Topics on Steiner Systems (Annals of Discrete Mathematics, Volume 7)

ANNALS OF DISCRETE MATHEMATICS Managing Editor Peter L. HAMMER, University of Waterloo, Ont., Canada Advisory Editors...

Author: Charles C. Lindner

178 downloads 1091 Views 14MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

ANNALS OF DISCRETE MATHEMATICS

Managing Editor Peter L. HAMMER, University of Waterloo, Ont., Canada Advisory Editors C. BERGE, UniversitC de Paris, France M.A. HARRISON, University of California, Berkeley, CA, U.S.A. V. KLEE, University of Washington, Seattle, WA, U.S.A. J.H. VAN LINT, California Institute of Technology, Pasadena, CA, U.S.A. G.-C. ROTA, Massachusetts Institute of Technology, Cambridge, MA, U.S.A.

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NEW YORK. OXFORD

ANNALS OF DISCRETE MATHEMATICS

TOPICS ON STEINER SYSTEMS

Edited by C.C. LINDNER Auburn University, Auburn, AL 36830, U.S.A.

and A. ROSA McMaste r University , Hamilton, Ont ., Canada.

1980

NORTH-€1OLLAND PUDLISHINC COMPANY

-

AMSTERDAM NEW YORK OXFORD

f

D NORTH-HOLLAND PUBLISHING COMPANY - 1980 .411 rights resewed. KO part of this publication mny be reproduced, stored in a retrieval system, or transmitted, in any f o r m or by any means, electronic, mechanical. photocopying. recording or otherwise, without the prior permission o f the copyright owner.

Suhrnisrioii t o this jourrral o f a paper witails the author's irrevocable arid exclusive authorizatiori o / the pirblislier l o collect ariv s u m or corisidcrations for eop.virig or rcproditction paJ.ahlc h!. tliirrl parries (as tncritioried in article 1 7 paragraph 2 of thc Dutch Copyriglit Act 01'1 91-7 and in f l i p Royal Decree of Julie 20. 1 9 7 4 [S. 351) pitrsuarit to article 16 b 0 ~ ' t l i cI)irtch CoiiJ,ri~c/ir Act 01 1912) atid/or to act in or out o f Court in coriricctiori r h cwwitli.

PRINTED IN T H E NETHERLANDS

PREFACE

The concept of a Steiner Triple System is so simple that any reasonably intelligent grade school student could understand it. A mathematician not familiar with the notion and its history, may well have concluded that the Kirkman paper of 1847, which settled the existence question, said all there was to say about this “puzzle”. Instead of this we find that in the intervening 132 years a large and impressive literature has grown up. The subject is still a lively one and there are no signs of a diminished interest in it. The interest is drawn from several sources. birstly, there is a large body of mathematics which has been used in the study of Steiner Triple Systems and their natural extensions. What immediately comes to mind are such things as cyclotomy and Galois Field Theory and other areas of algebraic numbers theory. Secondly, there is a connection between problems in combinatorial systems and other areas of mathematics. For instance, the work of Trevor Evans has shown that the problems of embedding partial systems in complete systems is equivalent to the solution of a word problem in certain algebras. One of the equivalents of the axiom of choice is the existence of Steincr Triple Systems on certain sets. Furthermore, the general Steiner Systcms have an interesting algebraic structure in themselves and their automorphisd groups are an attraction to many mathematicians particularly those interested in sporadic simple groups. Again, they have been useful in settling problems in other areas of mathematics, notably in lattice theory and universal algebra. It is of some interest to speculate on why such apparently diverse areas of mathematics are related to each other and why Steiner Systems play such an important role. How can such rich connections occur if mathematicians merely play a meaningless game with arbitrary axiom systems. Of course, mathematicians may concede that in form what they do may be described as a meaningless game. but in actuality what they achieve is a rich structure far removed from the sterility implied by the formal description. There is a mystique that mathematicians have so far avoided sterility because their axiom systems are not arbitrary or random, but are derived initially by abstracting from occurrences in “nature”. This mystique, in fact, has a sound rational basis. When a mathematical system originates in a concrete natural model the mathematician has available to him the ability to construct examples and counterexamples which guide him to conjectures on which he builds a cohesive structure. In fact, the mathematician operates as if he were an experimental scientist developing his theory from observations. For example, if one examines the way in which theorems in the theory of numbers were first conjectured by an experimental examination of the integers, the process bears a striking resemblance to the way in which a physical theory is built up from V

\i

Preface

experimental observation. And number theory has not yet lost any of its vitality. The same can be said from the present state of Steiner systems. Since Steiner Systems have a rich and cohesive structure let us examine their roots. The prime source of any combinatorial structure is geometry. The notion of a finite plane was derived from the “physical” planes of experience. It turns out that the Steiner Triple System of order 7 is the smallest finite projective plane while the system of order 9 is an afine plane of the next higher size. The use of the Galois Field for construction of systems thus becomes natural. The more general Steiner Systems which are associated with the Mathieu groups also have arisen naturally in more than one way. While attempts of mathematicians to use such systems in the construction of further quadruply and quintuply transitive groups have failed, the analogous use of doubly and triply transitive groups to construct cornbinatorial designs has been highly successful. This long preface to a volume on Steiner Systems was motivated by the depth and breadth of the papers included. Virtually no area of activity was overlooked. The volume gives an excellent survey of the state of the art and the direction in which research is moving. It is a most welcome addition to the literature and the editors are to be congratulated for their selection of participants, and the participants are to be congratulated for their contributions. N.S. Mendelsohn

FROM THE EDITORS In 1844 W.S.B. Woolhouse asked: for which integers 1, k, and u is it possible to construct a collection B of k-subsets of a set S of size u so that every t-subset of S is contained in exactly one member of B? In today’s terminology such a pair ( S , B )is called a Steiner system S(?,k, u ) . Three years later in 1847 the Rev. T.P. Kirkman obtained necessary and sufficient conditions for a Steiner system S(2,3, u ) to exist. One might wonder why, if the systems S(t, k, u ) were introduced by Woolhouse and the first major result was obtained by Kirkman, are they called “Steiner systems”. In 1853 J. Steiner unaware of the papers by Woolhouse and Kirkman, asked for the existence of the systems S(t, t + 1, u ) . The fact that the systems S(t, k, u ) still carry the commonly accepted name “Steiner systems” (a term coined by E. Witt) is most certainly due to the fact that writers in the late nineteenth and early twentieth century, just as Steiner, were unaware of the papers by Woolhouse and Kirkman and referred to the paper by Steiner. Regardless of the origin of the name “Steiner systems”, it is safe to say that the subject has grown to the point where today it occupies a significant position in combinatorial mathematics. To-date over 700 papers on Steiner systems have been written, the majority in the past 25 years. Obviously it would take an encyclopedia to even begin surveying all the aspects of Steiner systems which have been studied since Woolhouse’s day, and this volume is no such attempt. What we have tried to do here is bring together a collection of articles (both survey and research) which reflect a large (even if not exhaustive) portion of current research on Steiner systems. We would like to thank all the contributors for responding to our invitation and for their contribution to this volume. All articles were thoroughly refereed, and we would also like to extend our thanks to the referees for their effort.

C.C. Lindner and A. Rosa

Vii

CONTENTS

Preface (N.S. MENDFI.SOHN)

V

From t h e Editors (C.C. LINDNER and A. ROSA)

vii 1

1. Algebraic aspects of Steiner systems

B. GANTER and H. WERNER, Co-ordinatizing Steiner systems R.W. QUACKENRUSH, Algebraic speculations about Steiner systems L. BARN.Almost all Steiner triple systems are asymmetric

3 25 37

11. Steiner systems with higher value of r

41

P . J . CAVFRO\,External results and configuration theorems for Steiner systems R.H.F. D E N N I S I O N . The problem of the higher values of t

65

43

111. Steiner systems with given properties, collections of Steiner systems,

and their relationship to other combinstorial conligurstions

71

A.J.W. HIi:rm and L. TEIRLINCK, Dimension in Steiner triple systems E. MFNDF.I.SOHN. Perpendicular arrays of triple systems R.C. MUI.LIN and S.A. V A N S T O N E , Steiner systems and Room squares K.T. PriFi.ps, A survey of derived triple systems '4. ROSA.Intersection properties of Steiner systems

73 89 95 105 115

IV. Resolvability and embedding problems for Steiner systems

129

J. HA

1..

On identifying PG(3.2) and t h e complete 3-design on

seven points A. HARTMAN. A survey on resolvable quadruple systems M. Limos. Projective embeddings of small "Steiner Triple systems" C'.C'. L .I'U'I)NER. A survey of embedding theorems for Steiner systems D . E . WOOI.RRIGITI. O n the size of partial parallel classes in Steiner systems

203

V. Isomorphism problems and enumeration of Steiner systems

213

M.J.COI.HOUKN and R.A. MATHON.On cyclic Steiner ?-designs

215

R . H . F . DENNISION. Non-isomorphic reverse Steiner triple systems o f order 19

255

...

v111

131 143 151 175

Contents

ix

L.P. PETRENJUK and A.J. PETRENJUK, An enumeration method for nonisomorphic combinatorial designs K.T. PHELPS, On cyclic Steiner systems S(3,4,20) I. DIENER, On cyclic Steiner systems S(3,4,22)

265 277 30 1

VI. Bibliography and survey of Steiner systems

315

J . DOYENand A. ROSA, An updated bibliography and survey of Steiner systems

317

This Page Intentionally Left Blank

PART I

ALGEBRAIC ASPE .

(P,B)

A L G E B R A S

Let ( P , B ) b e a S t e i n e r ' T r i p l e System.

Let

i

= Q,{O,lj. ~~~~~

are defined.

(F;*).

B O O L E A K

N E A R

1''

choose

(P;o) i s a block algebra over

J

B

0 y := %l(%(x)*%(~)) f o r a l l x , y E P f o r which

i s a ( , i , q ) - S t e i n e r Systen

such t h a t

t

define a binary operation

%(x),

Let ( Q ; v , " , ' , O , l )

b

h:b--pF.

x

o by

g r o u p o i d g c n c r a t e d by x a n d y .

I'

(x-y).a.

+

(F;*).

~ x , y >denotes t h e sub-

(P,B)

F \ ( 0 , 1 ) an

be a (2,q)-Steiner

For e v e r y b l o c k

i d e n t i t i e s i n two v a r i a b l e s

rilierc

(P,B)

E

System.

i . c . a groupoid s a t i s f y i n g a l l

Define R : =

x * y := y

by

Let

he a b l o c k a l g e b r a

(F;*),

over

a 2).

>

tions

V , A

i n s u c h a way t h a t

f o r every block

h c R

( b u b ' u {O,ll;v,~,',O,l) i s a boolean algebra.

( Q ; v , ~ , ' , o , l ) is a

is a Steiner Triple

near boolean a l g e b r a .

System. 1Q1

=

ZIP1

+

2

Co-ordinatizing Sfeiner systems

S Q S

-

Let ( P ; q ) b e a n S Q S - s k e in , i.e. q is a t e r n a r y operation on P s a t i s f y i n g t h e i d e n t i t i e s

Y q(x,y,z) = q(x,z,y) q ( x , y , q ( x , y , z ) ) = 2. q(x,x,y)

S K E I N S Let ( P ,B) b e a S t e i n e r Q u a d r u p l e S y st em .

=

=

7

q(y,x,z)

D e f i n e on

P

a ternary

operation

q

by

q(x,x,y)

:=

q(x,y,x)

q(y,x.x) q(x,y,z)

:= Y

:=

4 t h p o i n t on t h e block through X , Y , a n d 2, f o r a l l x,y,zcP, XLYLZZX.

i s a S t e i n e r Quadruple

(P,B)

(P;q)

:=

i s a n S O S - sk ei n .

S y s t em

1. Co-ordinatizations With the abbreviation [x, y]" := x, [ x , y ] " + ' : = [ x , y]"oy,

n=0,1,2, ...

we can, for example, give defining identities for each variety of block algebras where t h e nearfield 9 is a field and t h e element a is primitive: x o x = x, [x, y]" = [y.

[x, yp--1 = x,

XI" whenever

1 s n, rn < q - 1 are such that a" + a"'

=

1.

We shall call such a variety a variety of block algebras ouer GF(q) although there is danger of confusion: The identities depend on t h e choice of the element a. There are, for example, two different varieties of block algebras over GF(S), corresponding to t h e primitivc elements a = 2 and a = 3:

(((x 0 y )o Y ) 0 Y )o Y = x.

For a = 3 the operation

0

is commutative, for a

=2

it is not.

B . Canter. H . Werner

8

The identities obtained in this way are not necessarily independent. Of course there are always several sets of identities defining a particular variety. N.S. Mendelsohn proved that the variety of all sloops can be defined by the single identity x ~ ( ( y ~ y ) ~ ~2 , ) ~ x ) =

D. Johnson has shown that one of the varieties of block algebras over GF(11) is definable by the two identities (y0x)ox = x o y

and

((xoy)o(yox))o(xoy)= y.

Each of t h e preceding examples establishes a correspondence between all Steiner Systems of a fixed type (1, k ) and a class of algebras' defined by a set of identities. Every such algebra co-ordinatizes a Steiner System of the respective type, and vice versa, each (t, k)-Steiner System is the underlying system of some (sometimes many) algebras in the class. The co-ordinatization is unique only for sloops, squags, and SQS-skeins (for squags and SQS-skeins it is even functorial). Except for near boolean algebras every co-ordinatizating algebra can be obtained from a Steiner System via the described construction. Note that squags and Stein quasigroups are special block algebras ( q = 3, q = 4) whereas sloops are near vector spaces over GF(2). We have already mentioned that all the classes of algebras defined in this paragraph are equational classes or uariefies, i.e. they are defined by identities (For block algebras we have an equational class for every particular choice of the element a E 9.). Varieties of algebras are closed under the formation of subalgebras (see Section 2), Cartesian producrs (Section 3). and quotient algebras (Sections 4-5). In some of the examples the identities have only implicitly been given, e.g. for block algebras. But for every choice of 9 and a E 3 an explicit set of (finitely many) identities defining t h e respective variety can be given.

2. Subalgebras A subspace of a Steiner System (P, B ) is a set S c P which is closed under forming blocks. A subalgebra of an algebra ( A ;F ) is a set S c A which is closed under the fundamental operations.

-

' W e LISC the word "algebra" as a generic term for algebraic structures like quasigroups. vector\pace\. lattices. ctc More precisely: An n-ary operarion f on a set A is a mapping f : A" A. An afRehrtr is a pair ( A where F = (jl, f,, . . .) is a family of ("fundamental") operations on A. The q u e n c e In,. P I ? . . . . ) of the arities of these operations is the type of the algebra ( A ; F ) . We shall consider o n l y classes in which all algebras have the same type. A groupoid ( A ; e.g. is of type (2). it has a single hinary ( = 2-ary) fundamental operation. In this case it is common to write xoy instead of ' I H . y 1 Another example: Near boolean algebras are of type ( 2 , 2 , I , 0.0) (constants are treated as nullarv operations).

;m.

0 )

9

Co-ordinatizing Steiner systems

If (P,B) is a Steiner System of type (t, k), then a subset S E P is a subspace iff JbflSlst j ~ G S .

VbEB

Trivially each subset of cardinality less than t is a subspace, and so is each block and the set P itself. If S is a subspace of (P,B), then (S, B f l P(S)) is a subsystem of (P,B). A subset S is a subalgebra of an algebra (A; F) iff

n,, . . . , x,,,ES j fi(x,,..., x , , ) E S .

V f i ~ F v x,..., ,

For example, a subset S E A is a subalgebra (subgroupoid) of a groupoid ( A; iff s, xoy E s. Varieties are closed under the formation of subalgebras: Every subalgebra of a squag is a squag, of a sloop is a sloop, etc. There is a 1-1-correspondence between the subalgebras of a coordinatizing algebra and the subspaces of the underlying Steiner System, except for near boolean algebras where some of the blocks may not correspond to subalgebras. A subalgebra (subspace) S is generated by a set G (in symbols: (G) = S ) if S is the smallest subalgebra (subspace) containing G. S is called n-generated if it is generated by an n-element set and strictly n-generated if, in addition, it is not generated by a set of smaller cardinality. The co-ordinatizing varieties can be characterized by their ?-generated algebras: An algebra (of appropriate type) is a 0)

vx, y E

Near vector space Every 2-generated subalgebra is a vector space over GF(q) over GF(q) Block algebra over ( F ;

Every 2-generated subalgebra is isomorphic to ( F ;

Near boolean algebra

Every 2-generated subalgebra is boolean

SQS-skein

Every 3-generated subalgebra is isomorphic to the 4-element SQS-skein

0)

0)

In particular, a groupoid is a squag iff every two distinct elements generate a 3-element subgroupoid isomorphic to the one displayed below.

I

x

Y

XOY

The 3-element squag

Moreover, a groupoid is a sloop iff every strictly 2-generated subgroupoid is isomorphic to the Klein four group A,.

I0

8 . Canter.

H. Werner

The four element SQS-skein (sometimes called Swierdzkowski-algebra) is Q,:=(A,;x+V+z). The next table shows the connection between “small” subalgebras and their underlying subspaces:

II

algebra

n

I

i sloop

1 2

2 4

point hlock

1

1 3

point hlock

j squag I

,

1 nearvector

1

j

&

:

1

3

nearboolcan algehre

I

4

2

X o r I6

I 2

3

SOS-skein

I

1

aleehra

j

I

point hlock

space

rbGz1

cardinality of a strictly n-generated subalgebra underlying suhspacc

hlock

1

pmnr block or S(Z. 3 . 7 ) point 3 points

3. Products The Cartesian (or direct) product of two algebras ( A :F) and ( 5 ;F) is the algebra ( A :F ) x

( B ;F ):= ( A x B ; F ) ,

where the operations f, E F are defined on A x R “componentwise” by

For example. the direct product of two groupoids ( A; 0) and ( B ;0) is t h e groupoid ( A x B : . ) detined by

Cartesian products are defined only for algebras of the same type. Varicties are closed under products: The direct product of sloops is a sloop, etc. The cardinality of the direct product is the product of the cardinalities of the factors. Thus t h e existence of a single finite algebra with at least two elements in a iarietv 3’guarantees the existence of infinitely many finite algebras in s‘.

Co-ordinatizing Sreiner sysrems

11

Fig. 1. The direct square of the 3-element squag.

An analogous construction for Steiner Systems in general does not exist: A direct product of an S ( 2 , 6 , 6 ) with itself would be an S ( 2 , 6 , 3 6 ) ; the parameter 25 = 5 . 5 is not even admissible for (4,5)-Systems.’ But in the cases where we have co-ordinatizations (that is, for ( t , k ) = ( 2 , q ) , ( 2 , q + l ) , or (3,4)), the Cartesian product of the co-ordinatizing algebras can be used to construct new Steiner Systems from old. The procedure is as follows: We start with two Steiner Systems ( P , , B , ) and ( P 2 , B2) of appropriate type (t, k ) , co-ordinatize (in the same variety, of course), and obtain the new system as the underlying of the Cartesian product of the two co-ordinatizations. (Pi, B,)

(PZ.B’)

I

co-ordinatization

factor systems

I

co-ordinatization

I

I

factors (algebras)

\

/

product algebra

underlying system

(P, B)

product system

As the Steiner System and the co-ordinatizing algebra usually do not have the same number of elements, the cardinality of the product system is not always the product of the cardinalities of the factor systems. The following table shows which ’The authors wonder if there is a reasonable way to turn the class of Steiner Systems of type (r, k ) into a category with products for some ( r , k ) # ( 2 . 3 ) , ( r , k ) # (3.4).

B. Ganter. H. Werner

12

cardinalities are obtained forming the product of an S(t, k, u ) and an S(r, k, w):

I

1 variety sloops squags near vector spaces block algebras ncar boolean algebras SOS-skeins

(1.

k)

(2.3) (2.3) ( 2 , q + 1) (2.4) (2, 3) (3.4)

cardinality of the product system

(u + I ) ( w + 1 ) - 1 uw

(q - 1)uw + u + w uw

2(u + I ) ( w + 1 ) - 1 uw

Of course, all these constructions can be performed without any coordinatization, in fact, most of them occur in Witt’s paper. Many of the attempts to co-ordinatize Steiner Systems were motivated by the hope to find further product constructions (the admissible values for (5,6)-Systems e.g. are closed under products), but n o new values were found in this way (so far). Nevertheless, in the algebraic setting the known product constructions are much more natural and easier to handle.

Fig. 2. The direct product of the 4-element sloop with the 2-element sloop.

We list their elementary properties: The product of subalgebras is always a subalgebra of the product. Furthermore a product of an algebra ( A; F ) with a one-element algebra is trivially isomorphic to ( A :F ) . Thus if one of t h e factors contains a one-element subalgebra then the other factor is (isomorphic to) a subalgebra of the product. With the exception of near boolean algebras all co-ordinatizing algebras have one-element subalgebras. So for each co-ordinatization except possibly for near boolean algebras, the factor systems are subsystems of the product system. SQS-skeins and block algebras (in particular, squags) are idernporent i.e. each singleton is a subalgebra. In a direct product of two such algebras t h e set of all pairs with fixed second component is a subspace whose subsystem is isomorphic to t h e first factor system, and the analogous situation holds for the second factor. The product system thus can in the case of idempotent co-ordinatization be partitioned i n t o subsystems isomorphic to a given factor.


13

As points and blocks are subspaces, they lead to subalgebras of the coordinatizations (with exceptions again in the case of near boolean algebras) and their products cause numerous subsystems of the product system. The following table shows which systems occur:

I variety

factor 1 factor 2

subsystem of the product

I

point point block

point block block

projective plane projective 3-space

point

point

point

spaces

point point block

point block block

block projective plane projective 3-space

block algebras

point point block

point block block

point block f i n e plane (over the nearfield 9 )

sloops

squags affine olane

point point point 2 points SQS-skeins point block 2 points 2 points 2 points block block block

point

2 points block block S(3,4,8) S(3.4, 16)(affine)

It seems funny that for near vector spaces the projective spaces can be generated “from nothing”; they are just the Cartesian powers of a single point. The same happens for SQS-skeins if we start with 2 points. We close this chapter with an example of a structure theorem for direct products which goes back to Birkhoff:

Theorem. If “Ir is a variety of algebras such that each member of “Ir has permutable congruences and a singleton subalgebra, then each finite algebra in Q has a decomposition into a direct product of directly indecomposable factors which is unique up to isomorphism and permutations of the factors. As we shall see in Section 5 this theorem applies to all co-ordinatizing varieties except to near boolean algebras (which do not have singleton subalgebras). But it can be generalized to near boolean algebras, thus all these products have the unique factorization property. 4. Morphisms A morphism between two Steiner Systems is a mapping which preserves subspaces.

8.Canter, H . Werner

14

A hornotnorphism between two algebras is a mapping which preserves the

fundamental operations. More precisely: A mapping q : PI

-

P2

is a morphism from the Steiner System (f,, €3,) to the Steiner System ( P 2 ,B 2 ) if the systems are of the same type and the image of each subspace of (PI,B , ) is a subspace of (P2.Bz). (An equivalent condition is that the image of each block is a block or has less than r elements). A mapping

q:A-+B is a homomorphism from the algebra ( A ;F ) to the algebra (€3; F ) if t l f , ~ F V a ,. . . . u , E A v ( f i ( a l s .. .. 4,,))=f,(cp(a1). . . ..v(a,,)).

Homomorphisms preserve subalgebras. The combinatorial interpretation of the homomorphisms is quite different for the co-ordinatizing varieties. We discuss the cases separately, excepting the case of near boolean algebras about which no results came to our knowledge. Syuags and SOS-skeins. A mapping between two squags (SQS-skeins) is a homomorphism if and o n l y if it is a morphism of the underlying Steiner Triple Systems (Steiner Quadruple Systems).

Block ulgehras. q > 3 . Every homomorphism between two block algebras is a morphism between the underlying systems, but not vice versa. Nevertheless. if a morphism CC, from ( P I .R , ) t o ( P 2 .B,) and a co-ordinatization of (P?. B2) in a \ arietg of block algebras are given. then (PI.B,)can be co-ordinatized in such a way that q is a homomorphism. ( A little bit more can be said about automorphisms. see Section 7 . ) Sloops. Ekery injective sloop-homomorphism induces an injective morphism of the underlying triple systems. and vice versa. If q : ( S ;0. e ) + (7'; 0, e ) is not injective t h e n q ' ( { e } is ) a nontrivial "normal" subsloop of (S:.. e ) . Such subdoops are treated in the next paragraph. The effect on the underlying Triple System is: ( a ) The underlying subspace N of the normal subsloop cp ' ( { e } )disappears. ( b ) Two points which are not in N are identified by cp iff the block joining them intersects N. ( c ) Blocks n o t intersecting N are mapped onto blocks.

N e w wcror spaces. Here t h e situation is even more involved and no theory has lieen developed. Every 1- 1-homomorphism induces a 1- 1-rnorphism and every I - I-morphism is induced by some (but not every) co-ordinatization o f the s y t e m s . For homomorphisms which are not 1-1. the situation should be similar to that i n thc case of sloops.


1s

Fig. 3.

5. Congruence relations and normal subalgebras If cp : A + B is a mapping, the kernel ker cp is the equivalence relation on A defined by (a, b ) E ker cp

a

cp(a)= cp(b).

If q : (P, 13)-+ (Q, C ) is a morphism between Steiner Systems, ker cp is called a congruence relation on (P, B ) . If cp : ( A ,f;) + ( B ;F ) is a homomorphism between algebras, ker cp is called a congruence relation on ( A ; F ) . If 8 is a congruence relation, the classes of 8 are called congruence classes. The congruence class containing the element x is denoted by [ X I & Instead of (a, b) E 8 we also write a8b. If S is the underlying set, denote S/0 := {[x]8 I x E S}. Let ( A ; F ) be an algebra and 8 an equivalence relation on A. 8 is a congruence relation on ( A ;F ) iff Vfi E F V a , , . . , , a,,,b l , . . . , b,,, E A a,86,..

. .. a,,Ob,,,

3

f ( a , , . . . , a,,)8f(bl.. . . , bq).

For example, an equivalence relation 8 on A is a congruence on the groupoid (A;o) iff

vX,y, xi, yi E A

X w , YOY

3

x 0 yex'o y r .

If 8 is a congruence relation on the algebra ( A ; F),then .rr, :( A; F ) + (A/8; F )

is a homomorphism with ker T, = 8, where vfi€FVx,,...,x,EA

and vx

EA

.rr,(~) :=

XI^.

fi(Cx,le , . . . , C ~ l e ) : = C f i ( X l ., . . , x,)18

B. Canter. H . Wenier

I6

( A / O ;F) is the quotient algebra of ( A ; F ) modulo 8.

Varieties are closed under the formation of quotient algebras (i.e. homomorphic images): Quotient algebras of sloops are sloops, etc. Let ( P . 5 ) be a Steiner System of type (2, k ) and 8 be an equivalence relation on P. 8 is a congruence relation on ( P . B ) iff VS. v. y ' ,

yay'

Z G P

3

((x.

y ) n [ ~ ] ~ = p ( s ( ,xq. n [ ~ ] e = p ) .

Here (x. y ) denotes the subspace generated by x and y. If 8 is a congruence relation of a Steiner System (P. B ) of type (2. k ) . then 7 , :(

is

P. R

-+ (

P/8. B/B)

a morphism with ker nn = 8. where r e ( x ) = [ x ] O for all x E Y and R/8:={nn(h)I b E B . lne(b)\>l}.

( P / f l .B/O) i5 called the quotient system of ( P , B ) modulo 6. Now we list some properties which congruence relations may or may not have: i Pernnurubilify. If 6 and

I

4 are congruence relations, then 8 0 4=~ $08.

I

j

j Regularity. If two congruence relations have a common class then they are

j equal. ~

Uniforriiiry. All classes of a congruence have the same cardinality.

11 Coherence. ~

If a subalgebra (subspace) contains a congruence class, then it is the union of congruence classes.

In particular t h e property of permutability has many structural consequences. For example, if two congruence relations 6 and t,b permute then 8ot,b is also a congruence. In t h e case of varieties of algebras, permutability implies that the lattice of congruences is modular, and many further properties. Block

Near vector spaces (sloops)

algebras (squags) Permutability Lattice of congruences Regularity Uniformity Coherence

