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Lattices and codes : a course partially based on lectures by F. Hirzebruch

Author: Wolfgang Ebeling; Friedrich Hirzebruch
Publisher: Braunschweig/Wiesbaden : Vieweg, ©1994.
Series: Advanced lectures in mathematics.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
The purpose of coding theory is the design of efficient systems for the transmission of information. The mathematical treatment leads to certain finite structures: the error-correcting codes. Surprisingly problems which are interesting for the design of codes turn out to be closely related to problems studied partly earlier and independently in pure mathematics. This book is about an example of such a connection:  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Ebeling, Wolfgang.
Lattices and codes.
Braunschweig/Wiesbaden : Vieweg, ©1994
(DLC) 94193344
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Wolfgang Ebeling; Friedrich Hirzebruch
ISBN: 9783322968791 3322968790
OCLC Number: 828774955
Language Note: English.
Description: 1 online resource (xii, 178 pages) : illustrations.
Contents: 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 --
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 .
Series Title: Advanced lectures in mathematics.
Responsibility: Wolfgang Ebeling.

Abstract:

This volume is about the relation between codes and lattices. Lattices are studied in number theory and in the geometry of numbers. Many problems about codes have their counterpart in problems about  Read more...

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