Finite Geometry and Character Theory
Difference sets are of central interest in finite geometry and design theory. One of the main techniques to investigate abelian difference sets is a discrete version of the classical Fourier transform (i.e., character theory) in connection with algebraic
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1601
Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen
1601
Alexander Pott
Finite Geometry and Character Theory
Springer
Author Alexander Pott Institut fur Mathematik der Universitat Augsburg Universitatsstralle 14 D-86135 Augsburg, Germany
Mathematics Subject Classification (1991): 05B05, 05B 10, 05B20, 05E20, 11R04, llT24, 20B25, 51E15, 51E30, 94A55, 94B15
ISBN 3-540-59065-X Springer-Verlag Berlin Heidelberg New York CIP-Data applied for This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer- Verlag. Violations are liable for prosecution under the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1995 Printed in Germany Typesetting: Camera-ready TEX output by the author SPIN: 10130289 46/3142-543210 - Printed on acid-free paper
Preface It is the aim of this monograph to demonstrate the applications of algebraic number theory and character theory in order to solve problems in finite geometry, in particular problems about difference sets and their corresponding codes and sequences. The idea is the following: Assume that D is an element in the integral group ring Z[G] which satisfies a certain "basic" equation like DD(-l) = M. (We will always assume that G is a finite group.) If G is abelian, we can apply characters X (or I-dimensional representations) to compute X(M). Using results from algebraic number theory, it is then (sometimes) possible to get informations about X(D). This might be helpful to construct D or to prove that no such D can exist which satisfies the basic equation (non-existence). These are the two fundamental problems concerning difference sets, and they will be investigated in this monograph. The classical paper where this approach has been first used is Turyn [167]. In the first chapter we will present the algebraic approach to attack these two problems. In the second chapter we will describe several constructions of difference sets and state some non-existence results. We do not present all known results in complete generality. But the proofs of many theorems about difference sets use ideas which are very closely related to the techniques developed in this monograph, in particular those presented in the first two chapters. I hope that these chapters are particularly useful for those readers who want to become acquainted with "difference set problems" . In most cases our groups are abelian. Of course, one can also ask for solutions of the "basic" equation in Z[G] if G is non-abelian. Sometimes our techniques are also applicable if G has a "large" homomorphic image which is abelian. But even i
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