Algebraic Function Fields and Codes

362 Pages·2008·6.369 MB·English

by  Henning Stichtenoth

254
Graduate Texts in Mathematics
Editorial Board
S. Axler K.A. Ribet
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2 OXTOBY. Measure and Category. 2nd ed. 35 ALEXANDER/WERMER. Several Complex
3 SCHAEFER. Topological Vector Spaces. Variables and Banach Algebras. 3rd ed.
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4 HILTON/STAMMBACH. A Course in Topological Spaces.
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5 MAC LANE. Categories for the Working 38 GRAUERT/FRITZSCHE. Several Complex
Mathematician. 2nd ed. Variables.
6 HUGHES/PIPER. Projective Planes. 39 ARVESON. An Invitation to C*-Algebras.
7 J.-P. SERRE. A Course in Arithmetic. 40 KEMENY/SNELL/KNAPP. Denumerable
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9 HUMPHREYS. Introduction to Lie Algebras 41 APOSTOL. Modular Functions and
and Representation Theory. Dirichlet Series in Number Theory.
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15 BERBERIAN. Lectures in Functional Analysis Dimensions 2 and 3.
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17 ROSENBLATT. Random Processes. 2nd ed. 49 GRUENBERG/WEIR. Linear Geometry.
18 HALMOS. Measure Theory. 2nd ed.
19 HALMOS. A Hilbert Space Problem Book. 50 EDWARDS. Fermat’s Last Theorem.
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20 HUSEMOLLER. Fibre Bundles. 3rd ed. Geometry.
21 HUMPHREYS. Linear Algebraic Groups. 52 HARTSHORNE. Algebraic Geometry.
22 BARNES/MACK. An Algebraic Introduction 53 MANIN. A Course in Mathematical Logic.
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24 HOLMES. Geometric Functional Analysis 55 BROWN/PEARCY. Introduction to Operator
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25 HEWITT/STROMBERG. Real and Abstract 56 MASSEY. Algebraic Topology: An
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30 JACOBSON. Lectures in Abstract Algebra I. 61 WHITEHEAD. Elements of Homotopy
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Linear Algebra. of the Theory of Groups.
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Theory of Fields and Galois Theory. 64 EDWARDS. Fourier Series. Vol. I. 2nd ed.
33 HIRSCH. Differential Topology. 65 WELLS. Differential Analysis on Complex
Manifolds. 2nd ed.
(continued after index)

Henning Stichtenoth
Sabanci University
Faculty of Engineering & Natural Sciences
34956 Istanbul
Orhanli, Tuzla
Turkey
[email protected]

Editorial Board
S. Axler K.A. Ribet
Mathematics Department Mathematics Department
San Francisco State University University of California, Berkeley
San Francisco, CA 94132 Berkeley, CA 94720-3840
USA USA
[email protected] [email protected]