SQS-skeins

Steiner Systems t y p e (2, k ) yes

yes modular Yes yes Yes Each congruence class is a subalgebra ~

* For

Near boolean algebras

modular Yes Yes Yes The congruence class containing 0 is a subalgebra

distributive Yes Yes yes No congruence class is a subalgebra (except e = Q x Q )

modular Yes yes Yes Each congruence class is a subalgebra

* yes yes Yes Each congruence class is a subspace

~~~~~

Steiner Systems of type (2. k ) , k > 3. the intersection of two congruence relations necessarily a congruence relation.

IS

not

i i

Co-ordinalizing Steiner systems

17

A subalgebra is called normal if it is a congruence class. From the above table it follows that for the co-ordinatizing varieties the congruence relations are uniquely determined by the normal subalgebras. In other words: For squags, sloops, etc. the normal subalgebras play the same r81e as the normal subgroups for groups. We give some characterizations of normality for sloops, squags and SQS-skeins due to R.W. Quackenbush and M. Armanious. The first one can be derived from the more general result that a subloop N of a commutative loop ( L ;0, e ) is normal iff vx, y € L

xo(y"N)=(xoy)oN.

Note that whether or not a subsystem ( S , C)of (P, B ) corresponds to a normal subalgebra does not depend on the structure of C but only on B \ C. x = y (mod N ) abbreviates that (x, y ) is in the congruence corresponding to N.

(AS ;subsloop 0. e) is normal N of iff a sloop for all x. y. z E S

x o yN ~ J ( x o z ) o ( y o ~ ) EN.

i..\J

;;;yN)iff

A subsquag N of a squag

( S : is normal iff for all x, y, z E S and w E N (xow)oy~N+ ( ( x ~ z ) ~ w ) ~ (Ny ~ z ) ~ 0 )

x-y(modN)iff xoa

3a, b E N

=

y 0b

x=y(rnodN)iff 3 a E N q(a,x, y ) E N (iff q(x, Y. N ) c N ) .

As immediate consequences we obtain: Every subsloop of ( S ; 0, e ) of cardinality :IS- 11 is normal. If two disjoint subsquags of ( S ; . ) both have cardinality $IS], then both are normal. Every sub-SQS-skein of (Q; q ) with cardinality

is normal.

18

R . (;aitrrr, H . Wenier

One-element subalgebras of an algebra ( A ;F ) and the algebra A itself are always normal, they correspond to the trivial congruence relations id, and A x A. If an algebra has no other congruence relations it is called simple. From the uniformity of the congruence relations we can deduce that every co-ordinatking algebra of prime cardinality is simple. even more: If 11 is t h e cardinality of a finite nnnsimple sloop then there are admissible numbers s and t greater than 0 with t i = ( s + I ! ( r + 1 J . If t i is t h e cardinality of a finite nonsimple squag or SQS-skein then there arc admissible numbers s. r > 1 with t i = sr. Quackenbush proved that for every finite Steiner Triple System either the co-ordinati7ing sloop or t h e co-ordinatizing squag is simple. He has also shown that t h e onlv nonsimple finite planar sloop has ti elements and the only nonsimple finite planar squag has 9 elements. Moreo\.er. he proved that everv finite simple sloop with more than 2 elements a n d e\ery finite simple squag with more than 3 elements is functionally complete. kl. Mendelsohn has t h e result that t h e sloop o f the smallest non-derived finite triple system must be simple.

6. Subvarieties So far we were concerned with the variety o f all sloops. the variety of all squags. etc. By imposing further identities we can form subvarieties. E.g. wc might consider t h e variety of all squags satisfying the associative law x . o ( V o z ) = (s0 y 1 0 2 . and would easily find that this variety is t h e rriuial subvariety (because

can be deri\ ed from these identities). I t is not always that disappointing, there are nontrivial suhvarieties of the varieties under consideration. In fact. there are very many as the following theorem (due to Quackenbush) shows:

Theorem. The Ititrice of carieties of sloops (squags) contains as a cowr preserving sirhiartice the larrice of all subsers of a coutlrahle ser. I f M C take any algebra ( A :F ) i n one of o u r varieties we may consider the \ ariet! genertcfed by rhis algebra. i.e. the variety defined by all identities that hold in ( A: F ) . This is always a subvariety. in most cases ( e . g .when ( A :F ) is finite) it is a proper subvariety. Each of the co-ordinatizing varieties has a stnallest nontrivial subvariety. that means a subvariety which is contained in all other subvarieties. Each of thew whvarietie\ is generated by the smallest nontrivial algebra in t h e respective class:

Co -ordinatizing Sfeiner systems

19 ~

~~

The variety generated by the 2-element sloop is the variety of all boolean groups. It is characterized among all sloops by the associative law x o ( y 0 z ) = (x0y)oz. The underlying triple systems are the projective spaces over GF(2). The variety generated by the 3-element squag is the variety of all groupoids ( V; o ) , where V is a vector space over GF(3) and x y : = 2x + 2y. It is characterized among all squags by the medial (or entropic) law ( W O X ) ~ (yoz)= ( w ~ y ) ~ ( x ~ z ) . The underlying triple systems are the affine spaces over GF(3). 0

The variety generated by t h e q-element near vector space over GF(q) is the variety of all vector spaces over GF(q). It is characterized among all near vector spaces by the associative law. The underlying Steiner Systems are the projective spaces over GF(q). The variety generated by a q-element block algebra over GF(4) is the variety of all groupoids ( V ; 0) where V is a vector space over GF(4) and t h e operation 0 is defined by x o y = y + (x - y)a. It is characterized among all block algebras over the same algebra by the medial law (wox)o(yoz)= ( w o y ) o ( x o z ) . The underlying Steinet Systems are the affine spaces over GF(4). The variety generated by the 2-element near boolean algebra is the variety of all boolcan algebras. It is characterized among all near boolean algebras by the associative law (X A y ) z ~= x A ( y A z ) . The underlying triple systems are the projective spaces over GF(2). The variety generated by the 2-element SQS-skein is the variety of all algebras ( V ; 4). where V is a vector space over GF(2) and q ( x , y, z ) = x + y + z . It is characterized among all SQS-skeins by the identity 4 ( u , x, q(u, y, 2 ) ) = 4 ( x . Y, z). The underlying Steiner Quadruple System of such an SQS-skein has as blocks the affine planes of the vector space V.

For block algebras over proper nearfields the situation is not yet clarified. The next step would be to consider varieties which are generated by other co-ordinatizing algebras. The authors know of only one case where this has been done: R.W. Quackenbush has studied the variety generated by the 7-element squag.

20

R . Ganter, H . Werner

Some work has been done concerning block algebras over GF(q). For every q > 3 the two distributive identities

together imply t h e medial law

(Soublin) and thus force the underlying Steiner System to be an afine space. This is not true for q = 3: The underlying triple systems of the squags satisfying both distributive laws are exactly the afine Triple systems, i.e. triple systems in which every triangle generates an affine plane. The algebraic properties of these systems are well studied, see Klossek and Young. T. Kepka (private communication) was able to construct for each q 3 5 finite block algebras which satisfy one of t h e distributive laws but not the other. This is a rather surprising result and the underlying Steiner Systems should be of conibinatorial interest because of their (transitive) automorphism group: Every point is the (only) fixed point of some automorphism. In (91 block algebras over G F ( q " ) are studied which have t h e property that the operation * defined by

is medial (this can. o f course be formulated as an identity for t h e operation 0, these algebras thus form a subvariety).The underlying Steiner Systems are exactly those of type (2. q " ) whose blocks can be represented by subspaces of a vectorspace over GF(q). If * satisfies. in addition,

thc bariety obtained co-ordinatizes exactly the (2, q")-translation spaces (cf. Rarlotti & C'ofnian). I n [9] some applications to translation planes and spreads are mentioned.

7. Some remarks on automorpbisms One might hope that there is an algebraic description of t h e automorphism group of a Steiner System, at least if the system was constructed via the direct product of co-ordinatizing algebras. In fact one has:

Co-ordinnrizing Steiner systems

21

Lemma. Let -Y. be a variety of block algebras or the variety of SQS-skeins. If ( A ; F ) , (B; F ) E ?f are simple algebras which are not embeddable into each other, then Aut((A; F ) x ( B ;F))=Aut((A; F))xAut((B; F ) ) .

In the case of squags and SQS-skeins the algebraic automorphisms coincide. For example, E. Mendelsohn used the language of squags and SQS-skeins in his proof that every finite group is the automorphism group of some finite STS and some finite SQS. For other block algebras the situation is quite different: Suppose (P, €3) is a Steiner System of type (2, q), q >4, and let cp be an automorphism of (P, B ) not fixing all blocks. Then there is a block algebra co-ordinatizing (P, B) for which q is not an automorphism. In [9] some attempts were made to obtain an algebraic description of the automorphisms of (P, B) by introducing the concept of a semilinear mapping: A mapping cp : P+ Q between two block algebras over GF(q) is called semilinear if there is an integer n < q such that

Theorem. Let ( P ; " ) and (Q;.) be nontrivial block algebras over GF(q) and suppose that every desarguesian plane in the underlying systems is co-ordinatized medially. Then every automorphism of the Steiner System co-ordinatized by ( P ;0) x ( Q ; is semilinear. 0)

Corollary. Every automorphism of the desarguesian plane of order q is a semilinear self-map of (GF(q); 0 ) x (GF(q); 0).

8. Derived operations Quackenbush has observed that there is a very simple connection between near vector spaces and block algebras over GF(q): If (Q; +, 0) is a near vector space over GF(q), define a groupoid (Q;.) by x o y := y

+ (x - y ) - a

for some suitable element a. Then (Q;.) is a block algebra. As an immediate consequence one has a construction3 which produces from every S ( 2 , q + 1, u ) an S ( 2 , q, ( q - l ) u + 1). This is an algebraic version of Quackenbush's "Idempotent Reduct Theorem". For more details cf. Quackenbush's article in this book.

_71

B. Gamer, H . Werner

The tZ.q)-system constructed in this way contains a central point, i.e. a point with t h e property that every triangle containing this point generates a (desarguesian) affine plane. I t seems likely that from a block algebra with a central point (1 near vector space can be reconstructed. This ha5 been studied in the case q = 3, where the following holds:: 1-et ( P ; 0 ) be a squag and c E P be a central point. Define o n P a binary operation @ by

Then ( P ;@. c ) is a near vector space o v e r GF(3). This near vector space is usually called the internal loop of t h e triple system. The internal loop satisfies t h e moufang identity ( . r @ ( y @ z ) ) @ x= ( x @ y ) @ ( z @ x ) iff the corresponding squag is self-distributive. and is an elementary abelian 3-group iff t h e squag is medial. Young and Klossek have applied Bruck's general t h e o r y o f nilpotency for loops to t h e internal loops of affine triple systems. and obtained rather sophisticated results on the lattice of subvarieties o f these systems. Nothing similar is known for the other co-ordinatizations, except that from each co-ordinatizing algebra a loop structure can be derived: neutral element of the loop

loop multiplication x c Y

t*

s F P arhitrary (I

s F I' arbitran

\ + Y

[[x. 51'. [v. sy1'. u - I - ( I h . (1

'

whcre -

a"'.

(I \

c I' arhitrar!

I t is completely unknown if thcsc or similar structures lead to further coordinatizations o!' Steiner Systems. lt seems likelv that there are such. We close the paper with t w o theorems showing that there are n o straightforward generalizations o f near vector spaces and of block algebras:

Theorem (Quackenhush ). Let 'I' he ari equational class with an esseritial binury polyriomial. Suppose that I ' has the property that euery 0-generared algebra h a s exactly 1 -elernenr. every strictly 1-generated algebra has exactly i)i elernerirs and evrry srrictly 2-gerierated algebra has exactly ti elements. Then Pn is a prinie power U t l d I1 = ) ? I 2 .

Theorem. 1-rt ( A ; F ) and ( R ;F ) be nontricial algebras such that the set of r -gerirrared subalgebras of ( A; F ) :< ( R ; F ) forms a Steiner System S(r, k . LI) with r -< L L. Theri these algebras are block algebras or SQS-skeins.

Co-ordinafizingSrcinrr systems

23

9. Some open questions ( 1 ) R.W. Quackenbush remarked that it should be straightforward to dcfine and study near vector spaces over nearfields. Is it? (2) Describe the variety generated by t h e smallest nontrivial block algebra over a proper nearfield 9. Are the underlying Steiner Systems just "affine spaces over 2P" (e.g. in the sense of Andre)? (3) Give a combinatorial description of t h e "near boolean algebra product" of Steiner Triple Systems. ( 3 ) Do the congruence relations of a Steiner System in general form a lattice'?

( 5 ) Give a simpler characterization of normal sub-SQS-skeins. (6) Is the assumption about the subplanes in t h e theorem in Section 7 dispensable? (7) Find a common generalization o f the two theorems in Section 8 ( 8 ) Is there a generalization of the co-ordinatization via near boolean algebras'?

References hot all references listed here are quoted in the text. We have also included some which arc of historical interest or which may be useful for further research in this field. [ I J J . Andri.. Affine Geometrien uber FastkBrpcrn. Mitt. Math. Sem. Univ. Gie.;scn 1 1-1 (1975,

I-w. [ 21 1 l.-J. Arnold. Zur Algebraisierung allgemeiner aHincr und zugehariger projektivcr Strukturen mit Hilfe eines vektoriellen Kalkuls. I n : Arnold et. al., cds.. Beitrage 7ur Gconietri\chen Algebra. l3irkhauser Verlag ( 1977) 25-29. 131 A. Rarlotti and J . Cofman. Finite Sperner spaces constructed from projective and aRnc spaces. Ahh. Math. Sem. Hamburg 40 (1973) 731-241. 1-11 R.H. Bruck. A survey of binary systems (Springer Verlag. Hcidelbcrg. 1971). [5] R.H. Bruck. What is a loop? in: A.A. Albert, ed.. Studies in Modern Algebra (Prcntice Ilall. Englewood Cliffs, NJ. 1063) 59-09. [ 6 ] T. Evans. Universal-algebraic aspects of combinatorics. ['reprint ( 1977). [7] B. Gantcr. Comhinatorial designs and algebras. t o appear in Algebra Unibersalis. [ X I 6 . Ganter, Kombinatorische Algebra (1.ecturc Notes. Darmstadt, 107-1). 191 B. Canter and R. Mety. Kombinatorische Algebra: Koordinatisierung von Hlockplanen. in: Arnold e t al., eds.. Beitrage zur Geometrischen Algebra (Rirkhauser Verlag, Rasel, 1977) I 11-124. [ l o ] B. Ganter and H. Werner, Equational classes of Steiner Systems. Algebra IJniv. 5 (197.5) 125-140. 1111 6. Canter and H. Werner, Equational classes o f Steiner Systems [I. Proc. Conf. Algebraic Aspects of Combinatorics IJniv. of Toronto (1975) 283-285. [I21 D. Johnson, A (2, 11) combinatorial groupoid, Discrete Math. 19 (1977) 265-271. [13] S. Klossek, Kommutative Spiegelungsraume, Mitt. Math. Sem. LJniv. Giessen 117 (1975) [ 141 E. Mendelsohn. The smallest non-derived triple system is simple as a loop. Algebra Univ. X (1978) 2 5 6 2 5 9 . [ 151 N.S. Mendelsohn. Combinatorial designs as models of universal algebras. Recent Progress in Combinatorics. Proc. Third Waterloo Conf. on Comb. (1968) 123-132.

24

B. Ganter,

H. Werner

[ 161 J . M . Oshorn. Vector loops, Illinois I . Math. 5 (1961) 565-584. [ 171 N.K Pukhare\.. Construction of At-algebras. Siberian Math. J . 7 (1966) 577-579. I1 X I R.W. Ouackenhush. Alprhraic aspects of Steiner Quadruple Systems. Proc. Conf. Algebraic Aspects of Comhinatorics. LJniv. of Toronto 11975) 265-268. 1141 R.W. Ouackenbush. Linear spaces with prime power block sizes. Arch. Math. 18 (1977) 38 1-386. [?!I] R W Ouackenhush. Near boolean algebras I: Comhinatorial aspects. Discrete Math. 1 0 i1074) 30 1-3OX. [ Z l R.W. Ouackenhush. Near vector spaces o \ e r GF(q) and (t.q + 1. I)-HIHL>s. Linear Algehra and Appl. 1 0 ( 1 0 7 5 , 259-266. [??I R.W. Quackenhush. Varieties o f Steiner loops and Steiner quasigroups. Canad. J . Math. 28 11976) 11x7-119s. (2.31 S.K. Stein. Homogeneous quasigroups. Pacific J. Math. I4 ( IO64I I(t91-I 102. [?-I] 1.. Szamktgowici. Sulla generalizzazione del concetto delle algehre A:. Atti Accad. Naz. Lincei Rend. Cl. Sci. Pis. Mat. Natur. ( 8 ) 78 (1965) 810-813. j 25 1 1. S 7 a m k c ~ o w i c t 0. 1 1 Steiner manifolds. Colloq. Math. 20 (1969) 45-5 I 12hj tf Werner. A unique factorization iheorein for Steincr Triple Systems. Preprint. (271 E. Witt.
R. W. Quackenbush

Roof. Let ( S , €2)

E P(q, u ) with distinguished element 0. Let a , , a2 E S and let (0, a , } € b,; we must describe the block 6 2 containing a , parallel to b , . But this is just the block containing a, in the affine plane generated by (0,a , , a,}. It is straightforward that this procedure gives a resolution of B into parallel classes.

What does this tell us about R ( q ) ? Of course, P ( q ) c R ( q ) cS ( q ) with each containment being proper. We know that u E S ( q + 1) implies u = 1 mod ( 4 ) and u ( u - 1 ) = 0 mod ( q 2+ q ) and that these conditions are asymptotically sufficient. Also, the number of residue classes mod ( q 2 +q ) which satisfy these two conditions is 2' where r is the number of distinct primes dividing q + 1; this number is less that q + 1. Thus when we translate to P ( q ) ,not all residue classes mod ( q 3 - q ) which are multiples of q occur. But for R ( q ) we know that the condition c = q mod (4'- q ) is asymptotically sufficient, so each residue class mod ( q 3- q ) which is a multiple of q occurs in R ( q ) . Hence Sp ( P ( q ) c ) Sp ( R ( q ) )c Sp ( S ( q ) ) with each containment being proper. For q = 3 we know that u = 1, 4 mod (12) is sufficient for S(4, u ) # $4. Hence c = 3 , 9 mod (24) is sufficient for P ( 3 , u ) # $3, We also know that u = 3 mod ( 6 ) (equivalently, u = 3 , 9 , 15,21 mod (24)) is sufficient for R ( 3 , u ) # $3. Thus when H. Hanani [ 5 ]showed that u = 1, 4 mod (12) was sufficient for S(4. u ) # fl, he solved half of Kirkman's Schoolgirl Problem. It would be nice to find some recursive constructions on R ( 3 ) (and more generally R ( q ) ) to generate designs with ) Sp ( P ( 3 ) ) . For instance, if u i E P ( q ) , uz E R ( q ) , then cardinalities in Sp ( R ( 3 ) L:,U~E R ( q ) . However, it seems difficult to find natural constructions on P ( q ) which yield members in R ( q ) but not in P ( q ) .This is analogous to the difficulty of finding constructions on P(6, u)'s with u = 1,6 mod (30) which yield a P ( 6 , u) with D = 16 or 21 mod (30). 4. Projective hyperplanes, a n e hyperplanes and maximal homomorphic images A proper subdesign (S', B')=( S , B ) E S ( q + 1) is a projective hyperplane if for every b E B. b n S'# $4. A proper subdesign ( S ' , B') c ( S , B ) E S ( q ) is an affine hyperplane i f for every p E S - S', (S,, BP)is a subdesign such that every b E B with exactly one point in S' has a point in S,, (here B,,= { b I P E b, b n S ' = $4) and S,, = { a I a E b E B,}). We next remind the reader that S ( q + 1) corresponds to the variety of all algebras such that every 2-generated subalgebra is a vector space over GF(q). This variety, V2(q), has a unique minimal non-trivial subvariety, namely "Ir(q),the Lector spaces over GF(q). Of course, V ( q ) is generated by V(q), the 1dimensional vector space over GF(q).See the article by B. Ganter and H. Werner i n this volumn. If q 2 3 , then we can have non-isomorphic A, A' in V , ( q ) corresponding to the same ( S , €3). For q = 2 however, if A and A' both correspond to (S. B), then A and A' are isomorphic. This distinction will become important later in this section.

Algebraic speculations about Sfeiner sysfems

29

Theorem. Let (S’, B‘) be a subdesign of ( S , B)E S(q + 1); (S’, B’)is a projective hyperplane iff there is an A E V2(q)corresponding to (S, B) and a homomorphism, h, from A onto V(q) such that (S’, B’)corresponds to h-’(O).

Proof. Let h : A + V ( q ) be onto and let (S‘,B‘) correspond to h-’(0); clearly (S’,B’)is proper. Let b E B ;thus b corresponds to a 2-dimensional subalgebra of A, A,. Hence h is not 1-1 on A, and so some 1-dimensional subalgebra of A, is mapped to 0 by h. This means that b nS’#F, and so (S’, B’)is a projective hyperplane. Conversely, let (S’, B’)be a projective hyperplane; we must define A and h : A+ V ( q ) so that h-’(O) corresponds to (S’, B’).As above, we let S = {a, I i E I}, V ( a , )= (0, ai, a:, . . . , a:-’} where V ( a i )is isomorphic to V(q)with 0 as zero; let A = U { V ( a i ) Ii E I } . For u , E S ’ , define h(V(a,))=O.For a i E S - S ’ , define h : V ( a , )+ V(q)to be an isomorphism. Since (S’,B‘) is proper, h maps A onto V ( q ) . For b E B, let V ( b )= U{V(a,)1 ai E b}. Since the operations of ‘V,(q) are at most binary, h will be a homomorphism provided it is a homomorphism on each V ( b ) . If b E B’,then h(V(b))= 0, so n o matter how we define V(b)to be isomorphic to V(q)2,h will be a homomorphism. Let {ao, a,, . . . , aq}= b E B - B’. Since (S’,B’)is a projective hyperplane, there is a unique element of b, say a,, contained in S’. Thus h( V(a,)) = 0 and h( V ( a , ) )= V(q)for 1S i s q . In this case we can also make V ( b ) isomorphic to V(q),such that h is a homomorphism on V ( b ) .Thus A E V2(q)corresponds to ( S , B) and h is a homorphism from A onto V ( q ) such that h-’(O) corresponds to (S’,B‘). Thus we see that the projective hyperplanes of (S, B)are just the subdesigns of (S, B)corresponding to “kernels” of homomorphisms of algebras in Sr,(q) corresponding to (S, B) onto V(q). Corollary. Let q = 2 (i.e., we are dealing with Steiner loops). Let ( S , B ) E S ( ~ ) correspond to A E ‘V2(2). Then there is a 1-1 correspondence between the projective hyperplanes of ( S , B ) and the homomorphisms of A onto V(2).

Proof. This follows from the fact that there is a unique A E V2(2)corresponding to (S, B). Problem. Is it true that for every ( S , B)E S(q + l),there is an A E ‘V2(q) such that the projective hyperplanes of ( S , B ) are in 1-1 correspondence with the homomorphisms of A onto V(q)? For A E V2(2),we let A, be the maximal quotient (homomorphic image) of A in ‘V(2); i.e., if A’ is any quotient of A in ‘V(2), then A’ is a quotient of A,. Such an A, always exists and is unique. Since A, E ‘V(2), it is a vector space and in some sense is the largest vector space which can be modeled in A. Thus, for instance, the dimension of A is the dimension of A,; the sublattice of subspaces of (S, B) generated by the projective hyperplanes is the projective geometry corresponding to A,; in particular, (S, B) is a projective space (i.e., A, = A ) iff

30

R.W. Ouackenbush

the intersection of all projective hyperplanes of ( S , B) is empty. The reader is referred to L. Teirlinck [12] where these and other variations are proved in the more general context of 2-coverings (where two points may lie in more than one block and where all blocks need not have t h e same size). A completely analogous situation occurs for affine hyperplanes. Recall that S ( q ) corresponds to dB?(q), the variety of all algebras such that every non-trivial 2-generated subalgebra is isomorphic to A(q), the 1-dimensional affine algebra over GF(q). Let d(q)be the variety of all affine spaces over GF(q) (vector spaces over GF(q) where the only operation is a x +(1 - a ) y ) ; d(q)is generated by A(q). Hence d ( 4 ) is the unique minimal non-trivial subvariety of d,(q).For qz-4,we again may have two non-isomorphic algebras in 7rz(q) corresponding to the same ( S , B ) E S(q). For q = 3 , the algebra corresponding to ( S , B ) is unique.

Theorem. Let ( A ' ,B') be a subdesign of ( A ,B ) ES ( q ) ; ( A ' ,B') is an amne hyperplane i f l there is an A E d,(q)corresponding lo ( A ,B ) and an a E A(q) and a homomorphism, h, from A onto A ( q ) such that A ' = h -'(a).

CoroUary. Let q = 3 (i.e., we are dealing with Steiner quasigroups). Let ( A ,B ) E S ( 3 ) correspond to A E d 2 ( 3 ) and let a E A. Then there is a 1-1 correspondence between the affine hyperplanes of ( A ,B )containing a and the homomorphisnis of A onto A(3).

The proofs are similar to the projective case and are omitted. Maximal quotients in d(3)and affine hyperplanes play an analogous role to maximal quotients in V ( 2 ) and projective hyperplanes. Also, for q > 4 we have an analogous problem.

Problem. Is it true that for every ( A ,B ) ES ( q ) , there is an A E sP2(q)corresponding to (A, B) such that for every a E A there is a 1-1 correspondence between the afine hyperplanes containing a and the homomorphisms of A onto A(q)'?

5. Almost all Steiner systems What percentage of Steiner triple systems have non-trivial automorphism groups'? To be more precise, let T ( n )be the number of (isomorphism classes of) triple systems of size s n and N ( n ) the number with non-trivial automorphism groups: what is the behaviour of N ( n ) / T ( n )as n - x ? Since every finite group is t h e automorphism group of a finite triple system it seems reasonable to expect N ( n ) / T ( n ) to tend to 1. Yet almost surely N ( n ) / T ( n )tends to 0. In fact, we conjecture that almost all triple systems have the following two properties: ( 1 ) (S. B )is planar; i.e., it is generated by every triangle (three points not in a block).

Algebraic speculations about Steiner systems

31

(2) ( S , B) is rigid; i.e., it has a trivial automorphism group. Evidence for this conjecture comes from a theorem of V.L. Murskii [8] about finite groupoids (algebras whose only operation is binary). Murskii proved that almost all finite groupoids have these three properties: (i) they are simple, (ii) their proper subalgebras are all trivial, (iii) they are rigid. We can consider triple systems as either idempotent quasigroups or loops; in either case they are groupoids. Hence we can speculate about the extent to which triple systems inherit these properties of groupoids. Since almost no groupoids are quasigroups, we can make no direct application of Murskii's result. Indeed, neither Steiner quasigroups or Steiner loops satisfy (ii) since every triple (together with 0 in the case of Steiner loops) forms a subalgebra. However, these are the only non-trivial proper subdesigns that a triple system must have; hence we conjecture (1). What about simplicity? Since we do not associate homomorphisms with triple systems and simplicity is concerned with homomorphisms, we can ignore condition (i). But perhaps triple systems which are simple as Steiner loops or Steiner quasigroups have certain properties which we should consider. Fortunately, a planar STS is simple both as a Steiner quasigroup and as a Steiner loop [lo] (with one exception in each case). Thus (1) and (2) are the analogues for triple systems of (i)-(iii). Note that (1) and (2) make no reference to block size; hence we shall extend the conjecture to S(n): Conjecture. Almost all members of S(n) are planar and rigid.

6. Derived triple systems

In another chapter of this book, K. Phelps has given a survey of derived triple systems which we assume the reader to have read. In this section we will speculate about how one could prove that every triple system is derived. The reader will have noticed that t h e general results of Phelps' article are recursive in nature (e.g.. the direct product of derived triple systems in again derived). However, it is clear that a proof that every triple system is derived is unlikely to be recursive in nature. A direct construction seems needed, but it is totally unclear how to make such a construction. We will describe a model-theoretic approach which may facilitate thinking along the lines of a direct construction. The first step is to prove that every infinite triple system is derived. The block designer, accustomed to considering only finite designs, may consider this result uninteresting or irrelevant. But this construction has two useful features; it shows that there is some hope for a direct construction, and it enables us to prove that

32

R . W. Quackenbush

the class of derived triple systems is axiomatic (i.e., it can be described by a set of first order sentences; what we would like to show is that this set can be taken to be just the sentences defining all triple systems).

Theorem. Every infinite triple system is derived. Proof. Let (S, T ) be an infinite triple system with O & S. Let S’= S U{O} and Q’= {I U{O} 1 I E T). Let T‘ be the set of triangles of S (i.e., 3-element subsets not in T); well-order T’ so that T’ = { f a .1 a ’ < a }where a = 19 .1. We assume inductively that for a”q is less than exp (-n'log

n(l/6-(1+3/q)/24+0(1))= n-(1-11q-cd1))n2'8

References [l] V.E. Aleksejev, 0 M e sistem trojek Stejnera, Mat. Zametki 15 (1974) 767-774. (English translation: On the number of Steiner triple systems, Math. Notes 15 (1974) 461-464.) [2] S. Friedland, A lower bound for the permanent of a doubly stochastic matrix, Ann. Math. 110 (1979) 167-176. [3] C.C. Lindner and A. Rosa, On the existence of automorphism free Steiner triple systems, J. Algebra 34 (1975) 430-443. [4] R.W. Quackenbush, Algebraic speculations about Steiner systems. [S] R.M. Wilson, Nonisomorphic Steiner triple systems, Math. Z. 135 (1974) 303-313.


PART I1

STEINER SYSTEMS WITH HIGHER VALUE OF

t


Annals of Discrete Mathematics 7 (1980) 43-63 @ North-Holland Publishing Company

EXTREMAL RESULTS AND CONFIGURATION THEOREMS FOR STEINER SYSTEMS Peter J. CAMERON Menon College, Oxford OX1 4JD. England

1. Introduction In the study of Steiner systems, most effort has been directed towards questions of existence and, to a lesser extent, enumeration. The point of view of this article is different, though related. The theme is structural properties of Steiner systems, and especially the characterization of special systems by structural properties. A prototype of this is the characterization of classical projective planes by the “theorems” of Desargues and Pappus. Two particular systems, those with parameters S(5,8,24) and S ( 5 , 6 , 12), will be very important in this paper. They and their first contractions were until recently the only known S(f, k, u ) with r >3; and they have a good claim to be regarded as “classical”. They have many striking properties. One of the best ways to illustrate these properties is through different constructions; and Section 2 is devoted to sketching several of the known constructions of these systems. Throughout the sequel, they reappear as the subject matter of various characterizations. They also crop up in other areas of mathematics, such as the theory of error-correcting codes [31], and the construction of finite simple groups [ 121. The next two sections discuss inequalities connecting the parameters of a Steiner system, and the determining of systems which attain the bounds. In Section 3, two simple inequalities are proved. In the first, the systems attaining the bound are projective planes and their extensions; the latter are completely known with the possible exception of one parameter set. In the second, both S(5,8,24) and S(5,6,12) meet the bound, but the situation is less well understood. The work of Gross on the numbers of blocks having prescribed intersection with a given block is described in Section 4. The material of the next three sections concerns some very geometric classes of Steiner systems: projective and affine spaces, projective, affine and inversive planes, and unitals. In the spirit of this paper, several classical characterizations of these systems are given. Another topic is the embeddability of Steiner systems in projective spaces or planes. In Section 8, we describe a simple property, the symmetric difference property, 43

