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Lattices and Codes: A Course Partially Based on Lectures by F.Hirzebruch (Advanced Lectures in Mathematics)
  
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Lattices and Codes: A Course Partially Based on Lectures by F.Hirzebruch (Advanced Lectures in Mathematics) (Hardcover)

by Wolfgang Ebeling (Author)
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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. It 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 lattices and sphere packings. The book starts with the basic definitions and examples of lattices and codes. A central theme is a fundamental correspondence between binary linear codes and certain integral lattices. The theta function of a lattice is introduced and it is shown that it is a modular form. Several applications of the theory of modeular forms to weight enumerators of codes are discussed. The classification of even unimodular lattices up to dimension 24 is studied using theta functions with spherical coefficients. Special attention is devoted to the Leech lattice, its constructions and the sphere covering determined by it. Finally, the book contains a detailed account on recent results of G. van der Geer and F. Hirzebruch concerning a generalization of some of the relations studied earlier in the book to self-dual codes over certain finite fields with more than two elements and lattices over the integers of certain algebraic fields. The book can serve as a text for a course. It should also be of use for students and mathematicians working in number theory, geometry or coding theory.

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