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Second Edition Volume I
Second Edition Thomas Beth Universitat¨ Karlsruhe Dieter Jungnickel Universitat¨ Augsburg Hanfried Lenz Freie Universitat¨ Berlin Volume I PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK www.cup.cam.ac.uk 40 West 20th Street, New York, NY 10011-4211, USA www.cup.org 10 Stamford Road, Oakleigh, Melbourne 3166, Australia Ruiz de Alarc´on 13, 28014 Madrid, Spain First edition c Bibliographisches Institut, Zurich, 1985 c Cambridge University Press, 1993 Second edition c Cambridge University Press, 1999 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1999 Printed in the United Kingdom at the University Press, Cambridge Typeset in Times Roman 10/13pt. in LATEX2ε[TB] A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication data Beth, Thomas, 1949– Design theory / Thomas Beth, Dieter Jungnickel, Hanfried Lenz. – 2nd ed. p. cm. Includes bibliographical references and index. ISBN 0 521 44432 2 (hardbound) 1. Combinatorial designs and configurations. I. Jungnickel, D. (Dieter), 1952– . II. Lenz, Hanfried. III. Title. QA166.25.B47 1999 5110.6 – dc21 98-29508 CIP ISBN 0 521 44432 2 hardback Contents I. Examples and basic definitions .................... 1 §1. Incidence structures and incidence matrices ............ 1 §2. Block designs and examples from affine and projective geometry ........................6 §3. t-designs, Steiner systems and configurations ......... -
Balanced Generalized Weighing Matrices and Their Applications
LE MATEMATICHE Vol. LIX (2004) – Fasc. I–II, pp. 225–261 BALANCED GENERALIZED WEIGHING MATRICES AND THEIR APPLICATIONS DIETER JUNGNICKEL - H. KHARAGHANI Balanced generalized weighing matrices include well-known classical combinatorial objects such as Hadamard matrices and conference matrices; moreover, particular classes of BGW-matrices are equivalent to certain rel- ative difference sets. BGW-matrices admit an interesting geometrical inter- pretation, and in this context they generalize notions like projective planes admitting a full elation or homology group. After surveying these basic con- nections, we will focus attention on proper BGW-matrices; thus we will not give any systematic treatment of generalized Hadamard matrices, which are the subject of a large area of research in their own right. In particular, we will discuss what might be called the classical param- eter series. Here the nicest examples are closely related to perfect codes and to some classical relative difference sets associated with affine geometries; moreover, the matrices in question can be characterized as the unique (up to equivalence) BGW-matrices for the given parameters with minimum q-rank. One can also obtain a wealth of monomially inequivalent examples and deter- mine the q-ranks of all these matrices by exploiting a connection with linear shift register sequences. The final section of this survey will consider applications to construc- tions of designs and graphs. We will divide this section into six parts. The first of these will deal with the work of Ionin. The second part will be devoted to Bush–type Hadamard matrices and twin designs. In the third part we will Keywords: Balanced generalized weighing matrix, Relative difference set, Simplex code, Symmetric design, Regular Hadamard matrix, Productive regular Hadamard matrix, Bush–type Hadamard matrix 226 DIETER JUNGNICKEL - H. -
Circulant Weighing Matrices
Circulant Weighing Matrices by Goldwyn Millar A Thesis submitted to the Faculty of Graduate Studies In Partial Fulfillment of the Requirements for the Degree of MASTER OF SCItrNCtr Department of Mathematics University of Manitoba Winnipeg, Manitoba Copyright @ 2009 by Goldwyn Millar THE UNIVBRSTTY OF MANTTOBA FACULTY OF GRADUATB STUDIBS COPYRIGHT PBRMISSION Circulant Weighing Matrices B--v Goldwyn Millar A Thesis/Practicum submitted to thc Faculty of Graduate Studies of The Univer.sity of Mnnitob¿r in ¡rartitl fulfillure nt of thc requirernent of the degree of Master of Science Goldrvvn MillarO2009 Permission h¿rs been grirntetl to the Univcl'sit],of M¿rnitoba Libraries to le¡rd a copy of this thesis/¡rracticttm, to Librar¡' antl Arctrives C¿rn¿rtla (LAC) to lencl a copy of this thesiii¡rracticum, and to LAC's âgent (UMIiProQuest) to microfilm, sellcopies and to pi t lirtr an abstr.act of this thesis/prncticum. This reproductiotl or copy of this tllesis h¿rs been rn:rde available by authority of the copyright owner solely for the pur¡rose of ¡rrivate study and reseârch, *nd may onl¡, 1¡" reprottucert aria copiea as perrnitted b¡' co¡rvright larvs or rvith ext)rcss n'l'itten ¿ruthoriz¿rtion frour thà copyrig¡t on,nér. Abstract A ci,rculant matrin is a matrix such that each of its rows, after the first, can be obtained from the row above it by a right cyclic shift. A ci,rculant wei,gh'ing matri,r of order n and weight w (a CW(n,w)) is a n x n circuiant (0, t1)-matrix trZ such that WWr : wI. -
Applications of Finite Geometries to Designs and Codes
