Characters and Cyclotomic Fields in Finite Geometry

106 Pages·2002·0.525 MB·English

by  Bernhard Schmidt (auth.)

1797
Lecture Notes in Mathematics
Editors:
J.-M.Morel,Cachan
F.Takens,Groningen
B.Teissier,Paris
3
Berlin
Heidelberg
NewYork
HongKong
London
Milan
Paris
Tokyo
Bernhard Schmidt
Characters and
Cyclotomic Fields in
Finite Geometry
1 3
Author
BernhardSchmidt
InstitutfürMathematik
UniversitätAugsburg
Universitätsstrasse14
86135Augsburg,Germany
E-mail:[email protected]
Cataloging-in-PublicationDataappliedfor.
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Schmidt, Bernhard:
Characters and cyclotomic fields in finite geometry / Bernhard Schmidt. -
Berlin ; Heidelberg ; New York ; Hong Kong ; London ; Milan ; Paris ; Tokyo
: Springer, 2002
(Lecture notes in mathematics ; 1797)
ISBN 3-540-44243-X
MathematicsSubjectClassification(2000):
05B10,05B20,05B25
ISSN0075-8434
ISBN3-540-44243-xSpringer-VerlagBerlinHeidelbergNewYork
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Acknowledgement
IwouldliketothankSiuLunMaandClintonWhiteforfruitfulcooperation.
Special thanks go to Dieter Jungnickel for help and encouragement.

