Second Edition - assets.cambridge.orgassets.cambridge.org/.../sample/9780521444323wsc00.pdf · First edition °c Bibliographisches Institut, Zurich, 1985 °c Cambridge University

Second Edition

Thomas BethUniversitat Karlsruhe

Dieter JungnickelUniversitat Augsburg

Hanfried LenzFreie Universitat Berlin

Volume I

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE

The Pitt Building, Trumpington Street, Cambridge, United Kingdom

C A M B R I D G E U N I V E R S I T Y P R E S S

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First edition c© Bibliographisches Institut, Zurich, 1985c© Cambridge University Press, 1993

Second editionc© Cambridge University Press, 1999

This book is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,

no reproduction of any part may take place withoutthe 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 LATEX 2ε [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 1999511′.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. . . . . . . . . . 15§4. Isomorphisms, duality and correlations. . . . . . . . . . . . . 20§5. Partitions of the block set and resolvability. . . . . . . . . . . 24§6. Divisible incidence structures. . . . . . . . . . . . . . . . . . 32§7. Transversal designs and nets. . . . . . . . . . . . . . . . . . 37§8. Subspaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . 44§9. Hadamard designs. . . . . . . . . . . . . . . . . . . . . . . 50

II. Combinatorial analysis of designs . . . . . . . . . . . . . . . . . 62

§1. Basics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62§2. Fisher’s inequality for pairwise balanced designs. . . . . . . . 64§3. Symmetric designs. . . . . . . . . . . . . . . . . . . . . . . 77§4. The Bruck–Ryser–Chowla theorem. . . . . . . . . . . . . . . 89§5. Balanced incidence structures with balanced duals. . . . . . . 96§6. Generalisations of Fisher’s inequality and

intersection numbers. . . . . . . . . . . . . . . . . . . . . . 101§7. Extensions of designs. . . . . . . . . . . . . . . . . . . . . . 111§8. Affine designs. . . . . . . . . . . . . . . . . . . . . . . . . . 123§9. Strongly regular graphs. . . . . . . . . . . . . . . . . . . . . 136§10. The Hall–Connor theorem. . . . . . . . . . . . . . . . . . . 146§11. Designs and codes. . . . . . . . . . . . . . . . . . . . . . . 152

xiii

xiv Contents

III. Groups and designs . . . . . . . . . . . . . . . . . . . . . . . . 162

§1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . 162§2. Incidence morphisms. . . . . . . . . . . . . . . . . . . . . . 163§3. Permutation groups. . . . . . . . . . . . . . . . . . . . . . . 167§4. Applications to incidence structures. . . . . . . . . . . . . . . 173§5. Examples from classical geometry. . . . . . . . . . . . . . . 184§6. Constructions oft-designs from groups. . . . . . . . . . . . . 190§7. Extensions of groups. . . . . . . . . . . . . . . . . . . . . . 198§8. Construction oft-designs from base blocks. . . . . . . . . . . 206§9. Cyclic t-designs. . . . . . . . . . . . . . . . . . . . . . . . . 218§10. Cayley graphs. . . . . . . . . . . . . . . . . . . . . . . . . . 225

IV. Witt designs and Mathieu groups . . . . . . . . . . . . . . . . . 234

§1. The existence of the Witt designs. . . . . . . . . . . . . . . . 234§2. The uniqueness of the small Witt designs. . . . . . . . . . . . 237§3. The little Mathieu groups. . . . . . . . . . . . . . . . . . . . 243§4. Properties of the large Witt designS(5, 8; 24). . . . . . . . . . 244§5. Some simple groups. . . . . . . . . . . . . . . . . . . . . . . 252§6. Witt’s construction of the Mathieu groups and Witt designs. . . 259§7. Hussain structures and the uniqueness ofS2(3, 6; 12) . . . . . . 262§8. The Higman–Sims group. . . . . . . . . . . . . . . . . . . . 270

V. Highly transitive groups . . . . . . . . . . . . . . . . . . . . . . . 277

§1. Sharplyt-transitive groups . . . . . . . . . . . . . . . . . . . 277§2. t-homogeneous groups. . . . . . . . . . . . . . . . . . . . . 283§3. Concluding remarks:t-transitive groups . . . . . . . . . . . . 291

VI. Difference sets and regular symmetric designs . . . . . . . . . . 297

§1. Basic facts . . . . . . . . . . . . . . . . . . . . . . . . . . . 298§2. Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . 303§3. Group rings and characters. . . . . . . . . . . . . . . . . . . 311§4. Multiplier theorems. . . . . . . . . . . . . . . . . . . . . . . 319§5. Difference lists . . . . . . . . . . . . . . . . . . . . . . . . . 330§6. The Mann test and Wilbrink’s theorem. . . . . . . . . . . . . 335§7. Planar difference sets. . . . . . . . . . . . . . . . . . . . . . 344§8. Paley–Hadamard difference sets and cyclotomy. . . . . . . . 353§9. Some difference sets with gcd(v, n) > 1 . . . . . . . . . . . . 363§10. Relative difference sets and building sets. . . . . . . . . . . . 369

Contents xv

§11. Extended building sets and difference sets. . . . . . . . . . . 382§12. Constructions for Hadamard and Chen difference sets. . . . . 388§13. Some applications of algebraic number theory. . . . . . . . . 410§14. Further non-existence results. . . . . . . . . . . . . . . . . . 419§15. Characters and cyclotomic fields. . . . . . . . . . . . . . . . 435§16. Schmidt’s exponent bound. . . . . . . . . . . . . . . . . . . 441§17. Difference sets with Singer parameters. . . . . . . . . . . . . 455

VII. Difference families . . . . . . . . . . . . . . . . . . . . . . . . . 468

§1. Basic facts . . . . . . . . . . . . . . . . . . . . . . . . . . . 468§2. Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . 472§3. More examples. . . . . . . . . . . . . . . . . . . . . . . . . 476§4. Triple systems. . . . . . . . . . . . . . . . . . . . . . . . . . 481§5. Some difference families in Galois fields. . . . . . . . . . . . 488§6. Blocks with evenly distributed differences. . . . . . . . . . . 499§7. Some more special block designs. . . . . . . . . . . . . . . . 502§8. Proof of Wilson’s theorem . . . . . . . . . . . . . . . . . . . 509