44

P.J. Cameron

shared by S(5,8,24) and S(5,6, 12); it seems that it may characterize these systems along with the binary affine spaces. The last section discusses possible generalizations and further directions for research: subspaces, matroid designs, coordinatizations, association schemes, and automorphism groups.

2. Constructions of S(5,8,24) and S66,U )

The existence and uniqueness of the remarkable Steiner systems S(5,8,24) and S(5.6, 12) has been known for nearly fifty years, while their automorphism groups were discovered a century ago. Since these systems figure prominently in the rest of this paper. This section is devoted to outlining a number of constructions of them. The constructions also illustrate various properties of the systems, and many of them can easily be modified to give uniqueness proofs. The groups now known as M,, and M,,, were discovered by Mathieu [35], who simply wrote down permutations which generate the groups. A more satisfying construction was given by Witt [49]. He observed that the stabilizer of three points in M,, is PSL(3,4), the unique simple normal subgroup of Aut S(2,5,21) (the projective plane of order 4). Witt proved a sufficient condition for a doubly transitive group to have a transitive extension, and showed that this condition could be applied three times to PSL(3,4), yielding MZ4. Witt also proved a sufficient condition for a 1-fold transitive group to act on a Steiner system S(r, k, u ) , and deduced the existence of S(5,8,24). He subsequently proved the uniqueness of this system [50]. The first part of Witt’s programme can also be used to construct MI,. However, it is not possible to deduce the existence of S(5.6,12)in quite the same way. This is because Witt identifies a block as the set of fixed points of a subgroup; in M,,, only the identity fixes a block pointwise. Nowadays, the usual procedure is to construct a combinatorial object and then deduce properties of its automorphism group. Witt’s construction has been rewritten in this spirit by Luneburg [33]. He shows that the projective plane ll of order 4 can be extended three times, in a unique manner, to give S(5,8,24).If x, y, z are the added points, then the blocks containing x, y and z must be the sets of the form {x, y, z } U L, where L is a line of Il. The blocks containing two, one or none of x, y, z meet Il in sets of 6, 7 or 8 points respectively, which are identified as geometric configurations within Il: hyperovals (complete arcs), Baer subplanes, and symmetric differences of pairs of lines respectively. (Thus, there are 168 hyperovals, falling into three classes of 56, such that the hyperovals in one class meet in 0 or 2 points, while hyperovals in different classes meet in 1 or 3 points. Each pair of the extra points x, y, z is adjoined to all the hyperovals in one class to form blocks. A similar procedure applies to Baer subplanes. Finally, each symmetric difference of two lines is a block.) Luneburg also shows that, if U is a unital in n, then U U { x , y, z} carries an

Exnemal results and configuration theorems for Steiner systems

45

S(5,6, 12). Alternatively, the latter system can be constructed by extending S(2,3,9) (the affine plane of order 3) three times in similar fashion. Given a set X of 24 points, the subsets of X form a vector space V over GF(2), where the sum of two subsets S,, S, is their symmetric difference S , A S,. Choosing the basis consisting of singletons, any subset is represented by its characteristic function, a 24-tuple of zeros and ones; the cardinality of a subset is the weight of the corresponding vector (the number of ones in it). A remarkable 12-dimensional subspace W of V was discovered by Golay [19]. It has the property that all its nonzero vectors have weight 8, 12, 16 or 24. This condition determines the numbers of vectors of each weight; for example, there are 759 = ('p)/(!) vectors of weight 8. The corresponding subsets are called octads. Now five points can lie in at most one octad; for if there were two such octads, their sum would have weight 6 or less. A counting argument now shows that any five points lie in a unique octad; that is, the octads are the blocks of S(5,8,24). There are a number of constructions for the binary Golay code W, each of which gives implicity a construction for S(5,8,24): van Lint [31] gives some of these. It is possible to find S(5,6,12) in the Golay code also. Let Y be a dodecad (a vector of weight 12 in W). For any octad B, lBAYI=8, 12 or 16, and so J Bn YI = 6, 4 or 2. Any five points of Y lie in a unique octad B, and hence in a unique set B n Y of size 6. So the intersections of size 6 of octads with Y are the blocks of S(5,6, 12). There is also a ternary Golay code, a 6-dimensional subspace of a 12dimensional vector space over GF(3), to which S(5,6,12) bears the same relation as S(5,8, 12) does to the binary Golay code. Some properties of the Steiner systems follow easily from this construction.

Proposition 2.1. (i) Let B, and B, be blocks of S(5,8,24). Then IB1n B,I = 0, 2 or 4. Moreover, i f IB,n B,I = 4, then B,A B2 is a block; and if B,nB2= 8, then the complement of B,U B2 is a block. (ii) In S(5,6,12), the complemenf of a block is a block; and if B , and B, are blocks with IB,n B,1 = 3, then B,AB, is a block. Proof. Take the sums of appropriate elements of W. Note that, strictly speaking, this proposition applies only to the systems we have constructed. Until uniqueness has been proved, we cannot assume that the result holds for all such systems; and the conclusions of Proposition 2.1 are required in the uniqueness proofs, so direct proofs by counting are still required. Some of these proofs appear in a more general context in Section 4. Another construction is related to Proposition 2.1 (ii) and the analogous fact that the complement of a dodecad is a dodecad. It produces S(5,6,12) on the union of two disjoint 6-sets, and S(5,8,24) on the union of two disjoint 12-sets. A 1-fractorizafion of the complete graph on a set A of n points is a partition of

46

P.J. Cameron

the edges (the 2-subsets of A ) into n-1 classes or 1-factors, each of which is a set of in disjoint edges (covering A ) . When n = 6, there are precisely six different 1-factorizations, any two isomorphic, and any two sharing a unique 1-factor. If X is the set of six 1-factorizations, we thus have a bijection between the set of 1-factors and the set of edges (2-subsets) of X . The correspondence is reciprocal-given an edge in A, the three 1-factors including it correspond to three edges forming a 1-factor in X.Furthermore, given a 3-subset of X , removal of the corresponding threel-factors in A leaves a disjoint union of two triangles; and reciprocally. So 3-subsets of X correspond to partitions of A into two bets, and reciprocally. Now the following sets are the blocks of S ( S , 6 , 12) on A U X : A, X ; A - { a , b } U { x ,y}, X - { x , y } U { a ,b } , where {a, b } is an cdge of the 1-factor {x. y ) (and reciprocally); A - { a , b, c}U { x , y. z } , where {ab, c } is a 3-set in the partition corresponding to (x. v, z } (and reciprocally). I n group-theoretic terms, there is an outer automorphism of the symmetric group S6. under which a transposition is mapped to a product of three transpositions, and a 3-cycle to a product of two 3-cycles. The above construction is conveniently written in terms of this automorphism. Now let A* be the set of 12 points carrying S(S,6, 12). We must construct a 12-set X" carrying a reciprocal S ( 5 , 6 , 12). By Proposition 2.1, the 132 blocks fall into 66 disjoint pairs. Let R be the set of such pairs. From a graph with vertex set 0, in which { R , R ' } and {C,C'} are joined if and only if IB n Cl = 3. (Note that two blocks meet in 0 , 2 , 3 or 4 points.) This graph is isomorphic to the graph T(12) whose vertices are the 66 pairs from a set X * of size 12, two vertices adjacent if thev contain a common element of X * . It is now possible to construct an S(S.6. 12) on X * . The corresponding group-theoretic fact is the existence of an outer automorphism of M , * . A quick construction for the reciprocal pair of Steiner systems uses a Hadamard matrix of order 12, a 12 x 12 matrix H with entries f 1 satisfying HH' = 121. (Such a matrix is unique, up to permutations and sign changes of rows and columns. L.et D be the 11 x 11 matrix with rows and columns indexed by G F ( I 1 ), and ( i , j ) entry + 1 if I - j is a nonzero square, -1 otherwise. Then H is obtained by bordering D with a row and column of + l s . Note in passing that the rows of H, taken m o d 3 , span the ternary Golay code.) Given two rows of H , there are by definition six columns where they agree and six where they disagree; call these 6-sets blocks. These 66 pairs of blocks constitute S(5,6, 12) o n the set of columns of H . Since also HTH= 121, a similar construction gives the reciprocal system on the rows of H . The bijection between the pairs of points of one system and the disjoint pairs of blocks of the other is apparent. Given four points of A * , the four blocks containing them meet pairwise in four points, so t h e corresponding four point-pairs in X* are disjoint; the complement of their union is thus a 4-subset of X * . The reciprocal construction leads back to

Extremal results and configuration theorems for Steiner systems

47

the original 4-set in A * . So there is a bijection between 4-subsets of A* and X * . Now the blocks of S(5,8,24) on A*, X* are of three types: a block of A* and the corresponding point-pair in X * ; a block of X * and the corresponding point-pair in A * ; the union of corresponding 4-subsets of A * and X * . The sequence of constructions breaks off here, since has no outer automorphism. Finally, mention should be made of Todd’s list [47] of all the blocks of S(5,8,24), Curtis’ “miracle octad generator” or MOG [ 131, and Mason’s construction using a dodecahedron [34].

3. Bounds and extremal systems Let S be a Steiner system S ( t , k, u); we almost always assume t < k < u. The diuisibility conditions, asserting that (f-:) divides (y-:) for 0s is t, are well known. An ith contraction of S is obtained by taking those blocks which contain a given set of i points, and deleting those points. It is an S ( t - i, k - i, u - i); the number of blocks in such a system is simply the quotient of the numbers occurring in the divisibility conditions. Suppose t = 2 . By hypothesis, there is a block €3 and a point x $ B . For any y E B, there is a unique block By containing x and y ; the blocks By, as y runs through B,are all distinct. Since x lies in just ( u - l ) / ( k - 1) blocks altogether (the number of blocks in the first contraction of S), we have u - 1 2 k ( k - 1). For arbitrary t 2 2, this bound can be applied to the ( t - 2)nd contraction of S, yielding

Proposition 3.1. The parameters of S ( t , k, u ) , with 2 S r < k < u, satisfy u - t + 12 ( k - t + 2)(k - t + 1). When does equality hold in Proposition 3.1? Inspection of the proof shows that, for t = 2, we have u - 1 = k(k - 1) if and only if every block containing x meets B. Since x and B were arbitrary, this is equivalent to the assertion that any two blocks meet. Such a system is called a projectiue plane. Putting k = n + 1, we have u = n 2 + n + 1; n is called the order of the plane. We note in passing that the dual of a projective plane, obtained by interchanging the labels “point” and “block”, is a projective plane of the same order.

Proposition 3.2. Suppose S(t, k, u ) attains the bound of Proposition 3.1. Then (i) i f t = 2, then S is a projective plane; (ii) if t > 2 , then ( t , k , u ) = ( 3 , 4 , 8 ) , (3,6,22), (4,7,23), (5,8,24), or (3, 12, 112). Proof. We have seen (i) already. Suppose t = 3 and the first contraction is a

48

P.J. Cameron

projective plane of order n. Then k = n + 2, u = n 2 + n + 2 . The divisibility conditions assert that (n;Z) divides (”*+;+*), that is, n + 2 divides (n’+ n +2)(n2+n + 1). By the Remainder Theorem, n + 2 divides 12, whence n = 2, 4 or 10. The divisibility conditions also show that S(4,5,9), S(6,9,25) and S(4, 13, 113) do not exist. This completes the proof. Systems with the first five parameter sets in Proposition 3.2 (ii) are unique (up to isomorphism). The existence of S(3, 12, 112) is undecided. We refer to [ l l ] for several variations on Proposition 3.2. The case t = 2 of Proposition 3.1 is a special case of a more general result. It asserts that there are at least u blocks, with equality if and only if any two blocks meet in a point. The generalization, known as Fisher’s inequality, holds in any 2-design (a collection of blocks or k-subsets of a u-set, any two points lying in just A blocks), with 2 s k s u - 1. Fisher’s inequality asserts that such a design has at least u blocks, with equality if and only if any two blocks meet in A points. (The matrix D of Section 2 is the incidence matrix of such a design, with u = 11, k = 5, A = 2.) Fisher’s inequality has been generalized by Ray-Chaudhuri and Wilson [41]. They showed that a t-design with t = 2s and t s k =su - s has at least (1) blocks. They also characterized the case of equality by intersection properties of blocks. It is worth noting that, up to complementation, the only known design meeting the bound with s 3 2 is the Steiner system S(4,7,23).However, Gross [20] has shown that the Ray-Chaudhuri and Wilson inequalities give no new nonexistence results for Steiner systems, and that no other Steiner system attains the bound. We will discuss his work further in the next section. Another inequality connecting the parameters of a Steiner system is the following.

Proposition 3.3. The parameters of S(t, k, u ) , with 1 s t < k < u , satisfy u s ( t + I)(k - t + 1). Proof. Let B be a block, x,, . . . ,x, E B, and x,, a point outside B. Then no block contains {x,. . . . , x,+,}; so the r + l blocks B,,. . . , B,,, containing t of these points are all distinct. Moreover. any two such blocks have t - 1 of the points x,, . . . , x,+, in common, and so share no further point. So B,,. . . , B,,, include ( t + l)(k - t ) points other than xI, . . . , x , + ~ giving , the desired inequality. As a simple corollary of Proposition 3.3, we note that u 3 2 k , with equality only if t = l or k = l + l . When can the bound of Proposition 3.3 be met? If u = ( t + l ) ( k - t + l), then the proof shows that any additional point lies in one of the blocks B,,. . . , that is. given a n y t + 2 points, there is a block containing at least t + 1 of them. Known systems meeting the bound with t > 1 are S(3,4,8), S ( 5 , 6 , 12) and S(5,8,24). It is conjectured that, apart from the unusual case u = 2 k (to be discussed further in the next section), the only system meeting the bound is S(5,8,24). Some information on this question follows.

,


49

Let B be a block of S ( t , k , v ) , where u = ( t + l ) ( k - t + l ) and l < t < k - 1 . Given x, y $ B and zl, . . . , z, E B, there is a unique block containing t + 1 of the points x, y , zl,. . . , 2 , ; this block must contain x and y . So the sets B-B’, where 1B nB’(= t - 1 and x, y E B’, form a Steiner system S ( k - t, k - r + 1, k ) on B, where x and y are fixed. Also, the sets B ’ - B , where B’ contains t - 1 of the points zl,. . . , z,, form a Steiner system S(2, k - t + 1, v - k ) on the complement of B. This system is resolvable: the blocks containing a given t - 1 of zl, . . . , z, form a parallel class. There exist sets X of t + 4 points, no t + 2 of them in a block. Any t + 2 of these points have the property that a unique block contains t + 1 of them. Thus the sets X-B, where I B n X l = r + l , are the blocks of an S(2,3, t+4) on X. These Steiner systems give further divisibility conditions. For example, the ( k - t - 2)nd contraction of the first system is an S(2,3, t + 2) (note k 3 f + 2 by assumption); so t = 5 (mod 6). The analysis produces, as a by-product, further insight into the rich structure of S(5,8,24). 4. Intersection triangles

Given a block B of a Steiner system S(t, k, v ) , and a set I of i points of B, the number of blocks B’ for which B nB’ = I depends only on i, t, k and v. The easiest proof of this fact includes an algorithm for computing this number. The intersection triangle of S ( t , k, u ) is a triangular array { p i j10 < j s is k } defined by the rules

and pi+l,i + p , + l , , +--lpij for O s j s i s k.

It is easily proved, by induction on i - j , that this is a good definition. Furthermore, the same proof shows the following: if B = {xl,. . . , x,,} is a block, then pijis the number of blocks B’ for which B ’ n { x l , .. . , x i } = { x I , . . . , xi}. This establishes the assertion. As an example, we give the intersection triangle for S(5,8,24).

759 506

253 176 330 210 120 56 80 40 130 78 52 28 12 46 32 20 8 30 16 16 4 4 30 0 16 0 4

77 21

5

16

1

4 4 0

1

0 0

1

0 0

0

1

50

P.J. Cameron

Our discussion of the intersection triangle is based on Gross [20]. It is clear that ps = 0 whenever t S j S i : these are the trivial zeros. In the triangle for S(5,8,24), p8,1and pLs,3 are nontrivial zeros, providing the information that no two blocks

meet in 1 or 3 points, a fact remarked on in Proposition 2.1. By Proposition 3.3, S(t, k, v ) satisfies u a 2 k , with equality only if either t = 1 or k = t + 1. The first case is trivial. For the parameters S( t, t + 1 , 2 t + 2), the intersection triangle is symmetric about its vertical axis. The reflection of p k k = 1 yields p k o = 1, or (since v = 2k) the complement of any block is a block. Moreover, the reflection of the trivial zero p k t gives a non-trivial zero P k 1 : no two blocks have just one common point. We see from this that any S ( t - 1, t, 2 t + 1) is uniquely extendable, if at all. For, in making the extension, we must add the new point to all existing blocks, and then include the complements of existing blocks as new blocks; this completes the system. Moreover, this construction always produces S(t, t + 1 , 2 t + 2 ) . So every S ( t - 1, t, 2 t + 1) is uniquely extendable. Systems with the parameters S(t, t + 1 , 2 t + 2 ) are known only for t = 3 and 5 . In any such system, t + 2 must be prime. (For, by the divisibility conditions, i ! divides ( t + 3)(t+ 4) * * * ( t + i + 1) for 2 < i < t + 1; so no prime less than t + 2 can divide t +2 . ) Mendelsohn and Hung [37] have shown that t = 9 is impossible; other values are undecided. There are some nonexistence proofs under additional assumptions, e.g. [2]. The entries in the intersection triangle are all non-negative integers. Does this place any further restrictions on the parameters t, k , v? Also, which systems have a non-trivial zero? These questions were considered by Gross [20]. First, if the divisibility conditions are satisfied, then all p,# are integers; by induction, all ptl are integers. Next, note that pk,r-= ( v - t - 1)/(k - t - 1)- 1> 0. The number p k , r - 2 is particularly interesting. It is the number of blocks disjoint from a given block in the ( t - 2)nd contraction. Now the inequality v - t + 1 3 ( k - t + 2)(k - t + 1) of Proposition 3.1 is equivalent to the non-negativity of P k . r - 2 , and Proposition 3.2 describes the systems with p k , l - 2 = 0 . Gross showed that, from the divisibility conditions and the inequality of Proposition 3.1, it is possible to deduce that every entry in the triangle is a non-negative integer; so no new inequalities are obtained. Obviously the inclusionexclusion principle provides a formula for pS as an alternating sum of the p,, with appropriate coefficients; but such a form is not well adapted for proving nonnegativity. The Heart of Gross' argument is the formula

For example, it is immediate that pij> 0 if j f t (mod 2). The formula is proved, as usual, by induction on i - j. Gross also proves the following result.

Extrernal resulfs and configuration theorerris for Steiner systems

51

Proposition 4.1. The only Steiner systems S(t, k , u ) , for 2 s t < k < u, with nontrivial zeros in their intersection triangles, are the following: (i) those of (3.2), with pk,,..*= 0 ; (ii) s(4, 7,23) and s(5,8,24), with P k , , - d = 0; (iii) S( t, t + 1, 2t + 3), with Pk.0 = 0 ; (iv) S(t, t + 1,2t + 2), with P k . 1 = 0 . In particular, the only Steiner systems in which any two blocks meet are projective planes, S(4,7,23), and S(t, t + 1,2t + 3). For related results, see Hubaut [25], Noda [39]. A slight variant of the argument which began this chapter proves a stronger result: given a block B of S ( t , k , a ) , an i-subset I of B, and a point x$ B. the number of blocks B' with x E B' and B n B'= I depends only on i, r, k , u. This number is clearly p k i ( k- i ) / ( u - k ) ; it can also be found from an intersection triangle, as we illustrate for S(5,8,24). 253 176 77 120 56 21 80 40 16 52 28 12 4 33 19 9 3 21 12 7 2 1 15 6 6 1 1 15 0 6 0 1 0

5 1

0

1

0 0

0 0

0

0 0

0

The analogous result for two or more points outside B is false in general; but it would be interesting to know in which systems it holds with i = 0. We will have a little more to say about this later. Intersection triangles can be used in other situations too. Fow example, the following triangle shows that, if X is a set of 5 points of S(3,4, 10) containing no block, then the complement of X also contains no block.

30 18 12 10 8 4 5 5 3 1 2 3 2 1 0 0 2 1 1 0 0 See also Conway [12].

52

P.J. Cameron

5. Projective and a5ne spaces The subject-matter of this section and the next is “classical” geometry; we merely outline some of the theory in the context of Steiner systems. We are also concerned with embedding Steiner systems in the classical ones. Let F = GF(q) denote the finite field with q elements, and V a vector space of rank d + 1 over F ( d 2 2). Define a configuration whose points are the subspaces of rank 1 of V, and whose lines are the subspaces of rank 2, incidence being defined by set-theoretic inclusion. Clearly any two points lie on a unique line. Easy counting arguments show that we have a Steiner system S ( 2 , 4 + 1, (4‘‘ ’ I - l)/(q - 1)). We call this the d-dimensional projective geometry over F, and denote it by PG(d, 4). Note that PG(2, q ) is a projective plane of order 4. If x and y are distinct points, we denote the line joining them by x y . If W is any subspace of V, of rank e + 1, then the points and lines contained in W form PG(e, q ) (degenerate if e d 1). In particular, three non-collinear points x , , x2. x3 are contained in a projective plane 11. If L is a line meeting x , x 2 and x I x 3 but not containing x , , then L G 11, and so L meets x z x J . This property virtually characterizes projective geometries:

Theorem 5.1. Let S ( 2 , k , v ) ( 2 < h < v ) have the property that a line meeting two sides of a triangle (not at a vertex) must meet the third side also. Then either (i) k - 1 is u prime power q. and S = PG(d. q) for some d b 3 ; or (ii) u = k ’ - k + 1 and S is a projective plane of order k - 1. An alternative form of this theorem is often convenient:

Theorem 5.1’. The conclusions of Theorem 5.1 hold i f any three non-collinear points lie in u projective plane S ( 2 , k , k ’ - k + 1). For a proof, see Veblen and Young [48]. Consider a subspace PG(d - 1 , 4 ) in PG(d, 4). The rank formula for subspaces of a vector space shows that any line not contained in PG(d - 1, q ) meets it in a unique point. So these lines are the blocks of S ( 2 , 4 , q J ) on the complement of PG(d - 1.4). This system is the d-dimensional afine geometry A G ( d . q ) . Two lines of A G ( d , 4 ) are said to be parallel if they meet the “hyperplane at infinity” PG(d - 1. q ) in the same point. Parallelism is an equivalence relation satisfying Euclid’s postulate: through any point x there is a unique line parallel to a given line L. An alternative description of A G ( d , q ) is as follows: t h e points are the vectors o f a vector space of rank d over F; the lines are the cosets of subspaces of rank 1 : two lines are parallel if they are cosets of the same subspace. An ufine plane of order n is an S ( 2 , n, n’). (Thus AG(2, q ) is an affine plane of


53

order q.) Call two lines of an arbitrary affine plane parallel whenever they are equal or disjoint. (This agrees with the previous definition for A G ( 2 , q).) If x # L, then x lies on n + 1 lines, n of which meet L ; so x lies on a unique line parallel to L. Thus, parallelism is an equivalence relation satisfying Euclid’s postulate. Now let S be a system S(2, k , u ) in which any three non-collinear points lie in an affine plane S(2, k, k’). We may call two lines parallel if they are contained in an affine plane and are parallel there. Then Euclid’s postulate holds; but this parallelism may not be an equivalence relation. In fact, we have the following:

Theorem 5.2. Let S(2, k, u ) (2< k < u ) have the properties (a) any three non-collinear points lie in an afine plane; (b) parallelism (defined above) is an equivalence relation. Then either (i) k is a prime power q, and S is AG(d,q ) for some d 2 3; or (ii) S is an afine plane S(2, k , k’). For proofs see Lenz [30], Hall [23]. Hall gives a class of examples to show that condition (b) is necessary. In view of this, the following result of Buekenhout [6] is remarkable.

Theorem 5.3. Condition (b) of Theorem 5.2 may be deleted if k > 3. Thus, Hall’s examples are Steiner triple systems. It follows from a result of Bruck and Slaby [4] (on commutative Moufang loops of exponent 3) that in such systems u is a power of 3. The smallest example has u = 81. Note that affine geometries, as we have defined them, are trivial when q = 2. In this case, Steiner systems can be defined by using planes (instead of lines) as blocks. We can do this most economically by letting AG,(d, 2) ( d 2 3) have as points the vectors of a vector space of rank d over GF(2), and as blocks the quadruples of vectors with sum 0. It is an S(3,4, 2d).Note that, if B, and B, are blocks of AG,(d, 2) with IB, nB,I = 2, then B,AB, is a block (cf. Proposition 2.1). This property characterizes AG,(d, 2) among S(3,4, u ) s : see Lenz [30], or Corollary 7.3. We can translate Theorem 5.1 into a different form, one which suggests many open problems. See also Hubaut [25].

Proposition 5.4. Suppose S(2, k, u ) has disjoint blocks (i.e. u > k 2 - k + 1). Then S is PG(d, q ) (d a 3 ) i f and only if, given disjoint blocks B,, B, and a point x+!BIUB,, there is at most one block containing x and meeting B, and B,. Moreover, S is PG(3, q ) if and only if the same condition holds with “at most” replaced by “exactly”.

54

P.J. Cameron

We mention just three open problems.

(1) Find a similar characterization of A G ( d , 4). (2) What if “at most” is replaced by “at least”? (It can be shown that v = s k ( k 2 - 2 k + 2 ) , with equality precisely for PG(3,q). Moreover, if o s 2 k 2 3 k + 2, then the condition necessarily holds.) (3) Find similar results concerning common transversals to intersecting pairs of blocks.

6. Projective, ffie and inversive planes Projective and affine planes are intimately related. Given a projective plane of order n, deleting a line (the “line at infinity”) and all of its points yields an affine plane of the same order. Conversely, given an affine plane, adjoin an ideal point to all the lines of each parallel class, and an ideal line incident with all the ideal points; the projective plane is recovered. PG(2, q ) and AG(2, q ) are the classical projective and affine planes. However, there are very many non-classical planes; the exceptions in Theorems 5.1 and 5.2 are genuine. It should be noted, though, that all known planes have prime power order. For orders n with 2 s n 8. the planes are unique (except for n = 6 , where n o plane exists): for n = 9 there are at least four nonisomorphic planes; and for t i = 10 the existence question is unsettled. The strongest nonexistence theorem is due to Bruck and Ryser [5]:

Theorem 6.1 lf a projective plane of order n = 1 or 2 (mod 4) exisrs, then 11 = a ’ + h’ for some integers a and h. The classical theorems of Desargues and Pappus are “configuration theorems” which characterize PG(2, q ) and AG(2, q ) among projective and affine planes. We state these for projective planes, together with a theorem of Gleason [18], referring to Demhowski [ 151 for variations and elaborations.

Tbeorem 6.2. A project plane of order n is PG(2, q ) (with n = q ) if and only i f ,

wheneuer a , a 2 , bib, and c,cz are concurrent lines and a , b , n a , b , = z , u Ic2f~ a,c, = y. h,c, f3 b2c, = x , the points x , y, z, are collinear.

Theorem 6.3. A projective plane of order n is PG(2, q ) (with n = q ) if and only i f . whenever a , b , c , and a,h,c, are collinear triples of points and a , b , n a , b , = z, u , c z f l a,cl = y. h , c , n b,c, = x , the points x, y, z are collinear.

Extremal results and configuration theorems for Steiner sysrems

55

Fig. 1.

Theorem 6.4. A projective plane of order n is PG(2,2‘) (with n = 2‘) if and only i f , whenever a,, a,, a3, a4 are points with no three collinear and a,a,n a3a4= z, a,a,na,a,= y, a,a,na2a3=x, the points x, y , z are collinear. The forward implications in Theorems 6.2 and 6.3 are the theorems of Desargues and Pappus: see Arthin [ 11 for the converses. There is an elementary proof that Pappus’ theorem implies Desargues’: van Lint [32] gives a proof for the case where no degeneracies occur. The converse is also true in finite projective planes, but the problem of finding an “elementary” proof remains open. The question of extendability of projective planes is virtually settled by Proposition 3.2. The situation is different for affine planes: the divisibility conditions are always satisfied by the parameters S(3, n + 1, n 2 + 1). A system with these parameters is called an inversive plane of order n. An ovoid is a set of q 2 + 1 points in PG(3, q ) , no three collinear. Any ovoid is the point set of an inversive plane of order q, whose blocks are the non-trivial plane sections. An inversive plane constructed in this way is called egglike. The classical inversive plane I ( q ) is obtained from the elliptic quadric in PG(3, q ) , the set of points spanned by zeros of the quadratic form Q(x,,

~ 2~ , 3 =) U

XI,

+ hxox, + CX: + ~ 2 x 3 ,

X ~

