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Lattices and Codes

A Course Partially Based on Lectures by F. Hirzebruch

  • Wolfgang Ebeling

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

Table of contents

  1. Front Matter
    Pages i-xv
  2. Wolfgang Ebeling
    Pages 1-35
  3. Wolfgang Ebeling
    Pages 37-81
  4. Wolfgang Ebeling
    Pages 83-103
  5. Wolfgang Ebeling
    Pages 105-126
  6. Back Matter
    Pages 167-180

About this book

Introduction

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).

Keywords

Dimension algebra classification code coding coding theory design error-correcting code history information mathematics modular curve number theory

Authors and affiliations

  • Wolfgang Ebeling
    • 1
  1. 1.Institut für MathematikUniversität HannoverHannoverGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-322-96879-1
  • Copyright Information Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden GmbH, Wiesbaden 1994
  • Publisher Name Vieweg+Teubner Verlag
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-528-06497-6
  • Online ISBN 978-3-322-96879-1
  • Series Print ISSN 0932-7134
  • Buy this book on publisher's site