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On the Theory of Point Weight Designs Royal Holloway
On the theory of Point Weight Designs Alexander W. Dent Technical Report RHUL–MA–2003–2 26 March 2001 Royal Holloway University of London Department of Mathematics Royal Holloway, University of London Egham, Surrey TW20 0EX, England http://www.rhul.ac.uk/mathematics/techreports Abstract A point-weight incidence structure is a structure of blocks and points where each point is associated with a positive integer weight. A point-weight design is a point-weight incidence structure where the sum of the weights of the points on a block is constant and there exist some condition that specifies the number of blocks that certain sets of points lie on. These structures share many similarities to classical designs. Chapter one provides an introduction to design theory and to some of the existing theory of point-weight designs. Chapter two develops a new type of point-weight design, termed a row-sum point-weight design, that has some of the matrix properties of classical designs. We examine the combinatorial aspects of these designs and show that a Fisher inequality holds and that this is dependent on certain combinatorial properties of the points of minimal weight. We define these points, and the designs containing them, to be either ‘awkward’ or ‘difficult’ depending on these properties. Chapter three extends the combinatorial analysis of row-sum point-weight designs. We examine structures that are simultaneously row-sum and point-sum point-weight designs, paying particular attention to the question of regularity. We also present several general construction techniques and specific examples of row-sum point-weight designs that are generated using these techniques. -
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 ......... -
Invariants of Incidence Matrices
Invariants of Incidence Matrices Chris Godsil (University of Waterloo), Peter Sin (University of Florida), Qing Xiang (University of Delaware). March 29 – April 3, 2009 1 Overview of the Field Incidence matrices arise whenever one attempts to find invariants of a relation between two (usually finite) sets. Researchers in design theory, coding theory, algebraic graph theory, representation theory, and finite geometry all encounter problems about modular ranks and Smith normal forms (SNF) of incidence matrices. For example, the work by Hamada [9] on the dimension of the code generated by r-flats in a projective geometry was motivated by problems in coding theory (Reed-Muller codes) and finite geometry; the work of Wilson [18] on the diagonal forms of subset-inclusion matrices was motivated by questions on existence of designs; and the papers [2] and [5] on p-ranks and Smith normal forms of subspace-inclusion matrices have their roots in representation theory and finite geometry. An impression of the current directions in research can be gained by considering our level of understanding of some fundamental examples. Incidence of subsets of a finite set Let Xr denote the set of subsets of size r in a finite set X. We can consider various incidence relations between Xr and Xs, such as inclusion, empty intersection or, more generally, intersection of fixed size t. These incidence systems are of central importance in the theory of designs, where they play a key role in Wilson’s fundamental work on existence theorems. They also appear in the theory of association schemes. 1 2 Incidence of subspaces of a finite vector space This class of incidence systems is the exact q-analogue of the class of subset incidences. -
7 Combinatorial Geometry
7 Combinatorial Geometry Planes Incidence Structure: An incidence structure is a triple (V; B; ∼) so that V; B are disjoint sets and ∼ is a relation on V × B. We call elements of V points, elements of B blocks or lines and we associate each line with the set of points incident with it. So, if p 2 V and b 2 B satisfy p ∼ b we say that p is contained in b and write p 2 b and if b; b0 2 B we let b \ b0 = fp 2 P : p ∼ b; p ∼ b0g. Levi Graph: If (V; B; ∼) is an incidence structure, the associated Levi Graph is the bipartite graph with bipartition (V; B) and incidence given by ∼. Parallel: We say that the lines b; b0 are parallel, and write bjjb0 if b \ b0 = ;. Affine Plane: An affine plane is an incidence structure P = (V; B; ∼) which satisfies the following properties: (i) Any two distinct points are contained in exactly one line. (ii) If p 2 V and ` 2 B satisfy p 62 `, there is a unique line containing p and parallel to `. (iii) There exist three points not all contained in a common line. n Line: Let ~u;~v 2 F with ~v 6= 0 and let L~u;~v = f~u + t~v : t 2 Fg. Any set of the form L~u;~v is called a line in Fn. AG(2; F): We define AG(2; F) to be the incidence structure (V; B; ∼) where V = F2, B is the set of lines in F2 and if v 2 V and ` 2 B we define v ∼ ` if v 2 `. -
