This monograph contributes to the existence theory of difference sets, cyclic irreducible codes and similar objects. The new method of field descent for cyclotomic integers of presribed absolute value is developed. Applications include the first substantial progress towards the Circulant Hadamard Matrix Conjecture and Ryser`s conjecture since decades. It is shown that there is no Barker sequence of length l with 13<1<4x10^(12). Finally, a conjecturally complete classification of all irreducible cyclic two-weight codes is obtained.
Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris 1797 3 Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo Bernhard Schmidt Characters and Cyclotomic Fields in Finite Geometry 13 Author Bernhard Schmidt Institut für Mathematik Universität Augsburg Universitätsstrasse 14 86135 Augsburg, Germany E-mail:
[email protected] Cataloging-in-Publication Data applied for. Die Deutsche Bibliothek - CIP-Einheitsaufnahme Schmidt, Bernhard: Characters and cyclotomic fields in finite geometry / Bernhard Schmidt. Berlin ; Heidelberg ; New York ; Hong Kong ; London ; Milan ; Paris ; Tokyo : Springer, 2002 (Lecture notes in mathematics ; 1797) ISBN 3-540-44243-X Mathematics Subject Classification (2000): 05B10, 05B20, 05B25 ISSN 0075-8434 ISBN 3-540-44243-x Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and sto