VIII. Further direct constructions . . . . . . . . . . . . . . . . . . . 520

§1. Pure and mixed differences. . . . . . . . . . . . . . . . . . . 520§2. Applications to the construction of resolvable block designs. . 528§3. A difference construction for transversal designs. . . . . . . . 531§4. Further constructions for transversal designs. . . . . . . . . . 544§5. Some constructions using projective planes. . . . . . . . . . . 564§6. t-designs constructed from graphs. . . . . . . . . . . . . . . 584§7. The existence oft-designs for large values ofλ . . . . . . . . . 588§8. Higher resolvability oft-designs . . . . . . . . . . . . . . . . 595§9. Infinite t-designs . . . . . . . . . . . . . . . . . . . . . . . . 598§10. Cyclic Steiner quadruple systems. . . . . . . . . . . . . . . . 600

Notation and symbols. . . . . . . . . . . . . . . . . . . . . . . . . 1005Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093

Contents of Volume II

IX. Recursive constructions . . . . . . . . . . . . . . . . . . . . . . 608

§1. Product constructions. . . . . . . . . . . . . . . . . . . . . . 608§2. Use of pairwise balanced designs. . . . . . . . . . . . . . . . 617§3. Applications of divisible designs. . . . . . . . . . . . . . . . 621§4. Applications of Hanani’s lemmas. . . . . . . . . . . . . . . . 627§5. Block designs of block size three and four. . . . . . . . . . . 636§6. Solution of Kirkman’s schoolgirl problem. . . . . . . . . . . 641§7. The basis of a closed set. . . . . . . . . . . . . . . . . . . . 644§8. Block designs with block size five. . . . . . . . . . . . . . . 651§9. Divisible designs with small block sizes. . . . . . . . . . . . 660§10. Steiner quadruple systems. . . . . . . . . . . . . . . . . . . . 664§11. Embedding theorems for designs and partial designs. . . . . . 673§12. Concluding remarks. . . . . . . . . . . . . . . . . . . . . . . 681

X. Transversal designs and nets . . . . . . . . . . . . . . . . . . . . 690

§1. A recursive construction . . . . . . . . . . . . . . . . . . . . 690§2. Transversal designs withλ > 1 . . . . . . . . . . . . . . . . . 693§3. A construction of Wilson. . . . . . . . . . . . . . . . . . . . 696§4. Six and more mutually orthogonal Latin squares. . . . . . . . 703§5. The theorem of Chowla, Erdös and Straus . . . . . . . . . . . 706§6. Further bounds for transversal designs and

orthogonal arrays . . . . . . . . . . . . . . . . . . . . . . . 708§7. Completion theorems for Bruck nets. . . . . . . . . . . . . . 713§8. Maximal nets with large deficiency. . . . . . . . . . . . . . . 725§9. Translation nets and maximal nets with small deficiency. . . . 731

xvii

xviii Contents

§10. Completion results forµ>1 . . . . . . . . . . . . . . . . . . 749§11. Extending symmetric nets. . . . . . . . . . . . . . . . . . . . 758§12. Complete mappings, difference matrices and maximal nets. . . 761§13. Tarry’s theorem. . . . . . . . . . . . . . . . . . . . . . . . . 772§14. Codes of Bruck nets. . . . . . . . . . . . . . . . . . . . . . . 778

XI. Asymptotic existence theory . . . . . . . . . . . . . . . . . . . . 781

§1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 781§2. The existence of Steiner systems withv in

given residue classes. . . . . . . . . . . . . . . . . . . . . . 783§3. The main theorem for Steiner systemsS(2, k; v) . . . . . . . . 787§4. The eventual periodicity of closed sets. . . . . . . . . . . . . 790§5. The main theorem forλ= 1 . . . . . . . . . . . . . . . . . . . 793§6. The main theorem forλ>1 . . . . . . . . . . . . . . . . . . . 796§7. An existence theorem for resolvable block designs. . . . . . . 801§8. Some results fort ≥ 3 . . . . . . . . . . . . . . . . . . . . . . 805

XII. Characterisations of classical designs. . . . . . . . . . . . . . . 806

§1. Projective and affine spaces as linear spaces. . . . . . . . . . 806§2. Characterisations of projective spaces. . . . . . . . . . . . . . 808§3. Characterisation of affine spaces. . . . . . . . . . . . . . . . 821§4. Locally projective linear spaces. . . . . . . . . . . . . . . . . 828§5. Good blocks . . . . . . . . . . . . . . . . . . . . . . . . . . 833§6. Concluding remarks. . . . . . . . . . . . . . . . . . . . . . . 841

XIII. Applications of designs . . . . . . . . . . . . . . . . . . . . . . 852

§1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . 852§2. Design of experiments. . . . . . . . . . . . . . . . . . . . . 856§3. Experiments with Latin squares and orthogonal arrays. . . . . 874§4. Application of designs in optics. . . . . . . . . . . . . . . . . 880§5. Codes and designs. . . . . . . . . . . . . . . . . . . . . . . 892§6. Discrete tomography. . . . . . . . . . . . . . . . . . . . . . 926§7. Designs in data structures and computer algorithms. . . . . . . 930§8. Designs in hardware. . . . . . . . . . . . . . . . . . . . . . 937§9. Difference sets rule matter and waves. . . . . . . . . . . . . . 946§10. No waves, no rules, but security. . . . . . . . . . . . . . . . . 956

Appendix. Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 971

§1. Block designs. . . . . . . . . . . . . . . . . . . . . . . . . . 971§2. Symmetric designs. . . . . . . . . . . . . . . . . . . . . . . 981

Contents xix

§3. Abelian difference sets. . . . . . . . . . . . . . . . . . . . . 990§4. Small Steiner systems. . . . . . . . . . . . . . . . . . . . . . 997§5. Infinite series of Steiner systems. . . . . . . . . . . . . . . . 999§6. Remark ont-designs witht ≥ 3 . . . . . . . . . . . . . . . . 1001§7. Orthogonal Latin squares. . . . . . . . . . . . . . . . . . . 1002

Notation and symbols. . . . . . . . . . . . . . . . . . . . . . . . . 1005Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093

I

Examples and Basic Definitions

It’s elementary, Watson(Conan Doyle)

§1. Incidence Structures and Incidence Matrices

The most basic notion in (finite) geometry is that of an incidence structure.It contains nothing more than the idea that two objects from distinct classesof things (say points and lines) may be “incident” with each other. The onlyrequirement will be that the classes do not overlap. We now make this moreprecise.

1.1 Definitions. An incidence structureis a tripleD = (V,B, I )whereV andB are any two disjoint sets andI is a binary relation betweenV andB, i.e.I ⊆ V × B. The elements ofV will be calledpoints, those ofB blocksandthose ofI flags. Instead of(p, B)∈ I , we will simply write pI B and use suchgeometric language as “the pointp lies on the blockB”, “ B passes throughp”,“ p andB are incident”, etc.

