Current Research Topics on Galois Geometrics PDF
Preview Current Research Topics on Galois Geometrics
MATHEMATICS RESEARCH DEVELOPMENTS C R T URRENT ESEARCH OPICS G G IN ALOIS EOMETRY No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services. MATHEMATICS RESEARCH DEVELOPMENTS Additional books in this series can be found on Nova‘s website under the Series tab. Additional E-books in this series can be found on Nova‘s website under the E-books tab. CRYPTOGRAPHY, STEGANOGRAPHY AND DATA SECURITY Additional books in this series can be found on Nova‘s website under the Series tab. Additional E-books in this series can be found on Nova‘s website under the E-books tab. MATHEMATICS RESEARCH DEVELOPMENTS C R T URRENT ESEARCH OPICS G G IN ALOIS EOMETRY LEO STORME AND JAN DE BEULE EDITORS Nova Science Publishers, Inc. New York Copyright © 2012 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. 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In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book. Library of Congress Cataloging-in-Publication Data Storme, Leo. Current research topics on Galois geometrics / Leo Storme and Jan de Beule. p. cm. Includes index. ISBN 978-1-62081-363-8 (eBook) 1. Galois theory. 2. Geometry, Algebraic. I. Beule, Jan de. II. Title. QA214.S76 2011 516'.11--dc22 2 0 1 1 0 0 3 5 6 7 Published by Nova Science Publishers, Inc. †New York CONTENTS Preface vii Chapter1 ConstructionsandCharacterizationsofClassicalSetsinPG(n,q) 1 FrankDeClerckandNicolaDurante Chapter2 SubstructuresofFiniteClassicalPolarSpaces 35 JanDeBeule,AndreasKleinandKlausMetsch Chapter3 BlockingSetsinProjectiveSpaces 63 AartBlokhuis,Pe´terSziklaiandTama´sSzo˝nyi Chapter4 LargeCapsinProjectiveGaloisSpaces 87 Ju¨rgenBierbrauerandYvesEdel Chapter5 ThePolynomialMethodinGaloisGeometries 105 SimeonBall Chapter6 FiniteSemifields 131 MichelLavrauwandOlgaPolverino Chapter7 CodesoverRingsandRingGeometries 161 ThomasHonoldandIvanLandjev Chapter8 GaloisGeometriesandCodingTheory 187 IvanLandjevandLeoStorme Chapter9 ApplicationsofGaloisGeometrytoCryptology 215 Wen-AiJackson,KeithM.MartinandMauraB.Paterson Chapter10 GaloisGeometriesandLow-DensityParity-CheckCodes 245 MarcusGreferath,CorneliaRo¨ßingandLeoStorme Index 271 PREFACE Galois geometry is in our mind a field of mathematics that deals with structures living inprojectivespacesoverafinitefield. Thisisaveryroughdescription,butaswithanyfield inmathematics,itsbordersandcontentsarenotclearlydefined. Several facts have influenced the list of topics that are covered by the chapters in this volume. A wide list of topics are fundamental in the sense that many results in Galois geometry rely on them, not only in the field itself, but also in the wider field of finite (incidence) geometry. We especially think of those structures living in a projective space thatareusedtobuildmodelsofinteresting(non-classical)incidencestructures. Topicsthat arerelatedtoGaloisgeometryandthestudyofsomeofitssubstructures, but thatcanalso be seen as research topics in algebra, have been included. This brings us to the list of relatedtopics. Twofieldsplayaspecialrole: codingtheoryandcryptography. Thereason, forus,thattheyplayaspecialrole,isthatresearchinthesefieldsdoesnotonlyuseresults from Galois geometry, but is also inspiring and influencing research in Galois geometry. Therefore,thelistofcoveredtopicshasaratherlargeintersectionwiththesetwofields,but wetookcarethatthelinkswithGaloisgeometrywerealwaysundoubtedlypresent. Withtheseideasinmind,wecansurveythelistoftopicspresentinthedifferentchap- ters. The first two chapters each discuss a variety of substructures in Galois geometry. They are specifically intended for readers wishing to obtain a broad overview of Galois geometry. In particular, Chapter 1 presents results on classical objects in the projective space PG(n,q), such