Lattices and Codes

A Course Partially Based on Lectures by Friedrich Hirzebruch

verfasst von: Wolfgang Ebeling

Verlag: Springer Fachmedien Wiesbaden

Buchreihe : Advanced Lectures in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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. In this book, examples of such connections are presented. The relation between lattices studied in number theory and geometry and error-correcting codes is discussed. The book provides at the same time an introduction to the theory of integral lattices and modular forms and to coding theory.

In the 3rd edition, again numerous corrections and improvements have been made and the text has been updated.

Inhaltsverzeichnis

Frontmatter

1. Lattices and Codes

Abstract

In this section we introduce the basic concept of a lattice in Rn. For references see [81],[61], [9], [45], and [72].

Wolfgang Ebeling

2. Theta Functions and Weight Enumerators

Abstract

Let Γ ⊂ Rn be a lattice. We associate to Γ a function which is defined on the upper half plane H = {τ ∈ C | Im τ > 0} ⊂ C.

Wolfgang Ebeling

3. Even Unimodular Lattices

Abstract

In this section we study modified theta functions, namely theta series with spherical coefficients, and their behavior under transformations of the modular group. The results of this section are due to E. Hecke [32] and B. Schoeneberg [78, 79]. Our presentation follows [71, Chap. VI] and [83].

Wolfgang Ebeling

4. The Leech Lattice

Abstract

This chapter is devoted to this important lattice. We shall first show the uniqueness of this lattice. We shall prove the following theorem.

Wolfgang Ebeling

5. Lattices over Integers of Number Fields and Self-Dual Codes

Abstract

Up to now, we have only considered binary codes. In this chapter we want to present a generalization of the results of Chapter 2 to self-dual codes over Fp, where p is an odd prime number. The results which we want to discuss are due to G. van der Geer and F. Hirzebruch [36, pp. 759-798]. In Sect. 1.3 we associated an integral lattice in Rn to a binary linear code of length n. In Sect. 5.2 we shall generalize this construction by associating a lattice over the integers of a cyclotomic field to a linear code over Fp. In this section we shall study lattices over integers of cyclotomic fields. For the background on algebraic number theory see also [76] and [87].

Wolfgang Ebeling

Backmatter

Titel: Lattices and Codes
verfasst von: Wolfgang Ebeling
Verlag: Springer Fachmedien Wiesbaden
Electronic ISBN: 978-3-658-00360-9
Print ISBN: 978-3-658-00359-3
DOI: https://doi.org/10.1007/978-3-658-00360-9

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

1. Lattices and Codes

2. Theta Functions and Weight Enumerators

3. Even Unimodular Lattices

4. The Leech Lattice

5. Lattices over Integers of Number Fields and Self-Dual Codes

Backmatter

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