ISSN: 0072-5285
ISBN: 978-3-540-76877-7 e-ISBN: 978-3-540-76878-4
Library of Congress Control Number: 2008938193
Mathematics Subject Classification (2000): 12xx, 94xx, 14xx, 11xx
(cid:176)c 2009Springer-VerlagBerlinHeidelberg
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Preface to the Second Edition
15 years after the first printing of Algebraic Function Fields and Codes, the
mathematics editors of Springer Verlag encouraged me to revise and extend
the book.
Besides numerous minor corrections and amendments, the second edition
differs from the first one in two respects. Firstly I have included a series
of exercises at the end of each chapter. Some of these exercises are fairly
easy and should help the reader to understand the basic concepts, others are
more advanced and cover additional material. Secondly a new chapter titled
“Asymptotic Bounds for the Number of Rational Places” has been added.
This chapter contains a detailed presentation of the asymptotic theory of
function fields over finite fields, including the explicit construction of some
asymptotically good and optimal towers. Based on these towers, a complete
andself-containedproofoftheTsfasman-Vladut-ZinkTheoremisgiven.This
theorem is perhaps the most beautiful application of function fields to coding
theory.
The codes which are constructed from algebraic function fields were first
introducedbyV.D.Goppa.AccordinglyIreferredtotheminthefirstedition
as geometric Goppa codes. Since this terminology has not generally been ac-
cepted in the literature, I now use the more common term algebraic geometry
codes or AG codes.
I would like to thank Alp Bassa, Arnaldo Garcia, Cem Gu¨neri, Sevan
Harput and Alev Topuzog˘lu for their help in preparing the second edition.
Moreover I thank all those whose results I have used in the exercises without
giving references to their work.
I˙stanbul, September 2008 Henning Stichtenoth
Preface to the First Edition
An algebraic function field over K is an algebraic extension of finite degree
overtherational functionfieldK(x)(thegroundfieldK maybeanarbitrary
field). This type of field extension occurs naturally in various branches of
mathematics such as algebraic geometry, number theory and the theory of
compactRiemannsurfaces.Henceonecanstudyalgebraicfunctionfieldsfrom
very different points of view.
In algebraic geometry one is interested in the geometric properties of an
algebraic curve C = {(α,β) ∈ K ×K|f(α,β) = 0}, where f(X,Y) is an
irreducible polynomial in two variables over an algebraically closed field K.
It turns out that the field K(C) of rational functions on C (which is an
algebraicfunctionfieldoverK)containsagreatdealofinformationregarding
the geometry of the curve C. This aspect of the theory of algebraic function
fields is presented in several books on algebraic geometry, for instance [11],
[18], [37] and [38].
One can also approach function fields from the direction of complex anal-
ysis. The meromorphic functions on a compact Riemann surface S form an
algebraicfunctionfieldM(S)overthefieldCofcomplexnumbers.Hereagain,
thefunctionfieldisastrongtoolforstudyingthecorrespondingRiemannsur-
face, see [10] or [20].
In this book a self-contained, purely algebraic exposition of the theory
of algebraic functions is given. This approach was initiated by R. Dedekind,
L.KroneckerandH.M.Weberinthenineteenthcentury(overthefieldC),cf.
[20];itwasfurtherdevelopedbyE.Artin,H.Hasse,F.K.SchmidtandA.Weil
in the first half of the twentieth century. Standard references are Chevalley’s
book‘IntroductiontotheTheoryofAlgebraicFunctionsofOneVariable’[6],
whichappearedin1951,and[7].Thecloserelationshipwithalgebraicnumber
theory is emphasized in [1] and [9].
Thealgebraicapproachtoalgebraicfunctionsismoreelementarythanthe
approachviaalgebraicgeometry:onlysomebasicknowledgeofalgebraicfield
extensions, including Galois theory, is assumed. A second advantage is that
VIII Preface to the First Edition
someprincipalresultsofthetheory(suchastheRiemann-RochTheorem)can
be derived very quickly for function fields over an arbitrary constant field K.
Thisfacilitatesthepresentationofsomeapplicationsofalgebraicfunctionsto
coding theory, which is the second objective of the book.
An error-correcting code is a subspace of IFn, the n-dimensional standard
q
vector space over a finite field IF . Such codes are in widespread use for the
q
reliabletransmissionofinformation.AsobservedbyV.D.Goppain1975,one
can use algebraic function fields over IF to construct a large class of inter-
q
esting codes. Properties of these codes are closely related to properties of the
corresponding function field, and the Riemann-Roch Theorem provides esti-
mates, sharp in many cases, for their main parameters (dimension, minimum
distance).
While Goppa’s construction is the most important, it is not the only link
betweencodesandalgebraicfunctions.Forinstance,theHasse-WeilTheorem
(which is fundamental to the theory of function fields over a finite constant
field) yields results on the weights of codewords in certain trace codes.
A brief summary of the book follows.
The general theory of algebraic function fields is presented in Chapters 1,
3 and 4. In the first chapter the basic concepts are introduced, and A. Weil’s
proof of the Riemann-Roch Theorem is given. Chapter 3 is perhaps the most
important. It provides the tools necessary for working with concrete func-
tion fields: the decomposition of places in a finite extension, ramification and
different, the Hurwitz Genus Formula, and the theory of constant field ex-
tensions. P-adic completions as well as the relation between differentials and
Weil differentials are treated in Chapter 4.
Chapter5dealswithfunctionfieldsoverafiniteconstantfield.Thischap-
tercontainsaversionofBombieri’sproofoftheHasse-WeilTheoremaswellas
someimprovementsoftheHasse-WeilBound.Asanillustrationofthegeneral
theory, several explicit examples of function fields are discussed in Chapter 6,
namely elliptic and hyperelliptic function fields, Kummer and Artin-Schreier
extensions of the rational function field.