as arcs, ovals, ovoids and unitals, and Chapter 2 presents results on substructuresofclassicalpolarspaces. Polarspacesareincidencestructuresdescribedby different axioms than projective spaces, but the classical ones are represented completely by symplectic, quadratic and sesquilinear forms on a projective space, and as such, their study fits completely in the field of Galois geometry. Results on (partial) ovoids, (partial) spreads, covers and blocking sets are presented. The next two chapters discuss specific substructures in projective spaces. Chapter 3 discusses results on blocking sets in projec- tive spaces. Blocking sets occur within many problems in Galois geometry, thus giving them a central place within Galois geometry. Chapter 4 presents results on large caps in projective spaces; here a topic is discussed which has a well-known link to coding theory, i.e.,tothecap-codes. Thenexttwochaptersdivergeabitfromthefourprecedingchapters viii J.DeBeuleandL.Storme to introduce two important related topics. Chapter 5 is entitled The polynomial method in Galoisgeometries,discussesapowerfultechniquewithinGaloisgeometry,andpresentsre- sultsthathaveastrongalgebraicnature,butthathaveimmediateconsequencesforsomeof thementionedstructuresinprojectivespaces,especiallyblockingsetsofprojectivespaces. Chapter6presentsresultsonfinitesemifields;thesearealgebraicstructuresthatarerelated to e.g. spreads of projective spaces. Also this chapter has a more algebraic nature, but its connections with Galois geometry are clearly described. These first 6 chapters can be thoughtofasthepartofthecollectionthatdealswithfundamentalsofGaloisgeometry. Chapter7presentsresultsoncodesoverringgeometries. Chapter8presentsresultson linearcodes,andinparticularonthosegeometricobjectsrelatedtolinearcodes. Similarly, Chapter 9 presents applications of Galois Geometry to Cryptology. Finally, Chapter 10 presents results on LDPC codes and on their links to Galois geometry. These last four chapters can be thought of as the part of the collection that deals with applications, but as we explained, many of the topics presented here have influenced and inspired research in Galoisgeometry,andthisfactcanbefoundthroughoutthesechapters. Undoubtedly,(many)moretopicscouldhavebeenincludedinthisvolume. During the editorial process, cross references between chapters originated, but each chapter of this volume is a self contained paper, and it can be read independently of the others. We assume that the reader is familiar with the basic concepts of Galois Geometry. A thoroughintroductiontoGaloisgeometrycanbefoundinthethreefundamentalvolumesof HirschfeldonGaloisgeometry,ofwhichThasistheco-authorforthethirdvolume;[1–3]. The aim and hope of this collected volume on Galois geometry is to give the readers a survey of current important research topics in Galois geometry, describing to them the main results, main techniques and ideas that led to these results, and to present them open problems for future research. We hope that the chapters of this collected work inspire and motivate the readers to contribute to Galois geometry, and encourage them to continue or initiate research on Galois geometry. There is something for everybody’s taste in Galois geometry! JanDeBeuleandLeoStorme April10,2010 References [1] J.W.P.Hirschfeld,Finiteprojectivespacesofthreedimensions,OxfordMathematical Monographs, TheClarendonPressOxfordUniversityPress,NewYork, 1985. Oxford SciencePublications. [2] , Projectivegeometriesoverfinitefields, OxfordMathematicalMonographs, The ClarendonPressOxfordUniversityPress,NewYork,seconded.,1998. [3] J. W. P. Hirschfeld and J. A. Thas, General Galois geometries, Oxford Mathematical Monographs, TheClarendonPressOxfordUniversityPress,NewYork, 1991. Oxford SciencePublications.
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