A Generalization of Singer's Theorem: Any Finite Pappian A-Plane Is
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 71, Number 2, September 1978 A GENERALIZATIONOF SINGER'S THEOREM MARK P. HALE, JR. AND DIETER JUNGN1CKEL1 Abstract. Finite pappian Klingenberg planes and finite desarguesian uniform Hjelmslev planes admit abelian collineation groups acting regularly on the point and line sets; this generalizes Singer's theorem on finite desarguesian projective planes. Introduction. Projective Klingenberg and Hjelmslev planes (briefly K- planes, resp., //-planes) are generalizations of ordinary projective planes. The more general A-plane is an incidence structure n whose point and line sets are partitioned into "neighbor classes" such that nonneighbor points (lines) have exactly one line (point) in common and such that the "gross structure" formed by the neighbor classes is an ordinary projective plane IT'. An //-plane is a A-plane whose points are neighbors if and only if they are joined by at least two lines, and dually. A finite A-plane possesses two integer invariants (/, r), where r is the order of IT and t2 the cardinality of any neighbor class of n (cf. [3], [6]). Examples of A-planes are the "desarguesian" A-planes which are constructed by using homogeneous coordinates over a local ring (similar to the construction of the desarguesian projective planes from skewfields); for the precise definition, see Lemma 2. If R is a finite local ring with maximal ideal M, one obtains a (/, r)-K-p\ane with t = \M\ and r = \R/M\. The A-plane will be an //-plane if and only if A is a Hjelmslev ring (see [7] and [»])• We recall that a projective plane is called cyclic if it admits a cyclic collineation group which acts regularly on the point and line sets. -
Certain Properties of Orbits Under Collineation Groups*
JOURNAL OF COMBINATORIAL THEORY 2, 546-570 (1967) Certain Properties of Orbits under Collineation Groups* DAVID A. FOULSER AND REUBEN SANDLER University of lllinois at Chicago Circle, Chicago, lllinois Communicated by Gian-Carlo Rota l. INTRODUCTION" In this paper, the authors examine some of the combinatorial proper- ties of orbits of projective planes under collineation groups. Dembowski, in [5], has studied in some detail the relations which can hold between a point orbit ~ and a line orbit ~. Our approach here is to associate with the set s of all lines incident with two or more points of ~, and to look at the numerical relationships which occur. In Section 2, the basic concepts are defined, and some necessary condi- tions are derived which must hold if a collineation group with certain properties can exist. Section 3 discusses the insufficiency of these condi- tions and gives some results yielding sufficiency in some special cases; in addition, further necessary conditions are developed. In Section 4, a more detailed analysis is made of the nature of the interaction between and ~(~) under special conditions on the collineation group G. These special conditions are analogous to requiring that G be Abelian -- class (A)--or of prime order -- class (I-I)--and very detailed information is available under these hypotheses. Finally, Section 5 contains a num- ber of examples illustrating the principles discussed in the earlier sections. Standard notation for permutation groups is used throughout this paper. See for example [11]. * This work was partially supported by NSF contract No. GP 5276. 546 CERTAIN PROPERTIES OF ORBITS UNDER COLLINEATION GROUPS 547 2. -
Masterarbeit