where the quadratic ax2+ bx + c is irreducible over GF(q). In the spirit of Theorems 6.2-6.4, there is a configuration theorem (Miquel’s theorem) characterizing the classical inversive planes, and another (the bundle theorem) which holds in all egglike planes and is thought to characterize them: see Dembowski [15]. The most important theorem is due to Dembowski [14]:

Theorem 6.5. A n inversive plane of even order n is egglike (and so n is a power of 2). This result is very satisfactory, since there are non-classical egglike inversive planes of even order. By contrast, the only known inversive planes of odd order

56

P.J. Cameron

are the classical ones, but the only known restriction is the Bruck-Ryser theorem (Theorem 6.1). Extensions of inversive planes are considered in Section 7. We conclude this chapter with some remarks on embeddings. Let S = S(2, K. V). and let T be a subset of S of cardinality u with property (*)

any line of S meets T in 0, 1 or k points.

Such a set is called a {u, k}-arc in S (or a u-arc if k = 2). Obviously the nontrivial intersections of T with blocks of S give T the structure of S ( 2 , k, u ) . Considering lines through a point of T, we see that ( u - l ) / ( k - 1) 2) there are eight possible values of n. Using Dembowski’s theorem (Theorem 6.5), the values n = 10, 18,28 and 58 are eliminated, leaving n = 3, 4, 8 or 13 for consideration. The case n = 8 is especially interesting. The divisibility conditions alone merely show that S(t, f + 6, t + 62) must satisfy t s 12. Using the inequality of Proposition 3.3 we can improve this to t s 9 ; this turns out to be far from best possible. Dembowski’s

P.J. Camerori

58

theorem shows that S(3,9,65) is represented by an ovoid in PG(3,8). Kantor [27] has proved that a similar statement holds for the hypothetical extension S(4, 10,66): it would be represented by a set S of 66 points in PG(4,8), no four coplanar, the blocks being the nontrivial 3-space sections. Clearly two lines meeting S in disjoint pairs of points would be skew. Thus there would be 6 6 t i . 6 6 . 6 5 . 7 = 1501.5 points lying on such lines; but there are only (8’- 1)/(81 ) = 4681 points in PCi(4,8). Extending the argument, Kantor shows:

Proposition 7.1. If an inuersiue plane of order n > 2 is extendable, then n = 3 or possibly 13. In the latter case, at least one contraction i s a non-egglike inversive plane.

(Ofcourse, the inversive plane of order 3 is twice extendable.) More generally. Kantor’s result shows that if every contraction of S(t, k, u ) (for t 2 4 ) is embeddahlc as a “sufficiently large” piece of PG(t - 1, q), then the whole system is similarly embcddable in PG(t, q). Even this does not fully describe his theorem. which applies to “geometries” (more precisely, geometric lattices) with n o restrictions on the cardinalities of subspaces. We refer to Kantor [27, 281 for precise statements. For example, S(4,5, 11) and S ( 5 , 6, 12) are embedded in PG(4,3) and PG(S.3) respectively. Note the restriction t 2 4. Thus, for example, S(3,6. 22) is not embeddable in PG(3,4), even though every contraction is PG(2,4). However. the extensions of projective spaces can be characterized as the systems S ( 3 , k . L?) satisfying the following condition: (i, Zf B,. B 2 , B, are blocks for which B , n B , and B , n B , are disjoint 2-subsets of B , , then B,nBB, is empty or a 2-set.

Thus we have the following theorem and corollary:

Proposition 7.2. An S(3, k, v ) satisfying (t) must be A G 2 ( d ,2 ) , S(3,6,22) or S ( 3 , 12, 112).

Corollary 7.3. Suppose S(3,4, t.) has the property that, whenever B, and B, ure blocks with 16,nB,I = 2 , then B,AB, is a block. Then S is A G , ( d , 2). Cameron [ 101 considered a related property. (++)

With the hypotheses of (t),I€?,

n B,I s 1.

He showed that a system S(3, k , u ) satisfying (tt) has u z 2 + i ( k - 1)’(k-2). Moreover, (W)holds with u = 2 + f ( k- 1)’(k -2) if and only if the hypotheses of the following result are fulfilled.

Exrremal results and configwarion theorems for Steiner systems

59

Proposition 7.4. Suppose S(3, k, u ) has the property that, given any block B, the blocks meeting B in two points form an S(2, k - 2, u - k ) on the complement of B. Then S has parameters S(3,5,26), S(3,23,5084) or S(3,105,557026). The existence of these systems is undecided; none of the known S(3,5,26) satisfy the hypothesis, and systems with the other parameters are unknown. The hypothesis, reminiscent of the relation between projective and affine planes, suggests variations. For example, S(4,7,23) has the property that, for any block B, the blocks meeting B in three points form an S(3,4, 16) on the complement of B (actually AG2(4, 2)). Does such a property characterize it among S(4, k , u)s? What about other values of t‘! Compare also the remarks at the end of Section 4.

8. The symmetric difference property Motivated by Proposition 2.1 and Corollary 7.3, we say that a Steiner system S ( t , k, u ) , with k even and k < 2 t , has the symmetric difference property (SDP) if, whenever B, and B2 are blocks with IB2nB21= $ k , then B , A B , is a block. (The conditions on k and f are obviously necessary to ensure that such a pair of blocks exists, and are also sufficient, by Proposition 4.1.) It is striking that both S ( 5 , 6 , 12) and S(5,8,24) have the SDP. The only other known systems with this property are the AG,(d,2). A complete classification is not yet known, but the most dramatic result is due to Cameron [9].

Theorem 8.1. The only S ( t , k , u ) , with k A G , ( d , 2) and S(5,8,24).

= 22 - 2 > t,

having the SDP, are the

A feature of this theorem is the proof, which uses the determination of perfect binary error-correcting codes in an unexpected way: the code is a relation module for a certain group constructed from the Steiner system. At the same time, a characterization on the contractions is obtained. Though the hypotheses appear complicated, they should be compared with those of Theorem 5.1: the cases t = 2 of Theorem 8.2 and k = 3 of Theorem 5.1 are identical.

Theorem 8.2. Suppose S = S ( t , k , u ) , with k = 2t - 1, has the property that whenever p and p‘ are points and Q,, . . . , Q4 are ( t - 1)-sets for which { p } U Q, U QZ, {p’}U Q, U Q, and {p’}U QZU Q4 are blocks, then { p } U Q, U Q4 is a block. Then S = PG(d, 2) or S(4,7,23). The only further results known about the SDP are also due to Cameron [ lo], and are summarized below.

60

P.J. Cameron

Proposition 83. Suppose S ( t , k, u ) , with k even and k < 2t, has the SDP. Then (i) if k = 2t -4, or if k = t + 1 with t > 3, then k = 2 (mod 4); (ii) if t = 5 and k = 6 , then u = 12. Of this, only (ii) is non-trivial. The proof contains a uniqueness proof for S(5,6, 12) based on the construction via 1-factorizations of K6 (see Section 2). Since Theorem 8.1 and Proposition 7.2 have a common specialization (Corollary 7.3), it is natural to look for a common generalization. Such a result was found by Cameron [ll]. Somewhat surprisingly, the conclusions are just the disjunction of those of Theorem 8.1 and Proposition 7.2.

Theorem 8.4. Suppose S = S ( t , k, tl), with k 3 2t - 2, t 2 3, has the property that, whenever B , , B,, B, are blocks for which B1n B , and B , n B , are disjoint ( t - 1)subsets of B,,then IB, n B,I = 0 or t - 1. Then either (i) t = 3, S is AG,(d, 2), S ( 3 , 6 , 2 2 ) or S(3,12,112); or (ii) k = 2t - 2, S is AG,(d, 2) or S(5.8,24).

Proof. Note that the conditions

t 3 3 and k 3 2t - 2 avoid the possibility that the hypotheses are vacuously fulfilled. A simple counting argument shows t = 3 or k = 21 - 2; now Proposition 7.2 and Theorem 8.1 apply. Theorems 5.1 and 8.2 also have a common specialization, but no common generalization is known.

9. Further directions

In this section, some other lines of investigation will be briefly mentioned. A feature of t h e projective and affine spaces is that they are equipped with large numbers of subspaces. Teirlinck [45] and Doyen, Hubaut and Vandensaval [ 171 have studied the analogues of hyperplanes in arbitrary Steiner triple systems. The existence of hyperplanes (with properties like those in projective and affine spaces) is closely related to the rank (mod 2 or 3) of the incidence matrix, and so to the considerations of coding theory. Hall [24] and Teirlinck [44] have looked at planes, generalizing Theorems 5.1' and 5.2 by allowing different types of planes to occur. Buekenhout [7] has introduced a general theory of subspaces in "block spaces". with potential applications to designs (and Steiner systems in particular). Another generalization starting with the same observation is the theory of perfect rnatroid designs [38]. These are geometries with subspaces of various dimensions, in which an i-space and a point outside it determine a unique ( I + I)-space. and the cardinality of a subspace depends only on its dimension. (In S(I. k , v ) , the i-spaces for i < t - 1 are the ( i + 1)-sets, while the hyperplanes are the blocks.) For example, a 3-dimensional perfect matroid design consists of an S ( 2 , k , t i ) , whose blocks are the lines, with' a collection of subsystems called planes, three non-collinear points lying in a unique plane, and any plane having

Extremd resulfs and configuration theorems for Sfeiner sysrems

61

cardinality p. If k = 2, it is simply an S(3, p, u ) . For k > 2 , the cases p = kZ- k + 1 and p = k 2 are covered by Theorems 5.1’ and 5.3. Two open problems: Are there any examples with k > 3 apart from projective and affine spaces? If any two planes meet, must the geometry be a projective 3- or 4-space? Steiner systems have been coordinatized in various ways. On the one hand, the familiar introduction of coordinates in PG(2, q ) has been generalized to arbitrary projective planes; the system of coordinates is a “planar ternary ring”. (See Hall [21], Dembowski [15].) On the other, there is a familiar connection between Steiner triple systems and certain varieties of quasigroups (or loops). (The exact method of coordinatization depends on whether we want PG(d, 2) or AG(d, 3) to be coordinatized by the additive group of the underlying vector space.) This too has been extended to other systems, leading to a close relation between combinatorics and universal algebra: see Ganter and Werner’s and Quackenbush’s article in this volume. In all cases, structural properties of a Steiner system are reflected by algebraic properties of the coordinatizing structure. If the blocks of S(2, k, u ) are regarded as the vertices of a graph, two vertices adjacent if the corresponding blocks intersect, then the resulting graph is “strongly regular” in the sense of Bose [3]. That author has also given sufficient conditions (involving only the parameters) for a strongly regular graph to be obtained from a Steiner system (and, more generally, from a “partial geometry”). A good example of the connection between structural properties of graph and Steiner system occurs in the work of Sims [42] on graphs satisfying the 4-vertex condition. Cameron [lo] investigated similar connections between Steiner systems with t = 3 and “association schemes” (generalizations of strongly regular graphs). For r>3, little is known except an isolated result of Ito and Patton [26] characterizing S(4,5, 11). An important invariant of a Steiner system is its automorphism group. Mendelsohn [36] has shown that any group is the automorphism group of a Steiner triple or quadruple system. Much work has been done on finding, for example, Steiner triple systems with prescribed automorphisms. From the point of view of this article, the most interesting systems are those with large automorphism groups and, in particular, those S(r, k, u ) which have f-fold transitive groups. The systems S(5,8,24) and S(5,6,12) and their contractions, together with the projective and affine spaces, classical inversive planes, (and higher-dimensional “circle geometries”), and classical and Ree unitals, all fall into this category, and it is believed that there are no others. However, the limitations of our knowledge are shown clearly by the fact that not even the doubly transitive Steiner triple systems have been determined! Hall [22] has shown that a Steiner triple system whose automorphism group is transitive on non-collinear triples of points must be PG(d, 2) or AG(d, 3). (This is shown by identifying the planes as projective or affine.) Other results in this vein are surveyed by Kantor [29]. A large automorphism group often forces a system to have structural properties of the kind discussed in this paper.

62

P.I. Cameron

References [ l ] E. Artin, Geometric Algebra (Interscience, New York, 1957). [ 2 ] E.F. Assums Jr. and M.T. Hermoso, Non-existence of Steiner systems of type S(d - 1, d, Zd), Math. Z. 138 (1974) 171-172. [3] R.C. Bose, Strongly regular graphs, partial geometries, and partially balanced designs, Pacific J. Math. 13 (1963) 389-419. [4] R.H. Bruck, A Survey of Binary Systems (Springer-Verlag, Berlin, 1958). [5] R.H. Bruck and H.J. Ryser, The nonexistence of certain finite projective planes, Canad. J. Math. I (1949) 88-93. [6] F. Buekenhout, Une caracterisation des espaces affines basee sur la notion d e droite, Math. Z, 111 (1969) 367-371. [7] F. Buekenhout, What is a subspace? in: P.J. Cameron, ed.. Combinatorial Surveys (Academic Press, London, 1977) 1-22. 181 F. Buekenhout, Existence of unitals in finite translation planes of order q2 with a kernel of order q. to appear. [Y] P.J. Cameron, Characterisations of some Steiner systems, parallelisms and biplanes, Math. Z. 136 (1974) 31-39. [ 101 P.J. Cameron. Two remarks on Steiner systems, Geometriae Dedicata 4 (1975) 403-418. 11 11 P.J. Cameron. Extensions of designs: variations on a theme, in: P.J. Cameron, ed., Combinatorial Surveys (Academic Press, London, 1977) 23-43. Conway. A group of order 8.315,553.613.086,720.~00, Bull. London Math. Soc. 1 (1969) [12] J.H. 79-88. [13] R.T. Curtis. A ncw comhinatorial approach to M24. Math. Proc. Camb. Phil. SOC. 79 (1976) 25-42. [ 141 P. Dembowski, Mobiusehenen gerader Ordnung. Math. Ann. 157 (1964) 179-205. I151 P. Dembowski. Finite Geometries (Springer-Verlag. Berlin, 1968). [ 161 R.H.F. Denniston. Some maximal arcs in finite projective planes, J. Combinatorial Theory 6 (1969) 317-319. 1171 J. Doyen, X. Hubaut and M. Vandensaval. Ranks of incidence matrices of Steiner triple systems, Math. Z. 163 (1978) 251-259. [lX] A.M. Gleason. Finite Fano planes, Amer. J. Math. 78 (1956) 797-807. [I91 1M.J.E. Golay. Notes on digital coding, Proc. I R E 37 (1949) 657. 1201 B.H.Gross, Intersection triangles and block Intersection numbers for Steiner systems, Math. Z . 139 (1974t 87-104. 1211 M. Hall Jr.. Projective planes, Trans. Amer. Math. SOC.5 4 (1943) 229-277. 1221 M. Hall Jr.. Group thcory and block designs. in: L.G. Kovacs and B.H. Neumann. cds.. Proc. Intern. Conf. Theory of Groups (Gordon & Breach, New York, 1967) 11.3-134. 1231 M. Hall Jr.. Incidence axioms for affine geometry, J. Algebra 21 (1972) 535-547. [24] J.1. Hail. Steiner systems with geometric minimally generated subsystems, Quart. J. Math. Oxford ( 2 ) 25 (1974) 41-SO. [25] X. Huhaut. Systemes de Steiner minimaux, Bull. SOC.Math. Belgique 2 3 (1971) 411-415. [26] N. Ito and W.H. Patton. On a class of 4-(c,5, 1) designs, to appear. [27] W.M. Kantor. Dimension and embedding theorems for geometric lattices, J . Combinatorial Theory ( A ) 17 (1974) 173-195. 1781 W.M. Kantor. Envelopes for geometric lattices. J. Combinatorial Theory ( A ) 18 (1975) 12-26, [29] W.M. Kanror. ?-transitive designs, in: M. Hall Jr. and J.H. van Lint, eds.. C'omhinatorics ID. Keidcl. Dordrecht, 1975) 365-418. 1301 H . Lcnz. Z u r Bcgrundung der analytischen Geometrie. Sitz.-Ber. Bayer. Hayer Akad. Wiss. I 1054) 17-71. [31] J . H . van l i n t . Coding Theory, Lccturc Notes in Math. 201. (Springer-Verlag, Berlin, 1971). [ 3 2 ] J.H. van Lint, Combinatorial Theory Seminar Eindhovcn, 1-ccturc Notes in Math. 382 (SpringerVcrlag, Bcrlin. 1974). [ 3 3 ] H . Lunehurg. Transitive Erweiterungen endlicher Perrnutationsgruppen, Lccturc Notes in Math. S4 (Springer-Verlag, Berlin, 1969).


63

[34] D.R. Mason, On the construction of the Steiner system S(5,8,24), J. Algebra 47 (1977) 77-79. [35] E. Mathieu, Sur la fonction cinq fois transitive de 24 quantitts, J. Math. Pures Appl. (Liouville) (2) 18 (1873) 24-46. [36] E. Mendelsohn, On the groups of automorphisms of Steiner triple and quadruple systems, In: E. Mendelsohn, ed., Proc. Conf. Algebraic Aspects of Combinatorics, Utilitas Math., Winnipeg, (1975) 255-264. [37] N.S. Mendelsohn and S.H.Y. Hung, On the Steiner systems S(3,4, 14) and S(4,5, IS), Utilitas Math. 1 (1972) 5-95. [38] U.S.R. Murty, J. Edmonds and H.P. Young, Equicardinal matroids and matroid designs, Proc. 2nd Chapel Hill Conf. Combinatorial Mathematics, Chapel Hill, NC (1970) 498-541. [39] R. Noda, Steiner systems which admit block transitive automorphism groups of small rank, Math. Z. 125 (1972) 113-121. [40] M.E. O’Nan, Automorphisms of unitary block designs, J. Algebra 20 (1972) 495-511. [41] D.K. Ray-Chaudhuri and R.M. Wilson, On [-designs, Osaka J. Math. 12 (1975) 737-744. [42] C.C. Sims, On graphs with rank 3 automorphism groups, Unpublished. [43] D.E. Taylor, Unitary block designs, J. Combinatorial Theory (A) 16 (1974) 51-56. [44] L. Teirlinck, Planes and hyperplanes of 2-coverings, Bull. SOC.Math. Belgique 29 (1977) 73-81. [45] L. Teirlinck, On projective and affine hyperplanes, to appear. [46] J.A. Thas, Some results concerning {(q + l)(n - 1); n}-arcs and {(q + l)(n - 1)+ 1; n}-arcs in finite projective planes of order q, J. Combinatorial Theory (A) 19 (1975) 228-232. [47] J.A. Todd, A representation of the Mathieu group MZ4as a collineation group, Ann. di Mat. Pura ed Appl. (4) 71 (1966) $?9-238. [48] 0. Veblen and J.W. Young, Projective Geometry (Ginn, Boston, 1916). [49] E. Witt, Die 5-fach transitiven Gruppen von Mathieu, Abh. Math. Sem. Hamburg 12 (1938) 256-264. [50] E. Witt, Uber Steinersche Systeme, Abh. Math. Sem. Hamburg 12 (1938) 265-275.



THE PROBLEM OF THE HIGHER VALUES OF

t

R.H.F. DENNISTON Department of Mathematics, University of Leicester, Leicester LEI 7RH, England

About systems with parameters ( t , k, u ) , where t is 2 or 3, a vast body of knowledge has grown up, and is reviewed elsewhere in this volume. There is also a review by Cameron of the classical systems (5,6,12) and (5,8,24), with their contractions - another subject about which a formidable amount is known. The contrast is remarkable when we consider our ignorance about even the existence of other systems for which r is greater than 3. It is curious that Witt [ll] should have chosen to extend the name “Steiner systems” to these finite geometries. In fact, Steiner [lo] proposed a sequence of problems, the first two of which amount to the construction of a system (2,3, N) and of its extension (3,4, N + 1).However, his third problem is not, as might have been supposed, equivalent to the construction of a “Steiner system” (4,5, N + 2). And the strange thing is that, whereas Steiner had in one case solved his own fifth problem, nobody has yet constructed a ‘‘Steiner system” for which r = 6. (I am, of course, disregarding the trivial cases t = k and k = u, as well as the designs and tactical configurations for which h > 1.) We have two significant necessary conditions for existence, one being the familiar divisibility condition:

The other comes from Fisher’s theorem that a design, with t equal to 2, must have at least as many blocks as points; when we consider an extension to a higher value of r, we have the condition that u - t + l >(k-t+2)(k-

t+l).

Kramer and Mesner [6] review some attempts that have been made to strengthen these necessary conditions -but, apart from some extensions of the non-existent system (4,10,66), they find no “admissible” set of parameters that can be excluded in such a way. Positive results are the discoveries of systems (5,6,12) and (5,8,24), which (as already mentioned) can be regarded as classical; of systems (5,6,24), (5,7,28), (5,6,48) and (5,6,84) 131; and of a system (5,6,72) [S]. These all have in 65

66

R.H.F. Denniston

common the property of being invariant under a group PSL (2, u - 1). Each has, of course, a contraction for which t = 4 : but there is no case in which a system with t equal to 4 has been discovered, and its extension with t equal to 5 has not. Negative results begin with the non-existence [ l l ] of a system (4,6, 18). A computer search [8] for a system (4,5,15) was exhaustive and unsuccessful. And the non-existence of a system (4,10,66) is shown by Cameron [l, p. 351 to be implicit in work by Kantor [5]. The extensions of these designs, which have admissible parameters up to (9, 10,20) and (12, 18,74), are of course also non-existent. Another result in [ 5 ] is that, if a system (4,15,171) exists, at least one of its contractions is a non-Miquelian inversive plane of order 13: it would be surprising if this latter geometry were found to exist. General theorems about lower values of t are not usually helpful for our purpose; one exception is Dembowski’s result [2,6.2.14] that an even number, not a power of 2, cannot be the order of an inversive plane. Admissible sets of parameters, ruled out by this theorem, begin with (4, 12,102) and (4,20,326). This leaves, as we can see from the table in [6], plenty of admissible sets of parameters for which the question of existence is undecided. It may be of interest, or may at least save people from wasting effort on hopeless searches, if we consider the feasibility of methods with which this problem can be attacked. We might have hoped that exhaustive searches, in the style of [8], could be made in other cases. However, it was possible for Mendelsohn and Hung to succeed because, up to isomorphism, there are only (as was known) two systems (2,3, 13), and (as they themselves found) four systems (3,4,14). The situation in the very next case is entirely different [7]; and this seems to close the prospect of a similar method ever being used again for our problem. There remains the possibility of requiring the unknown system to be invariant under some group. It might be objected that the classical systems can be described i n various ways without their groups being explicitly mentioned, and that this raises the hope of constructing a new system by some ingenious trick. But in fact any short description of one of these big systems is sure to have, implicit in it, the existence of a good-sized group that leaves the system invariant. And it will be realistic for us to d o our research, not by hoping for a trick, but by thinking about some group from the beginning. The smallest undecided set of parameters is (4,5, 17). Here, I have considered all the primes which might possibly have served as periods for automorphisms, 2nd disposed of them by machine searches. So I can assert that, if a system (4,5, 17) exists, its automorphism group must be trivial. But, once more, there is hardly any larger case in which a very small group could be exhaustively considered; we soon find that, say, a doubly transitive group, or something even larger, will be needed if the search is to be feasible. On the other hand, if the group is too large, the hypothesis that it leaves the supposed system invariant may be easy to disprove. Let us look at some simple propositions which are often useful in this way.

The problem of the higher values of t

61

Theorem 1. Let G be a group of permutations of a u-set, and let a t-subset T be fixed by a subgroup H. Let { U } be the collection of all (k - t)-subsets disjoint from T. Then, in any system (t, k, u ) inuariant under G, there is a block T U U, where U E{U}, and U is fixed by H.

Proof. The definition of a “Steiner system” tells us that U exists. Suppose if possible that U is not fixed by H; then some permutation in H fixes T, but sends U to a different subset U*. By the hypothesis that G fixes the system, not only T U U but also T U U* is a block containing the t-subset T, and the definition is contradicted. Corollary 1. If H fixes no element of { U } , there is no system (t, k, v ) inuariant under G. Corollary 2. If H fixes just one element U , of { V } ,any system (t, k, v), inuariant under G, must include T U U1 as a block. Although I have not tried many cases, my impression is that these Corollaries will rapidly dispose of most hypotheses that large permutation groups could be applied to our problem. The best hope seems to be offered by the groups of the projective line, and particularly by the group PSL(2,v-l), which (as we have seen) has already had some successes. Even this group does not seem to work when t = 4 (that is, for the small proportion of admissible parameter-sets (4, k, u ) that have a prime power plus one as u). In fact, any 4-subset of a projective line is fixed by three involutions in the PGL group; and I find in practice (though I have not tried to prove) that the PSL group always includes at least one of these three. This means that, in the supposed system (4,k, u), every block would (by Theorem 1) be invariant under enough involutions to fix each of its 4-subsets at least once-which seems hopeless. If we move on, accordingly, to the cases where t = 5 , we come across some more propositions.

Theorem 2. No system (4,5,p ) , on the residues modulo the prime p as points, can be fixed by the dihedral group (x --j x + 1, x ---* -x>.

Proof. In Corollary 2, put T={-2, -1,1,2} and H =(x 4-x); since 0 is the only fixed point, we must have a block {-2,-1,0,1,2}. But then, since the supposed system is fixed by x + x + 1, we have another block {-1, 0, 1,2,3}. This is a contradiction, since one t-subset ( t = 4) is in two blocks. Corollary 3. If p = l m o d 4 (so that -1 is a quadratic residue), no system + 1) can be inuariant under PSL (2, p ) .

(5,6,p

R.H.F. Denniston

68

Corollary 4. In the contrary case (p =--1 mod 4), Q system (5,6, p + 1) cannot be fixed by PGL (2, p), though the possibility of its being fixed by PSL (2, p) remains open. These propositions, though I have stated them as above to fix the ideas, can be immediately extended to systems (2j- 1,2j, p” + l), where the prime p is greater than 2j. So there are various admissible sets of parameters: (5,6, 18), (7,8,30), ( 5 , 6 , 3 0 ) ,(7.8,38), . . . , for which it can be said that not even a PSL group fixes a system. Now suppose, if possible, that a system (6,7,65) is invariant under PSL (2, 26). Let i be a primitive mark of GF (2“), so that i6’ = 1. Take T in Theorem 1 to be the 6-subset (‘9

.12, ; X I

1 .1

.33

7

I-

.51

,1

.T4

, I‘

1,

and H to be ( x -+i 2 ’ x , x -+ l / x ) . Then Corollary 1 applies, since H leaves T invariant but has no single fixed point. We can use the same T and H to prove that no system (6,9,65) is invariant under the same group. In fact, the only 3-subset fixed by H is {io, i 2 ’ , i42}; so, by Corollary 2, the supposed system has a block {i” .9 . I 2 . 2 1 i”’ i 3 3 , i 4 2 , i ” , p 4 } , .I , I , , 3

1

Multiplying by i ” , we get another block i”, i 3 } .

Once more, we now have the impossible situation of two blocks with t points in common (though t is now 6). Likewise, PSL (2, 5 2 ) , which has a 6-subgroup with orbits of sizes 6, 3, and 2 (but not 1). cannot leave a system (9, 10,26) invariant. There is another line of argument by which we can rule out various possibilities of systems being fixed by groups. We know how to calculate the number, b say, of blocks in any system such as we are considering. And we know that, if it is to be invariant under a group G, the system will have to be the disjoint union of a suitable set of orbits, chosen out of the whole collection of orbits of G on k-subsets of our u-set. Now a prime p, if it divides the order of G, will also divide the cardinalities (or shall we say “sizes”) of most orbits in the collection; the only exceptions are orbits made up of k-sets, each of which is fixed by some permutation of period p in G. This may enable us to prove, by reducing modulo p, the impossibility of the assumption that b is the sum of the sizes of an appropriate set of orbits. The hypothesis of a system (4,5, 17) fixed by PSL (2, 24), though we have already seen arguments against it, may serve as an easy example to illustrate this method. One 5-subset of PG (1, 24), consisting of the point at infinity together with the subfield GF(2*) of GF(z4), has in PSL(2,z4) a stabiliser which is

The problem of the higher oalues of r

69

precisely PSL (2, 22). So we can find the size of the orbit to which this 5-subset belongs; namely,

-

17 16 15 = 68 = 3 (mod 5 ) . 5.4.3 We find that any 5-subset, if it is fixed by a permutation of period 5 in the group, must be one of these 68 (it suffices, by the Sylow theorems, to look at the three 5-orbits of a single such operation). Therefore, of the sizes of orbits on 5-subsets, all are congruent modulo 5 to 0 except the one congruent to 3. And n o combination of these sizes will add up to b, since b=(147)/(i)=22.7.17-1

(mod5).

Likewise, of the sizes of orbits of 8-subsets under PSL (2, 26), all are congruent modulo 7 to 0 except one congruent to 4. So this group does not fix a system (6,8,65), for which b = 6. For 8-subsets under PSL (2,43), two orbits have a size congruent modulo 7 to 6. It follows that this group fixes neither (5,8,44), for which b =3, nor (6,8,44), for which b = 4. And likewise, PSL (2,127), under which two orbits of 8-subsets have a size that =4 mod 7, fixes neither (4,8, 128) (b =3), (5,8,128) ( b = 2), nor (6,8, 128) (b = 5 ) . There are other cases in which the impossibility of a system, fixed by a PSL group, can be established, even though neither of the simple arguments given above will apply. One usually begins by seeing what k-subsets are forced into the supposed system by Corollary 2, and then reaches some sort of impasse after going a little further, without an exhaustive search being necessary. The fact that (7,8,20) is one such case has been established, not only by me, but independently by researchers at Preston Polytechnic [4]. Other cases are (11,12,24), (7,8,24), (7,8,26), (5,6,28), (7,9,30) and (7,8,44). I should like to conclude by answering a question that was asked after one talk I gave on the problem. Is it feasible to add one more point A to PG (1, q), and find a system (having q + 2 as its u ) which is invariant under PSL (2, q ) , the point A being fixed under all permutations? This method may have worked for other problems, but it seems hopeless for the present one. Suppose, for instance, that a system (6,7,25) could be constructed in this way: we note that PSL(2,23) includes many permutations of period 2 or 3, but that no point of PG (1,23) is fixed by any of these. So a 6-subset fixed by such a permutation must, according to Corollary 2, go into a block with the point A. But this means that, relative to A, we have a system (5,6,24), whose blocks include all the 6-subsets fixed by permutations of period 2 or 3 - and there are far too many of them. Likewise €or the possibility of extending the other systems (5,6,12u) that have been found. Finally, I have tried to extend the (unique) system (5,7,28) that is invariant under

70

R.H.F. Dennisfon

