators.

4 Testing for error locators.

5 Consistency.

6 Majority voting.

7 The Duursma error processing algorithm.

8 Verification.

9 A complete example.

10 Exercises.

II FIELDS OF ALGEBRAIC FUNCTIONS

(95)

8 Introduction: the algebraic approach

(2)

1 Function fields.

2 Pros and cons.

3 Chapter layout.

9 Function fields and places

(10)

1 Function fields of plane curves.

2 Algebraic functions.

3 The field defined by an irreducible polynomial.

4 Places of a function field F.

5 Places of the field of rational functions.

6 Structure of places.

7 The existence of places.

8 Exercises.

10 Valuations

101

(10)

1 Discrete valuations.

2 Valuations and point orders.

3 Valuation rings and places.

4 Valuation rings determine valuations.

5 Places are valuation rings.

6 The approximation theorem.

7 Places of F over a given place of K(x).

8 Exercises.

11 Divisors

111

(14)

1 The degree of a place.

2 Divisors.

3 Relative dimensions.

4 Elements of F as functions on places.

5 The space L(D).

6 Finiteness of the rank of a divisor.

7 The divisor of zeros of a function.

8 A lower bound for the rank of a divisor of zeros.

9 The degree theorem.

10 Principal divisors.

11 Riemann's theorem and the genus.

12 Rational function fields.

13 Exercises.

12 Repartitions and differentials

125

(17)

1 Classical integration.

2 Repartitions.

3 Relative dimensions.

4 Differentials.

5 Relative dimensions again.

6 The index of a divisor.

7 The weak Riemann-Roch theorem.

8 Differentials over D.

9 Properties of the index.

10 The divisor of a differential.

11 Differentials as multiples.

12 The strong Riemann-Roch theorem.

13 The strong approximation theorem.

14 Residues.

15 The residue theorem.

16 The residue representation of geometric Goppa codes.

17 Goppa codes are dual Goppa codes.

18 Exercises.

13 Extensions of function fields

142

(19)

1 Change of base field: the genus.

2 Change of base field: differentials.

3 Constant field extensions.

4 The genus of constant field extensions.

5 Places in an extension field.

6 Extensions of a place.

7 Extending divisors.

8 Affine rings of extension fields.

9 Coordinate tests.

10 Integral bases.

11 Repartitions of an extension.

12 The trace.

13 Extending differentials.

14 The genus of a separable extension.

15 The different.

16 Inseparable extensions.

17 Subfields of rational function fields.

18 Exercises.

14 Curves and function fields

161

(13)

1 Basic definitions.

2 Points and places.

3 The field of constants of K(C).

4 Algebraic extensions.

5 Points and ideals.

6 Places over points.

7 The zeros theorem and singular points.

8 Simple points.

9 The horizon theorem.

10 Extending the base field.

11 Plucker's formula for the genus.

12 Plucker's formula is exact for smooth curves.

13 Exercises.

15 More on Goppa codes

174

(10)

1 The volume of a ball of radius r.

2 The simple Gilbert bound.

3 The relative minimum distance.

4 The entropy function.

5 The relation between entropy and volume.

6 The asymptotic Gilbert bound.

7 Geometric Goppa codes and the Gilbert bound.

8 The theorem of Tsfasman, Vladut, and Zink.

9 Towards a good family.

10 The curves of Garcia and Stichtenoth.

11 Exercises.

12 Research problems.

References

184

(3)

Index

187

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Codes and Algebraic Curves

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