Characters and Cyclotomic Fields in Finite Geometry

Frontmatter

1. Introduction

Abstract

• 1.1 The nature of the problems
• 1.2 The combinatorial structures in question

  • 1.2.1 Designs
  • 1.2.2 Difference Sets
  • 1.2.3 Projective planes and planar functions
  • 1.2.4 Projective geometries and Singer difference sets
  • 1.2.5 Hadamard matrices and weighing matrices
  • 1.2.6 Irreducible cyclic codes, two-intersection sets and sub-difference sets
• 1.3 Group rings, characters, Fourier analysis
• 1.4 Number theoretic tools
• 1.5 Algebraic-combinatorial tools

Bernhard Schmidt

2. The field descent

Abstract

• 2.1 The fixing theorem
• 2.2 Prescribed absolute value
• 2.3 Bounding the absolute value
• 2.4 The modulus equation and class groups

  • 2.4.1 Class groups of cyclotomic fields
  • 2.4.2 Class groups of CM-fields
  • 2.4.3 p-ranks and class fields towers

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3. Exponent bounds

Abstract

• 3.1 Self-conjugacy exponent bounds

  • 3.1.1 Turyn’s exponent bound
  • 3.1.2 The coset intersection lemma
  • 3.1.3 McFarland difference sets
  • 3.1.4 Semiregular relative difference sets
  • 3.1.5 Two recent families of difference sets
  • 3.1.6 Chen difference sets
  • 3.1.7 Davis-Jedwab difference sets
• 3.2 Field descent exponent bounds

  • 3.2.1 A general exponent bound for difference sets
  • 3.2.2 Difference sets with \({\rm gcd}(v,n) > 1\)
  • 3.2.3 Towards Ryser’s conjecture
  • 3.2.4 Circulant Hadamard matrices and Barker sequences
  • 3.2.5 Relative difference sets and planar functions
  • 3.2.6 Group invariant weighing matrices

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4. Two-weight irreducible cyclic codes

Abstract

• 4.1 A necessary and sufficient condition
• 4.2 All two-weight irreducible cyclic codes?

  • 4.2.1 Subfield and semiprimitive codes
  • 4.2.2 The exceptional codes
• 4.3 Partial proof of Conjecture 4.2.4
• 4.4 Two-intersection sets and sub-difference sets

  • 4.4.1 Two-intersection sets in PG(m–1,q)
  • 4.4.2 Sub-difference sets of Singer difference sets

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Bibliography

Abstract

Abstract not available

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Index

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Abstract not available

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Backmatter