For reasons of convenience, we will usually not state whether a given objectis a point or a block; this will be clear from the context and we will alwaysuse lower case letters (e.g.p,q, r, . . .) to denote points and upper case letters(e.g.B,C, . . .) to denote blocks. Now let us look at some examples! Of course,familiar (euclidean) geometry provides examples, e.g. taking points and lines(as blocks) in the euclidean plane or points and planes (as blocks) in 3-space.One reason for choosing the term “block” instead of “line” is that we will veryoften consider planes or hyperplanes as blocks. These classical examples are ofcourse infinite; in this book we will almost exclusively deal withfiniteincidencestructures (i.e. bothV andB are finite). According to our definition, any binaryrelation between two disjoint finite sets will give an example. Naturally, thisis much too general to be of any interest in itself and, guided by a series

1

2 I. Examples and basic definitions

of examples, we will single out classes of incidence structures having moreinteresting properties. Before doing so, let us introduce some notation. Ifp isany point,(p) will denote the set of blocks incident withp, i.e.

(p) := {B∈B : pI B}(1.1.a)

and more generally for any subsetQ of the point set

(Q) := {B∈B : pI B for eachp∈ Q}.(1.1.b)

Instead of({p1, . . . , pm}), we simply write(p1, . . . , pm)when it is clear that thissymbol is not intended to denote the orderedm-tuple of these points. Similarly,we write

(C) := {p∈V : pI B for eachB∈C}(1.1.c)

for any subsetC of the block setB. For a pointp, the number|(p)| is called thedegreeof p, and similarly for blocks. Distinct blocksG andH may well be inci-dent with the same point set; thus(V,B, I )with V ={1, 2, 3}, B={B1, B2, B3}and(B1)= (B2)={1, 2}, (B3)={1, 3} is a perfectly respectable incidence stru-cture. But in this case one may not just list the sets of points incident with ablock but one has to give their multiplicities too. If there are distinct blockswith the same point set one speaks of “repeated blocks”. Often we will con-sider incidence structures where distinct blocks have distinct point sets.

1.2 Definition. An incidence structure is calledsimpleif (B) 6= (C) wheneverB andC are distinct blocks. Here thetrace(B) of a blockB is the set{x ∈V :x I B} of its points, cf. (1.1.c).

1.3 Example. Take as point setV = {0, . . . ,6}, as block setB = {{0, 1, 3},{1, 2, 4}, {2, 3, 5}, {3, 4, 6}, {4, 5, 0}, {5, 6, 1}, {6, 0, 2}} and as incidence rela-tion the membership relation∈.

For any simple incidence structure, we can (and usually will) identify each blockB with the corresponding point set(B) and the incidence relationI with themembership relation∈. However, the way we have written our example is notintuitively appealing. So one can sometimes draw a picture, representing pointsby points in the real plane and blocks by point sets, usually drawn somewhat like

§1. Incidence structures and incidence matrices 3

Venn diagrams or alternatively represented by curves in the plane. For instance,we may draw Example 1.3 as follows.

Figure 1.1

Later in this book, we will usually simplify notation and omit the symbolI forthe incidence relation. Thus we just writeD = (V,B) instead ofD = (V,B, I ),even ifD is not simple. ThenB should be considered as a “multiset” or “list”; seeIII.9.2 for a formal definition. Finally, we introduce another way of representingan incidence structure, which will later allow us to use methods from linearalgebra.

1.4 Definition. Let D = (V,B, I ) be a finite incidence structure and labelthe points asp1, . . . , pv and the blocks asB1, . . . , Bb. Then the matrixM =(mi j ) (i = 1, . . . , v; j = 1, . . . ,b) defined by

mi j :={

1 if pi I B j

0 otherwise

is called anincidence matrixfor D. The column ofM belonging to a blockBis called theincidence vectorof B.

Instead ofmi j one may writeM(p, B)(p∈V, B∈B). ThenM is a mapping ofV × B into {0, 1}.Of course,M depends on the labelling used, but up to column and row permuta-tions it is unique. Conversely, every matrix with entries from{0, 1} determinesan incidence structure.


1.5 Example. The incidence structure of 1.3 yields the following incidencematrix.

1 0 0 0 1 0 11 1 0 0 0 1 00 1 1 0 0 0 11 0 1 1 0 0 00 1 0 1 1 0 00 0 1 0 1 1 00 0 0 1 0 1 1

.

An interesting question one may ask about finite incidence structures in generalis to find necessary and sufficient conditions for the existence of an incidencestructure with given point degrees and block degrees (or in matrix language,for the existence of a (0, 1)-matrix of typev × b with given row and columnsums). The answer to this question is not trivial and its proof requires themethods of transversal theory. Proofs may be found in the books of Mirsky(1971) and Jungnickel (1999). Here we will only give one necessary (in generalnot sufficient) condition, as its proof is an example of a very important methodin finite geometry.

1.6 Proposition. Assume the existence of an incidence structure with pointdegrees r1, . . . , rv and block degrees k1, . . . , kb. Then necessarily

v∑i=1

ri =b∑

j=1

kj .(1.6.a)

Proof. We count all flags (i.e. pairs(pi , Bj ) with pi I B j ) in two ways. Usingthe fact that thei -th point is on exactlyri blocks, the left-hand side of (1.6.a)counts the total number of flags. Similarly, since thej -th block Bj containsexactlykj points, the right-hand side of (1.6.a) also counts the total number offlags. Hence these two quantities agree.¥

1.7 Corollary. Let D be an incidence structure withv points, b blocks, allpoint degrees equal to r and all block degrees equal to k. Then

vr = bk. ¥(1.7.a)

The principle of counting in two ways will be used throughout this book andis – in spite of its simplicity – an extremely powerful tool (especially whencombined with statistical ideas, like computing means and variances).

§1. Incidence structures and incidence matrices 5

We conclude this section with a first non-trivial result, namely by settling theexistence problem for simple incidence structures with constant point degreesrand constant block degreesk; in the terminology we shall introduce in Definition3.1 below, such a structure will be called asimple1-design. The simple proofwe present is due to Billington (1982). We begin with the following lemmaconcerning the case of not necessarily uniform point degrees.

1.8 Lemma. Suppose the existence of a simple incidence structureD on vpoints with b blocks, constant block degrees k and point degrees r1, . . . , rv. Ifone has ri > r j for a pair of indices i, j (with i > j , say), then there alsoexists a simple incidence structureD′ onv points with b blocks, constant blockdegrees k and point degrees r1, . . . , ri − 1, ri+1, . . . , r j−1, r j + 1, . . . , rv.