Contents
1 Introduction 1
1.1 The nature of the problems . . . . . . . . . . . . . . . . . . . 2
1.2 The combinatorial structures in question . . . . . . . . . . . . 4
1.2.1 Designs . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Difference Sets . . . . . . . . . . . . . . . . . . . . . . 6
1.2.3 Projective planes and planar functions . . . . . . . . . 7
1.2.4 Projective geometries and Singer difference sets . . . . 9
1.2.5 Hadamard matrices and weighing matrices . . . . . . 10
1.2.6 Irreducible cyclic codes, two-intersection sets and sub-
difference sets . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Group rings, characters, Fourier analysis . . . . . . . . . . . . 14
1.4 Number theoretic tools . . . . . . . . . . . . . . . . . . . . . . 19
1.5 Algebraic-combinatorial tools . . . . . . . . . . . . . . . . . . 24
2 The field descent 27
2.1 The fixing theorem . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Prescribed absolute value . . . . . . . . . . . . . . . . . . . . 31
2.3 Bounding the absolute value . . . . . . . . . . . . . . . . . . . 36
2.4 The modulus equation and class groups . . . . . . . . . . . . 37
2.4.1 Class groups of cyclotomic fields . . . . . . . . . . . . 39
2.4.2 Class groups of CM-fields . . . . . . . . . . . . . . . . 44
2.4.3 p-ranks and class fields towers. . . . . . . . . . . . . . 48
3 Exponent bounds 53
3.1 Self-conjugacy exponent bounds . . . . . . . . . . . . . . . . . 53
3.1.1 Turyn’s exponent bound . . . . . . . . . . . . . . . . . 54
3.1.2 The coset intersection lemma . . . . . . . . . . . . . . 55
3.1.3 McFarland difference sets . . . . . . . . . . . . . . . . 57
3.1.4 Semiregular relative difference sets . . . . . . . . . . . 58
3.1.5 Two recent families of difference sets . . . . . . . . . . 61
3.1.6 Chen difference sets . . . . . . . . . . . . . . . . . . . 62
3.1.7 Davis-Jedwab difference sets . . . . . . . . . . . . . . 66
3.2 Field descent exponent bounds . . . . . . . . . . . . . . . . . 67
VIII CONTENTS
3.2.1 A general exponent bound for difference sets . . . . . 68
3.2.2 Difference sets with gcd(v,n)>1 . . . . . . . . . . . . 69
3.2.3 Towards Ryser’s conjecture . . . . . . . . . . . . . . . 71
3.2.4 Circulant Hadamard matrices and
Barker sequences . . . . . . . . . . . . . . . . . . . . . 73
3.2.5 Relative difference sets and planar functions . . . . . . 74
3.2.6 Group invariant weighing matrices . . . . . . . . . . . 77
4 Two-weight irreducible cyclic codes 79
4.1 A necessary and sufficient condition . . . . . . . . . . . . . . 80
4.2 All two-weight irreducible cyclic codes? . . . . . . . . . . . . 83
4.2.1 Subfield and semiprimitive codes . . . . . . . . . . . . 83
4.2.2 The exceptional codes . . . . . . . . . . . . . . . . . . 84
4.3 Partial proof of Conjecture 4.2.4 . . . . . . . . . . . . . . . . 85
4.4 Two-intersection sets and sub-difference sets . . . . . . . . . . 88
4.4.1 Two-intersection sets in PG(m−1,q) . . . . . . . . . 88
4.4.2 Sub-difference sets of Singer difference sets . . . . . . 88
Bibliography 91
Index 99
Chapter 1
Introduction
Thismonographcontributestotheexistencetheoryofcombinatorialobjects
admitting certain types of automorphism groups. We will investigate (rel-
ative) difference sets, planar functions, group invariant weighing matrices,
andtwo-weightirreduciblecycliccodes. Allthesecombinatorialobjectshave
in common that they can be studied in terms of group ring equations, see
Section 1.3. We use Fourier analysis on abelian groups to derive necessary
conditions on the existence of the objects in question. This approach was al-
readyusedinthefundamentalworkofTuryn[127]anddozensofsubsequent
papers on difference sets. However, all these results rely on severe technical
conditions, the self-conjugacy assumption being the most infamous one, see
Remark 1.4.4. The main merit of this monograph is the development of a
method free from such severe assumptions and thus providing nonexistence
theorems of broader applicability than all previously known results. We will
obtainsubstantialprogresstowardsthreemajorconjectureswhichpreviously
hadseemedoutofreach: ThecirculantHadamardmatrixconjecture,Ryser’s
conjecture and the Barker conjecture, see Sections 3.2.3, 3.2.4.
Theseresultswillbeprovedbythenewmethodofthe“fielddescent”which
will be developed in Chapter 2. Roughly speaking, the field descent means
thatcyclotomicintegersX forwhich|X|2 isrationalusuallyarecontainedin
amuchsmallercyclotomicfieldthanaprioriexpected. Thefielddescentnot
onlycanbeusedtoprovenonexistenceresultsforcombinatorialobjects, but
alsoprovidestheprobablymostelementaryapproachtoclassgroupestimates
forCM-fields. Ourresultsonclassnumberfactorsandboundsonp-ranksof
class groups are comparable – in some cases even seem stronger – than those
obtained by the usual methods of class field theory and Galois cohomology.
Though our number theoretic results are only a by-product of our work, we
believe that they establish an interesting connection between combinatorial
and number theoretic questions.
Besides the field descent, this monograph contains two further major contri-
B.Schmidt:LNM1797,pp.1–25,2002.
(cid:1)c Springer-VerlagBerlinHeidelberg2002
2 CHAPTER 1. INTRODUCTION
butions: An improvement of Turyn’s self-conjugacy exponent bounds in Sec-
tion 3.1 and a classification of two-weight irreducible cyclic codes in Chapter
4. The self-conjugacy exponent bounds are obtained by a substantial refine-
ment of Turyn’s method. In some cases, we obtain a dramatic improvement
of Turyn’s bounds and are able to provide necessary and sufficient condi-
tions on the existence of certain difference sets. In Chapter 4, we use the
Fourieranalysisapproachfortheinvestigationoftwo-weightirreduciblecylic
codes. Though these objects have been studied in many papers (see [18]), a
classification had not even been attempted yet. We will give a conjecturally
complete classification and provide evidence for the completeness through
theoretical results and computer searches.
Thestructureofthismonographisasfollows. InChapter1,weintroducethe
combinatorial objects we will study and provide the necessary algebraic and
numbertheoreticbackground. AllresultslistedinChapter1werepreviously
known. In the following chapters, almost all results are new; previously
known work is quoted as “Result xyz”. In Chapter 2, we develop the
methodofthefielddescentandgiveitsapplicationstoclassgroupestimates.
Chapter3containsthevariousexponentboundswederivebyFourieranalysis
together with the field descent and algebraic-combinatorial methods. The
results of Sections 3.1.2 3.1.3 and 3.1.4 are joint work with Siu Lun Ma.
Chapter 4 on two-weight irreducible codes is joint work with Clinton White.
1.1 The nature of the problems
Good news! This section is readable for anyone who knows what a root
of unity is. Through a hopefully well chosen example, we intend to give
an impression of the nature of the combinatorial problems as well as of the
typical methods the reader will be confronted with later. We will use only a
minimumofterminologysothatthematerialshouldbeveryeasilyaccessible.
What we mainly will study is the existence problem for combinatorial struc-
tures invariant under certain operations, often called automorphisms. Such
an existence problem usually is – if ever – decided in one of two ways: Pos-
itively through a construction of the desired object or negatively through
a nonexistence proof. This monograph mainly contributes to the negative
world though we also will have some petite positive news in the chapter on
two-weight codes.
Let us illustrate the typical questions by an example. The combinatorial
structure we consider is Hadamard matrices. An Hadamard matrix of
order v is a v ×v-matrix H with entries ±1 any two rows of which are
orthogonal. The operation under which we require the Hadamard matrix to
be invariant is something like cyclic shifting. More precisely, writing H =
(h )v−1 , we require h = h for all i,j where the indices are taken
i,j i,j=0 i+1,j+1 i,j
modulo v. A matrix satisfying this condition is called circulant. It has the
form