Ebeling Lattices and Codes

A Course Partially Based on Lectures by F. Hirzebruch
2. revidierte Auflage 2002
ISBN: 978-3-322-90014-2
Verlag: Vieweg & Teubner
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 188 Seiten, eBook

Reihe: Advanced Lectures in Mathematics

In the 2nd edition numerous corrections have been made. More basic material has been included to make the text even more self-contained. A new section on the automorphism group of the Leech lattice has been added. Some hints to new results have been incorporated. With several new exercises.

Zielgruppe

Upper undergraduate

Autoren/Hrsg.

Ebeling, Wolfgang

Inhaltsverzeichnis

1 Lattices and Codes.- 1.1 Lattices.- 1.2 Codes.- 1.3 From Codes to Lattices.- 1.4 Root Lattices.- 1.5 Highest Root and Weyl Vector.- 2 Theta Functions and Weight Enumerators.- 2.1 The Theta Function of a Lattice.- 2.2 Modular Forms.- 2.3 The Poisson Summation Formula.- 2.4 Theta Functions as Modular Forms.- 2.5 The Eisenstein Series.- 2.6 The Algebra of Modular Forms.- 2.7 The Weight Enumerator of a Code.- 2.8 The Golay Code and the Leech Lattice.- 2.9 The MacWilliams Identity and Gleason’s Theorem.- 2.10 Quadratic Residue Codes.- 3 Even Unimodular Lattices.- 3.1 Theta Functions with Spherical Coefficients.- 3.2 Root Systems in Even Unimodular Lattices.- 3.3 Overlattices and Codes.- 3.4 The Classification of Even Unimodular Lattices of Dimension 24.- 4 The Leech Lattice.- 4.1 The Uniqueness of the Leech Lattice.- 4.2 The Sphere Covering Determined by the Leech Lattice.- 4.3 Twenty-Three Constructions of the Leech Lattice.- 4.4 Embedding the Leech Lattice in a Hyperbolic Lattice.- 4.5 Automorphism Groups.- 5 Lattices over Integers of Number Fields and Self-Dual Codes.- 5.1 Lattices over Integers of Cyclotomic Fields.- 5.2 Construction of Lattices from Codes over p.- 5.3 Theta Functions over Number Fields.- 5.4 The Case p = 3: Ternary Codes.- 5.5 The Equation of the Tetrahedron and the Cube.- 5.6 The Case p = 5: the Icosahedral Group.- 5.7 Theta Functions as Hilbert Modular Forms (by N.-P. Skoruppa).

Über Autor(innen)

Prof. Dr. Wolfgang Ebeling, Department of Mathematics, Universität Hannover, Germany.