Proof. The hypothesisri > r j implies thatD has more blocks containingthe point pi but not the pointpj than blocks containingpj but not pi . Thisguarantees the existence of some blockB of D which containspi but not pj

and such thatB∗ := (B \ {pi }) ∪ {pj } is not a block ofD. We now replace theblock B by B∗ and obtain the desired incidence structureD′. ¥

1.9 Theorem. A simple incidence structure onv points with b blocks, constantblock degrees k and constant point degrees r exists if and only if

vr = bk and b≤(v

k

).(1.9.a)

Proof. In view of Corollary 1.7 and the simplicity requirement, condition(1.9.a) is obviously necessary for the existence of the desired incidence struc-ture. Conversely, assume the validity of condition (1.9.a) for some integersk, r, v,b. We first define a simple incidence structureD0 by choosing any set ofb pairwise distinctk-subsets{B1, . . . , Bb} from somev-setV ={p1, . . . , pv}.Denote the point degrees ofD0 by r1, . . . , rv; obviously,

vr = bk= r1+ . . .+ rv.(1.9.b)

If all point degrees happen to agree withr , we are finished. Otherwise, we mayapply Lemma 1.8 to reduce one degree larger thanr by 1 while simultane-ously enlarging some other degree smaller thanr by 1. Applying this processrecursively yields the desired simple incidence structureD. ¥

1.10 Definition. Thecomplementary structureof an incidence structureD =(V,B, I ) is the incidence structureD = (V,B, J) with J = (V × B) \ I , that


is,

x J B⇐⇒ x –I B for all x ∈V and allB∈B.(1.10.a)

If M is an incidence matrix forD, then an incidence matrix forD is obtainedby interchanging 0’s and 1’s inM .

1.11 Exercise.Let D be an incidence structure for which each point is onexactlyr blocks and any two points are on exactlyλ common blocks. Showthat any two points are on exactlyb− 2r + λ common blocks ofD.

1.12 Remark. For simplicity of notation, we will very often write incidencestructures in the formD = (V,B), even if D is not simple. ThenB is a mul-tiset of blocks. Thus the incidence relation will not be mentioned explicitly ingeneral.

§2. Block Designs and Examples From Affine andProjective Geometry

We now study some geometric examples which will lead us to the funda-mental concept of a block design. We first look at (finite) projective and affineplanes and prove some of their basic properties, although we will not makeany detailed geometric investigations. The interested reader may consult thestandard text books by Pickert (1975) and by Hughes and Piper (1982). Let usdefine a projective plane:

2.1 Definition. An incidence structureD= (V,B, I ) is called aprojectiveplaneif and only if it satisfies the following axioms:

Any two distinct points are joined by exactly one line.1(2.1.a)

Any two distinct lines intersect in a unique point.(2.1.b)

There exists aquadrangle, i.e. four points no three of which are(2.1.c)on a common line.

We have already met an example in the previous section: Figure 1.1 representsa projective plane with seven points and seven lines. (In fact this is the smallest

1 In geometry texts one usually speaks of “lines” instead of “blocks”. In design theory, this willonly be done for incidence structures for which any two points are joined by at most one block.

§2. Block designs and affine and projective geometry 7

possible projective plane as the reader may check by considering the axioms.)This example shows a remarkable degree of uniformity: each point is on threelines, each line contains three points, and the number of points and lines agrees.Before giving more examples, we show that this is not a coincidence:

2.2 Proposition. Let D = (V,B, I ) be a finite projective plane. Then thereexists a natural number n, called theorderof D, satisfying:

|(p)| = |(G)| = n+ 1 for all p ∈V and G∈B;(2.2.a)

|V | = |B| = n2+ n+ 1.(2.2.b)

Proof. Consider any pointp and any lineG with p –I G. By (2.1.a) and (2.1.b),the mappingπ : (G)→ (p)with qπ : = pq for all q I G is a bijection (herepqdenotes the unique line joiningp andq). Hence|(p)| = |(G)|wheneverp –I G.Thus (2.2.a) follows if we can show that for any two distinct linesG, H thereis a pointp with p –I G, H . But by (2.2.c) there exists a quadrangleq, r, s, t .If each of these points is onG or H , we may assume thatq, r I G ands, t I H .Then we can choosep to be the intersection of the linesqs andr t . To verify(2.2.b) we again use counting in two ways. We choose a fixed pointp andcount all flags(q,G) with pI G andp 6=q. By (2.1.a), we obtain|V | − 1 suchflags; and by (2.2.a) we have(n + 1)n such flags. Thus|V | = n2 + n + 1.The assertion on|B| follows similarly (or using (2.2.a), the value for|V |and (1.7.a)).¥

Now let us construct some more examples:

2.3 Proposition. For each prime power q, there exists a projective plane oforder q.

Proof. Let F be the Galois field onq elements andW the vector space ofdimension 3 overF . Choose as points all 1-dimensional subspaces and aslines all 2-dimensional subspaces ofW. Using the dimension formula of linearalgebra, one checks that the axioms (2.1.a) and (2.1.b) are satisfied. For (2.1.c),one may choose the pointse1F, e2F, e3F and(e1+ e2+ e3)F wheree1, e2, e3

is any basis ofW. This argument works in fact for any 3-dimensional vectorspace. The fact thatF is the field onq elements is only needed to show that theresulting projective plane has orderq: the number of 1-dimensional subspacesof W is then(q3− 1)/(q − 1)=q2+ q + 1. ¥


The projective plane of orderq thus constructed will be denoted by the symbolPG(2,q). These are the most important projective planes. However there areothers. It is possible to tell just from the geometry of the plane whether it arisesfrom this construction, and below in 3.6 we shall mention such a characterisa-tion. Now let us look at affine planes.

2.4 Definition. An incidence structureD = (V,B, I ) is called anaffine planeif and only if it satisfies the following axioms:

Any two distinct points are joined by exactly one line.(2.4.a)

Given any pointp and any lineG with p –I G, there is precisely(2.4.b)one lineH with p I H and not intersectingG.

There is atriangle, i.e. three points not on a common line.(2.4.c)

We say that two linesG andH areparallel if G = H or |(G, H)| = 0, and wewrite G ‖ H . Thus (2.4.b) is Euclid’s parallel axiom.

2.5 Proposition. LetD = (V,B, I ) be an affine plane. Then parallelism is anequivalence relation onB. If D is finite, there exists a natural number n (calledtheorderof D) satisfying:

|(p)| = n+ 1 for all points p;(2.5.a)

|(G)| = n for all lines G;(2.5.b)

|V | = n2, |B| = n2+ n.(2.5.c)

Proof. We only need to check the transitivity of‖. Thus assumeG‖H andH‖K . W.l.o.g. (without loss of generality) we may suppose thatG, H andKare mutually distinct. IfG ∦ K there would be a pointpI G, K ; but thenG andK would be two parallels throughp to H , contradicting (2.4.b). The remainingassertion follows as in the proof of Proposition 2.2 (or from Propositions 2.2and 2.7 below). We leave the details to the reader.¥

We can provide examples of affine planes by showing that they are essentiallythe same as projective planes. To state the connection precisely, we need adefinition.