TheChapters2,8and9aredevotedtoapplicationsofalgebraicfunctions
to coding theory. Following a brief introduction to coding theory, Goppa’s
construction of codes by means of an algebraic function field is described
in Chapter 2. Also included in this chapter is the relation these codes have
withtheimportantclassesofBCHandclassicalGoppacodes.Chapter8con-
tainssomesupplements:theresiduerepresentationofgeometricGoppacodes,
automorphisms of codes, asymptotic questions and the decoding of geomet-
ric Goppa codes. A detailed exposition of codes associated to the Hermitian
function field is given. In the literature these codes often serve as a test for
the usefulness of geometric Goppa codes.
Chapter 9 contains some results on subfield subcodes and trace codes.
Estimates for their dimension are given, and the Hasse-Weil Bound is used
Preface to the First Edition IX
to obtain results on the weights, dimension and minimum distance of these
codes.
For the convenience of the reader, two appendices are enclosed. Appendix
A is a summary of results from field theory that are frequently used in the
text. As many papers on geometric Goppa codes are written in the language
of algebraic geometry, Appendix B provides a kind of dictionary between the
theory of algebraic functions and the theory of algebraic curves.
Acknowledgements
First of all I am indebted to P. Roquette from whom I learnt the theory
of algebraic functions. His lectures, given 20 years ago at the University of
Heidelberg, substantially influenced my exposition of this theory.
I thank several colleagues who carefully read the manuscript: D. Ehrhard,
P. V. Kumar, J. P. Pedersen, H.-G. Ru¨ck, C. Voss and K. Yang. They sug-
gested many improvements and helped to eliminate numerous misprints and
minor mistakes in earlier versions.
Essen, March 1993 Henning Stichtenoth
Contents
1 Foundations of the Theory of Algebraic Function Fields ... 1
1.1 Places.................................................. 1
1.2 The Rational Function Field .............................. 8
1.3 Independence of Valuations ............................... 12
1.4 Divisors ................................................ 15
1.5 The Riemann-Roch Theorem ............................. 24
1.6 Some Consequences of the Riemann-Roch Theorem .......... 31
1.7 Local Components of Weil Differentials..................... 37
1.8 Exercises ............................................... 40
2 Algebraic Geometry Codes ................................ 45
2.1 Codes.................................................. 45
2.2 AG Codes .............................................. 48
2.3 Rational AG Codes ...................................... 55
2.4 Exercises ............................................... 63
3 Extensions of Algebraic Function Fields ................... 67
3.1 Algebraic Extensions of Function Fields .................... 68
3.2 Subrings of Function Fields ............................... 77
3.3 Local Integral Bases ..................................... 81
3.4 The Cotrace of Weil Differentials and the Hurwitz Genus
Formula................................................ 90
3.5 The Different ........................................... 99
3.6 Constant Field Extensions................................112
3.7 Galois Extensions I ......................................120
3.8 Galois Extensions II .....................................130
3.9 Ramification and Splitting in the Compositum of Function
Fields..................................................137
3.10 Inseparable Extensions ...................................143
3.11 Estimates for the Genus of a Function Field ................145
3.12 Exercises ...............................................150
XII Contents
4 Differentials of Algebraic Function Fields ..................155
4.1 Derivations and Differentials ..............................155
4.2 The P-adic Completion ..................................161
4.3 Differentials and Weil Differentials.........................170
4.4 Exercises ...............................................179
5 Algebraic Function Fields over Finite Constant Fields .....185
5.1 The Zeta Function of a Function Field .....................185
5.2 The Hasse-Weil Theorem .................................197
5.3 Improvements of the Hasse-Weil Bound ....................208
5.4 Exercises ...............................................212
6 Examples of Algebraic Function Fields.....................217
6.1 Elliptic Function Fields ..................................217
6.2 Hyperelliptic Function Fields .............................224
6.3 Tame Cyclic Extensions of the Rational Function Field.......227
6.4 Some Elementary Abelian p-Extensions of K(x),
charK =p>0..........................................232
6.5 Exercises ...............................................235
7 Asymptotic Bounds for the Number of Rational Places ....243
7.1 Ihara’s Constant A(q)....................................243
7.2 Towers of Function Fields ................................246
7.3 Some Tame Towers ......................................259
7.4 Some Wild Towers.......................................261
7.5 Exercises ...............................................285
8 More about Algebraic Geometry Codes....................289
8.1 The Residue Representation of C (D,G)...................289
Ω
8.2 Automorphisms of AG Codes .............................290
8.3 Hermitian Codes ........................................292
8.4 The Tsfasman-Vladut-Zink Theorem.......................297
8.5 Decoding AG Codes .....................................303
8.6 Exercises ...............................................308
9 Subfield Subcodes and Trace Codes........................311
9.1 On the Dimension of Subfield Subcodes and Trace Codes .....311
9.2 Weights of Trace Codes ..................................315
9.3 Exercises ...............................................325
Appendix A. Field Theory.....................................327
Appendix B. Algebraic Curves and Function Fields............335
List of Notations ..............................................345