PSL ( 2 , Z3); and, even if no assumption is made that a group fixes the extension, I find that it does not exist.

References [ 11 P.J. Cameron, Extensions of designs: variations on a theme, in: Combinatonal Surveys, Proceedings of the Sixth British Combinatorial Conference (Academic Press, London, 1977) 23-43. [2] P. Dembowski, Finite Geometries (Springer, Berlin, 1968). [3] R.H.F. Denniston. Some new 5-designs. Bull. London Math. SOC. 8 (1976) 263-267. [4] T.S. Griggs, M.J. Grannell, and D.A. Parker, Personal communication. [ S ] W .M. Kantor, Dimension and embedding theorems for geometric lattices, .I.Combinatorial Theory ( A ) 17 (1974) 1 7 S 1 9 5 . [6] E.S. Kramer and D.M. Mesner, Admissible parameters for Steiner systems S(r. k . u) with a table for all (u-f) 3, can they be adapted to provide examples

Dimension in Steiner triple systems

81

of spaces of dimension 3? The theorem in this section, due to Teirlinck [14,15], shows that they can. Its proof is essentially very simple: clearly if, in a space (S, d) of dimension d, one replaces a subspace (T,B) of dimension 4, then replacing a subspace of length i by a non-degenerate plane would again give the contradiction that d(S, d)S 3. Let ( U ,V) be another STS and let du= (d\B) UV. We shall show that d(S, d,)3 d - 1. Let u,, . . . , u d - l be d - 1 points of S and let (V, D ) and (V,, 0,)be the subspaces of (S, d) and (S, d,) respectively generated by ul,. . . , Ud-1. Then V # S since d ( S , d ) = d . We shall prove that d ( S , d , ) s d - l by proving that V, # S. Let K , be the set of unordered pairs of points of U (so that ( U ,K,) is the complete graph on the points of V). Let dK= (d\U )U Ku. Let ( V K ,D K ) be the linear subspace of ( S , dK)generated by { u l , . . . ,Ud-l}. [These are the only instances in this section of linear spaces which are not STS’s]. We show that V, # S by considering various cases. Case 1. l U f 7 V K l S l . Then lUnV,l=sl so V , = V # S . Case 2. IU n VKl3 2 . Let xl, x2e U f l V,. Let x3E S be the point such that {xl, x2, x3} is a line of du.Let (U3,B3) denote the subspace of ( U , B) generated by { X l , X2r x31.

AJ. W.Hilton. L. 7eirlinck

89-

Case 2i. U = U 3 .Then vLJ

[Ul,.

..

7

Ud-11

u]du= [ u l , . . .

I

u d - 1 , u ] d = [ U l , .. *

1

Ud-1,

x31d.

But the subspace of ( S , d) of which [ u , , . . . , q - 1 , x,] is the ground-set has Consequently V,# S. dimension S d - 1 and thus is a proper subspace of (S, a). Case 2ii. U # U,. Then U,C U. Case 2iia. B 3 contains exactly one line of B, and ( U ? ,B,) is a maximal proper subspace of ( U , €3). Let X,E U\{x,, x2, x3). Then

so V" # s. Case 2iib. Either B , contains more than one line of B or (U3,B3) is not a maximal proper subspace of ( V .B ) . Let (T,9) be a maximal proper subspace ( U , B ) with x , . x?, x , E T , and let ( T . G ) be a non-degenerate plane with {x,. x?. x 3 } e G. Let d., = ( d \ 9 ) U (3. T h e n d ( S , & & , . ) a dsince the length of (T,.9) is Gi and, by assumption, the dimension of ( S , d ) cannot be reduced by replacing a subspace of length Gi by any other subspace. Let (V.,., DT)be t h e subspace o f ( S . d T )generated by { u , . . . . vd ,}. Consider the following two subcases of Case 2iib. Case 2iib 1. V , f' U ={x,, x2. x,}. Then (V,, DT) is a proper subspace of (S,.dT).Since { X , , X ~ , X , } E S ~ , it follows that ( V T , D T ) = ( V u , D L JSO) that (V,, D,) is a proper subspace of ( S , d,),and so V , # S . Case 2iib 2 . V, f?U # { x , , x2, xD}. For u E U let the subspace generated by {v,, . . . , v d - , , u ) in ( S , d T ) be denoted by (W,,, 8,").Then d(W,,, % T u ) s d - l . Case 2iib 2i. V , n U .3 T. Let y E U\ T. Since the dimension of (S,d-,.is) at least d, it follows that ( W,,, S T , ) #is, d T ) .Since ( T ,9)was a maximal proper subspace o f (U. R ) it follows that U c W,,, so W , , = l u , . u 2 . . . . , cd. ,,UIdL,.Thus

.

v,

~

5 s.

wTy

Case 2iib 2ii. V, n U # T. Since x,, X ~ V E , c V, and {x,, x2, x3} is a line of G c d, it follows that { x , , x 2 , x , } c V, n T. In fact V, n T = {x,, x2, x3}, for otherwise. since (T. 3 ) is a non-degenerate plane, it would follow that T = V, n Yc V, fl U , a contradiction. Since V, f' U # { x , , x2, x,} there is a point x E ( U \ T) f l Vp Let z E T\{x,, x 2 , x,}. Then x E W,, and T c WTz SO it follows that U c WT,. Thus W,, is the ground-set of Some subspace of (S, Sa,). Since d ( S ,d o )= d, W, # S. But V , c W,,, SO V, # S, as required. It now follows that d ( S , &,)a d - 1 . Now, given (S, 4)of dimension d, if we replace a maximal proper subspace by a non-degenerate plane, we can construct an STS of dimension S 3 . Thus, for some least value & a 2 of i it is true that replacing a subspace of length i by any other subspace cannot reduce the dimension, but replacing a subspace of length i + 1 by another subspace can reduce the dimension. Then, in that case, the dimension will be d - 1. The theorem now follows.

Dimension in Sreiner triple systems

7. The existence of STS’s of dimension d

83

23

In this section we show that STS’s of dimension d 5 3 exist for all sufficiently large orders n. The key to this is the singular direct product construction for STS’s. First we give some preliminary definitions. A latin square L = ( l i i : 1 S i S n , l S j s n ) of order n on the set X = {x,, . . . ,a}, where x i # x i if i#j, is an n x n matrix such that each element of X occurs exactly once in each row and column. A tricouer of order n on X is an n 2 x 3 matrix M =(pi : 1 S i S n2, 1S j s 3) such that each ordered pair (xv, x ~ ) , where 1 svsn and l s p s n , occurs exactly once as a row in each n 2 x 2 submatrix of M. It is well-known that a tricover corresponds to a latin square, since corresponding to L we have the matrix M given by x,

if j = 1,

lye, if j = 3, where i = ( y - 1)n + S and 1Si3S n, which is a tricover of X . Clearly a tricover exists for each order n 3 1. We now describe the singular product construction. Let ( S ’ , d ’ ) , ( S , d ) and ( T ,B ) be STS’s of orders u3, u2 and u1 respectively, and let (S‘, d’) be a subsystem of (S,d).Let R be the set of rows of a tricover of S\S’. Let (T, B, 6) denote the STS (T,B) with each triple of B assigned an ordering. Let (S‘U{(S\S’)x T}, ( B , 6 ) 0 ( d , d ‘ ,R ) denote the STS on S ‘ U { ( S \ S ’ ) X T } of order u , ( u 2 - u 3 ) + u 3 whose triples are (i) {a,?aj?ak} for {&, a,, a k ) E d‘, (ii) {a,, ( 4 ,b,), ( a k , b,)} for all {a,, a,, ak}E d with ai E s’, 4,a k E s\s’ and for all 6, E T, (iii) {(U,, b,), (aJ,b,), (ak, b,)} for all {U,, U,, ak}E d with U,, a,, uk E s\s’ and for all b, E T, (iv) {(U,, br), (aJ,bs),(ak, 4))for all {(U,, 4. ak)ER, and for all (br, b,, b t ) E ( B ,6 ) . Then (S’U{(S\S‘)x T } ; (B, 6 ) 0(d, d‘, R ) ) is the singular direct product of ( S , d),(S’,d’), (T, B) with respect to R and 6. This construction was essentially due to Moore [8]. Notice that we can be sure that the singular direct product contains (T, B) as a subsystem by requiring that (s, s, s) E R for some s E S \ S’.

Theorem 5. Ler ISI>JS’I. If (T,B ) has dimension d , then

has dimension at least d. Proof. Let e l , . . . ,c, be any d points in S’U((S\S’)x T}. We may suppose that for some e, with O S e S d , {cl , . . . , c , } c S ’ and { c e + ,] . . . , cd}c_(S\S‘)xT.

84

A.I. W. Hilton. L. Teirlinck

Suppose c,=(a,,b,), where a,eS\S', b , c T ( e + l a i C d ) . Then {be+,,..., bd} generates a proper subsystem ( T ' , B ' ) of (T,B), and so {c,,. . . , c J generO')U(d,d', R)).(Here ates a subsystem contained in (S'U{(S\S')x T'}, (B', (T', B', 0') is the restriction of (T. €3.0) to T'.) This proves Theorem 5. We have immediately from Theorems 4 and 5: Tbeorem 6. Ler ( T ,B ) be an S T S of dimension d a 2 and order u , . Let ( S , d)be an STS of order u2 with a proper subsysfem ( S ' d ' ) of order u,( 15 the problem becomes computationally intractable. Cole, White and Cummings [4]first determined that there are exactly 80 nonismorphic triple systems of order 15. This was later reconfirmed by Hall and Swift [9]. A (corrected) listing of all 80 triple systems can be found in Bussemaker and Seidel [2]. The author [18] has determined that 38 of these are derived: #1-34, 61-64 (Bussemaker and Seidel [2]). P. Gibbons [8] has added #59, 70, and 76 to this list. Finally the author (unpublished) has determined that #35 and #53 are also derived. As has been amply demonstrated, a brute force computer attack on this problem is not feasible. Perhaps the most promising approach is via transformations of a given quadruple system of order 16. This idea is discussed by P. Gibbons [8] in relation to another problem.

References I.S.0. Aliev, SimmetriEeskjje algebry i sistemy Stejnera Dokl. Akad. Nauk SSR, 174 (1967) 5 11-5 13; English translation: Symmetric algebras and Steiner systems, Soviet Math. Dokl. 8 (1967) 651-653. [2] F.C. Bussemaker and J.J. Seidel, Symmetric Hadamard matrices of order 36, T.H.-Report 70WSK-02, Dept. of Mathematics Tech. University Eindhoven, Netherlands (1970). [3] P.T. Cameron, Extensions of designs: Variations on a theme, in: P.T. Cameron, ed., Combinatorial Surveys: Proceeding of the Sixth British Combinatorial Conference, (Academic Press, London, 1977) 23-43. [41 F.N. Cole, A.S. White and L.D. Cummings, Complete classification of triad systems on 15 elements, Mem. Nat. Acad. Sci., XIV (2) (1925). [ 5 ] D. Corneil, R. Mathon and P. Gibbons, Computing techniques for the construction and analysis of block designs, Utilitas Math. 11 (1977) 161-192. [6] M. Dehon, Un ThCorkme d’extension de l-designs, J. Combinatorial Theory (A) 21 (1976) 93-99. [7] P. Dembowski, Finite Geometries (Springer, Berlin, 1968). 181 P. Gibbons, Computing techniques for the construction and analysis of block designs, Ph.D. Thesis, University of Toronto 1976 (Department of Computer Science, University of Toronto, Technical Report No. 92 May 1976). [9] M. Hall Jr. and J.D. Swift, Determination of Steiner triple systems of order 15, Math. Tables Aids Compat. 13 (1959) 146-152. [lo] H. Hanani, On quadruple systems, Canad. J. Math. 12 (1960) 145-157. [l 13 C.C. Lindner, On the construction of nonisomorphic Steiner quadruple systems, Colloq. Math. 29 (1974) 303-306. [12] C.C. Lindner, On the structure of Steiner triple systems derived Steiner quadruple systems, Colloq. Math. 34 (1975) 137-142. [13] C.C. Lindner and A. Rosa, There are at least 31,021 nonisomorphic Steiner quadruple systems of order 16. Utilitas Math. 10 (1976) 61-64. 1143 C.C. Lindner and A. Rosa, Steiner quadruple systems - a survey, Discrete Math. 22 (1978) 147-181. [ 151 H. Luneburg, Fahnenhomogene Quadrupelsysteme, Math. Z. 89 (1965) 82-90. [I61 E. Mendelsohn, The smallset non-derived Steiner triple system is simple as a loop, Alg. Universalis 8 (1978) 256-259. [l]

I14

K.T. Phelps

[17] N.S. Mendelsohn and S.H.Y. Hung, O n the Steiner systems S(3.4, 14) and S(4.5, 15). Utilitas Math. 1 (1972) 5-95. [IS] K.T. Phelps. Derived triple systems of order 15, M.Sc. Thesis, Auburn University, AL. 1975, S6 pp. [ 191 K.T. Phelps, Some sufficient conditions for a Steiner triple system to be a derived triple system, J. Combinatorial Theory (A) (1976) 393-397. [20] K.T. Phelps, Some derived Steiner triple systems, Discrete Math. 1 6 (1976) 343-352. [21] K.T. Phelps. Rotational Quadruple systems, A n Combinatoria, 4 (1977) 177-185. [ 2 2 ] B. Rokowska. Some new constructions of 4-tuple systems, Colloq. Math. 17 (1967) 111-121. [23] A. Sade, Produit direct singulier de quasigroups, orthogonaw et anti-atkliens, Annales de la Societe Scientifique de Bruxelles, Ser I, 74 (1960) 91-99. [24] E. Witt. Uber Steincrsche Systeme, Abh. Math. Sem. Univ. Hamburg, 12 (1938) 265-275.


INTERSECTION PROPERTIES OF STEINER SYSTEMS Alexander ROSA Department of Mathematical Sciences, McMasier Unioersiry, Hamilton, Ontario, Canada, L8S 4KI

1. Introduction Given two Steiner systems of the same type S(r, k, u ) on the same u-set, how many blocks in common can they have? Can one find two such systems with no blocks in common at all? And if yes, what is the largest number of such systems with (pairwise) no common blocks that one can find? It seems that questions like these have intrigued researchers ever since the infant stages of the subject of Steiner systems. Cayley, for instance, has established in 1850 [9] that there exist two but no more disjoint Steiner triple systems on a given 7-element set, and Kirkman found in the same year that the largest number of disjoint Steiner triple systems of order 9 is seven [24]. But virtually all results of substance in this area that we summarily label “intersection properties of Steiner systems” have been obtained in the last decade. In this paper we attempt to survey the present state of affairs in this area. We also deal briefly with related questions, such as disjoint triple systems with h > 1, perpendicular Steiner systems (a subject dealt with in more detail in a paper by R.C. Mullin and S.A. Vanstone elsewhere in this volume), and applications to the existence of designs with larger A. We do not deal with questions concerning finite embeddings of (sets of) partial Steiner systems with given intersection properties; this is the subject of a paper by C.C. Lindner elsewhere in this volume.

2. Disjoint Steiner systems Two Steiner systems of type S ( t , k, u ) , say, (V, Bl), (V, B2) are disjoint if B , nB2= @, i.e. if they have no blocks in common. The symbol d(r, k, u ) will denote the maximum number of pairwise disjoint S ( t , k, u)’s. Since each S(t, k, u ) has (:)/(:) blocks, and there are altogether (ti) k-subsets of a u-set, we have d ( t , k, u)S(EI:). If the equality sign holds, the corresponding d(t, k, u ) Steiner systems are said to form a large set of disjoint S(r, k, u)’s. 2.1. Steiner triple systems The earliest results on disjoint S(2,3, u)’s are due to Cayley [9] who has shown that d(2,3,7) = 2, and Kirkman [24] who proved d(2,3,9) = 7. This result was 115

A. Rosa

116

reproved by Sylvester [58,59], Walecki [40], Bays [4] and Emch [18]. However, Bays [4] was apparently the first to show that there exactly two nonisomorphic sets of seven disjoint S(2,3,9)’s, a fact verified anew in [29]. One of these sets is given by the 7 square arrays

124 128 125 129 123 126 127 378 943 983 743 469 357 346 956 765 476 586 785 489 598 while the other one is given by the square arrays

139 192 127 174 148 186 163 275 745 485 865 635 395 925 486 863 639 392 927 274 748 (the 12 triples of each system are the three rows, three columns and the six products in the expansion of the determinant of each array). The first paper that offered a significant progress on the issue in a long time and that served, in our opinion, as an impetus for further development, was that of Doyen [ 161. Although for u B 13 a few results on d(2,3, u ) date back to Kirkman (such as d(2,3. 1 3 ) a 3, d(2,3, 1 5 ) ~ [25]) 2 and a few other results were implicit in the literature (such as, say, that implied by the existence of cyclic S(2, 3, u)’s, namely that for u = 1 (mod 6), d(2,3, u ) >2), Doyen was first to offer nontrivial lower bounds for d(2,3, u ) . Let d*(t, k, u ) denote the maximum number of pairwise disjoint and isomorphic S(1, k, u)’s. Then

d*(2,3,6m + 3) 2

4m + 1 if rn = 0 , 2 (mod) 3), 4rn - 1 if rn = 1 (mod 3);

the proof is by direct construction [I61 which, in turn, uses as its basis a well-known construction due to Bose [8]. Similarly,

I’

d*(2,3,6rn + 1)2 2rn rn

if rn = O (mod 2), if rn = 1 (mod 2);

this is also shown by a direct construction based this time on a construction due to Skolem [%I. Doyen’s paper [ 161 contains also a recursive construction showing

d(2,3,2u + 1 ) a d(2,3, u ) + 2 for u 3 7

(3)

which has as its corollary

d(2.3,6rn + I ) 3 2rn - 1 for rn = 1 (mod 2).

(4)

The lower bound (1) was subsequently improved in [5] to

d*(2,3,6rn + 3) a 4m + 2.

(5)

In [26]. the inequality (4) was strengthened to

d ( 2 , 3 , 6 r n + l ) 2 3 r n + l for r n = l (mod2).

(6)

Intersection properties of Steiner systems

117

The above bounds can certainly be improved. Already in 1917, Bays [4] conjectured that

d(2,3, u ) 3 ( u - 1)/2 for all u > 7.

(7)

This has now been shown true for all u except when u = 1 (mod 12); the smallest open case is u = 3 7 . But more recently, it has been repeatedly conjectured [16,60] that d(2,3, u ) = u -2 for u 3 9 . Teirlinck [60] was the first to establish that the conjecture is true for infinitely many values of u. He proved the inequality

d(2,3,3u) 3 2u + d(2,3, u ) for every u 3 3

(8) by providing a simple and elegant recursive construction whose immediate corollary is that d(2,3,3”’)= 3“‘- 2 for all m

3 1 [60].

(9) Teirlinck’s result thus enables one to “triplicate” large sets of disjoint S(2,3, u)’s. Simultaneously with Teirlinck’s result, several direct constructions of large sets of disjoint Steiner triple systems were suggested by Denniston, Schreiber and Wilson. The most natural approach appears to be the following. Let V = Zv-;U(ml,m2} and let p = (0, 1 * * u - 3)(col)(mz) be permutation of V. The 3-subsets of V are partitioned into orbits under the action of ( p ) ; these orbits are all of full length u - 2. Suppose one can construct an S(2,3, u ) (V, B) containing exactly one triple from each orbit. For instance, if u = 13, and

B = ((0,1, X I ) , I07 2,317 (0,4951, {0,6,8), {0,7, mz), {0,9. 101, { 1 , 2 , ~ z ) , {1,3,8), {1,4,9), {1,5,10), {1,6,7), {2,4,10), {2,5,6), {2,7,m1), {2,8,9), (37 4,719 (39 5 , (39 6,101, {3,9,mz), {4,6,mz), {4,8,m i l , (5,77919 (5989 xzl, (6799 mi), (7789 101, (10, mi, mz)) m ~ ) y

(cf. [30]) then (V, B)has this property. Then obviously (V, P’B), i E &-2, is also an S(2,3, u), and p ’ B n P’B = $7 for if j . It follows that {( V, p ’ B )1 i = 0, 1, . . . , u - 3) is a large set of isomorphic disjoint S(2,3, u ) k Denniston [ 121, working by hand, succeeded in constructing in this way large sets of u - 2 disjoint S(2,3, u)’s for u = 13, 15, 19, 21,25,31,33,43,49,61,69. Moreover, an exhaustive computer search showed that there are exactly two nonisomorphic sets of 11 disjoint S(2,3, 13)’s that can be obtained in this way [12,30]. Schreiber [53] and Wilson [66] have independently shown how to construct an S(2,3, u ) with the above property provided all prime divisors p of u - 2 are such that the order of -2 modulo p is congruent t o 2 modulo 4; this condition is satisfied, for example, if p =7 (mod 8). The first few orders for which this method works are u = 9, 25,33,49, 51, 73,75, 81,91, 105, 129, 153, 163, 169, 193,201,

and for all these values of u, we have d*(2,3, u ) = u - 2. Another recursive construction - a purely combinatorial one, using properties

A. Rosa

1I X

of special 1-factorizations of K,, -appears

in [52] where it is shown that

d ( 2 , 3 , 2 u + 1 ) a u + 1+ d(2,3, u ) for u 2 7 .

(10)

This enables one to construct large sets of disjoint S ( 2 , 3 , 2 u+ 1)’s whenever there is a large set of disjoint S(2,3, u)’s. A further result on d(2,3, u ) is due to Teirlinck [62]: If u is the product of primes p for which the order of -2 ( m o d p ) is congruent to 2 (mod4) and if d(2,3, w ) = w - 2, then

d(2,3, u ( w - 2) + 2) = U( w - 2).

(1 1)

Since Teirlinck’s construction contained in [62] is not as readily available as the others discussed here, we reproduce it here in full. For u as above, the graph whose vertex-set is Z: and whose edges are the 2-subsets {g, -2g}, g E Z:, can be decomposed into two 1-factors, say F, and F2. Let {(W, B,) 1 i = 1 , 2 , . . . , w -2) be a large set of disjoint $(2,3. w)’s with W = Z,-,U{~,,~,}. Define Y = { ( V , C , j ) l i = ,l . . . , u ; j = 1 , . . . . w-2) as follows: V = ( Z , X Z , - , ) U { ~ ~ , ~ ~ } , C,,= C:,” U Cl:’ u C!;) where

C,”={P,. 3c2, (i, r)I I {=,, a;,,

Bj} U k , ( i , T ) ~ ) , (i, r2)l I k = 1 . 2 ; h,rl, r 2 } 6 Bj} U{{(i, s,), (i. sz), (i, s3)} I Is,, s2,SJEB,},

C!:)= {{xk,( r + i, s), (-2r

+ i, s + j ) } I r E Zb, s E Z,

.2 ,

( r , -2r)

E F,,

k

=

1,2}

U{{r+ i, sI),( r + i. s,), (-2r+ i, (sl + s2)/2+j)JI re Zb, s l , S,E Z ,

2 r sl

# ~217

C ! : ) = { { ( r ~ . ~ 1 ) . ( ~ ~ . ~ ~ ) , (r lr, ~ r 2,, sr 3~E+Z su ~ + j ) ) r, + r2 + rJ = 3i, r’, < r; < r;, r:, r;, r: unique integers with0sr:Su-1

and r ; + u Z = r , , k = 1 , 2 , 3 ; ~ , , s , E Z ~ - ~ } .

Then Y is a large set of disjoint S(2,3, u ( w - 2) + 2)’s. Combining all the above results and constructions, we get d(2,3, u ) = II - 2 for all u = 1 , 3 (mod 6), 9 s u ~ 2 0 5except , possibly for u = 37, 85, 97, 109, 133, 141, 145, 157, 159, 181, 195. One could require additional properties from the disjoint Steiner systems involved. Let d,(t, k. u ) be the maximum number of disjoint cyclic S ( t , k, u)’s (an S(r, k, u ) is cyclic if it has an automorphism consisting of a single cycle of length u). and denote by dr(t, k, u) the maximum number of disjoint isomorphic cyclic S(t. k, u)’s. So,for example, df(2,3,7) = 2, d,(2,3,9) = 0 (since there is no cyclic S(2,3,9)). It is easily seen that dr(2,3,6m + 1)2 2 for every m > O [16, SO]. Doyen’s construction [ 161 shows df(2,3,6m + 3) 2 4m + 1 for t+ 1 (mod 3). But not much else seems to be known about d,(2,3, u ) . One could require, as it is done in [50], that the disjoint cyclic S(2,3, u)’s have the same cyclic automorphism p ; denote the maximum number of such disjoint

Intersection properties of Sreiner systems

119

cyclic S(2,3, u)’s by d,,(2,3, u ) . Then dCc(2,3,6m + 3) = 1 (dcc(2,3,9) = 0) since any cyclic S(2,3,6m +3) must contain the same “short” orbit (one of length 2m + 1) of triples. On the other hand, dcc(2,3 , 7 ) = 2, dc,(2, 3, 13) = 2, dcc(2,3, 19) = 6 but no general results on dcc(2,3, u ) are known. Let d,(2,3, u ) be the maximum number of disjoint S(2,3, u)’s with the same regular transitive automorphism group. It is shown in [ 191 that d,(2,3,3“)= 33”- 1). A problem in a slightly different direction was proposed by Doyen [16]: Given an S(2,3, u ) with u 7, say (V, B), is there always another Steiner triple system (V, B’) isomorphic to and disjoint from it? This was recently settled in the affirmative by Teirlinck [63] who proved a much more general theorem and the best possible result in this direction.

Theorem (Teirlinck [63]). If ( V1,Bl), (V2,B2)are any two S(2,3, u)’s u 2 7, and if V is any u-set, then there exist two disjoint S(2,3, u)’s (V, B;),(V, B;)such that (Vi, Bi)-(V,B;) and (V2,B2)=(V7B!d. 2.2. Kirkman triple systems Almost one-hundred and thirty years ago, Sylvester (cf. [9,57,59]) asked the following question: “Can fifteen schoolgirls walk out in five rows of three seven times a week for a quarter of thirteen weeks in such a way that any two girls are in the same row just once in each week, and any three just once in the term?” Of course, to find such an arrangement for just one week is the famous Kirkman’s problem of fifteen schoolgirls which has “excited some attention” [9,24] in the middle of the past century. (A nice account of the problem and its modifications is by Ahrens [l], a complete classification of solutions was given by Cole [lo].) A Kirkman triple system of order u (KTS ( u ) ) is a resolvable S(2,3, u). It was only recently that Ray-Chaudhuri and Wilson [49] have shown a KTS ( u ) to exist for every u = 3 (mod 6). Let d,(2,3, u ) be the maximum number of disjoint KTS (u)’s. Then Sylvester’s problem asks whether d,(2,3, 15) = 13. This problem was assumed to have no solution by Cayley [9] (cf. also [ l , 61). Denniston [13] finally gave a solution in 1974 that was found with the aid of a computer: (0, 1,9) (0727 7) {0,3,11I {0,4,6) {0,5,8) {0,10,12) (1,495)

(2949 121 (3949 8) (1,79121 {1,8, 11) (1,273) 1395, 9) (296, 11)

(5910, 111 (576,121 {6,8,10) R 9 , 10) {6,7,9) (47 7,111 (377, 10)

(738, xi) (9911, mi} {2,5, “1) (3, 12, x , ) (4, 10, E l ) (196, mi) {8,9, 12)

{3,6,4 {1,10, ~ 2 ) {4,9, ~ 2 ) 1 5 7 , a2) (11, 12, =*I {2,8, ~02) {O, mi, mz}

This is one of the 13 KTS(l5)’s. The remaining twelve KTS’s are obtained by developing the one above modulo 13.

A. Rosa

120

Clearly dK(2,3,9) = 7 as any S(2,3,9) is resolvable. Denniston[ 111 was first to show, by recursive methods, the existence of a large set of disjoint KTS's for infinitely many orders, by proving dK(2,3,3")= 3" - 2. Schreiber [ 5 5 ] has constructed a large set of disjoint KTS (33)'s, i.e. d,(2, 3, 33) = 31. Using essentially the same method as Schreiber, Denniston [ 151 has shown d,(2,3, u ) = u - 2 for u = 51, 75, 105, 129, and subsequently, by generalizing his own recursive method of [ 113. that dK(2,3,3"m) = 3"m - 2 for n 2 1 and m = 5,25,43 [14]. Although the existence of a large set of disjoint KTS (21)'s is still in doubt, it is reasonable to conjecture, in analogy with the conjecture in Section 2.1, that dK(2,3, u ) = u 2 for u = 3 (mod 6), u 2 9 . 2.3. Steiner quadruple systems One has obviously d(3,4, u ) =su - 3. Unfortunately, apart from the trivial case u = 4, not in a single instance is the equality known to hold. It is easily established that d(3,4,8)= 2 and any two pairs of disjoint S(3,4,8)'s are isomorphic. Kramer and Mesner [29] have determined d(3,4,10) = 5 , and any two sets of five disjoint S(3.4, 10)'s are isomorphic. Only partial results are known for u a 14. The first general result in this direction seems to be the inequality d ( 3 , 4 , 2 u ) s 1+ d(3,4, u )

obtained in [38] (cf. also [39]). This was subsequently improved by Lindner [34] to d(3.4,2u) 2 u.

His construction is simple and easy to describe. Let (V, B) be an S(3,4, u ) with u 3 8, V = { 1,2, . . . , u } ; and let L be a latin square of order u with no subsquare of order 2 (such squares are known to exist for all orders u except u = 2,4). Let PI be a permutation of V defined by xp, = y if and only if the cell (x, i) of L is occupied by y. Put S = V X{ 1,2), and for i = 1,2, . . . , u, define a collection B,of quadruples as follows: B,= B:U B','where

B: = {{(x, I), (Y, I), (2,I), ( W S I , 2%{(x, 3 , (y, 2), (2,2), ( w p ; ' , 1)),

3,( w , 1)1,{(x, 21, (Y. 2), ( z p ; ' , l), (w,2% ( ( 411, (YPI, 3,( 2 , I), (w, I)}, {(x,2)- o$;', 11, (2,2), ( w , 211, ( ( 4I), (Y, I), ( Z B ,

{(XPl,

21, (Y, 1 ) s

(2,11, (w, 1)),

{(xp;'. 11, (Y, 3,(2,a,(w, 2))I {x, y, 2, W I G B), B:'= U(X9 I), (Y, l)(XS,,a,(YSI, 2)) I {x, Y) c V).

I

Then {(S,B,) i = 1,2, . . . , u} is a set of u disjoint S(3,4,2u)'s. A different construction using special 3-quasigroups and due to Phelps [48] shows d(3,4,2 5") = 5" for n 3 1.


121

However, for other orders u = 2 or 10 (mod 12) it still remains to be shown that there exists at least a pair of disjoint S(3,4, u)’s! In order to construct large sets of disjoint S(3,4, u)’s, one could try an obvious extension of the method that worked well for Steiner triple systems and was described in Section 2.1. One would want to take V = Zv-3U{=1,=2,=3} as the set of elements and have p = ( ~ ~ ) ( o c ~ ) ( = ~ )1( O* * * u - 4) as an automorphism of the large set of u - 3 disjoint S(3,4, u ) k It was shown in [30] that one cannot find in this way a set of 11 disjoint S(3,4,14)’s. If one lets Cv3above to be replaced by an additive Abelian group G of order u -3 but adds the requirement that all 4-subsets {x, y, z , w} of G with x + y + z + w = 0 are contained in the initial S(3,4, u ) of a large set then it can be shown [7] that such a large set does not exist for any order u 2 8. 2.4. Orher Steiner systems. For the five Mathieu systems S(t, k, u ) (i.e. those with the Mathieu groups M , , , M,,, M22, M23, M2*, as their automorphism groups), Assmus and Mattson [2] have given a simple proof to show d(t, k, u ) 3 2 . Subsequently, Kramer and Mesner [29] have shown that for systems with the small Mathieu groups, this is the maximum number, i.e. d(4,5, 11)= 2, d(5,6, 12) = 2. Moreover, any two pairs of disjoint S(4,5, 11)’s (S(5,6, 12)’s) are isomorphic. For the systems with the large Mathieu groups, Kramer and Magliveras [28] used computer to improve the bounds substantially; they show d(5,8,24)3 9, d(4,7,23) = 24, d(3,6,22) 2 6 0 . They also show d(2,5,21)* 197; we must have d(2,5,21) O and

d , ( 2 , 3 , u ) = b(u - 2) for all u = 2 (mod 6), and so, for u even, dA,,,(2,3 , u ) is completely determined. For A(u) = 3 there are two partial results. First, there is a construction by Kramer [27] showing d 3 ( 2 ,3 , u ) = f ( u - 2 ) whenever u is a prime power, u = 5 (mod6). Another construction is due to Teirlinck [62]; he shows that if u is the product of primes p for which the order of -2 (mod p ) is congruent to 2 (mod 4), then d 3 ( 2 ,3 , 3 u + 2 ) = u. (Here necessarily 3u + 2 = 5 (mod 6) as for u as above, u = f 1 (mod 6).) The smallest values of u = 5 (mod 6) that are not covered by either of the two constructions, and for which, consequently, the question whether d3(2, 3, u ) = f(u-2) remains open, are u = 35, 65, 77, 119, 155, 161, 185, 203.

4.2. Existence of designs S,(r, k, u ) . A r-design S,(L k, u ) is a pair (V, B) where V is a u-set and B is a set of k-subsets of V called blocks such that any r-subset of V is contained in exactly A blocks. Clearly, the existence of A disjoint S ( t , k, u)’s implies the existence of an S,(r, k, u ) , and also of an S,dt, k, u ) where A ’ = (:::)-A. Thus, for instance, the results of [28] surveyed in Section 2.4 imply that there exists an S,(5, 8,24) for l s A s 9 , and S,(4,7,23) for 1 s A s 2 4 , an S,(3,6,22) for 1 s A s 6 0 and an S, (2,5,2 1) for 1s A S 197, and also for A ’ = 969 - A whenever A is in the above range. More generally, if there exist s disjoint S,(r, k, u)’s, then there exists an SsA( r, k, u ) . If a design S, (2,3, u ) exists then 1=sA =s u - 2. Therefore if we want to show that the designs S,(2, 3 , u ) exist for all values of A in this range, it suffices to show the existence of at least s, = [ ( v - 2)/(2A(u))] disjoint S,,,,(2, 3 , u)’s. It


125