2.6 Definition. Let D = (V,B, I ) be an incidence structure andQ ⊆ VandC ⊆ B. Then the incidence structureinducedby D on Q andC is D′ =


(Q,C, I | Q × C), andD′ is called aninduced substructureof D. Instead ofI | Q× C we will usually again writeI .

2.7 Proposition. Let D = (V,B, I ) be a projective plane and G a line ofD. Then the substructureDG = (V \ (G),B \ {G}, I ) is an affine plane. Con-versely, every affine plane may be obtained in this way from a projective plane.In the finite case, the orders ofD andDG are the same.

Proof. Let D be a projective plane and remove a lineU together with all itspoints. ThenDU satisfies (2.4.a), becauseD satisfies (2.1.a). Letp be a pointof DU andG a line ofDU with p –I G. If q is the point of intersection ofG andU in D, then the linepq is the desired parallel toG throughp. Certainly, it isunique. Now, (2.4.c) follows from (2.1.c) and the existence of a point which ison neither of two given lines (cf. the proof of Proposition 2.2).

Conversely, given an affine plane, we can obtain a projective plane. Extendthe point set by choosing an additional point corresponding to each parallelclass of lines. Then choose one additional line (“the line at infinity”). The newincidence extends the old incidence as follows: The line at infinity contains eachnew point and a new point is contained in each of the lines in the correspondingparallel class. (This can be thought of as decreeing that each parallel classmeets at some “point at infinity”.) Clearly the given affine plane arises from thenew structure by deleting the line at infinity. We leave the verification that theenlarged structure is indeed a projective plane to the reader.¥

We remark that this process of embedding an affine plane is in fact uniqueup to isomorphism, whereas non-isomorphic affine planes may in general beobtained from a given projective plane, depending on which line is deleted.

As an immediate consequence of Propositions 2.3 and 2.7 we have:

2.8 Corollary. For any prime power q, there exists an affine plane of order q.¥

These affine planes may, of course, be constructed directly, in a way similar tothe proof of Proposition 2.3. One then uses a 2-dimensional vector spaceW overthe fieldF and takes as points the vectors and as lines the cosets of 1-dimensionalsubspaces. The reader should verify this assertion and draw pictures of the affineplanes of orders 2 and 3. We now generalise the common combinatorial featuresof finite affine and projective planes and arrive (by replacing “exactly one” in(2.1.a) and (2.4.a) by “exactlyλ” whereλ is an arbitrary natural number) at thefollowing definition of a block design.


2.9 Definition. A finite incidence structureD = (V,B, I ) is called ablockdesignwith parametersv, k, λ (v, k, λ∈N) if it satisfies the following condi-tions:

|V | = v;(2.9.a)|(p,q)| = λ for all {p,q} ∈ (V

2

), i.e. any two distinct points(2.9.b)

are joined by exactlyλ blocks.

|(B)| = k for any blockB.(2.9.c)

For reasons which will be explained later we will often callD briefly anSλ(2, k; v) or in case ofλ= 1 simply anS(2, k; v). The letterS abbreviates“Steiner system”, cf. Definition 3.1.

Thus in this terminology, projective planes of ordernare examples ofS(2, n+ 1;n2 + n+ 1) and affine planes ofS(2, n; n2). In fact, these are the only exam-ples of block designs with these parameters. The reader should prove this as anexercise, see Exercise 2.21 below.

We have found examples of projective planes for all prime power values ofn.It is not known whether there exist projective planes of any other order.We will prove a famous non-existence result due to Bruck and Ryser (1949) inSection II.4.

The reader might wonder why we have postulated only the constancy of theblock degrees in Definition 2.9 and not also of the point degrees, which wasalso satisfied in our examples. The reason is that this property is a simpleconsequence of the next result.

2.10 Theorem. LetD be an Sλ(2, k; v). Then we have:

|(p)| = λ(v − 1)/(k− 1) =: r for all points p;(2.10.a)

|B| = λv(v − 1)/k(k− 1) =: b.(2.10.b)

Proof. Let p be a fixed point ofD. As in the proof of Proposition 2.2, wecount all flags(q,G)with pI G andq 6= p in two ways and obtain the equationλ(v − 1) = |(p)|(k − 1) by (2.9.a), (2.9.b), and (2.9.c). This yields (2.10.a);then (2.10.b) follows from (1.7.a) using (2.10.a).¥

As r andb must be natural numbers, we have the following


2.11 Corollary. Let v, k, λ∈N. Necessary conditions for the existence of anSλ(2, k; v) are

λ(v − 1) ≡ 0 (modk− 1);(2.11.a)

λv(v − 1) ≡ 0 (modk(k− 1)). ¥(2.11.b)

Much of this book is devoted to the problem of finding sufficient existenceconditions for block designs, the basic problem in design theory. We will infact prove in Chapter IX that fork = 3, 4, and 5 the conditions (2.11.a) and(2.11.b) are sufficient (Hanani), with one exception fork = 5. Also, by a deepresult of Wilson (1975), they are asymptotically sufficient, i.e. for all sufficientlylargev, given a fixed pair(k, λ). This will be proved in Chapter XI.

Givenv andk, there is a trivial block design with these parameters, where onetakes as blocks allk-subsets of av-set; hence in this caseλ = ( v−2

k−2

). More gen-

erally, we shall call any “multiple” of such a design (that is, in the terminologyto be introduced in §3, anyt-design withk= t) a trivial design. For obviousreasons, these designs are sometimes called “complete designs”; for instance,this terminology is used by Cameron (1976a). However, this contradicts theoriginal use of the term in the statistical literature on designs, where acompletedesign is a design withk= v; we shall use this term in the same sense. Notethat a complete design is a special type of trivial design.

Affine and projective spaces provide further families of block designs, gener-alising the constructions of projective and affine planes given above.

2.12 Definition. Let F be a field andW ann-dimensional vector space overF .Then the set of all cosets of subspaces ofW, ordered by inclusion, is calledthe n-dimensional affine spaceover F . If F is the field onq elements, itwill be denoted byAG(n,q). The cosets of{0} are calledpoints, those of1-dimensional subspaceslines, those of 2-dimensional subspacesplanes, those(n − 1)-dimensional subspaceshyperplanes, and in general the cosets ofi -dimensional subspaces are calledi-dimensional flatsor just i-flats. The pointsof AG(n,q) together with thed-dimensional flats ofAG(n,q)as blocks and inci-dence by natural containment form an incidence structure denoted byAGd(n,q).By abuse of notation, we often writeAG(n,q) instead ofAG1(n,q). A justifi-cation for this will be given in Section 8.

