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  • Textbook
  • © 2002

Lattices and Codes

A Course Partially Based on Lectures by F. Hirzebruch

Authors:

  1. Wolfgang Ebeling

    1. Institut für Mathematik, Universität Hannover, Hannover, Germany

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  • Modern Aspects in the Design of Codes

Part of the book series: Advanced Lectures in Mathematics (ALM)

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Table of contents (5 chapters)

  1. Front Matter

    Pages i-xvii
    PDF

  2. Lattices and Codes

    • Wolfgang Ebeling
    Pages 1-37

  3. Theta Functions and Weight Enumerators

    • Wolfgang Ebeling
    Pages 39-86

  4. Even Unimodular Lattices

    • Wolfgang Ebeling
    Pages 87-108

  5. The Leech Lattice

    • Wolfgang Ebeling
    Pages 109-134

  7. Back Matter

    Pages 175-190
    PDF
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About this book

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: 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 lattices and sphere packings. We give a detailed introduction to these relations including recent results of G. van der Geer and F. Hirzebruch. Let us explain the history of this book. In [LPS82] J. S. Leon, V. Pless, and N. J. A. Sloane considered the Lee weight enumerators of self-dual codes over the prime field of characteristic 5. They wrote in the introduction to their paper: "The weight enumerator of anyone of the codes . . . is strongly constrained: it must be invariant under a three-dimensional representation of the icosahedral group. These invariants were already known to Felix Klein, and the consequences for coding theory were discovered by Gleason and Pierce (and independently by the third author) . . . (It is worth mentioning that precisely the same invariants have recently been studied by Hirzebruch in connection with cusps of the Hilbert modular surface associated with Q( J5).
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Keywords

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Authors and Affiliations

  • Institut für Mathematik, Universität Hannover, Hannover, Germany

    Wolfgang Ebeling

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About the author

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

  • Book Title: Lattices and Codes

  • Book Subtitle: A Course Partially Based on Lectures by F. Hirzebruch

  • Authors: Wolfgang Ebeling

  • Series Title: Advanced Lectures in Mathematics

  • DOI: https://doi.org/10.1007/978-3-322-90014-2

  • Publisher: Vieweg+Teubner Verlag Wiesbaden

  • eBook Packages: Springer Book Archive

  • Copyright Information: Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden 2002

  • eBook ISBN: 978-3-322-90014-2Published: 06 December 2012

  • Series ISSN: 0932-7134

  • Series E-ISSN: 2512-7039

  • Edition Number: 2

  • Number of Pages: XVIII, 188

  • Topics: Algebra, Number Theory, Algebraic Geometry

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eBook USD 74.99
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  • Available as PDF
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