Michigan Technological University Digital Commons @ Michigan Tech Dissertations, Master's Theses and Master's Dissertations, Master's Theses and Master's Reports - Open Reports 2011 Applications of finite geometries ot designs and codes David C. Clark Michigan Technological University Follow this and additional works at: https://digitalcommons.mtu.edu/etds Part of the Mathematics Commons Copyright 2011 David C. Clark Recommended Citation Clark, David C., "Applications of finite geometries ot designs and codes", Dissertation, Michigan Technological University, 2011. https://doi.org/10.37099/mtu.dc.etds/199 Follow this and additional works at: https://digitalcommons.mtu.edu/etds Part of the Mathematics Commons APPLICATIONS OF FINITE GEOMETRIES TO DESIGNS AND CODES By David C. Clark A DISSERTATION Submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY (Mathematical Sciences) MICHIGAN TECHNOLOGICAL UNIVERSITY 2011 c 2011 David C. Clark This dissertation, "Applications of Finite Geometries to Designs and Codes," is hereby approved in partial fulfillment of the requirements for the Degree of DOC- TOR OF PHILOSOPHY IN MATHEMATICAL SCIENCES. Department of Mathematical Sciences Signatures: Dissertation Advisor Dr. Vladimir Tonchev Committee Member Dr. Donald Kreher Committee Member Dr. Stefaan De Winter Committee Member Dr. Steven Seidel Department Chair Dr. Mark Gockenbach Date Contents List of Figures ..................................... ix List of Tables ...................................... xi Preface .........................................xiii Acknowledgments ................................... xv Abstract .........................................xvii 1 Designs, codes, and finite geometries ....................... 1 1.1 Introduction .................................. 1 1.1.1 Designs ................................ 2 1.1.2 Error-correcting codes ........................ 4 1.1.3 The codes of a design and p-ranks .................. 5 1.2 Finite geometry designs and codes ..................... -
On Generalized Hadamard Matrices and Difference Matrices: $ Z 6$
On Generalized Hadamard Matrices and Difference Matrices: Z6 Bradley W. Brock∗, Robert Compton†, Warwick de Launey‡ and Jennifer Seberry§ October 26, 2015 We give some very interesting matrices which are orthogonal over groups and, as far as we know, referenced, but in fact undocumented. This note is not intended to be published but available for archival reasons. Keywords: Difference Matrices, Generalised/Generalized Hadamard Matrices; Bhaskar Rao Designs, Orthogonal Matrices; Cretan Matrices; Butson-Hadamard Matrices; 05B20. 1 Introduction (Caution: please note Jennifer Seberry has tried but been unable to contact the persons named as co-authors. Apolgogies for half finished results.) We note that the literature on this subject is very disorganized. Authors have not read the literature on their own key-words and papers thus ignored, on the other hand papers have been claimed to be published which have not. We have put together this note to make them accessible to all. We do not give examples of all the literature but point to references which may help. This work is compiled from journal work of the authors from old pieces of paper. We first note that the matrices we study here have elements from groups, abelian and non-abelian, and may be written in additive or multiplicative notation. The matrices may have real elements, elements ∈ {1, −1}, elements SnS≤ 1, elements ∈ {1, i, i2 = −1}, elements ∈ {1, i, −1, −i, i2 =−1}, integer elements ∈ {a + ib, i2 =−1}, nth roots of unity, the quaternions {1 and i, j, k, i2 = j2 = k2 =−1, ijk =−1}, (a + ib)+ j(c + id), a, b, c, d, arXiv:1510.06816v1 [math.CO] 23 Oct 2015 integer and i, j, k quaternions or otherwise as specified. -
On the History of Ring Geometry (With a Thematical Overview of Literature)
On the history of ring geometry (with a thematical overview of literature) Dirk Keppens Faculty of Engineering Technology, KU Leuven Gebr. Desmetstraat 1 B-9000 Ghent BELGIUM [email protected] Abstract In this survey paper we give an historical and at the same time the- matical overview of the development of “ring geometry” from its origin to the current state of the art. A comprehensive up-to-date list of literature is added with articles that treat ring geometry within the scope of inci- dence geometry. In questo documento di ricerca forniamo una panoramica storica e allo stesso tempo tematica dello sviluppo della “geometria sopra un anello” dalla sua origine allo stato attuale. E` aggiunto una lista di letteratura aggiornata completa di articoli che trattano la geometria degli anelli nel contesto della geometria dell’incidenza. In diesem Forschungsartikel geben wir einen historischen und gleich- zeitig thematischen Uberblick¨ ¨uber die Entwicklung der “Ringgeometrie” von ihrem Ursprung bis zum aktuellen Stand der Technik, mit einer Liste aktualisierter Literatur einschließlich Artikeln zur Ringgeometrie im Kon- text der Inzidenzgeometrie. arXiv:2003.02881v1 [math.HO] 5 Mar 2020 Dans ce document de recherche, nous fournissons un aper¸cu historique et `ala fois th´ematique du d´eveloppement de la “g´eom´etrie sur un anneau”, de son origine `al’´etat actuel des connaissances. Nous ajoutons une liste de la litt´erature actualis´ee comprenant des articles traitant la g´eom´etrie sur un anneau dans le contexte de la g´eom´etrie de