2.13 Proposition. AGd(n,q) is a block design with parametersv = qn, k =qd, r = [ n

d

]q, λ = [ n−1

d−1

]q

and b= qn−d[ n

d

]q. Here

[ ni

]q

denotes the number


of i -dimensional subspaces of an n-dimensional vector space over GF(q), theso-calledGaussian coefficients.

Proof. The values forv andk are trivial. Note that the number ofd-flats contain-ing a pointx equals the number ofd-dimensional linear subspaces containing0 = x − x, i.e.

[ nd

]q. Similarly, the number ofd-flats containing two pointsx

andy equals the number ofd-dimensional linear subspaces containing 0 andx− y. But this is the same as the number of(d− 1)-dimensional subspaces ofW/U , whereU is the subspace generated byx − y. Henceλ = [ n−1

d−1

]q. ¥

For the convenience of the reader, we will compute an explicit formula for theGaussian coefficients:

2.14 Lemma. Let q be a prime power and n and d positive integers with d≤ n.Then [n

d

]q= (qn − 1)(qn−1− 1) · · · (qn−d+1− 1)

(qd − 1)(qd−1− 1) · · · (q − 1).(2.14.a)

Proof. Let W be ann-dimensional vector space overGF(q). The number oforderedd-tuples of linearly independent elements ofW is

(qn − 1)(qn − q) · · · (qn − qd−1).

The first vector may be chosen as any of theqn − 1 non-zero vectors. Then a1-dimensional subspace ofW is excluded, and the second vector may be chosenin qn− q ways. Continuing in this way, one arrives at the number given above.Now eachd-tuple determines ad-dimensional subspace ofW, and each suchsubspace is determined by(qd−1)(qd−q) · · · (qd−qd−1) d-tuples, since thisis the number ofd-tuples of linearly independent elements in ad-dimensionalvector space. Thus

[ nd

]q

is the quotient given in the assertion.¥

2.15 Definition. Let F be a field andW an(n+ 1)-dimensional vector spaceover F . Then the set of all subspaces ofW, ordered by inclusion, is called then-dimensional projective spaceover F . If F is the Galois fieldGF(q), it willbe denoted byPG(n,q). The 1-dimensional (2-dimensional, 3-dimensional,n-dimensional) subspaces ofW are calledpoints (lines, planes, hyperplanes);in general, the(i + 1)-dimensional subspaces are calledi-flats. The points ofPG(n,q) together with thed-dimensional flats ofPG(n,q) as blocks and inci-dence by natural containment form an incidence structure denoted byPGd(n,q).


By abuse of notation, we often writePG(n,q) instead ofPG1(n,q). A justifi-cation for this will be given in Section 8.

2.16 Proposition. PGd(n,q) is a block design with parametersv = [ n+11

]q=

(qn+1− 1)/(q− 1), k = [ d+11

]q = (qd+1− 1)/(q− 1), r = [ n

d

]q, λ = [ n−1

d−1

]q

and b= [ n+1d+1

]q.

The proof is similar to that of Proposition 2.13 and will be left to the reader.

For more about the Gaussian coefficients (and for the Gaussian polynomials),we refer the reader to Constantine (1987) . We have only given the most basiccombinatorial properties of finite affine and projective geometries by showingthat they yield designs. More material on the combinatorics of finite geometrieswill be provided later in §VIII.5 (subplanes, ovals, unitals, maximal arcs) and in§X.9 (partial spreads). For a detailed treatment of the combinatorial aspects offinite geometries, we refer the reader to Batten (1986), Beutelspacher (1982a,1983) and Hirschfeld (1985, 1998), Hirschfeld and Thas (1991) .

We will conclude this section with some more definitions and notation. As willbe quite clear from what follows, sometimes the postulates for block designsare too restrictive. It is in fact useful to consider conditions (2.9.b) and (2.9.c)separately. This yields the following notions:

2.17 Definition. Let λ be a natural number andK ⊆ N. A finite incidencestructureD = (P,B, I ) is called apairwise balanced design(PBD) with blocksizes fromK iff it satisfies

|(p,q)| = λ for any two pointsp,q,(2.17.a)

|(B)| ∈ K for any blockB.(2.17.b)

If D hasv points, it is called anSλ(2, K ; v). Forλ = 1, we also use the simplernotationS(2, K ; v) and the termlinear space(instead ofPBD). Note that notall numbers inK actually need occur as block sizes. In particular,K may beinfinite. If K is a singleton{k}, we arrive at the notion of a block design again.

2.18 Definition. Letk be a natural number andD = (P,B, I )a finite incidencestructure.D is called ak-hypergraphiff it satisfies

|(B)| = k for any blockB.(2.18.a)


If k= 2, we speak of agraph. Then the points are calledverticesand the blocksare callededges. (This is often called a “multigraph” in the literature; then“graph” means “simple graph” in our terminology.)

2.19 Notation. Letλ be a natural number andK ⊆ N. Then the set of allv ∈Nfor which anSλ(2, K ; v) exists is denoted bySλ(2, K ) or alternativelyB(K , λ)(which is Hanani’s notation). In caseλ = 1, we use the simpler notationB(K ).If K is a singleton{k}, we just writeSλ(2, k) or B(k).

Thus we may considerB as an operator on 2N, mappingK ⊆ N onto B(K ).This idea will play a fundamental role in the existence theory for block designsandPBD’s in later chapters. In fact,B will turn out to be what is known as aclosure operator. For now we will only translate the results of this section intoour new notation and collect the following facts.

qn ∈ B(q) for all prime powersq;(2.19.a)

qn + qn−1+ . . .+ q + 1∈ B(q + 1) for all prime powersq;(2.19.b)

qn ∈ B

(qd,

[n− 1d − 1

]q

)for all prime powersq and alln > d;(2.19.c)

qn + qn−1+ . . .+ q + 1∈ B

(qd + . . .+ q + 1,

[n− 1d − 1

]q

)(2.19.d)

for all prime powersq and alln > d.

In particular, we have

7, 9, 15, 27, . . . . ,2n − 1, 3n ∈ B(3) for all n;(2.19.e)

13, 16, 40, 64∈ B(4);(2.19.f)

21, 25, 85, 125∈ B(5).(2.19.g)

2.20 Exercise.Show thatr ≥ k, and henceb ≥ v, in anyS(2, k; v). This willbe generalised in Sections II.2 and II.6.

2.21 Exercises.(a) Prove that anyS(2, n + 1; n2 + n + 1) with n ≥ 2 is aprojective plane of ordern. Hint: Computer and assume the existence of twonon-intersecting blocks to obtain a contradiction.