follows from what was said in Section 2.1 and Section 4.1 that this has already been proved for u even, and for u = 3, 7 or 9 (mod 12) (this time including also u = 7!) but not for u = 1 (mod 12) (when h ( u ) = 1) or for u = 5 (mod 6) (when h ( u ) = 3). Observe that assuming the existence of at least s, disjoint SA,,,(2,3, u)’s in these two cases is a much weaker assumption than either the conjecture of Section 2.1 or Teirlinck’s conjecture of Section 4.1. 4.3. Perpendicular Steiner svstems One can put additional requirements on disjoint Steiner systems; the motivation for this may come from an area outside of the theory of block designs. Two Steiner systems (V, Bl), (V, B2) of type S(t, k, u ) are perpendicular if (i) they are disjoint, (ii) whenever Q, Q are two blocks of Bl intersecting in a (k - t)-subset P, and R, R‘ are such that ( Q \ P ) U R E B ~ (, Q t \ P ) U R ’ ~ B 2then R f R’. Perpendicular Steiner systems (older term: orthogonal) were introduced in [45] as a generalization of perpendicular S(2,3, u)’s that are dealt with in [21,35,42,43,44,46,47,51]. The relationship between the perpendicular S(r, k, u)’s and the existence of Room squares and their generalizations is discussed in detail in a paper by R.C. Mullin and S.A. Vanstone elsewhere in this volume. The main open problem here is that of the existence of pairs of perpendicular S(2,3, u)’s. Let dp(2,3, u ) be the maximum number of pairwise perpendicular S(2,3, u)’s. A construction in [43] shows dp(2,3, u ) =- 2 whenever u = p” = 1 (mod 6) is a prime power. The set of orders u = 1 (mod 6) for which dp(2,3, u ) 3 s is PBD-closed [65]; this was shown for s = 2 (and any u ) by Lawless [31] and for any s by Gross [21]. Different constructions for pairs of perpendicular S(2,3, u)’s where u = p a = 1 (mod6) is a prime power were given in [42]. It was shown in [35] that the existence problem for pairs of perpendicular S(2,3,u)’swith u = 1 (mod 6) can be reduced to solving the problem for the case u = p * q where p and q are primes = 2 (mod3). The smallest order u = l (mod6) for which a pair of perpendicular S(2,3, u)’s is not known, is u = 55. As for orders u = 3 (mod 6), there exists no pair of perpendicular S(2,3,9)’s (cf. [44]). It was conjectured [45,46] that there does not exist a pair of perpendicular S(2,3, u)’s for any u = 3 (mod 6). However, in [51] a pair of perpendicular S(2,3,27)’s was constructed. It follows easily that dp(2,3, u ) a 2 for u = 1 or 3 (mod 6) with finitely many exceptions, i.e. there exists a constant u,(2) such that dp(2,3, u ) b 2 for all orders u > u,(2), u = 1 or 3 (mod 6). The smallest orders u = 3 (mod6) for which it is not decided whether a pair of perpendicular S(2,3, u)’s exists, are u = 15,21,33,39,45,51,57,63,69,75. Similarly, there exists a constant u;(s) such that for all orders u = 1 (mod 6), u > u;(s), dp(2,3, u ) 2 s. A table for orders u =s175 with best known lower bounds for 4 ( 2 , 3 , u ) appears in [21]. In particular, dp(2,3,31) 2 6 ; u = 31 is the smallest order for which a set of more than two perpendicular S(2,3, u)’s is known to

-

A . Rosa

126

exist. Recently, Zhu [68] has provided a new direct construction for sets of perpendicular S(2.3. u ) ‘ s . In particular. h e shows d,(2,3.2” - 1 )3 3 whenever (n,6) = 1, and dJ2.3. 127)5 6. Clearly. d,(2. 3 . u ) s u-2. and we have d,(2,3,7)= 2, d,(2, 3, 9 ) = 1. It is widely believed that d,(2, 3, u ) s i ( u - 1 ) but n o nontrivial upper bounds on d,(2. 3. u ) are known. Not much is known about perpendicular Steiner systems other than S(2,3, 111’s. The following condition is necessary for the existence of a pair of perpendicular S(1, k, u)’s [45]:

For instance, when r = 3. k = 4, this forces u S 8 ; there exists indeed a pair of perpendicular S(3,4,8)’s [45]. Thus, d,(3,4, u ) =

2

if u = 8,

1

if v

2 or (mod 6), u f 8.

Similarly, d,(4, 5, u ) = 1 for all u for which an S(4,5, u ) exists [45]. On the other hand, d,(4.7,23) 2 9 (cf. [28,45]). Observe that directly from the definition of perpendicular Steiner systems follows that if k 2 2 t , any two disjoint S(r. k . u)’s are perpendicular.

References W. Ahrens. Mathematische Unterhaltungen und Spiele (B.G. Teubner. Leipzig, 19 18). E.F. Assnius Jr. and H.F. Mattson Jr.. Disjoint Steiner systems associated with the Mathieu groups. Bull. Amcr. Math. Soc. 72 (1966) 843-845. R.D. Baker. Partitioning the planes of AG2,,,(2) into 2-designs. Discrete Math. 15 (1976) 20.5-2 1 I . S. Bays. llne question de Cayley relative au probleme des triades de Steiner, Enseignement Math. 19 (1917) 57-67. G.F.M. Beenker. A.M.H. Gerards and P. Penning, A construction of disjoint Steiner triple systems, T H Report 78-WSK-01, Department of Mathematics, Technological Univ. Eindhoven. C. Berge. ThCorie des Graphes et ses Applications (Dunod, Paris. 19.58). T. Beth. On resolutions of Steiner systems. Dissertation. Erlangen 1978. K.C. Boss. On the construction of balanced incomplete block designs. Ann. Eugenics 9 (1939) 353-399. A . Cayley. On the triadic arrangements of seven and fifteen things, London, Edinhurgh and Dublic Philos Mag. and J. Sci. (3) 37 (18.50)50-53. F.N. Cole. Kirkman parades. Bull. Amer. Math. SOC.28 (1922) 43.5-437. R.H.F. Denniston. Double resolvability of some complete 3-designs. Manuscripta Math. 12 11974) 105-1 12. R.H.F. Denniston. Some packings with Steiner triple systems, Discrete Math. 9 (1974) 213-227. R.H.1:. Denniston, Sylvester’s problem of the 15 schoolgirls, Discrete Math. 9 (1974) 220-23-1. R.H.F. Denniston. Further cases of double resolvability, J. Combinatorial Theory ( A ) 26 (1Y7Y) 79X-3(13 K . H . F . Dennistun. Four doubly resolvable complete designs, Ars Combinatoria 7 ( 1979) 265272.


127

[16] J. Doyen, Constructions of disjoint Steiner triple systems, Proc. Amer. Math. SOC.32 (1972) 409-4 16. [17] J. Doyen, Recent developments in the theory of Steiner systems, Teorie Cornbinatorie I, Colloq. Roma 1973, Atti dei Convegni Lincei 17, Roma (1976) 277-285. [18] A. Emch, Triple and multiple systems, their geometric configurations and groups, Trans. Amer. Math. SOC.31 (1929) 25-42. [ 191 G. Ferrero, Su un problema relativo ai sistemi di Steiner disgiunti, Rend. 1st. Mat. Univ. Trieste 7 (1975) 1-7. [20] B. Ganter, J. Pelikan and L. Teirlinck, Small sprawling systems of equicardinal sets, Ars Combinatoria 4 (1977) 133-142. [21] K.B. Gross, On the maximal number of pairwise orthogonal Steiner triple systems, J. Combinatorial Theory (A) 19 (1975) 256-263. [22] J.I. Hall and J.T. Udding, On pairs of Steiner triple systems intersecting in subsystems, Technological University Eindhoven Report 76-WSK-04, August 1976. [23] J.I. Hall and J.T. Udding, On intersection of pairs of Steiner triple systems, Indag. Math. 39 (1977) 87-100. [24] T.P. Kirkman, Note on an unanswered prize question, Cambridge and Dublin Math. J. 5 (1850) 255-262. [25] T.P. Kirkman, Theorems on combinations, Cambridge and Dublin Math. J. 8 (1853) 38-45. [26] A. Kotzig, C.C. Lindner and A. Rosa, Latin squares with no subsquares of order two and disjoint Steiner triple systems, Utilitas Math. 7 (1975) 287-294. [27] E.S. Kramer, Some triple system partitions for prime powers, Utilitas Math. 12 (1977) 113-1 16. [28] E.S. Kramer, S.S. Magliveras, Some mutually disjoint Steiner systems, J. Combinatorial Theory (A) 17 (1974) 39-43. [29] E.S. Kramer, D.M. Mesner, Intersections among Steiner systems, J. Combinatorial Theory (A) 16 (1974) 273-285. [30] E.S. Kramer, D.M. Mesner, The possible (impossible) systems of 11 disjoint S(2.3, 13)'s (S(3.4, 14)'s) with automorphism of order 11, Utilitas Math. 7 (197.5) 55-58. 13 13 J.F. Lawless, Pairwise balanced designs and the construction of certain combinatorial systems, Proc. 2nd Louisiana Conf. Combinatorics, Graph Theory and Computing, Baton Rouge (1971) 353-366. [32] C.C. Lindner, A simple construction of disjoint and almost disjoint Steiner triple systems, J. Combinatorial Theory (A) 17 (1974) 204-209. [ 3 3 ] C.C. Lindner, Construction of Steiner triple systems having exactly one triple in common, Canad. J. Math. 26 (1974) 225-232. [34] C.C. Lindner, A note on disjoint Steiner quadruple systems, Ars Combinatoria 3 (1977) 27 1-276. [35] C.C. Lindner and N.S. Mendelsohn, Construction of perpendicular Steiner quasigroups, Aequat. Math. 9 (1973) 150-156. [36] C.C. Lindner and A. Rosa, Construction of large sets of almost disjoint Steiner triple systems, Canad. J. Math. 27 (1975) 256-260. [37] C.C. Lindner and A. Rosa, Steiner triple systems having a prescribed number of triples in common, Canad. J. Math. 27 (1975) 1166-1175. Corrigendum: 30 (1978) 896. [38] C.C. Lindner and A. Rosa, Finite embedding theorems for partial Steiner quadruple systems, Bull. SOC.Math. Belg. 27 (1975) 315-323. [391 C.C. Lindner and A. Rosa, Steiner quadruple systems-a survey, Discrete Math. 22 (1978) 147-18 I . [40] E. Lucas, RCcrCations mathtmatiques, Vol. 2 (Gauthier-Villars, Paris, 1883). [41] S. S. Magliveras, Private communication. [42] N.S. Mendelsohn, Orthogonal Steiner systems, Aequat. Math. 5 (1970) 268-272. [43] R.C. Mullin and E. Nemeth, On furnishing Room squares, J. Combinatorial Theory 7 (1969) 266-272. [44] R.C. Mullin and E. Nemeth, On the nonexistence of orthogonal Steiner systems of order 9, Canad. Math. Bull. 13 (1970) 131-134. [45] R.C. Mullin and A. Rosa. Orthogonal Steiner systems and generalized Room squares, Proc. 6th

128

A. Rosa

Manitoba Conf. Numerical Math., Congressus Numerantium XVlIl (Utilitas Math., Winnipeg, 1977) 315-323. [46] C.D. OShaughnessy, A Room design of order 14. Canad. Math. Bull. 11 (1968) 191-194. [47] C.D. OShaughnessy, On Room squares of order 6 m +2, J. Combinatorial Theory (A) 13 (1972) 306314. 1481 K.T. Phelps. A construction of disjoint Steiner quadruple systems, Roc. Eighth S.-E. Conf. Combinatorics, Graph Theory and Computing, Baton Rouge 1977, Congressus Numerantium XIX (Utilitas Math., Winnipeg, 1977) 559-567. [49] D.K. Ray-Chaudhuri and R.M. Wilson, Solution of Kirkman's schoolgirl problem, Proc. Sympos. Pure Math. 19 (Amer. Math. Soc., Providence, RI, 1971) 187-203. [50] A. Rosa, P o z n h k a o cyklick9ch Steineroech systkmoch trojic, Mat.-Fyz. cas. 16 (1966) 28.5-290. [51] A. Rosa, On the falsity of a conjecture on orthogonal Steiner triple systems, J. Combinatorial Theory (A) 16 (1974) 126-128. [S2] A. Rosa, A theorem on the maximum number of disjoint Steiner triple systems, J . Combinatorial Theory (A) 18 (1975) 305-312. [S3] S. Schreiber, Covering all triples on n marks by disjoint Steiner systems, J . Combinatorial Theory (A) 15 (1973) 347-350. [54] S. Schreiber, Some balanced complete block designs, Israel J. Math. 8 (1974) 31-37. [55] S. Schreiber, Private communication. [56] Th. Skolem. Some remarks on the triple systems of Steiner, Math. Scand. 6 (1958) 273-280. [57] J.J. Sylvester, Note on the historical origin of the unsymmetrical six-valued function of six letters, London, Edinburgh and Dublin Philos. Mag. and J. Sci. 21 (1861) 369-377. [58] J.J. Sylvester, Remark on the tactic of nine elements, London Edinburgh and Dublin Philos. Mag. and J . Sci. (4) 22 (1861) 144-147. [59] J.J. Sylvester, Note on a nine schoolgirls problem, Messenger Math. (2) 22 (1892-93) 159-160, Correction: 192. [60] L. Teirlinck, On the maximum number of disjoint Steiner triple systems, Discrete Math. 6 (1973) 299-300. [61] L. Teirlinck, On the maximal number of disjoint triple systems, J. Geometry 6 (1975) 93-96. [62] L. Teirlinck, Combinatorial Structures, Thesis, Vrije Universiteit Brussel, Departement voor Wiskunde, 1976. [63] L. Teirlinck. On making two Steiner triple systems disjoint, J. Combinatorial Theory (A) 23 (1977) 349-350. [64] T.M. Webb, Some constructions of sets of mutually almost disjoint Steiner triple systems, M.Sc. Thesis, Auburn Univ. 1977. [65] R.M. Wilson, An existence theory for painvise balanced designs, I. J. Combinatorial Theory (A) 13 (1972) 220-245; 11: 13 (1972) 246-273. [66] R.M. Wilson, Some partitions of all triples into Steiner triple systems, Hypergraph Seminar, Ohio State Univ. 1972, Lecture Notes Math. 411 (Springer, Berlin, 1974) 267-277. [67] G.V. Zaicev, V.A. Zinoviev and N.V. Semakov, Interrelation of Preparata and Hamming codes and extension of Hamming codes to new double-error-correcting codes, Proc. 2nd Internat. Sympos. Information Theory, Tsahkadsor, Armenia, USSR, 197 1 (Akadkmiai Kiado, Budapest, 1973) 257-263. [68] L. Zhu, A construction for orthogonal Steiner triple systems, Ars Combinatoria (to appear).

PART IV

RESOLVABILITY AND EMBEDDING PROBLEMS FOR STEINER SYSTEMS



ON IDENTIFYING PG(3,2) AND THE COMPLETE 3-DESIGN ON SEVEN POINTS J.I. HALL Deparfmenf of Mathematics, Michigan Sfare Uniuersiry, East Lansing, MI 48824, USA

Let f be a graph whose 35 vertices are labelled by the distinct 3-element subsets of a set of size 7. We join two vertices of f whenever their labels intersect in a set of size 1. Let f* be a graph whose 35 vertices are labelled by the 35 distinct lines of a projective geometry of dimension 3 over GF(2). We join two vertices of f* whenever their labels are intersecting lines. Then as graphs f and r* are isomorphic. In this note we verify the graph isomorphism mentioned above and thereby identify the projective geometry PG(3,2) with the complete 3-design on 7 points. This identification then allows us in an elementary fashion to prove various results related to both designs. For instance, by investigating aut(r) = aut(f*) we verify the sporadic isomorphism A, = GL(4,2). By examining the complete 3-design on 7-points, we are able to catalogue all parallel classes and Kirkman parallelisms of PG(3,2). Conversely, the geometry of PG(3,2) aids us in finding various resolutions of multiples of the complete 3-design on 7 points. Much of what we shall discuss is relatively well-known (indeed Section 5 reproves several results given in [ 11). Our advantage is that the results are proved here in an elementary manner.

1. PG(2,2) and PG(3,2) We shall be concerned mainly with three designs - the projective spaces PG(2,2) and PG(3,2) and the complete 3-design on 7 points Here we may think of PG(2,2) as the design whose points are the non-zero vectors of a three dimensional vector space over GF(2) and lines are the sets of three non-zero linearly dependent vectors. PG(3,2) is the design whose points are the non-zero vectors of a 4-dimensional space over GF(2), lines are the sets of three non-zero linearly dependent vectors, and hyperplanes are the sets of seven non-zero vectors of subspaces of dimension 3. We shall let X = { 1,2,3,4,5,6,7}, so that the design is all subsets of X with cardinality 3 and has parameters t - ( u , k, A ) with t = k = 3, u = 7, and A = 1. General properties and definitions concerning designs and incidence structures can be found in [3], [4]. We shall sometimes consider our designs as subsets of the point set and other times as incidence structures, in the interest of clarity.

(T).

(T)

131

1.1. Hall

132

While the only properties of (f) and its automorphism group S, that we shall need are evident, certain well-known and easily verified properties of PG(2,2) and PG(3,2) will be assumed.

1.1. Aut(PG(2,2)) is the group G L ( 3 , 2 ) s A , and has order 168. GL(3,2) is doubly transitive on the points and lines of PG(2,2). No non-identity element of GL(3,2) fixes more than three points.

1.2. Aut(PG(3,2)) is the group GL(4,2) of order $(8!).

13. Let Y be a set of size 7 or 8. Further let d be a collection of subsets of Y , each containing three elements and maximal subject to having pairwise intersections of size 1. Then d is the set of lines of a design on 7 of the points of Y which is isomorphic to PG(2,2). In particular, up to isomorphism, there is a unique 2-(7,3, 1) design. Note that the points and lines of the hyperplanes of PG(3,2) form a design PG(2,2). Conversely, this characterizes PG(3,2).

1.4. A 2-( 15,3, 1) design is isomorphic to PG(3,2) if and only if each collection of three points of the design are points of a subdesign isomorphic to PG(2,2). This is a consequence of Pasch’s axiom “31, p. 24, (3)]. Alternatively, the information given is enough to reconstruct the underlying vector space easily. 2. Identifying (f)and PG(3,2)

(t)

In this section we define an incidence structure .9 on which is seen to be the structure of PG(3,2). Let G be the symmetric group S7 on { 1 , 2 , 3 , 4 , 5 , 6 , 7 } ,and let K be its unique subgroup of index 2, K = A 7 . We choose a particular design P from ):( and isomorphic to PG(2,2):

P = { ( I , 2,3), ( 2 , 4 , 7 ) ,(3,4,6), ( 1 , 4 , 5 ) ,(2,5,6), ( 3 , 5 , 7 ) ,(1,6,7)). Note that we are denoting members of (f) by triples (a, b, c). Now H = G{p)= GL(3,2) has order 168, hence P has 30 distinct images under G, that is, there are precisely 30 distinct PG(2,2) from (f). As H S K , the 30 distinct PG(2,2) fall into two K-orbits, each of length 15. Let one of these obrits be denoted +c, and call its members points. Let the other orbit be denoted Ye, and call its members hyperplanes. Call the 35 triples of lines. We define now an incidence structure 9 on /cU (T) U X If t‘E (f) and Y ejz U X, then 8 and Y are incident if and only if the triple 4 is contained in the PG(2,2) design Y. If YE/^ and Q E 2,then Y and Q are incident if and only if the

(T)

On identifying P G ( 3 , 2 )and the complete 3-design on 7 points

133

(T).

intersection Y n Q of the two PG(2,2) contains at least one line from No other incidences occur. Observe that K acts in a natural fashion on the incidence structure preserving /z, and X and respecting incidence. The group of all permutations of /zU U X which do this forms the automorphism group of 4 and so contains K. Note that K is transitive on each of 6, and X. A permutation of 4 which respects incidence and but exchanges the sets z/, and X is called a duality of 4. It is clear that all elements of G-K induce dualities of 4. We say that two lines meet whenever they are incident to a common point or a common hyperplane.

(t), (t)

(T)

(T),

2.1. Two lines t', and 8, meet if and only if It',nt',l

= 1. If two lines meet then there is a unique point and a unique hyperplane incident with both lines.

Proof. Clearly if 4, and t', meet, then It', fl[,I=

1. Now suppose It',fl t',l = 1. By the transitivity of K we may assume 8, = (1,2,3) and 4, = (1,4,5). These two lines can be completed to a PG(2,2) in exactly two ways - as the design P given above and as Pa where a = (6,7) E G. As a is a duality, exactly one of P and Pa is a point and the other is a hyperplane. 2.2. (1) Each line is incident with exactly 3 points and exactly 3 hyperplanes. (2) Any two points are incident to exactly one common line. Any two hyperplanes are incident to exactly one common line. (3) If Y is a point and Q is a hyperplane with Y and Q incident, then there are exactly 3 lines incident with both Y and 0. (4) Each hyperplane is incident with exactly 7 points and 7 lines.

Proof. By the transitivity of K we need prove (1) only for the line t'= (1,2,3). Let S be the set of all PG(2,2) from U X which contain 4'. As t'r~P and H is transitive on the lines of P, S ={PgI g e G{el}. Thus IS(= IG{,,I/JG{,,nHI= 144/24 = 6. We have in fact that S ={PIg E ((1,2,3), (4,5))}. This gives (1). Observe that P f l Pg= t' for g = ( 1 , 2 , 3 ) and g = (1,3,2); thus no pair of points is incident to more than one line, and dually no pair of hyperplanes is incident to more than one line. Using (l),each pair of points is incident on the average to 35-3/(:5)= 1 line. This and a dual observation give (2). For (3) assume that t'= (1,2,3)E Y n Q and P is one of Y and Q. If P is a point (or hyperplane) incident with 8 then the three hyperplanes (or points) incident with 't are P(a, 6) for (a, 6) E G, {a, 6 )G { 1,2,3}. We find that P f P( l a, 6) consists precisely of the three lines of P containing c where {a, b, c} = {1,2,3}, giving (3). As Ifi = 1x1= 15, (4) is a direct consequence of (1) and (3). 2.3. Each triple of points of 4 is incident with at least one hyperplane of 4. The points and lines incident with a given hyperplane form a PG(2,2).

J.I. Hall

134

Proof. If the triple of points is collinear (incident to a common line) then the first sentence follows from 2.2(1). Otherwise it is a consequence of 2.1 and 2.2(2). From 2.2(4) each hyperplane is incident with 7 points and 7 lines. 2.2(2) and 1.3 then imply that these points and lines form a PG(2,2), as desired. Now 1.4, 2.2(1), 2.2(2), and 2.3 give immediately 2.4. The incidence structure 9 is isomorphic to the incidence structure of points, lines, and hyperplanes of the projective geometry PG(3,2).

3. The graph

r

r

Recall that the graph is a graph with 35 vertices, each labelled by a member of ):( and two vertices joined whenever their labels have intersection of size 1. In are terms of our correspondence with PG(3,2), 2.1 states that two vertices of joined precisely when their labels meet, when considered as lines of PG(3,2). This displays the graph isomorphism of I' and I'* discussed at the beginning of the note. We observe that PG(3,2), or more properly the incidence structure 9, can be recovered from r. By 1.3, the maximal cliques of r are in one-to-one correspondence with the thirty distinct PG(2,2) contained in (:). Furthermore 2.2(2) and 2.2(3) show that the relation of intersecting in precisely one vertex is an equivalence relation on the maximal cliques of r having two equivalence classes each of size 15. Let these two classes be denoted fi* and X*. We form an incidence structure 9* on #z*U V ( r ) U X* where V(1') is the vertex set of 1'. A member of +* is incident with a member of %* if the two cliques concerned have non-trivial intersection. A member of V(T) is incident with each member of ,jt* and X* which contains it. N o other incidences occur. From the discussion of Section 2, it is clear that the two incidence structures 9 and 9" are isomorphic. We now calculate the automorphism group of the graph r, a u t( r ) . From our description of the incidence structure 9*, it is clear that any automorphism of I' induces either an automorphism or duality of 9 (isomorphic to .a*). Indeed, by our construction it is clear that G = S , acts naturally on X and inducing automorphisms of f.The elements of K = A, as before induce automorphisms of .a and those of G - K induce dualities of 9.As aut(9) = aut(PG(3,2))= GL(4,2), we have

r

(T)

3.1. Aut(f') has a subgroup of index 2 which is a subgroup of GL(4,2). We now define yet another graph. Let Y = {m} U X. The graph A has (;): = 35 vertices, each vertex labelled with 2 disjoint subsets of Y of size 4. Suppose that a

On identifying P G ( 3 . 2 )and the complete 3-design on 7 p i n t s

135

vertex of A has label {A,B} with A U B = Y and ( A (= IBI = 4 and that a second vertex has label {C, D}. Then the two vertices are joined in A precisely when \ An

CI

=

I A no1= IBnCI = IBnDl=2.

It is clear that, by its natural action on Y and so on the labels of A, S, acts as an automorphism group of A. We define a map C$ from the vertices of A to those of r. If a vertex x of A has label {A,B} with A U B = Y and IAl= IBI = 4 then the vertex C$(x)of will be that vertex with label (a, b, c) where either {m,a, b, c} = A or {z, a, b, c} = B. We can now quickly verify that C#J is in fact an isomorphism of the two graphs A and 1’. Now S,=saut(A)=aut(r); and by 1.2 and 3.1, ( a u t ( r ) l s 2IGL(4,2)1=8!. Therefore another application of 3.1 gives

r

3.2. aut(l-) = S,. As S8 has a unique subgroup of index 2, we have an immediate corollary to 3.1 and 3.2.

3.3. A, = GL(4,2). 4. Resolutions of PG(3,2) A parallel class of PG(3,2) is a subset S of the lines of PG(3,2) such that each point is incident with exactly one line of S. Note that 15.1 = 5 and that by duality each hyperplane is incident with exactly one line of S also. A Kirkman parallelism

of PG(3,2) is a partition of the lines of PG(3,2) into 7 disjoint parallel classes. Such parallelisms provide solutions to the famous “problem of the fifteen schoolgirls” originally proposed by the Rev. T.P. Kirkman in 1850 ( [ 5 ] . A solution was given by Cayley, [2].) In this section we use our identification of the lines of PG(3,2) with the triples of ( f )to give a description of all parallel classes and all Kirkman parallelisms of PG(3,2). The five lines of a parallel class of PG(3,2) correspond to five triples of ):( with pairwise intersections never of cardinality 1.

4.1. Let S be a set of 5 triples from (T) such that x, y E S and x f y implies I x n y J ~ { 0 , 2 }Then . we have one of (1) S is all members of (q) containing a given pair {i, j } s X , or (2) S is all members either equal to or disjoint from a given member (a, b, c) of

(3. Proof. If all intersections of pairwise distinct lines in S have size 2, we must have (1). If S contains a disjoint pair of triples then all further members of S are

J.I. Hall

136

disjoint from one member of the pair and have intersection of size 2 with the other. This leads to (2). Therefore we have exactly + 35 = 56 distinct parallel classes of PG(3,2), and under the action of A, these fall into two orbits of lengths 21 and 35. Of more relevance is the action of aut(PG(3,2))- A, on the parallel classes. Let the parallel class of 4.1(1) be denoted by the symbol (a,i, j ) , and let that of 4.1(2) be denoted by the symbol (a, b, c). Now the natural action of A, acting on Y = {a} U X induces action on PG(3,2) by way of the graph isomorphism 4 constructed in Section 3. Let be the associated map from pairs of disjoint subsets of given by cardinality 4 from Y into

(z)

4 (t)

where either {m, a, 6, c } = A or {m,a, b, c} = B. It can now be checked that for any element g E A, acting on Y, we have the parallel class equality ( a g ,be, cg) = ( a , b, c)&-'g&.

(This is in fact valid even for dualities from s8).A particular consequence of this is that under the action of A, = GL(4,2) the 56 parallel classes of PG(3,2) form a single orbit. Using the descriptions of 4.1, it is clear that the two parallel classes (a, b, c ) and (d, e, f) will have no common lines if and only if /{a,b, c } n { d , e, f}l = 1. Therefore the symbols for the classes of a Kirkman parallelism of PG(3,2) form a set of 7 subsets of Y with cardinality 3 with all pairwise intersections of cardinality 1. By 1.3 the symbols are therefore the lines of a projective plane PG(2,2) on 7 of the points of Y. Conversely, if we take the lines of any projective plane whose point set is contained in Y, we have the set of symbols for a Kirkman parallelism of PG(3,2). The remarks of the previous paragraph show that GL(4,2) permutes the parallelisms precisely as A, permutes the PG(2,2) composed of symbols. As before, since aut(PG(2,2))= GL(3,2)6 A,, there are 8!/168 = 240 distinct Kirkman parallelisms of PG(3,2) which under the action of GL(4,2) are permuted in two orbits of length 120. We summarize our results. 4.2. (1) PG(3,2) has 56 distinct parallel classes, all equivalent under the action of GL(4,2) = aut(PG(3,2)). (2) PG(3,2) has 240 distinct Kirkman parallelisms which fall into two dual orbits of length 120 under the action of GL(4,2).

From the construction, it is clear that the stabilizer of a given Kirkman parallelism in GL(4,2) is isomorphic to GL(3, Also, the stabilizer of a parallel class is isomorphic to a subgroup of index 2 in S3x Ss.

T).

On identifying P G ( 3 , 2 )and the complete 3-design on 7 points

137

5. Planar resolutions for multiples of ($) Let A(): denote the collection of the 35 3-subsets of X,each subset repeated A times. Thus we are interested in the complete 3-design with parameters 3(7,3, A). In [l], Brouwer has studied, for various A, partitions of):(A into sets of 7 triples, each set constituting the lines of a PG(2,2). We call such a partition a planar resolution of .):(A So for instance, 2.2(1) shows that the 30 distinct PG(2,2) from (f)form a planar resolution of 6 ( 3 . It was shown by Cayley ([2]) that there is no planar resolution of (q) itself (see 5.1). Brouwer proves that planar exist for all integral A at least 2. In view of the observation resolutions of A)(: above regarding the case A = 6, it suffices to give examples with A = 2 and A = 3. These contain, respectively, 10 and 15 copies of PG(2,2). In Section 5 we use our identification of the points and hyperplanes of PG(3,2) with the subsets of (q) forming PG(2,2) to reprove Brouwer’s results and in some cases answer questions which he left open.

5.1. If A,, A,, and A, are three distinct PG(2,2) from ($) then, for some pair i and j with 1s i <j s 3, Ai nA, contains at least one line. In particular there is no planar resolution of

(T).

Proof. This is a direct consequence of 2.2(2).

5.2. No PG(2,2) can be repeated in a planar resolution of A():

if A = 2 or 3.

Proof. For A = 2 this is clear by an argument similar to that of 5.1. For A = 3, we can suppose (if necessary by applying a duality of PG(3,2)) that a PG(2,2) which is repeated twice in the resolution is a point Y of PG(3,2). As before it cannot be repeated three times. As the resolution has 15 PG(2,2), it is easy to see that of the seven hyperplanes incident to Y,at most one appears in the resolution. This quickly leads to a contradiction. A helpful observation is one already used in the proof of 5.2. Namely, if we as a collection of points and hyperlanes of think of a planar resolution of ):(A PG(3,2), then its images under any automorphism of duality of PG(3,2) will also be a planar resolution of A(:). We first consider planar resolutions of 3 ( 3 . Such a resolution has been given by Lindner and Rosa [6], and Brouwer [l] observes that there are at least three isomorphism classes (under S,) of planar resolutions of 3($). In fact, there are exactly four. Note that a planar resolution of 3(:) contains 15 of the 30 distinct PG(2,2) from (T), hence those PG(2,2) not contained in the resolution themselves form a second complementary planar resolution of 3 ( 3 . In particular, in finding all planar resolutions of 3(:) we need only find those which have more points than the hyperplanes, the rest coming from complementation.

1.1. Hall

138

Let d be a planar resolution of 3 ( 3 . Suppose that under our correspondence x members of d are points of PG(3,2) and y are hyperplanes. Then we have x+y=15

and

(3.(3+3f=(3

*

35.

Here the first equation counts the members of d. The second equation counts pairs of duplicate triples occurring in d, where f denotes the number of incident point-hyperplane pairs contained in d. The only solutions in non-negative integers for x. y, f subject to x 2 y are (x, y, f ) = (15,0, O), (12, 3, 12), and (9,6, 18). One orbit of S, acting on all planar resolutions of 3 ( 3 is that which contains only two resolutions - one consisting of all points of PG(3,2) and the other consisting of all hyperplanes of PG(3,2) (under our correspondence). This is the resolution found by Lindner and Rosa [6]. We now describe all other planar resolutions of 3 ( 3 in terms of the corresponding configurations in PG(3,2). 5.3. Let d be a set of 15 points and hyperplanes from PG(3,2) consisting of 12 points and 3 hyperplanes. Then d corresponds to a planar resolution of 3 ( 3 if and only if d consists of all points of PG(3,2) not incident to a given line 8 and all three hyperplanes incident to t'.

Proof. We first observe that a configuration from PG(3,2) consisting of all points not incident to given line and all hyperplanes incident to the line furnishes a planar resolution of 3 ( 3 . Conversely, if 1 is a planar resolution of 3 ( 3 consisting of the twelve points not incident to a given line t' plus 3 additional hyperplanes, these hyperplanes must be those incident to t'. Thus it suffices to show that if a planar resolution d of 3 ( 3 contains exactly 12 points, then the points are those not incident to a given line of PG(3,2).Assume that the three deleted points p, q and r do not lie on a common line. Each pair of points from {p, q, r} is incident with a unique tine of PG(3,2) and each line as determined is incident with only one point of d. Therefore each of these lines must be incident with exactly 2 hyperplanes in d.This requires at least four hyperplanes, a contradiction which proves 5.3. There are 35 planar resolutions of 3(:) of the type described in 5.3, and in addition there are 35 complementary resolutions. As A, is transitive on the lines of PG(3,2). there 70 planar resolutions of 3(:) form a single orbit under t h e action of S,. 5.4. Let d be a set of 15 points and hyperplanes from PG(3,2) containing 9 points and 6 hyperplanes. Then d corresponds to a planar resolution of 3 ( 3 if and only if the points of PG(3,2) not in d are those incident with two disjoint lines t',and 8, and the hyperplanes of PG(3,2) in d are those incident to 8 , and t2.