Masterarbeit Solution Methods for the Social Golfer Problem Ausgef¨uhrt am Institut f¨ur Informationssysteme 184/2 Abteilung f¨ur Datenbanken und Artificial Intelligence der Technischen Universit¨at Wien unter der Anleitung von Priv.-Doz. Dr. Nysret Musliu durch Markus Triska Kochgasse 19/7, 1080 Wien Wien, am 20. M¨arz 2008 Abstract The social golfer problem (SGP) is a combinatorial optimisation problem. The task is to schedule g ×p golfers in g groups of p players for w weeks such that no two golfers play in the same group more than once. An instance of the SGP is denoted by the triple g−p−w. The original problem asks for the maximal w such that the instance 8−4−w can be solved. In addition to being an interesting puzzle and hard benchmark problem, the SGP and closely related problems arise in many practical applications such as encoding, encryption and covering tasks. In this thesis, we present and improve upon existing approaches towards solving SGP instances. We prove that the completion problem correspond- ing to the SGP is NP-complete. We correct several mistakes of an existing SAT encoding for the SGP, and propose a modification that yields consid- erably improved running times when solving SGP instances with common SAT solvers. We develop a new and freely available finite domain constraint solver, which lets us experiment with an existing constraint-based formula- tion of the SGP. We use our solver to solve the original problem for 9 weeks, thus matching the best current results of commercial constraint solvers for this instance. -
Incidence Structures Near Configurations of Type (N
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 19 (2020) 17–23 https://doi.org/10.26493/1855-3974.1685.395 (Also available at http://amc-journal.eu) Incidence structures near configurations ∗ of type (n3) Peter J. Dukes y , Kaoruko Iwasaki Mathematics and Statistics, University of Victoria, Victoria, Canada Received 24 April 2018, accepted 28 June 2020, published online 26 October 2020 Abstract An (n3) configuration is an incidence structure equivalent to a linear hypergraph on n vertices which is both 3-regular and 3-uniform. We investigate a variant in which one constraint, say 3-regularity, is present, and we allow exactly one line to have size four, exactly one line to have size two, and all other lines to have size three. In particular, we study planar (Euclidean or projective) representations, settling the existence question and adapting Steinitz’ theorem for this setting. Keywords: Geometric configurations, incidence structures. Math. Subj. Class. (2020): 52C30, 05C65 1 Introduction A geometric (nr; mk) configuration is a set of n points and m lines in the Euclidean (or projective) plane such that every line contains exactly k points (uniformity) and every point is incident with exactly r lines (regularity). Grunbaum’s¨ book [3] is an excellent reference for the major results on this topic, including many variations. Our study here is a variation in which uniformity (or, dually, regularity) is mildly re- laxed; see also [1] and [4, x6.8] for prior investigations in this direction. An incidence structure is a triple (P; L; ι), where P is a set of points, L is a set of lines, and ι ⊆ P × L is a relation called incidence. -
Pedal Sets of Unitals in Projective Planes of Order 9 and 16
SARAJEVO JOURNAL OF MATHEMATICS Vol.7 (20) (2011), 255{264 PEDAL SETS OF UNITALS IN PROJECTIVE PLANES OF ORDER 9 AND 16 VEDRAN KRCADINACˇ AND KSENIJA SMOLJAK Dedicated to the memory of Professor Svetozar Kurepa Abstract. Let U be a unital in a projective plane P. Given a point P 2 P n U, the points F 2 U such that FP is tangent to U are the feet of P , and the set of all such points is the pedal set of P . In this paper we study pedal sets of unitals embedded in projective planes of order 9 and 16. 1. Introduction A finite projective plane of order q can be defined as a block design with parameters (q2 + q + 1; q + 1; 1); that is, a finite incidence structure with q2 + q + 1 points, q + 1 points on every line, and a unique line through every pair of points. Similarly, a unital of order q is a (q3 +1; q +1; 1) block design. For more on finite projective planes we refer to the book [10], and for unitals to [2]. The most interesting unitals are the ones embedded in a projective plane P of order q2. In this setting a unital U is a set of q3 + 1 points of P with the property that every line of P meets U either in one point, or in q + 1 points. In the first case the line is tangent to U, and in the second case it is secant. The set of all tangents is a unital in the dual plane P∗, called the dual unital and denoted by U ∗. -
Novem~Er 1963