(b) Prove that anyS(2, n; n2) with n ≥ 2 is an affine plane of ordern.

§3. t-designs, Steiner systems and configurations 15

§3. t-Designs, Steiner Systems and Configurations

Let us reconsider one of our examples a bit more closely. In Proposition2.13, we observed thatAG2(n, 2) (i.e. the incidence structure formed by pointsand planes in then-dimensional affine space overGF(2)) is an Sλ(2, 4; 2n)

with λ = 2n−1 − 1. In particular, any two points are on 2n−1 − 1 planes. Butthere is an even more remarkable uniformity: any three points are on a uniquecommon plane. Here we use the property that no three points are on a commonline in a vector space overGF(2). This observation leads us to the followinggeneralisation of our previous concepts:

3.1 Definition. Let t andλ be positive integers andD = (V,B, I ) a finiteincidence structure. ThenD is calledt-balancedwith parameterλ iff

|(Q)| = λ for anyt-subsetQ ⊆ V.(3.1.a)

If |V | = v, |(B)| ∈ K for eachB∈B, thenD is called anSλ(t, K ; v). If D isalsok-hypergraph (Definition 2.18), then it is called at-designwith parametersk andλ. A t-design onv points is called anSλ(t, k; v). In caseλ = 1, it is calledaSteiner system S(t, k; v). This is the reason for the notationSλ(t, k; v).

Thus affine and projective planes are Steiner systems. ThePBD’s are 2-balancedincidence structures and block designs are 2-designs. The designsAG2(n, 2)are simultaneously 1-, 2- and 3-designs. This is accounted for by the followingresult.

3.2 Theorem. LetD be a t-design and let s< t be a positive integer. ThenDis also an s-design. More specifically, ifD has parametersv, k andλt (whereλt is the number of blocks through a t-set), then the parameterλs (the numberof blocks through an s-set) is given by

λs = λt

(v − st − s

)/(k− st − s

).(3.2.a)

Proof. Let S be ans-set of points. We count all pairs(X, B), whereX is a(t − s)-set of points withS∩ X = 6© and B is a block withS∪ X ⊆ (B)in two ways. By (3.1.a) we obtainλt

(v−st−s

)such pairs, as there are

(v−st−s

)ways

of choosingX. On the other hand, for givenB with S⊆ (B) there are( k−s

t−s

)ways of choosingX; hence we also obtain|(S)|( k−s

t−s

)such pairs. This yields

the desired formula withλs = |(S)|. ¥


An interesting improvement of this result will be given in Theorem II.6.1. Asin Section 2, we introduce some notation.

3.3 Notation. Let t andλ be positive integers andK ⊂ N. ThenSλ(t, K ), orsimply S(t, K ) whenλ= 1, denotes the set of allv ∈N for which at-balancedincidence structure withv points, parameterλ, and block sizes fromK exists.For K = {k}, we simply writeSλ(t, k) or S(t, k). For t = 2, we will retainthe earlier notation 2.19. Hanani’s notation is not used fort 6= 2. There are alsosome special names for particular types of Steiner systems:

3.4 Definition. An S(2, 3; v) is called aSteiner triple system STS(v). AnS(3, 4; v) is called aSteiner quadruple system SQS(v).

Our example above has shown thatAG2(n, 2) is in fact a Steiner quadruplesystem. Thus we have

2n ∈ S(3, 4) for all n > 1.(3.4.a)

We will meet more examples oft-designs witht ≥ 3 later. There are in factother “classical” examples (such as the Möbius planes), but their constructionneeds a knowledge of automorphisms and will be postponed to Chapter III. Wenow look at 1-designs.

3.5 Definition. A 1-designSr (1, k; v) is called atactical configuration(orsimply configuration) with parametersv, r, k, andb := vr/k.

In terms of the incidence matrix, a configuration is characterised by constantrow sumsr and constant column sumsk. A special case of Theorem 3.2 is thefact that every 2-design is a configuration (already proved in Theorem 2.10). Wemention some more examples: any graph of constant degreer is anSr (1, 2; v)(such graphs are usually calledr-regular). Next, we give two examples whichare of fundamental importance for classical projective geometry.

3.6 Example. The configuration with parametersv= b= 10,r = k= 3 givenby the incidence matrix below and drawn in Figure 3.1 is called theDesarguesconfiguration. Its geometric importance is a follows. A projective plane is iso-morphic to one constructed from a field as in the proof of Proposition 2.3 ifand only if for any two triangles{p5, p6, p7} and{p8, p9, p10}, which are “inperspective” from a pointp1 (as drawn), the intersection pointsp2, p3, p4 ofcorresponding sides are on a common lineG1, see e.g. Hughes and Piper (1982).


Figure 3.1

The corresponding incidence matrix is

0 1 1 1 0 0 0 0 0 01 0 0 0 1 0 0 0 0 11 0 0 0 0 1 0 0 1 01 0 0 0 0 0 1 1 0 0

0 1 0 0 0 0 0 0 1 10 0 1 0 0 0 0 1 0 10 0 0 1 0 0 0 1 1 0

0 0 0 1 0 1 1 0 0 00 0 1 0 1 0 1 0 0 00 1 0 0 1 1 0 0 0 0

.(3.6.a)

In view of the importance of this characterisation, planes arising from fields arecalledDesarguesianand all others are callednon-Desarguesian. In definingprojective planes in Proposition 2.3, we did not require that the field is com-mutative. Interestingly, whether or not the field is actually commutative can bediscerned from another configuration.


3.7 Example. The Pappos configuration, with parametersv = b = 9, r =k = 3, is given by the incidence matrix

1 1 1 0 0 0 0 0 00 0 0 1 1 1 0 0 00 0 0 0 0 0 1 1 1

1 0 0 1 0 0 1 0 00 1 0 0 1 0 0 1 00 0 1 0 0 1 0 0 1

1 0 0 0 0 1 0 1 00 1 0 1 0 0 0 0 10 0 1 0 1 0 1 0 0

(3.7.a)

and looks as follows (check this!):

Figure 3.2

Its geometric meaning is the following. A projective plane is isomorphic to oneconstructed from a commutative field as in the proof of Proposition 2.3 if andonly if for any six distinct pointsp1, . . . , p6 (with p1, p3, p5 and p2, p4, p6

resp. collinear) the intersection pointssi of pi pi+1 and pi+3 pi+4 (i = 1, 2, 3;indices modulo 6) are collinear, see e.g. Hughes and Piper (1982).