On identifying P G ( 3 , 2 ) and the complete 3-design on 7 points

139

Proof. We first observe that if SB contains the 9 points not incident to two given disjoint lines, then for d to correspond to a planar resolution of 3 ( 3 the six hyperplanes of d must be those incident to the two lines. Furthermore, such an d does produce a planar resolution of 3 ( 3 . Thus we need only prove that a planar resolution d of 3(:) containing 9 points and 6 hyperplanes has its points those not incident to two disjoint lines. Let p, q and r be three points not in d. We claim that for some line e incident with two of p, q and r the third point incident to 4 is also not in d. Assume otherwise that p, q, r are not collinear and so are incident with a unique common plane Q. Our assumption implies there is a line incident with Q all of whose points are in d. Thus Q is not in d and the hyperplanes of d are precisely those 6 incident with two of p, q and r. Now as Q is incident with at most 4 points of d, there are at least 21 incident point-hyperplane pairs in d.The contradiction verifies our claim. As the claim is valid for all triples of points not in d, we find that the points not in d either are those incident with two disjoint lines or are all incident to a common hyperplane. In this second case, counting incident point-hyperplane pairs in d again produces a contradiction and completes the proof of 5.4. A, has two orbits on disjoint pairs of lines from PG(3,2) corresponding to pairs of triples from (T)with intersections of size 0 and 2. The orbits have lengths 70 and 210 respectively. As these numbers are distinct, a planar resolution of the type described in 5.4 must be in the same orbit under S, as its complement. Therefore 5.5. There are exactly 632 distinct planar resolutions of 3 ( 3 falling into four orbits under S, of lengths 2, 70, 140 and 420.

(T)

5.6. There are exactly 316 different partitions of the 30 distinct PG(2,2) from into two sets of 15, each set providing a planar resolution of 3 ( 3 . The 316 fall into four orbits under S, of lengths 1, 35, 70 and 210.

We now consider planar resolutions of 2 ( 3 . These have been discussed rather thoroughly by Brouwer in [l], so we do not give all our proofs in detail. For our purposes, it is convenient to have the concept of an ovoid of PG(3,2). An ovoid is a set of 5 points of PG(3,2) no 4 of which are incident to a common hyperplane. Ovoids are easy to construct - any three points not incident to a common line can be extended to an ovoid in exactly 4 ways. From the construction, it is clear that GL(4,2) is transitive on the ovoids of PG(3,2), there being 168 distinct ovoids. (The global stabilizer of an ovoid is isomorphic to S5 and is generated by transvections.) Using the results of Section 2 we can check that the ovoids break into two orbits of length 42 and 126 under the action of A,. It is easy to see that for each point p of an ovoid there is a unique hyperplane tangent to the ovoid at p, that is, incident to p but to no other point of the ovoid. These 5 tangent hyperplanes in fact form a dual ovoid of PG(3,2).

J.I. Hall

140

5.7. Let d be a set of 10 distinct points and hyperplanes of PG(3,2). Then d corresponds to a planar resolution of 2 ( 3 if and only if d is the set of points and tangent hyperplanes of some ovoid of PG(3,2).

Proof. Suppose ‘t is a line of PG(3,2) incident with no point of an ovoid 6. As no 4 points of 6 are incident to a common hyperplane, one of the hyperplanes containing t‘ is incident with 3 points of 6 and the other two hyperplanes containing t‘ are tangent to 6. With this in mind, it is easy to see that the points of an ovoid and its tangent hyperplanes d o correspond to a planar resolution of 2 ( 3 as desired. Now assume that d is a collection of 10 points and hyperplanes of PG(3,2) which corresponds to a planar resolution of 2(3. Let x be points of PG(3,2) and y be hyperplanes. As before, we have x+y=lO

and

(2”) + (2’) +

3f=

(;)

*

35,

where f is the number of incident point-hyperplane pairs in d. The only non-negative integral solutions are (x, y, f ) = (2,8,2), (8,2,2) and (5,5,5). Any 2 distinct hyperplanes of PG(3,2) are incident with all but 4 points of PG(3,2). This and a dual observation immediately rule out the possibility of configurations d with (x, y,f)=(8,2,2) or (2,8,2). Therefore we must have (x, y,f)=(5,5,5). Clearly no three points in d can be incident to a common line of PG(3,2), therefore if 4 points of d are incident to a common hyperplane Q they must be those not incident to some given line contained in Q. But in that case, at least 3 hyperplanes of d would be incident to 2 or more points of d, contradicting f = 5 . Thus the points of d are those of an ovoid in PG(3,2). It now easily follows that the hyperplanes of d must be those tangent to the ovoid. This gives 5.7 and 5.8. (Brouwer, [l]). There are 168 planar resolutions of 2 ( 3 which fall into orbits of length 42 and 126 under the action of S,. Brouwer [l] describes all ways of partitioning the 30 distinct PG(2,2) from (t) into three disjoint planar resolution of 2(:). In PG(3,2) such a partition corresponds to 3 pairwise disjoint ovoids. Considering the PG(2,2) from ):( we see that we need only find two disjoint ovoids to insure that the remaining 5 points also form an ovoid. We have seen above that any hyperplane of PG(3,2) is either tangent to a given ovoid or is incident to three points of the ovoid, these three points of course being incident to no common line (they form a triangle). Thus for any given hyperplane Q and any given set of three disjoint ovoids, one of the ovoids has Q as a tangent hyperplane while the other two ovoids are incident to two disjoint triangles of points from Q. There are 42 pairs of disjoint triangles in the hyperplane Q, and each pair can be completed to two disjoint ovoids in 8 different ways. Hence

On identifying PG(3.2) and the complete 3-design on 7 points

141

5.9. (Brouwer, [l]).There are 336 different partitions of the 30 distinct PG(2,2) from (t)into 3 sets of 10, each of which is a planar resolution of 2(?).

Brouwer shows that the 336 partitions come in two orbits of lengths 126 and 210 under S,. Again, this can be checked by using the action of A, on PG(3,2) given in Section 2.

Note added in proof A similar identification, due to Gleason, may be found in Wagner, Math. Zeit. 76 (1961) 424.

References [l] A.E. Brouwer, A note on the covering of all triples on seven points with Steiner triple systems, Mathematish Centrum ZN 63/67, 1976. [2] A. Cayley, On the triadic arrangements of seven and fifteen things, London, Edinburgh, and Dublin Philos. Mag. and J. Sci., (3) 37 (1850) 50-53. (Collected Mathematical Papers, Vol. 1, 481-484). [3] P. Dembowski, Finite Geometries, Springer-Verlag, Berlin-Heidelberg-New York, (1968). [4] M. Hall, Jr., Combinatorial Theory, Blaisdell, Waltham-Toronto-London, (1967). [5] T.P. Kirkman, Query, Lady’s and Gentlemen’s Diary (1850) 48. [6] C.C. Lindner and A. Rosa, Construction of large sets of almost disjoint Steiner triple systems, Canad. J. Math. 27 (1975) 256-260.



A SURVEY ON RESOLVABLE QUADRUPLE SYSTEMS Alan HARTMAN Department of Mathematics, Uniuersify of Newcastle, New South Wales, 2308, Australia A Steiner system ( X . p), denoted $(I, k, 0). is a set X of poinfs, of cardinality u, and a collection p of k-subsets of X called blocks, with the property that every 1-subset of X is contained in precisely A blocks. A quadruple system is a Steiner system S,(3,4, 0). A triple (X,p . y ) is called an (s, p)-resoloable system if, for some s < I , it is a partition of an SA(r,k, c ) system (X, p ) into subsystems ( X , y , ) , each of which is an S,(s, k. v ) system, such that y = y , I y z I . . I yc is a partition of 0. A system is doubly resolvable if it is resolvable and each ( X , yi) is also resolvable. This article surveys the work done on the existence of (s, @)-resolvable and doubly-resolvable quadruple systems for (s, /A) = (2, 1). ( 2 , 3 ) and ( 1 , 1).

Introduction and terminology A Steiner system (X, p ) , denoted SA(r,k, u ) , is a set X of points, of cardinality u, and a collection p of k-subsets of X , called blocks, with the property that every 1-subset of X is contained in precisely A blocks. The number of points, u, is the order of the system. Where blocks are listed explicitly square brackets will be used. A quadruple system is a Steiner system S1(3,4, u ) . A system Sl(l,k, u ) is simply a partition of X into k-subsets, and is called a parallel class. A triple (X, p, y ) is called an (s, p)-resolvable system if, for some s < 1, it is a partition of an SA(f.k, u ) system (X, p ) into subsystems ( X , y,), each of which is an S,(s, k, u ) system, such that y = y1 1 y2 1 . . * 1 yc is a partition of p . The 7,’s are called resolution classes. A resolvable system ( X , p, y) is doubly resolvable if each (X, y I ) is resolvable. The existence of resolvable Steiner systems was first raised in 1847 by Kirkman [ 101 in his well-known “schoolgirl” problem. Kirkman asked for the determination of necessary and sufficient conditions on u for the existence of (1, 1)resolvable S,(2,3, u ) systems. This task was not completed until 1961 when Ray-Chaudhun and Wilson [12] showed that these systems exist if and only if u = 3 (mod 6). The literature on existence questions for resolvable Steiner systems is quite extensive: a bibliography is given in [7]. Necessary and sufficient conditions for the existence of quadruple systems were established only in 1960 by Hanani [5],when he showed that a quadruple system of order u exists if and only if u = 2 or 4 (mod 6). Although (1, 1)-resolvable quadruple systems of orders u=2”’ were known to Kirkman, the existence problems for resolvable quadruple systems have been tackled seriously only since 1976. This article is an attempt to summarize the research on these problems by 143

A. Hariman

144

Baker [ 11, Booth [2], Greenwell and Lindner [4] and the author [6,8,9]. These papers construct (s, p)-resolvable and doubly resolvable quadruple systems for which (s, p ) has the values (2, l),(2,3) and (1, 1).Baker [l]has constructed (2, 1) (1, 1)-doubly resolvable systems of all orders u =4". In [9], the author has constructed (2, 3) (1, 1)-doubly resolvable systems of orders u = 8, 20, 32, 44, 68, 80 and 104, and has also given a (2,3)-resolvable system of order u = 128. The other papers cited construct (1, 1)-resolvable quadruple systems. At this point we note a result of a more general nature on resolvable quadruple systems. Lindner [ 111 has shown that every quadruple system can be embedded in a (1, 1)-resolvable quadruple system.

1. Algebraic construction In this section we describe constructions due to Baker [l] and the author [8,9] which have an algebraic flavour. The quadruple system (X, p ) is constructed using an algebraic structure on the set X , and the algebraic properties of the system are used to facilitate resolution.

1.1. Afine consfrucfion Kirkman [lo] showed that for any n > 1, the planes of the affine space of dimension n over GF(2) form the blocks of a quadruple system of order u = 2". These systems are (1, 1)-resolvable by taking each parallel class to be a two dimensional subspace and all of its translates. Baker [ 11, in 1976, gave a slightly different presentation of these affine spaces for even n, and demonstrated a partition of their planes into S,(2,4,2") systems each of which is (1,1)-resolvable. This construction will now be described. Let X = GF(22m-1) x GF(2), then X can be considered either as the set of points (a, i) with a E G F ( ~ ~ " - ' )i,E ( 0 , 1) or as the set of vectors of length 2m over GF(2). Let p be the set of planes in the vector space over X. Writing the block BE^ as

B =[(a,, i,)(a2, i * k , iAa4, i4)1, define

Yb

by

where 6 is the Kronecker symbol and b E GF(22"-') \ (0). Some technical lemmas are needed to show that: (i) y = {yb :b E GF(22m-1)\ (0))is a partition of p ; (ii) ( X , yb) is an S,(2,4, Pm) system for each b; and (iii) if B E y b then B + ( a , i) E yb for all a, b and i.

Resolvable quadruple sysrems

145

The reader is referred to the original paper [ 11for proofs of these results. The first two points show that we have indeed constructed a (2, 1)-resolvable quadruple system, and the third shows that the partition y refines to the usual (1, 1)resolution of the affine space. 1.2. Projective construction In the 1930’s Carmichael [3] constructed two families of quadruple systems with highly transitive automorphism groups, on the projective line X = GF(q)U {m}.The first of these constructions has q = 3” and p is the orbit of the quadruple [m,0,1,2] under the action of the group GL(2,3“) defined by GL(2,3”) = {x

I+

(ax + b)/(cx + d ) :a, b, c, d E GF(3”), ad - b c f 0).

No work has yet been done on the resolution of these systems, although the field seems promising. The second construction, which is described below, has been investigated and shown to generate quadruple systems which are “easily” resolvable by computer search. Let q = 7(mod 12) be a prime power and let g be a primitive cube root of unity in GF(q). As before take X = GF(q) U {m},but now take 0 to be the orbit of the quadruple [m,0,1, g + 11 under the action of the group SL(2, q ) defined by

SL(2, q) = {x

-

(ax + b)/(cx+ d ) :a, b, c, d E GF(q), ad - bc = 1).

Carmichael [3] showed that (X, p ) is a quadruple system. Let e(m, c) denote the transformations of X defined by

Let w be a generator of the multiplicative group on GF(q) \ {0}, and define the following subsets of SL(2,q): Q={e(m, c ) : c ~ G F ( q ) m , =w2’,Osiwhich intersect x, we are able to find a block B , U{X,}EL,, which does not intersect N . Now suppose q < r and we have found a collection of disjoint blocks B, U { X , } E L,,. B 2 U { x 2 L }~ c j ,.. . , B, U{X,}ELcSwhich do not intersect N. Let F be the collection of blocks obtained by deleting x q + , from the blocks of Lcq.,. Again we recall that in a (partial) S ( k , k + 1, u ) Steiner system at most ((k + l ) / k ) ( L - : ) blocks can intersect any given k reflection shows that at most

+ 1-element set.

A bit of

blocks of F can intersect N U B, U . * * U B,. Since q < r S k. q + 1 s k and so

Therefore we can find a block B , , , U { ~ , + , } E L , ~ -such , that B , U { x , } , R, U {x?}. . . . , El,, I U { x q . ,} is a collection of disjoint blocks which d o not intersect JV. This implics we can extend t h e above collection is such a way that e l = B,U{~,JEL~ , , , = B , U { ~ , } E .L.,.~. . e , = l ? , U { x , } ~ L , ,isacollection of disjoint blocks which d o not intersect N. But t h e n .rrU{e,,e,, . . . e , } U { d } l { c ,c, 2 , .. . , c,} i \ a collection of disjoint edges in (S, 7)with a cardinality greater than I T \ . We have reached a contradiction and so our assumption that d = A U { x } c D' is falsc. In other words if d = A U { x } and A E D', then x $ D'. I f L' = { d E T I d = A U { x } , A E D'}. then obviously I

Suppose x, E P, c, E SF, dnd x, E c,. How many blocks of L' can intersect x,? For any C E T let L: = { d E L'l d n c f 0 ) . If E' is the collection of blocks obtained by deleting x, from the blocks of L : ) which interscct x,, then (D'. E') is a partial

On the size of partial parallel classes in Steiner systems

207

S ( k - 1, k, u - (k + 1)t + k a ) Steiner system and so

Equivalently at most

(

u - (k

+ l)f + k a k-1

blocks of L' can intersect each xi E P. Obviously at most

blocks of L' can intersect P. If b E p, how many blocks of L' can intersect b? Suppose there are at least

[( u

- (kk+-lit+

ka

blocks of L' which intersect b. Then there is one element of b, say y, which is incident with at least

blocks of L'. If Q is the collection of blocks obtained by deleting y from every block of L; which intersects y, then (D',Q) is a partial S ( k - 1, k, u - (k + 1)t + ka) Steiner system and so

We have assumed that at least

[(u

]

+ 111 + k a ) / k +1 k-1

- (k

blocks of L' are incident with b and so there must exist a block G U {z} E Lb with z E b and z # y. If

[(u-(kk+-1;r+ka

-( k - 1 k-2 k-1 k-1

)

(

and we are able to find a block Bp+,U{x,+,} such that B, U{X,}EL,,, B,U{X,}E Lc2,. . . , B,,, U { ~ , + , } E ~ is ~a , collection of disjoint blocks which do not intersect F or G. This implies we can extend the above collection in such a way that e l = B , ~ { x l } E L c , e2=B2U{x2}ELC2,.. , . , e S = B S U { x S } ~ Lisc , a collection of disjoint blocks which do not intersect F or G. But then .TrU{el, e , , . . . ,e , } u { F U { y } } U { G U { z } } l { cc,2, , .. . , c , } / { b ) is a collection of disjoint blocks in T with a cardinality greater than contradiction and so our assumption that

[(

u - ( kk+-l

1+ t

T. This

is a

ka )/kI+l

blocks of L’ can intersect b E P is false. We have shown that at most v -(k + l ) t + k a

k-1 blocks of L’ can intersect any x, E P and that at most

(

+l)t+ ka

u -(k

k-1

blocks of L’ can intersect an edge in p. These facts yield the inequality,

We now seek to relate inequalities (1) and (2). By substituting a for (r-a) for \PI in inequality (1) we see that v - (k + 1)f)

(

k

(;) ( v

s -

- ( k + 1)t)

k-1

+

(t-a)(2k2-k) ( k - 1)

(

V-(k+l)tk-2

and 1

+ l)f)(v - ( k + l ) t - k + 1) s a(u - ( k + 1 ) t )+ ( t - a ) ( 2 k 3 - k’), ( u - ( k + l ) t ) ( u - ( k + I ) ? - k + 1)-r(2k3- k 2 ) s a ( u - ( k + 1)r - 2 k 3 + (V

- (k

k2)

D.E. Woolbright

210

and finally, (U- ( k U S

+ l)r)(u - ( k + 1)t- k + 1 ) -

t(2k3- k2)

(3)

(u-(k+ l)t-2k3+k2)

By substituting u -(k

t

for

lcyl+lpl

+ 1)t+ ka

i

into inequality ( 2 ) we have 1 u - l ( k + 1)t+ k a

k-1

k

c - ( k + I ) t + k a - k + l s r , and, a s

( k + 2)t - u + k - 1 k

(4)

Combining (3) and ( 4 ) we have

( k + 2 ) t - ~ +k - l,(u k

/

( ( k+ 2 ) 1 -

t‘+

k

-

-(k+ l ) t ) ( ~ - (+ k 1)t-k+ 1)-t(2k3- k2) (u-(k+l)t-2k3+k2)

l)(u-(k

+ 1)r-2k3+ k 2 )

3k

( ~ - ( k +l ) t ) ( u - ( k + 1 ) t - k

+ 1 ) - t ( 2 k 4 - k”).

Multiplying and combining like terms we obtain the quadratic inequality below, 0>[k3

+ 3 k 2 + 4 k + 2)’

+ [ 5 k 3- k 2 - k

-

1 -(2k2+4k

+ 3)u]r

+[(k + l ) u 2 - ( 2 k 3 - l ) u + ( 2 k 4 - 3 k 3 + k2)]. ( 5 ) Obviously t is greater than or equal to the smallest root of inequality (5). In other words tz=[-b-(b2-3ac)”’]/2a where a = k 3 + 3 k 2 + 4 k + 2 , b = 5 k 3 - k 2 - k - 1 - ( 2 k 2 + 4 k + 3)u, and c = ( k + l ) u 2 - ( 2 k 3 - l ) u + ( 2 k 4 - 3 k 3 + k’). With a good bit of multiplying and combining like terms it is easy to show that b2-4ac

= U’

+ ( 8 k 6 + 4 k s - 4 k 4 -6k’

+ 6 k 2- 2 k - 2 ) ~ +(-8k7+ 13k6-10k’+11k4-5k2+2k+l).

Alternatively b2-4ac

=(u +4kh

+ 2 k 5 - 2 k 4 - 3 k 3 + 3 k 2 - k - 1)’

- ( 4 k 6 + 2 k 5 - 2 k 4 - 3 k 3 + 3 k 2 - k - 1)’

+ (-8k7 + 1 3 k 6 - 10k’+

1 l k 4 - 5 k 2+ 2 k

Since -(4k6+2ks-2k4-3k3-3k2-

k - 1)’

and (-8k7+ 13k6- 10ks+ l l k 4 - 5 k 2 + 2 k

+ 1)

are negative when k 2 2 we have b2- 4ac s (u + 4 k 6 + 2 k 5 - 2 k 4 - 3 k 3 + 3 k 2 - k - 1)2.

+ 1).

21 1

On rhe size of partial parallel classes in Sfeiner sysrems

Since t s [- b - ( b2 - 4ac)”’]/2a we have rs

-5k3

+ k 2 + k + 1 + ( 2 k 2 + 4 k+ 3 2 k 3 + 6 k 2 + 8 k+ 4 - [( u

) ~

+ 4 k6 + 2 k 5 - 2 k 4 - 3 k + 3 k 2 - k - l)’]’” 2 k 3 +6 k 2+ 8k + 4

13

( 2 k 2 + 4 k+ 2 ) ~ ( 4 k 6 + 2 k 5 - 2 k 4 + 2 k 3 + 2 k z -2k - 2 ) 2 k 3 + 6 k 2+ 8k + 4 2k3+ 6 k 2 + 8k + 4 -

IT I

= t 2 k2+

(k 2 + 2 k + I2) (L , ( 2k 3 k+l +

-

-5k2

9

+ 6 k - 1).

3. The results of Lindner and Phelps As stated in the introduction, C.C. Lindner and K.T. Phelps have shown that for any S ( k , k + 1, u ) Steiner System with u 3 k 4 + 3 k 3 + k 2 +1, there exists a partial parallel class containing at least ( u - k + l ) / ( k + 2) blocks of the system. As a corollary they were able to show that any triple system of order u 2 9 has a partial parallel class containing at least :(u - 1 ) blocks (except possibly for u = 15, 19, and 27). In the case of triple systems the results of this paper represent an improvement for all triple systems of order u s 141.

Reference [ I ] C.C. Lindner and K.T. Phelps, A note on partial parallel classes in Steiner systems, Discrete Math. 24 (1978) 109-112.


PART V

ISOMORPHISM PROBLEMS AND ENUMERATION OF STEINER SYSTEMS



ON CYCLIC STEINER 2-DESIGNS Marlene J. COLBOURN and Rudolf A. MATHON* Department of Computer Science, Uniuersify of Toronto, Toronto, Canada

1. Preliminaries and definitions For over a century, mathematicians have been investigating the existence and construction of block designs. One of the earliest and best known references is Kirkman’s school-girl problem posed in 1850 [K3]. The more general version of arranging n objects into triples such that each pair appears exactly once was posed by both Kirkman [K2] and Steiner [S3]. As Kirkman’s paper attracted very little attention [Cll], these systems have become known as Steiner triple systems (STS). A Steiner system S(t, k, u ) is a collection of k-subsets of a u-set such that every 1-subset of the u-set appears in exactly one of the k-subsets. Over the years, a vast amount of literature has been published concerning Steiner systems, as evidenced by the thorough bibliography of Doyen and Rosa [D7]. Herein, we survey the literature relevant to cyclic Steiner S(2, k, u ) systems. We shall partition our paper into three main sections: existence, enumeration and the determination of isomorphism. In preface, we give a brief historical overview and introduce required definitions and notation. It is appropriate to mention here some related investigations which are not discussed in this survey. The first notable exclusion is the study of cyclic Steiner systems with t > 2, such as Steiner quadruple systems which are S(3,4, u ) designs. The interested reader may consult [L2, P3]. As we will see later, the study of cyclic S(2, k, u ) designs is equivalent to examining difference families with A = 1 over cyclic groups. We have omitted all research concerning difference families which either have A > 1 or are represented by using non-cyclic groups. Pointers into this literature may be found in [B16, H1, H2, B1, Cl].

1.1. A n historical overview It is elementary to establish that a necessary condition for the existence of an S(2,3, u ) is that u = 1, 3 (mod 6). Kirkman [K2] and, later, Reiss [R3] established that this condition is also sufficient. It is straightforward to demonstrate that S(2,3,7) and S(2,3,9) are unique. The first is cyclic; the second is not. The systematic investigation of cyclic STS was initiated by Netto [ N l ] in 1893. He *Research supported by NSERC Grant A8651. 215

216

M.J. Colbourn, R . A . Marhon

demonstrated the existence of two infinite families of cyclic STS, including a cyclic S(2,3, 13) and S ( 2 , 3 , 15). One of Netto’s constructions was later generalized by Heffter [H7]. Denote by N ( u , k ) ( N C ( u , k)) the number of non-isomorphic (cyclic) S(2, k, u ) systems. At the turn of the century, it was shown that N(13,3) = 2 [ Z l , D2, B171; the unique cyclic S(2,3. 13) is that of Netto. Moore [M7] showed that for u > 13, N ( u , 3) ~ 2 Cole, . Cummings and White [C6, W2] determined that N(15;3) = 80. Hall and Swift [H4] later confirmed this. Netto constructed one cyclic S(2,3, 15); in fact, NC( 15,3) = 2. The existence question for cyclic S(2,3, u ) designs remained open until 1939. In that year, Peltesohn [ P l ] constructed cyclic S(2,3, u ) systems for all u = 1 , 3 (mod 6), u f 9. Continuing interest in these existence questions has involved restricted versions of the problem. In particular, Skolem [Sl, S2] examined an “integer partitioning” problem whose solutions correspond to cyclic STS. Various extensions of Skolem’s original work have been investigated by O’Keefe [Ol ] and Rosa [R4]. Very little is known about cyclic S(2, k, u ) , k >3. Hanani [H6] demonstrated that u = 1 . 4 (mod 12) is both a necessary and sufficient condition for the existence of an S(2,4, u ) : similarly u = 1 , s (mod 20) is both necessary and sufficient for the existence of an S ( 2 , 5 , u ) . Bose [B15] constructed an infinite family of cyclic S(2, k, u ) , k = 4, 5. Heretofore, existence was resolved only for relatively small values of u. The first asymptotic determination of NC(u, 3) was pursued by Bays [B6], who restricted his attention to the case when u is prime. He demonstrated that NC(u,3) tends to infinity when u is prime. Johnsen and Storer [ J l ] have since established an exponential lower bound for NC(u,3). Similar results are not known for k > 3. Some exact values for cyclic S(2, k, u ) , determined by Bays [B6, B9], Kaufmann [ K l ] and Colbourn [C3], will be presented later. In preparation for a more detailed survey, we next present the required definitions. 1.2. Definirions A 1-design r-(u, k, A ) , is a pair (V, B)where B is a collection of k-subsets called blocks of the u-set V. such that every 1-subset of V is contained in precisely A blocks of B.A balanced incomplete block design (BIBD) is a t-design with 1 = 2. A BIBD is said to be symmetric if u = b. A Sreiner system S(t, k, u ) is a t-design with A = 1. In this paper we are especially interested in S(2, k, 21) systems. It is elementary to show that for the existence of an S(2, k, u ) design it is necessary that u - I = O (mod k - 1) and c ( u - 1 )= 0 (mod k ( k - 1)). Two such systems (V,, B,)and (V,, B,) are isornorphic if and only if there exists a bijection f : V, + V, such that b E B, if and only if f ( b )E B,. A ( u , k, A ) (cyclic) difference set D = { d , ,. . . ,d k } is a collection of k residues

On cyclic Steiner 2-designs

217

modulo u such that for any residue x+ 0 (mod u ) the congruence di - di= x (mod u ) has exactly A solution pairs (4,4 ) with 4, diE D . Every (u, k, A ) difference set generates a cyclic symmetric BIBD, whose blocks are B ( i )= { d , + i, . . . , dk + i} (mod u ) , i = 0, . . . , u - 1. The difference set is often referred to as the starter or base block of the symmetric design. A (u, k, A ) difference family is a collection of such sets D,, . . . , O n each of cardinality k such that each residue x f 0 (mod u ) has exactly h solution pairs (di, dj) with di, diED,, for some m. A difference family is said to be planar or simple if A = 1. Each (u, k, A ) difference family generates a cyclic BIBD in the same manner as before. For example, the difference family (0,1,4), (0,2,7) generates the cyclic S(2,3,13) design with V={O, 1 , . . . , 12). Using this definition, an S(2,3, 15) design cannot be represented as a difference family. However, it is possible for the design to be generated by two starter blocks modulo 15, when one includes the five blocks generated by the extra starter block (0,5,10). There are two non-isomorphic cyclic S(2,3, 15) designs generated in this manner. For our purposes, we will call a S(2, k, u ) design cyclic if the design can be generated by m starter blocks modulo u, possibly with the extra starter block (0, m‘, 2m’, . . . ,( k - 1)m‘) where b = m u + m ’ , m’3) may have non-multiplier automorphisms. In fact, a stronger statement can be made. There exist isomorphic inequivalent circulants [El]. Let us return to cyclic designs. There exist cyclic STS which have non-multiplier automorphisms, e.g. the 2-transitive S(2,3,31). Despite the existence of these non-multiplier automorphisms no pair of inequivalent isomorphic designs is known. The Bays-Lambossy theorem [B9, L l ] which we prove next, guarantees that such a pair does not exist on a prime order.

Theorem 4.1 [B9, Part 111. Given 2 isomorphic cyclic structures on a prime number of elements, there exists a multiplier isomorphism transforming one to the other.

226

M.J. Colbourn, R.A. Marhon

Let C represent the cyclic group on u elements; S, the symmetric group. The metacyclic group M is the group of substitutions of the form x-ax + b (mod u ) , (a, u ) = 1, 0s b s ( u - 1). The normalizer of C in S is the group N , ( C ) = {s 1 s-'Cs = C, s E S}. It is easy to verify that N,(C) = M. Consider the following 4 systems of conjugates: AG={s-'Gs I SES},

BG = {n-'Gn 1 n E N s ( C ) } , DG={~-'CSIS-'C~ 13. This invariant is also insensitive. r sensitivity: size of largest class:

15

19

21

25

27

31

33

1

0.25

0.57

0.08

0.12

0.04

0.01

1

4

3

12

8

78

84

From a complexity standpoint this is a poor invariant; there is no known polynomial time algorithm for deciding whether one design is a subdesign of another. The corresponding problem for graphs is NP-complete [C7]. A further difficulty with subdesigns is that one must examine relatively large “pieces” of the design. A generalization called fragment analysis [G2] circumvents this difficulty. A specific instance of fragment analysis for STS will be discussed later. Gibbons employed another class of invariants, observing that invariants of the block intersection graph are in fact invariants of the design. Given a design D, we can define a series of block intersection graphs Gi,1 = 0 , . . . , k, defined as follows: The vertices of Gi are the blocks of D. Two vertices are adjacent if and only if the corresponding blocks contain exactly i elements in common. One effective invariant is to count the number of cliques of size c in G,; this will be referred to as (c, i)-clique analysis. Gibbons employed (4,O)-clique analysis with great success, indicating that the invariant is quite sensitive. For cyclic STS, c s 2 7 . (4, ())-clique analysis is a complete invariant. The complexity of this invariant is also appealing; an O(b4) algorithm for computing this invariant is immediate. When the design is transitive, we need only consider the number of cliques containing a particular element. Hence, an O(rb3)algorithm results. The practical difficulty, however, makes the invariant difficult to use. In a (4,O)-clique analysis of cyclic STS, we found the 2-transitive S(2,3, 15) contains 394 cliques, whereas one of the S ( 2 , 3 , 2 1 ) designs contains 24646 cliques. The number of cliques for relatively small values of u is enormous. Thus although the growth is


229

