Edited by. University of Augsburg

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1 DESIGNS AND FINITE GEOMETRIES Edited by Dieter J ungnickel University of Augsburg A Special Issue of DESIGNS, CODES AND CRYPTOGRAPHY An International Journal Volume 8, No. 112 (1996) KLUWER ACADEMIC PUBLISHERS Boston / Dordrecht / London

2 DESIGNS, CODES AND CRYPTOGRAPHY An International J oumal Volume 8, No. 112, May 1996 Special Issue Dedicated to Han/ried Lenz Guest Editor: Dieter lungnickel Preface... Dieter Jungnickel 7 A Life's Work in Geometry: An Homage to Hanfried Lenz Dieter Jungnickel and Gunter Pickert 9 Impossibility of a Certain Cyclotomic Equation with Applications to Difference Sets K. T. Arasu and Alexander Pott 23 On the Binary Codes of Steiner Triple Systems Alphonse Baartmans, Ivan Landjev, and Vladimir D. Tonchev 29 Orthogonal Partitions in Designed Experiments... R. A. Bailey 45 Regulus-free Spreads of PG(3, q).... R. D. Baker and G. L. Ebert 79 Designs, Codes and Crypts-A Puzzle Altogether... Thomas Beth 91 5-Cycle Systems with Holes Darryn E. Bryant, D. G. Hoffman, and C. A. Rodger 103 Stories about Groups and Sequences Peter J. Cameron 109 Groups Admitting a Kantor Family and a Factorized Normal Subgroup Dirk Hachenberger 135 Spreads in Strongly Regular Graphs Willem H. Haemers and Vladimir D. Tonchev 145 Codes Based on Complete Graphs Dieter Jungnickel, Marialuisa J. de Resmini, and Scott A. Vanstone 159 A Construction of Partial Difference Sets in 7l,p2 x 7l,p2 X... X 7l,p Ka Bin Leung and Siu Lun Ma 167

3 On the Characterisation of AG(n, q) by its Parameters as a Nearly Triply Regular Design Arlene A. Pascasio, Cheryl E. Praeger, and Blessilda P. Raposa 173 The Fundamental Theorem of q-clan Geometry... S. E. Payne 181 Extension of Gravity Centers Configuration to Steiner Triple Systems Gunter Pickert 203 Constructions of Partial Difference Sets and Relative Difference Sets Using Galois Rings D. K. Ray-Chaudhuri and Qing Xiang 215 m-systems and Partial m-systems of Polar Spaces E. E. Shult and J. A. Thas 229 Piotrowski's Infinite Series of Steiner Quadruple Systems Revisited Helmut Siemon 239

4 Distributors for North America: Kluwer Academic Publishers 101 Philip Drive Assinippi Park Norwell, Massachusetts USA Distributors for all other countries: Kluwer Academic Publishers Group Distribution Centre Post Office Box AH Dordrecht, THE NETHERLANDS Library of Congress Cataloging-in-Publication Data A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN-13: DOl: / e-isbn-13: Copyright 1996 by Kluwer Academic Publishers Softcover reprint of the hardcover 1 st edition 1996 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, mechanical, photo-copying, recording, or otherwise, without the prior written permission of the publisher, Kluwer Academic Publishers, 101 Philip Drive, Assinippi Park, Norwell, Massachusetts Printed on acid-free paper.

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6 Designs, Codes and Cryptography, 8, 7 (1996) 1996 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands. Preface The present special issue of Designs, Codes and Cryptography consists of some of the papers which have been submitted (by invitation) in honour of Professor Dr. Hanfried Lenz who celebrates his 80th birthday on April 22, 1996; one or two further such issues are to follow later. Professor Lenz is well-known for his fundamental work in all parts of geometry; in particular, his major interest in the last two decades has been in Finite Geometry and Design Theory. For a detailed description of Prof. Lenz' research, I refer to the joint article by Prof. G. Pickert and myself at the beginning of this issue. I take this opportunity to thank all the authors of these special Lenz issues for their contributions and all the referees for their (in most cases) speedy replies which made it possible to produce at least the first of these issues well in time for Prof. Lenz' actual birthday and thus also for his 80th birthday celebration to be held at the Freie Universitat Berlin on May 11, It only remains to express my hope that Prof. Lenz will like the celebration as well as this special tribute prepared for him and will continue to enjoy both good health and an ongoing interest in mathematics. Dieter lungnickel Augsburg, September 7, 1995