Planes satisfying this condition are calledPappian. By the remark about theDesargues configuration, the validity of the Pappos configuration must implythat of the Desargues configuration. Also, it is well known that all finite fieldsare commutative; hence every finite Desarguesian plane is actually Pappian.Some finite non-Desarguesian projective planes will be constructed in Coro-llary X.9.20.


We conclude this section with some interesting digressions. We have seen thatit is possible to represent the Pappos and Desargues configurations in the realplane using straight lines. However it is impossible to do this for linear spacesexcept for trivial spaces. Specifically:

3.8 Proposition (Sylvester’s Problem).An S(2, K ; n) with at least threepoints on every line cannot be represented in the real space (using straightlines only) unless all points lie on one line.

Proof. Assume the contrary and choose a lineL and a pointp 6∈ L such that theeuclidean distance fromp to L is minimal, sayd. As L has at least three points,there is a triangle{p,a, b} with a, b∈ L such that the angle ata is at leastπ/2.But then the distance ofa from pb is smaller thand, a contradiction.¥

Figure 3.3

Proposition 3.8 explains why the “circle” in Figure 1.1 is “necessary” whenrepresenting Example 1.3. In connection with Proposition 3.8, we refer thereader to Coxeter (1961).

3.9 Remark. Gropp (1992) gives a report on the birth of design theory inBritish India. Some comments on the early history of what is now (unfortu-nately) called Steiner systems can be found in the paper by de Vries (1984).An S(2, 3; 9) was already known to Plücker (1835), realised on the nine pointsof inflection of a cubic curve without singular points in the complex plane. Itshould be noted that Plücker’s purported construction of anS(3, 4; 28) on thedouble tangents of a curve of degree 4 (which is mentioned by de Vries) isincorrect, as pointed out by Felix Klein. Special cases of the existence problemfor Steiner systems were posed and studied by Woolhouse (1844) and Kirkman(1847, 1850a), well before the note by Steiner (1853). In particular, Kirkman(1847) already settled the existence problem for Steiner triple systems. Aninteresting paper on Kirkman’s mathematical work is due to Biggs (1981).


We finally give some general comments on the existence problem fort-designsSλ(t, k; v). Theorem 3.2 gives some necessary conditions, since the parametersλs are obviously integers. However, these conditions are in general not sufficient,as we shall see in Chapter II. One of the main research areas in design theory liesin finding necessary and sufficient conditions for the existence of anSλ(t, k; v),given the parameterst, k andλ. Of course,t-designs withk = t or k = v

always exist; we have called such examplestrivial . In spite of much effort,no non-trivial Steiner systemS(t, k; v) with t ≥ 6 has yet been found; theanalogous statement also applies fort-wise balanced designsS(t, K ; v) withmin K > t . The situation changes drastically if blocks of sizet are allowed;then there is an enormous number of such designs, as the following result ofColbourn, Hoffman et al. (1991) shows. We here anticipate the formal definitionof an isomorphism which will be given in the next section. Also, we use theLandau symbolo(n) to denote a quantityx depending on a natural numbernsuch that the ratiox/n tends to 0 asn tends to infinity.

3.10 Theorem. The number Nt (v) of non-isomorphic t-wise balanced designson av-set (allowing blocks of size t) withλ= 1 is given by the asymptotic formula

Nt (v) = v[( vt )/(t+1)](1+o(1)). ¥(3.10.a)

§4. Isomorphisms, Duality and Correlations

After having introduced some of the main objects of this book in the previ-ous few sections, we now consider the notion of an isomorphism. The formaldefinition is the following one:

4.1 Definition. Let D = (V,B, I ) andD′ = (V ′,B′, I ′) be incidence struc-tures and letπ : V ∪ B→ V ′ ∪ B′ be a bijection.π is called anisomorphismiff it satisfies:

Vπ = V ′ and Bπ = B′;(4.1.a)

pI B⇐⇒ pπ I ′Bπ for all p∈V and allB∈B.(4.1.b)

In this case,D andD′ are said to beisomorphic. If D = D′, thenπ is calledan automorphism(or, if D is 2-balanced withλ = 1, acollineation, since itpreserves lines).

4.2 Observation. In terms of their incidence matricesM andM ′, the structuresD andD′ are isomorphic if and only if there exist row and column permutations

§4. Isomorphisms, duality and correlations 21

transformingM into M ′, i.e. if and only if there are permutation matricesP, Qsatisfying

PMQ= M ′.(4.2.a)

Clearly the set of all automorphisms of a given incidence structureD forms agroup. ¥

We introduce the following

4.3 Convention. Let D be an incidence structure. Then the group of all auto-morphisms ofD is called thefull automorphism groupof D and denoted by AutD. Any subgroup of AutD will be calledan automorphism groupof D. Theterms “full collineation group” and “collineation group” are defined similarly.

As an example, let us consider the automorphisms of the projective plane oforder 2 (see Examples 1.3, 1.5 and Figure 1.1).

4.4 Proposition. LetD be the projective plane of order 2 (Example1.3). Then|Aut D| = 168andAut D is regular2 on ordered triangles.

Proof. We will sketch the proof and leave the details to the reader. Let(a1,a2,

a3) and(b1, b2, b3) be ordered triangles and assumeaαi = bi (i = 1, 2, 3) forα ∈ Aut D. Let a4 be the third point ona1a2 andb4 the third point onb1b2. Ifα is to be an automorphism, we have to haveaα4 = b4. Arguing similarly fora1a3,a2a3 anda3a4, one sees howα has to be defined. One then has to checkthat this definition indeed yields an automorphism ofD. To determine the orderof Aut D, we thus only have to count the number of ordered triangles(a, b, c)of D. This is easily seen to be 168, as there are seven possibilities for choosinga, then six possibilities for choosingb, and four possibilities for choosingc (thethird point onab is not allowed!).¥

By results from elementary group theory (Sylow theorems) we conclude theexistence of elements of order 3 and 7 and of a subgroup of order 8 of AutD. Itmay be useful to see them explicitly. In the equilateral triangle representationof Figure 1.1 an automorphism of order 3 is given by the rotation about the

2 This means that for any two ordered triangles(a1,a2,a3), (b1, b2, b3) there is precisely oneelementα ∈ Aut D with aαi = bi for i = 1, 2, 3. More details on permutation groups will begiven in Chapter III.

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Second Edition - assets.cambridge.orgassets.cambridge.org/.../sample/9780521444323wsc00.pdf · First edition °c Bibliographisches Institut, Zurich, 1985 °c Cambridge University