• ,UNIVERSITY OF NORTH CAROLINA Department of Statistics Chapel Hill" N. C. THE CONSTRUCTION OF TRANSLATION PLANES FROM PROJECTIVE SPACES by R. H. Bruck, University of Wisconsin and R. C. Bose, University of North Carolina Novem~er 1963 National Science Foundation Grant No. GP 16-60 and AF-AFOSR-Grant Noo 84-63 Can eve.ry (non-Desarguest.. an) projee.,tJVe p~' ~ imbedded (in some natural, g~anet.ric ff.hion) in af1J8sarguesian) projective space? The Ite,titn is ne...WI~.ut .. ,trnp.. · ortant, for, if the answer is yes, tfo entirely SEijifia'Wt fields of re- search can be united~ 1his paper ;p~1des. conceptually simple geometric construction 1vh1dti',Y1elds an affirmative answer for a broad class of pla.uer. A plane 1t is given by the construction precisely when 1t is a translation p1ar1e with a coordinatizing :right Veblen-Wedderburn system which is finite-dimensional over its left-operator skew field. The condition is satisfied by all known translation planes, including all finite translation p1aneso This research was supported by the National Science Foundation Grant No. GP 16-60 and the Air Force Office of Scientific Research Grant No. 84-63. Institute of Statistics Mimeo Series No. 378 THE CONSTRUCTION OF TRANSLATION PLANES • FROM PROJECTIVE SPACES by l R. H. Bruck, University of Wisconsin and R. C. Bose, University of North Carolina1 --------~--------------------------------.---------------------------------------- 1. Introduction. One might say, with some justice, that projective geometry, in so far as present day research is concerned, has split into two qUite separate fields. On the one hand, the researcher into the foundations of geometry tends to regard Desarguesian spaces as completely known. -
Codes from Incidence Matrices and Line Graphs of Paley Graphs
Advances in Mathematics of Communications doi:10.3934/amc.2011.5.93 Volume 5, No. 1, 2011, 93{108 CODES FROM INCIDENCE MATRICES AND LINE GRAPHS OF PALEY GRAPHS Dina Ghinelli Dipartimento di Matematica Universit`adi Roma `La Sapienza' I-00185 Rome, Italy Jennifer D. Key School of Mathematical Sciences University of KwaZulu-Natal Pietermaritzburg 3209, South Africa (Communicated by Ivan Landjev) Abstract. We examine the p-ary codes from incidence matrices of Paley graphs P (q) where q ≡ 1 (mod 4) is a prime power, and show that the codes q(q−1) q−1 q(q−1) q−1 are [ 4 ; q − 1; 2 ]2 or [ 4 ; q; 2 ]p for p odd. By finding PD-sets we show that for q > 9 the p-ary codes, for any p, can be used for permutation decoding for full error-correction. The binary code from the line graph of P (q) is shown to be the same as the binary code from an incidence matrix for P (q). 1. Introduction The codes from the row span of adjacency matrices of Paley graphs are the well- known quadratic residue codes and studied in many texts. We look here at the codes from the row span of incidence matrices of Paley graphs, following the ideas of some earlier papers that studied such codes for various classes of graphs and their line graphs, including Hamming graphs, complete graphs, and complete bipartite graphs: see [14, 16,6,7,8]. An incidence matrix G for a graph Γ = (V; E) is a jV j × jEj matrix with rows labelled by the vertices and columns by the edges and an entry 1 if a vertex is on edge, 0 otherwise. -
Computational Methods in Finite Geometry
Computational Methods in Finite Geometry Anton Betten Colorado State University Summer School, Brighton, 2017 Q: What makes a problem interesting? Maybe it is being difficult. If this is true, then Finite Geometry is full of interesting problems. In the words of John F. Kennedy: We choose to go to the moon and do the other things not because they are easy but because they are hard. In the words of Peter Cameron: That a problem is hard does not mean we should not solve it. Prologue The goal is this lecture sequence is to say something meaningful about the problem of constructing and classifying combinatorial objects. (with an emphasis on objects from finite geometry). A large number of problems can be reduced to classifying orbits of groups acting on sets. The sets are often very large and at times not readily available. We need to perform search and isomorph rejection at the same time. Terminology Let G be a group. Let G act on a finite set X. For x; y 2 X; say that x ∼G y if x and y belong to the same G-orbit. Terminology The isomorphism problem is the following: Given x; y 2 X; determine whether x ∼G y or not. Terminology The classification problem is the problem of determining the orbits of G on X, for instance by listing one element from each orbit. Such a list of orbit representatives is called a transversal for the orbits of G on X. From now on, assume that a transversal for the orbits of G on X has been fixed.