polynomial the computation is extremely expensive. Furthermore, it appears that in order to maintain high sensitivity, the size of cliques being examined must increase as a function of u. If this is indeed the case, the computation is extremely difficult from a complexity standpoint-it is, in fact, a special case of a #Pcomplete problem [Vl, Gl]. 4.2.2. Invariants for S(f, t f 1, u ) In 1913, White [Wl] introduced a method of distinguishing the two S(2,3, 13) designs. Given a STS D, consider a triad (x, y, z) which is not in D. (x, y, z) is transformed by replacing each pair (x, y), (x, z), (y, z) by the single element with which it appears in D. Another triad results. For example, let D contain the three triples (1,2,4), (1,3,5), (2,3,6); the triad (1,2,3) will be transformed into (4, 5 , 6 ) . If one continuously repeats this operation, one of two things must occur. Either a triad of D is encountered or a previous triad is again reached. For simplicity, White refers to triads of D as one term cycles. Hence, every triad not in D initiates a frain of triples which terminates in a periodic cycle. Examining these trains allowed White to differentiate the two S(2,3, 13) designs. Clearly, two isomorphic designs will have the same train structure. However, the converse is not known to be true. In order to determine a design's train structure, each triad must be examined once. Hence, an O(u3)algorithm results. White proposed this invariant simply for STS; however, the obvious extension allows one to investigate trains for S(t, f + 1, u ) designs in O(u'+')time. Another invariant introduced to distinguish STS is the graph of interlacing. (We will use the term cycle structure.) Cole [C5], Cummings [ClO], Hall and Swift [H4] employed this invariant to distinguish STS of small orders; we describe it here in a more general setting. For a given S ( f, f + 1, u ) Steiner system D = ( V, B ) , consider any set A c V such that IAl= t - 1. For convenience let A ={x,, . . . , x * - ~ } .We define a graph GA to be G(VA, EA) where

VA = V- A

and E A

= {(a,b )

I a, b E VA,

. . . ,x,- 1, a, b ) E B}.

(XI,

This graph is a 1-factor. Given D, consider two sets of elements A = { x l , . . . , x , - ~ } and C = {x,, . . . , x ~ - x~l }, . We define GA = (V,, EA)and G, = (V,, E,) as above. We now define the union of two such graphs GA U Gc to be G(V', E', L ) where v ' = VA n v C - { x

I (XI,.

..

9

XI, X ) E

B}

and

I

E' = {(a, b ) a, b E V', (a, b ) E EA or (a, b ) E Ec}

and L is a mapping of edges to labels. L(a, b) = A if (a, b ) E E A . Because every t-tuple must appear exactly once in D, each element x in V' appears once in a block with the set A and once with the set B. Hence, GA UG, is regular of degree 2; it is therefore a union of cycles.

M.J. Colbourn. R.A. Mathon

230

A compact notation for this graph is just the list of cycle lengths in ascending order. This is called the cycle list for the pair of ( t - 1)-sets A and C.Consider the cycle lists for every pair of ( t - 1)-sets, which have t-2 elements in common. This collection of lists. when ordered lexicographically, is called the cycle structure. For cyclic STS one only has to consider the cycle lists for the pairs (0, i), 1=sic ;(u - 1). For SQS, this invariant has been used by Phelps [P3] to distinguish the 29 S(3,4,20)designs. For STS, Petrenyuk and Petrenyuk [P2] define an equivalent invariant, T tables. They pose t h e question of determining the sensitivity of this invariant. In the case of cyclic STS, cycle structure is quite sensitive, although not complete. , sensitivity is approximately 0.9. For L' ~ 4 5 its

L'

sensitivity: F i x

15

19

21

25

27

31

33

37

39

43

45

I

1

0.714

0.917

I

0.913

0.940

0.911

0.915

0.901

0.911

1

1

3

2

1

4

5

of the

largest class:

50

36

464

S17

There is an elementary O(u3)algorithm for computing this invariant for STS. (O(u2)for cyclic STS). Its high sensitivity guarantees the existence of many classes containing a single design. It has the added attraction that even for classes containing more than one design, a subexponential isomorphism algorithm based on cycle structure can be employed to differentiate the designs [C3]. In order to appreciate the effectiveness of cycle structure, it is worthwhile to examine the diversity of the encountered cycle structures. Let us, therefore, consider potential cycle lists. Clearly, every cycle has even length 5 4. Hence, the number of potential cycle lists is exactly the number of partitions of ( u - 3 ) into even integers >2. In our studies we found that almost all potential cycle lists actually appear in a cyclic STS:

Li

31

33

31

39

43

4s

85.3

85.4

89.4

89.9

92.7

89.7

% of potential cycle lists which

appear in cyclic STS

It appears from this table that the asymptotic growth rate of the number of cycle lists agrees with that of the partitions. This rapid growth, together with the many ways of combining cycle lists to form cycle structures accounts, in some sense, for the sensitivity of cycle structure. It would be interesting to establish that asymptotically all potential cycle lists are realized in an STS. This would provide strong evidence that cycle structure remains sensitive for higher orders.


23 1

4.2.3. Invariants for Steiner systems In this section we consider arbitrary Steiner S(t, k, u ) systems. The invariants from the previous section are applicable only when k - t = 1. However, when this is not the case we can still define the graph GA, / A [ =t - 1. GA is a collection of disjoint ( k - t + 1)-cliques. We may again define the labelled graph GA U G,, as before. Any invariant of this graph is an invariant of the pair of ( 1 - 1)-sets A and C. For a given invariant Z, let Z(A, C) denote the value of Z on GA U GC An invariant of the design is the multiset {Z(A, C) 1 ( A l = t - 1, ICJ=t - 1, ( An CI= t-2, A c V, C C v>. One can see that cycle structure is an invariant of this form. Let us consider a specific graph GAU Gc When k - t > 1 there does not appear to be a convenient notation €or the graph (i.e. one similar to cycle lists). We will, therefore, examine weaker invariants of the graph. Let X be the ( k - t + 1)-clique common to both GA and G,. Initially, we arbitrarily order the ( k - t f 1)-cliques of GA- X. Now each ( k - t + 1)-clique K in G, - X can be represented as a ( k - t + 1)-set S ( K ) ; if u belongs to the ith clique of GA - X and u E K, i E S ( K ) . Observe that for u, w E K, vf w, v and w belong to different cliques in GA - X . Hence, S(K) is a ( k - t + 1)-set. For a given i, consider the k - t + 1 sets S ( K , ) ,. . . , S(Kk-,+,)which contain i. From this collection form T ( K j )= S ( K , ) - { i } . Now T(K,), . . . , T(Kk-r+,)form the edges of a ( k - 1)-uniform hypergraph, which we will denote Hi and call an ouerlap graph. Any invariant of the collection {Hi} is an invariant of GA UG,. Each overlap graph Hihas the same number of edges, so this invariant would result in no discrimination. However, they may have a different number of vertices. With this in mind, we define the overlap list of GA U G,, OL(A, C), to be the multiset {I V(Hi)l}.The overlap list is clearly invariant under isomorphism. The overlap structure of a design is the multiset {OL(A, C ) 1 JAl=r - 1, ICI= r - 1, IA n C J =t-2, A c V, C c

v}.

A seemingly more powerful invariant can be defined by enumerating all ( k - t)-uniform hypergraphs with (k - t + 1) edges and arbitrarily ordering them 1 through m. For such a hypergraph H denote by # ( H ) its index in this list. The The typed typed overlap list of GA U Gc,TOL(A, C), is the multiset {#(Hi)}. ouerlap structure is the obvious analogue of overlap structure. We will illustrate the above definitions by an example. Consider an S(2,4,40). Let A = ( 0 ) and C = { 1). GA is composed of disjoint triangles, similarly G,. One numbers all triangles of GA, excluding the triangle containing 1, one through r - 1. A given triangle K of G,, excluding the triangle containing 0, is then represented as a triple S ( K ) of numbers from one through r - 1. We have r - 1 such triples. Consider l G i G ( r - 1). Let S ( K , ) , S ( K , ) , S ( K 3 ) be the triples containing i. Deleting i from each, we obtain three 2-sets T ( K , ) , T ( K , ) , T ( K 3 ) . These form the edges of the 2-uniform overlap graph Hi. In determining the

232

M.1. Colbourn. R.A. Mothon

overlap list OL(A, C ) we examine only the number of vertices in each Hi, 1s i s ( r - 1). In computing the typed overlap list TOL(A, C), we first enumerate the eight multigraphs with three edges, number them 1 through 8, and represent each overlap-graph Hiby its index in the list. With respect to ease of computation there is an efficient algorithm for computing this invariant. Furthermore, in our investigations of cyclic designs we found the invariant to be extremely sensitive. In fact, for cyclic S(2,4,u ) , u s 6 4 , the overlap structure distinguishes all designs. Considering typed overlap lists, it is interesting to note that in the cyclic S(2,4, u ) designs investigated, all possible , structure is again 3-edge multigraphs appear. For cyclic S(2,5, u ) , u ~ 6 5overlap complete. To our knowledge, this invariant has not previously appeared in the literature.

4.3. Concluding remarks

Our investigation of cyclic S(2, k, u ) designs has indicated that overlap structure is a very sensitive invariant. Applying such an invariant to a catalogue of designs aids in distinguishing them. Isomorphism testing may, nevertheless, be required if the invariant is not complete. When u is prime, however, this is not the case, since the Bays-Lambossy theorem guarantees that we need only decide equivalence (via an elementary polynomial time algorithm). Any non-trivial generalization of this theorem could potentially obviate much of this isomorphism testing. Alspach and Parsons’ theorem suggests an appropriate first step: the resolution of the case when u is a product of distinct primes.

5. Open problems

We recall here open problems mentioned in the text 5.1. Existence

(1) Find other infinite families of cyclic S(2, k, u) designs for k 2 4. (2) Show that the necessary conditions for a cyclic S(2,4, u ) design are sufficient for all u b u ’ , u’ = 37. In other words, show that these necessary conditions are sufficient for all but a finite number of orders. (3) Find the smallest u‘ as above for any fixed k >4.

S.2. Enumeration (1) Does N C ( u , k ) grow exponentially for every fixed k?

(2) Can Skolem’s partitioning problems be generalized to yield cyclic S(2, k , u ) designs, k = 4,5?


233

5.3. Isomorphism (1) Do there exist isomorphic inequivalent cyclic S(r, k, u ) designs? (2) Generalize the Bays-Lambossy theorem to non-prime orders. (3) Can isomorphism of arbitrary Steiner systems be decided in subexponential time? (4) Given an arbitrary design, what is the complexity of deciding whether it is cyclic? ( 5 ) What is the asymptotic sensitivity of various invariants such as trains, cycle structure and overlap structure?

6. References [A l l V.E. Alekseev, Skolem method of constructing cyclic Steiner triple systems, Math. Notes 2 (1967) 571-576. [A21 B. Alspach and T.D. Parsons, Isomorphism of circulant graphs and digraphs, Discrete Math. 25 (1979) 97-108. [Bl ] L. Babai, On the isomorphism problem, unpublished. [B2] L.D. Baumert, Cyclic Difference Sets, Lecture Notes in Mathematics No. 182 (Springer-Verlag, Berlin, 1971). [B3] S. Bays, Sur les systkmes cycliques de triples de Steiner, C.R. Acad. Sci. Paris, SCrie A 165 (1917) 543-545. [B4] S. Bays, Sur les systemes cycliques de triples de Steiner, C.R. Acad. Sci. Paris, Strie A 171 (1920) 1363-1365. [BS] S. Bays, Sur les systtrnes cycliques de triples de Steiner, C.R. Acad. Sci. Paris, Strie A 175 (1922) 936-939. [B6] S. Bays, Recherche des systtmes cycliques de triples de Steiner difftrents pour N premier (ou puissance de nombre premier) de la forme 6n + 1, J. Math. Pures Appl. (9) 2 (1923) 73-98. [R7] S. Bays, Sur les systbmes cycliques de triples de Steiner, Ann. Sci. Ecole Norm. Sup. (3) 40 (1923) 55-96. [B8] S. Rays, Sur les systemes cycliques de triples de Steiner difftrents pour N premier (ou puissance de nombre premier) de la forme 6n + 1, Ann. Fac. Sci. Univ. Toulouse (3) 17 (1925) 23-61. [B9] S. Bays, Sur les systtmes cycliques de triples de Steiner diffkrents pour N premier (ou puissance de nombre premier) de la forme 6n + 1, I, Comment. Math. Helv. 2 (1930) 294-305. 11-111, Comment Math. Helv. 3 (1931) 22-41. IV-V, Comment Math. Helv. 3 (1931) 122-147. VI, Comment Math. Helv. 3 (1931) 307-325. [R10] S. Bays, Sur les systbmes cycliques de triples de Steiner differents pour N premier de la formes 6n + 1, Comment Math. Helv. 4 (1932) 183-194. [Bl I] S. Bays, Sur le nombre de systtmes cycliques de triples differents pour chaque classe W . Actes SOC.Helv. Sci. Nat. 116 (1935) 275-276. [Bl2] S. Bays, Sur les systtmes de caracteristiques appartenant a d = 3, Actes SOC.Helv. Sci. Nat. 116 (1935) 276-277. [B13] T. Beyer and A. Proskurowski, Symmetries in the graph coding problem, Proceedings of the Annual Northwest ACM/CIPS Conference (1976). [B 141 K.S. Booth and C.J. Colbourn, Problems polynomially equivalent to graph isomorphism, Technical Report CS-77/04, Dept. of Computer Science, University of Waterloo (1979). [B15] R.C. Bose, On t h e construction of balanced incomplete block designs, Ann. Eugenics 9 (1939) 353-399. [B16] R.H. Bruck, Difference sets in a finite group, Trans. Amer. Math. SOC.78 (1955) 464481. [B17] G. Brunel. Sur les deux systtmes de triads de treize elements, J. Math. Pures. Appl. 7 (1901) 305-330.

234


[Cl ] P. Camion. Difference sets in elementary abelian groups SCminaire de Mathtmatiques Suprieures, Montreal (1979). [C2] C.J. Colbourn and R.C. Read, Orderly algorithms for graph generation, Int. J. Comp. Math., to appear. [C3] M.J. Colbourn, An analysis technique for Steiner triple systems, Proceedings of Tenth Southeastern Conference on Combinatorin, Graph Theory and Computing (1979). [C4] M.J. Colbourn and C.J. Colbourn, Concerning the complexity of deciding isomorphism of block designs, submitted tor publication. [CS] F.N. Cole, The triad systems of thirteen letters, Trans. Amer. Math. SOC. 14 (1913) 1-5. [C6] F.N. Cole, L.D. Cummings and H.S. White, The complete enumeration of triad systems in 1.5 elements, Proc. Nat. Acad. Sci. U S A . 3 (1917) 197-199. [C7] S.A. Cook, The complexity of theorem-proving procedures, Proceedings of the Third ACM Symposium on the Theory of Computing (1971) 51-58. [C8] D.G. Corneil, Graph isomorphism, Technical Report 18. Dept. of Computer Science, University of Toronto (1968). [C9] D.G. Corneil, Recent results on the graph isomorphism problem, Proceedings of the Eighth Manitoba Conference on Numerical Mathematics and Computing (1978). [ClO] L.D. Cummings, On a method of comparison for triple-systems, Trans. Amer. Math. SOC.1.5 (1914) 311-327. [Cl I ] L.D. Cummings. An undervalued Kirkman paper, Bull. Amer. Math. SOC.24 (1918) 336-339. [Dl] R.O. Davies, On Langford's problem, Math. Gaz. 4 3 (1959) 253-255. ID21 V. DePasquale. Sue sistemi ternari di 13 elementi, Rend. R. ist Lombard0 Sci. e lett. 32 (1x99) I! 13-221. [D3] D.Z. Djokovic, Isomorphism problem for a special class of graphs, Acta Math. Acad. Sci. Hung. 21 (1970) 267-270. [D4] J. Doyen, On the number of non-isomorphic Steiner systems S(2. rn, n ) , in: Combinatorial Structures and their Applications, Proceedings of the Calgary International Conference (1969) 63-64. [DS] J. Doyen. Sur la croissance du nombre de systemes triples de Steiner non isomorphes, J. Combinatorial Theory 8 (1970) 42-41. ID61 J. Doyen and G. Valette, On the number of non-isomorphic Steiner triple systems, Math. 2. 120 (1971) 178-192. [D7] J. Doyen and A. Rosa, An extended bibliography and survey of Steiner systems, Proceedings of the Seventh Manitoba Conference on Numerical Mathematics and Computing (1977) 297-361. [ E l ] B. Elspas and J . Turncr. Graphs with circulant adjacency matrices, J. Combinatorial Theory 9 (1970) 297-307. [FI] M. Fontet. private communications (1979). [G l ] M.R. Carey and D.S. Johnson, Computers and Intractability; a Guide to the Theory of NP-Completeness (Freeman, San Francisco, 197Y). [GZ] P.B. Gibbons, Conputing techniques for the construction and analysis of block designs, Technical Report 92, Dept. of Computer Science, University of Toronto (1976). [G3] P.B. Gibbons, R.A. Mathon and D.G. Corneil, Computing techniques for the construction and analysis of block designs, Utilitas Math. 11 (1977) 161-192. [H l ] M. Hall, Jr., A survey of difference sets, Proc. Amer. Math. SOC.7 (1956) 97.5-986. [HZ] M. Hall. Jr.. The Theory of Groups (Macmillan, New York, 1959). [H3] M. Hall, Jr., Combinatorial Theory (Wiley. New York, 1967). [H4] M. Hall, Jr. and J.D. Swift, Determination of Steiner triple systems of order 1.5. Math. Tables Aids Comput. 52 (1955) 146-1.52. [ H S ] H. Hanani. A note on Steiner triple systems, Math. Scand. 8 (1960) 154-156. [H6] H. Hanani. The existence and construction of balanced incomplete block designs. Ann. Math. Statist. 32 (1961) 361-386. [H7] L. Heffter. Ueber Tripelsysteme. Math. Ann. 49 (1897) 101-1 12. [J I] E.C.Johnsen and T. Storer, Combinatorial Structures in Loops: IV Steiner triple systems in neofields. Math. Z. 138 (1974) 1-14. [ K I ] P.B. Kaufmann, Studien iiber zyklische Dreiersysteme der Form N = 6n +3, InauguralDissertation der Math.-Natur. Fakultat der Universitat Frieburg in der Schweiz, Sarnen (1926).


235

[K2] T.P. Kirkman, On a problem in combinations, Cambridge and Dublin Math. J. 2 (1847) 191-204. [K3] T.P. Kirkman, query, Lady’s and Gentleman’s Diary 48 (1850). [Ll] P. Lambossy, Sur une manibre de diffkrencier les fonctions cycliques d’une forme donnte, Comment. Math. Helv. 3 (1931) 69-102. [L2] C.C. Lindner and A. Rosa, Steiner quadruple systems-a survey, Discrete Math. 21 (1978) 147-1 8 1. [Ml] A.A. Markov, A combinatorial problem, Problemy Kibernetiki 15 (1965). [M2] R.A. Mathon, Sample graphs for isomorphism testing, Proceedings of the Ninth Southeastern Conference on Combinatorics, Graph Theory and Computing (1978) 499-5 17. [M3] R.A. Mathon, A note on the graph isomorphism counting problem, Inf. Proc. Lett. 8 (1979) 131-132. [M4] B.D. McKay, Backtrack programming and the graph isomorphism problem, M.Sc. Thesis, Math. Dept., University of Melbourne (1976). [M5] G.L. Miller, On the nlogn isomorphism technique, Proceedings of the Tenth ACM Symposium on the Theory of Computing (1978) 51-58. [M6] W.H. Mills, The construction of balanced incomplete block designs, Proceedings of the Tenth Southeastern Conference on Combinatorin, Graph Theory and Computing (1979). [M7] E.H. Moore, Concerning triple systems, Math. Ann. 43 (1893) 271-285. [Nl] E. Netto, Zur Theorie der Tripelsysteme, Math. Ann. 42 (1893) 143-152. [Ol] E.S. O’Keefe, Verification of a conjecture of Th. Skolem, Math. Scand. 9 (1961) 80-82. [Pl] R. Peltesohn, Eine Losiing der beiden Heffterschen Differenzenprobleme, Compositio Math. 6 (1939) 251-257. [P2] L.P. Petrenyuk and A.Y. Petrenyuk, An enumeration method for non-isomorphic combinatorial designs, Annals of Discrete Math. 7, this volume. [P3] K.T. Phelps, On cyclic Steiner systems S(3,4,20), Annals of Discrete Math. 7. [Rl] R.C. Read, Every one a winner, Annals of Discrete Math. 2 (1978) 107-120. [R2] R.C. Read and D.G. Corneil, The graph isomorphism disease, J. Graph Theory 1 (1977) 339-363. [R3] M. Reiss, Uber eine Steinersche combinatorische Aufgabe, welche im 45sten Bande dieses Journals, Seite 181, gestellt worden ist, J. reine angew. Math. 56 (1859) 326344. [R4] A. Rosa, Poznamka o cyklickych Steinerovych systemoch trojic, Math. Fyz. Cas. 16 (1966) 285-290. [Sl] Th. Skolem, On certain distributions of integers in pairs with given differences, Math. Scand. 5 (1957) 57-68. [S2] Th. Skolem, Some remarks on the triple systems of Steiner, Math. Scand. 6 (1958) 273-280. [S3] J. Steiner, Combinatorische Aufgabe, J. reine angew. Math. 45 (1853) 181-182. [Tl] J. Turner, Point-symmetric graphs with a prime number of points, J. Combinatorial Theory 3 (1967) 136145. [Vl] L.G. Valiant, The complexity of computing the permanent, Theor. Comp. Sci. 8 (1979) 189-20 1. [Wl] H.S. White, Triple-systems as transformations and their paths among triads, Trans. Amer. Math. SOC.14 (1913) 6-13. [W2] H.S. White, F.N. Cole and L.D. Cummings, Complete classification of the triad systems on fifteen elements, Memoirs Nat. Acad. Sci. U.S.A. 14, 2nd memoir (1919) 1-89. [W3] R.M. Wilson, Nonisomorphic Steiner triple systems, Math. Z. 135 (1974) 303-313. [Zl] K. Zulauf, Uber Tripelsysteme von 13 Elementen, Dissertation Giessen, Wintersche Buchdruckerei Darmstadt (1897).

Summary of the groups of cyclic S(2.3, v )

MULT

AUT

N(r)

c

7

1

168

21

13

1

39

39

15

1

1

60 20 160

60 60

1 2 1

19 57 171

19 57 171

1

1

21 47 I26 504 882 1oox

21 42 126 63 63 126

25

12

25

25

27

8

27

27

31

63 15 1 1

31 93 465 9999360

31 93 465 155

33

7x 3 2 1

33 66 165 330

33 66 330

37

777 42 I

37 111 333

37 111 333

39

730 4 55 2 4 2 1

39 78 117 156 234 468 3042

39 78 117 1S6 234 468 117

43

9377 129 1 1

43 129 30 1 903

43 129 30 1 903

11616

45

45

19

21

1

2 1 1

4s

165

Legend. N ( u ) is the number of cyclic S(2.3, u) with the specified AUT and MULT. AUT is the order of the automorphism group. MULT is the order of the multiplier group. 236

C y c l l c steiner S ( 2 , 3 , v )

v =

%?sLgns,

v = 7 t.?.roiigL 7 ?

7

1 1 1

0

1

3

I

1

I

1

I

I

II

11 II

v = ?1 1

I1

2 11 3 11 4 5 6 7

11 11 11 11

0

1 7 1

3

1

1

n

0 0 0 0

1 5 1 1 5 1

0 3

1

5

1

0

1 1

9 7

1 1

0

3

7

0

0

4

1 2 1

3

171

?

1 3 1

2 3 ?

1 0 1

-

D 0

5’11 5 ’ 3

I

3 ’ 1 7 1 5 1

1 3 1

0 3 3

5 1

3

U ’ 7 I

r I

3

1 1 ’ 7 1

7

‘ 1

I 1 I I I I I

II

50u

II II

21

II II I I II

120s

63

126 1 1

1

11 11 1I 126 1 1

1

12h “47 63 i? 4? !.?b

126

11

a

I I I I

7

II II II II

71 1 1

1

Legend. AUT is the order of the automorphism group, MULT is the order of the multiplier group, D, is the number of S ( 2 . 3 , u ) subdesigns.

I t

Y n

0

i;’

v = :> 11 I1

II

II II I I II

II II 11 II II

II II II 11 II I1 II

II v = 71

1 1

11 II 11

\ I II II

!I !I 1 1

I I II

m

n

O n cyclic Sreiner 2-designs

I-

239

240

m r(

3

m

.r

n

a

k3 .4


“ I

c

c

s

0 0 0 C 0 0

it c1

aa

s a j

as

J

fa

Y . J 3

r)


3 3

a

zi

r\i N

r ‘ r :r: c.” ri

PIP

24 1

r J ri

R

a

a

a

oooocooccoocooooooooooooooooooocYooo~O

zi

0 0

oooooooooo~oooooooooooooooooooooooooc

:Y

00

IJ N

I I I I

I I I I II II I I 11 I1 II II

I1

If II

I I II I I 11

I I I I I1 II 11 1 1 II 1 1 II

I I

ir

5

p

5 P

?

3

$ I

On cyclic Sreiner 2-designs

Cyclic s t e i n e r S ( 2 . 4 . v )

243

v = 17 t h r o u g h 5 2

Designs,

v = 13 111

0

1

0 0

3

3 2 4 1 1 3 2 4 1

0 0

4 9 1 5 1 4 3 6 3 7 1

0 0 0 0

1 1 1 1 1

0

2 2 2 2 2 ? 2

3

9 1

I

I

I

v = 37 111 2 1 1

3

0

7 1 7 2 5 1 ' 1 7 2 5 1

v = 40 111 2 1 1 3 1 1 4 1 1

5 1 1 6 1 1

711 8 1 1

9 1 1 1011

0

0 0 0 0 3

1 1 1 1

1 1 1 1 7 a 1 5 1 5 1 5 1 5 1

1 1 1 1 1

3 3 3 3 3

1

4 4 4 4 4

3 7 3 T 3 3 7 7 7 7

1 ) 1 I 1 1 1 1 1 1

0 0 0 0 0 0 0 0 0

1 1 1 I

7 7 7 7 8 S 9

2 2 2 2 2 3 1

4 4 4 U 5 5 5

1 1 1 I 1 1 1

O

I

S

I

0 0 0 0

2 2 7 3 3 1 2 2 7 3 3 1

0 0 0 0 0 0

4 1 8 2 ' 3 1 U I R 2 7 ( 0 4 7 8 2 9 1 0 4 1 8 2 3 1 0 4 2 0 3 0 ) 0 4 2 0 3 9 1 0 4 2 0 3 0 1 0 4 2 0 3 0 1 0 4 2 3 3 7 1 0 4 2 3 3 3 1 0 4 2 3 3 3 1 0 4 2 3 3 3 1 0 4 2 4 3 5 1 0 ' 4 2 4 3 5 1 0 4 2 4 3 5 1 0 a 2 4 3 5 1 ! I 4 1 6 3 5 I 0 4 1 6 3 5 1 0 4 1 6 3 5 1 0 4 1 6 3 5 I 0 4 1 8 3 7 1 0 4 1 8 3 7 ) 0 U 1 8 3 7 1 0 4 1 8 3 7 1 0 4 2 0 2 7 1 0 4 2 0 2 7 1 0 4 2 0 2 7 1 0 4 2 0 2 7 1 0 4 2 6 3 3 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

z

6 6 5 5 5 6 3 3 3 3

1 4 2 5 1 2 1 3 2 1 1 1 1 9 1 2 6 3 4 1 1 1 1 9 1 1 4 2 5 1 1 4 2 7 1 3 1 2 1 1 1 4 2 2 1 212:,1

b 6 6 5 6 6 0

? l 3 3 / ,71331 2 2 3 4 1 2 2 7 U 1 1 7 3 1 1 1 7 3 1 1 2 4 7 8 1

I I 1 1 I 1 I I

I

I

v = 49 111 2 1 1 3 1 1 4 1 1

5 6 7 0

1 1 1 1

1 1 1 1

9 1 1 1011 1111 1211 1311 1411 1511 1611

1711 1811 1911 '011

2111 2211 2311 2411 2511 2611 2711 2811 2911

0 0 0 C 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8 1 9 1

8 1

R I 8 1 R I

1

3

1 1 1 1 1 1 1 1 1 1 1 1 1

3 3 3 3 3 3 7 3 3 3 3 3 3

1

3

O

1 1 1 1 1

3 3 3 3 3 3 3 1 3

9 1

f

1 1 1

8 1 8 1 9 1

S l 8 1 R l 8 1

3 8 8 3

1 1 1 1

" 1 9 1 1

9 1

3 1

q I " J

9 1 ' I 9 1 9 1

0 0

3 1 9 3 2 1 0 3 2 h 3 3 1 0 9 1 9 3 7 1 0 3 2 6 3 3 1 0 9 2 1 3 6 1 0 0 2 2 3 7 1 0 9 2 1 3 6 1 6 z u 3 9 1 0 3 2 2 3 7 1 6 1 7 3 1 1 0 9 2 1 3 6 1 6 1 7 3 1 1 0 9 2 2 3 7 1 6 2 4 3 8 1 0 9 2 1 3 0 1 6 2 4 3 R 1 0 3 2 2 3 7 1 6 2 1 3 3 1 0 9 1 9 3 2 1 6 2 1 3 3 1 0 9 2 6 3 q 1 6 2 2 3 4 1 0 9 1 9 3 2 1 6 , 2 2 3 4 1 0 9 2 6 3 9 1 5 2 2 2 9 1 0 1 0 2 1 3 6 1 5 7 2 2 9 1 0 1 3 2 3 3 S 1 5 2 5 3 2 1 0 1 0 2 1 3 6 1 5 2 5 3 2 1 0 1 0 2 3 3 8 I 5 2 2 2 9 1 0 1 0 2 1 3 6 1 5 2 2 2 9 1 0 1 0 2 3 3 8 1 5 2 5 3 2 1 O l O 2 1 3 6 ( 5 2 5 3 2 1 0 1 0 2 3 3 8 1 5 1 7 3 5 1 0 1 0 2 1 3 4 1 5 1 7 3 5 1 0 1 0 2 5 3 8 1 5 1 ? 3 7 1 0 1 0 2 1 3 4 1 5 1 9 3 7 1 0 1 0 2 5 3 8 1 5 1 7 3 5 1 O f 0 2 1 3 4 1 0

244

M.J.Colbourn, R.A. Mothon

O n cyclic Steiner 2-designs

9 9 3 9

0 1 2 3

1 1 1 1

1 1 1 1

3 4 1 )

9511 36

11

9 7 1 ) QR11 9911 10011 10111 10211 103 j I lO4II 105 1 1 106(1 10711 108 1 1 10911 11011 11111 11211 11311 11U1i 11511 11611 11711 11811 11911 12011 12111 122 1 1 12311 12411 12511 12611 12711 '2811 12911 13011 171ll 13211 133 1 1 174 1 1 13511 136 1 1 137 1 1 13811 139 1 1 14011 1Ul

11

lu211 l u 3 1 1 l u o j j

14511 146 1 1 14711

14811 lU911

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

o

0 0 0 0 0 0 9

0 0

o

0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1

3 7 3 3 3 3

1 1 1 1 1 1

7 j 3 1 3 1 3 1 1 1 3 1 3 13 I 3 1 U 1 7 1 1 1 1 3 ' 4 1 3 l U l 3 1 4 1 3 1 4 1

1 1 1 1 1 1 1 3 14 I 1 3 1 1 1 1 1 3 14 I 1 1 1 4 1 1 3 1 4 1 1 3 1u 1 1 7 1 4 1 1 7 1 6 1 1 3 1 4 1 1 3 1 4 1 1 3 1 5 1 1 3 1 5 1 1 3 1 5 1 1 7 1 5 1 1 3 1 8 1 1 7 1 8 1 1 3 1 8 1 1 3 1 7 3 1 1 3 1 8 1 i 3 IR I 1 3 1 8 1 1 3 1 Q l 1 3 1 9 1 1 3 1 9 1 1 3 1 9 1 1 3 1 4 1 1 > 1 9 1 1 7 1 9 1 1 7 1 9 1 i 1 1 9 1 1 -4 20 I 1 3 29 I 1 7 2 3 1 1 7 20 I 1 23 I 1 3 2 0 1 1 7 20 I 1 3 2 0 1 1 3 21 I 1 3 2 1 1 1 3 2 1 1 7 3 2 1 1 1 3 2 1 1 1 3 21 I 1 7 2 1 1 1 3 2 1 1 1 3 3 7 1

0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

o

0 0 0 0 0 0 0 0 0

o

0

0 0

0

7 0 0 0 0

0 3 0 0 0 0 0 0

4 4

4

9 2 5 1 9 2 5 1 7 2 5 1

4 2 8 4 4 1

4 2 8 U U I 4 2 8 4 4 ) 4 2 5 4 4 1 4 9 1 2 1 4 9 3 2 1 4 9 3 2 1 4 9 3 2 1 4 2 1 3 0 1 U 7 1 7 0 1 U 21 30 I U 2 1 7 O I 4 21 4 U I 4 2 1 4 4 1 4 2 1 U U 1 4 21 u4 1 4 2 3 3 2 1 4 2 3 7 2 1 4 2 7 1 2 1 4 2 7 3 2 1 4 1 0 2 6 1 U 1 0 2 6 1 4 1 0 2 b ( 4 2 7 4 3 1 4 1 0 7 4 1 U 1 0 2 U ( 4 1 0 2 4 1 U 1 0 2 4 1 4 2 9 4 3 1 4 29 a i I 4 2 9 4 3 1 U 2 9 U 3 1 4 1 0 3 2 1 4 1 0 3 2 1 4 1 0 3 2 1 U 1 0 3 2 ( 4 2 1 4 3 1 4 2 1 4 3 1 u ? 1 4 3 1 u = i u 3 1 4 '1 37 I U 11 37 1 4 1 1 3 7 1 U '1 37 I 11 1 6 4 2 I U 1 6 4 2 1 U 16 U 2 1 4 1 6 4 2 1 u 1-4 1 7 I 4 1 3 1 9 1 U l 3 1 q 1 4 1 3 1 9 1 U 3 u u o 1 4 74 40 1 4 7 U U O j

4 3 4 4 0 1 U l l 4 1 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0

0 0

0 0 0 0 0 0

0 0 0 0 0 0

o

0

0 0 0 0 0 0 0 3

n 0 0

0 0 0 0 0 0 0 9 0

0 0 0 0 0 0

6 1 7 3 5 1 6 2 0 3 8 1 6 2 0 3 8 1 b 1 7 3 5 1 fi17351 6 2 0 3 8 1 6 2 0 3 8 I 6 1 6 3 1 1 6 l f i 3 1 1 6 2 4 3 9 1 6 2 4 3 9 1 5 1 2 3 4 1 5 1 2 3 4 1 5 20 42 I 5 2 3 4 2 1 6 16 31 1 6 1 6 3 1 1 6 2 4 3 9 1 6 2 4 39 I 5 1 2 3 U I 5 1 2 3 4 1 5 2 0 4 2 1 5 2 0 4 2 1 5 1 8 2 5 1 5 1 9 2 5 1 5 2 9 3 6 1 5 1 5 2 5 1 5 1 2 3 8 1 5 1 2 3 8 1 5 1 6 4 2 1 5 1 6 U 2 I 5 1 2 3 8 1 5 i ? 38 I 5 1 6 4 2 1 5 7 6 4 2 1 5 1 2 U l I 5 1 2 4 1 1 5 1 3 U 2 ( 5 1 3 4 2 1 5 1 2 4 1 1 5 1 2 4 1 1 5 1 3 4 2 1 5 1 3 4 2 1 5 17 U O 1 5 1 3 UO 1 5114411 5 14 41 I 5 13 40 I 5 1 3 U O I 5 1 U 41 I 5 1 4 4 1 1 1 7 38 I 5 7 2 3 8 1 5 1 6 U 2 I 5 1 6 4 2 1 S q ; ! 3 8 l 5 ' 2 38 I 5 1 6 4 2 ) 5 7 6 4 2 1 5 1 5 3 2 1

245

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0

0 0 0 0 0 0 0 0 0 0

o

0 0

0 0 0 0 0

0 0

o

0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

7 7 7 7 7 7 7 7 7 7 7 6 6 6 6 7 7 7 7 6 6 6 6 8 8 8 8 8 5 8

2 6 3 4 2 2 3 0 2 6 3 4 2 2 3 0 2fi34 2 2 3 0 1 2 6 3 4 I 1 9 2 7 1 2 9 3 7 1 1 9 2 7 1 2 9 3 7 1 1 6 2 U I 3 1 3 9 1 16 24 I 7 1 3 9 1 1 9 27 I 2 9 3 7 1 1 9 2 7 1 2 9 37 I 1 6 2 U I 3 1 3 9 1 1 6 2 U I 3 1 3 9 1 1 7 3 8 1 1 3 U O l 1 9 U O I 1 7 3 8 1 2 1 3 0 1 2 7 3 6 1 2 1 3 0 1

8 2 7 3 6 1

8 2 1 3 0 1 8 2 7 36 1 8 2 1 3 0 1 8 2 7 3 6 1 9 2 3 3 4 1 O Z U 3 5 1 9 2 3 3 U I 9 2 S 3 5 1 9 2 3 3 4 ) 9 2 U 3 5 ( 0 2 3 3 3 1 9 2 4 3 5 1 6 2 1 31 I 6 2 4 34 I 6 2 1 3 1 1 6 2U 3U I 6 2 1 31 I b 2 U 3 4 I 6 21 31 1 6 2 4 3 4 1 9 2 2 37 1

8 R 8 9 8 8 8 F

2 5 2 2 2 5 2 2 25 2 2 2 5 1 5

3 5 3 2 3 5 3 2 35 3 2 3 5 3 3

1 1 1 1

I 1 1 1

M.J.CoJbourn,R.A. Marhon

246 15011 lcl 1 1 152 1 1 15311 15411 15511 15611 157 1 1 158 1 1 153 1 1 160 11 151 1 1 16211 1F.311 16411 165 1 1 16611 16711 16811 163 1 1 17311 171 1 1 17211 171 1 1 17411 17511 17611 17711 17811 17711 1 R O I I 1 s l I I 1R?II 15311

?null l a 5 1 1 186 1 1 167 1 1 1nqII lSSII l ? O I I

1 9 l I l 13711 19711 17a11

19511 176 1 1 197 1 1 l?SII 1?911 200 1 1 20111

’0211 23311 20411 20511 23611 20711 309 1 1 20911

0

0 0 0 3 0 0 0 0

o

0 0

0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 3

1 3 2 4 1 3 79 1 7 2O 1 3 3 3 3 1 0

’

1 i

’ 1

1 1 1 1 1 1 1 1

1 1

1 1

I

o

3 3 0 J 3 32 I 1 3 2 1 3 3 2 1 3 3 2 1 3 32 I 3 3 2 1 3 7 2 ) 3 3 ? 1 3 33 I 3 3 2 1 3 33 1 3 3 2 1

0 0 0 0

7

70

0

0

0 0 0 0 0 0

1 3 3 2 1

0 0 0

3 3 ? 1

C

’ 1 1 1 1 1 1

’’ ’ i

0 0 0 0

1

0 0 0 0

0 0 0 0 0 0 0 0

1 3 3 1 3 37 I 3 33 1

1

1

0 0 0 0 0

0

I I

1 3 3 0 1 ’7

0

0 0 0 0 0 0

1

’

7 39

3 7 1 3 3 3

3 2 3 2 3 2 3 2 3 2 3 2 3 3 2 3 3 2 7 3 3 1 3 3 37 7 3“ 3 3 9 3 3 3 3 7 7

1 1 1 1 1 1 1 1 1 1 ~

I 1 1 1 1

1 1 1 3 1 S ( 1 3 3 3 1 ’ 4 Q 1

’

u

? I

1

4

? 1

1

4

3

I

’

4 4

3

1

‘ 1

4

’ 7 1 9 1

1

4

1

’ 4 1 4 U l U 1 4 1 U 1 4 1 u 1 4 1 4

1 1 1 1 )

’

0

0 0 3 3

0 3 3 3

1o 0 0 0

0 0 3

0 3 3 0 0

0 0 B 0 3 0 3 3

u r u 1

0 0

c

’ 7 4 1 4 l ’ u 14 I

0

4 ? 7 1

0 3 3

0 C

1

’

U 1 1 4 1 4 1 2 43 4 12 42 4 3 1 6 4 9 1 6 4 9 1 6 4 9 1 6 4 37 4 4 4 37 44 u 3 7 nu 4 3 7 44 U 1 1 37 4 1 1 3 7 u l l 1 7 4 1 1 3 7 4 16 27 4 1 6 2 7 U 1 6 7 7 4 1 6 2 7 4 16 27 4 1 6 2 7 11 1 6 2 7 4 1 6 2 7 U 16 4 2

j

r)

1 1

0 0 0 0 0 0 0 0

1 1 1 1

1 f

I I I 1 1 1

o

0 0 0 3 r)

I

0

1 1 1

1

0 0 0 3 0 0 3

I

0

J

1

I

u 2 6 3 7 1

0 0 0 0 0 0 0 0 0 3

U ? G 7 7 1

3

4 ’ 6 3 2 1

u ‘ F U 4 1 6 4 U 2 6 7 4 2 6 3

? I 1 1 7 1 7 1

U 2 6 3 7 (

4 2 6 3 7 1 4 2 6 3 7 1 4 2 6 3 7 1

~ 2 1 2 U ?1 2 2 4 2 1 ?? u 2 1 2 3 4 2 4 3 2 U ? 4 3 2 3 2 4 3 2

7 19 I 0 I 0 0 1 1 0 ) 0 1 3

4 2 4 3 2 1

2 1 4 3 2 ) ? l a 3 2 1 7 1 4 3 2 1 ? 111 3 2 I ? 19 77 I ? 1 3 3 7 (

7 2 2 2 2 2 2 7 2

1 9 19 2 1 2 1 2 1 2 1 2 2 7 2 2 2

3

3 0 0

0 0

1 7 1

D

I

0 3

37

2 2 2 2 3 3 3

9 7 9 9 0 3 0

1 1 1 1 1 1 1

2 22 7 0 I 2

0

3 3 b 1

0

3 0 0

0 0 0 0

5 1 8 5 19 5 13 5 1 4 6 1 4 6 2 3 6 2 3

3 2 1

14

I I I I I

6

6 19 6 21 6 23 5 15

32 I 32 3 2 3 2 4 1 4 1 32 32 ui 41 24

I 1 1 1 1

515241 5 5 5 5 5 S 5 S

3 0 3 3 3 0 3 9 1 4 24 1 4 2 4 1 5 2 4 1 5 2 U 7 0 .37 3 0 3 9 5 3n 40

1 1

I 1 1 I

I 1

I

5 3 9 4 0 1 5 15 2 4 I

5 1 5 2 4 1 5303’!1

5 5 5 5 5 5 5

3 3 3 ’ 3 1 4 7 U 1 4 2 4 1 5 2 4 1 5 2 4 3 0 3 9 3 3 3 9

1 ( 1 1 1 1 1

5 7 3 4 0 1 5 3 0 U O I 5 ’ 3 3 1 1 ci 1’7 31 I Z 2 3 35 I 5 2 1 3 5 1 5 1 7 3 1 1 5 1 ? 3 1 ( 5 2 3 3 5 1 5 2 1 3 5 1 6 2 2 2 9 ) 6 3 7 2 7 1 6 2 6 3 3 1 r. 2 6 3 3 1 6 2 2 29 I 6 2 2 2 3 1 6 2 6 3 3 1 6 26 3 3 I 5 ’ 1 3 7 1 5 1 1 3 7 1 5 1 7 4 3 1 S 1 7 4 3 I 5 1 1 3 7 1 5 1 1 3 7 1 5 1 7 4 3 1 5 1’ 4 7 I 1 , 1 7 2 8 1

0 0 0

6 1 6 4 0 1 6 1 5 39 I 6 1 6 40 I 0 1 0 2 1 3 4 1 0 1 0 2 5 3 8 1 O 1 0 2 1 3 4 ( 0 1 0 2 5 3 H I 0 10 2 1 34 I 0 1 3 2 5 38 I o 10 2 1 3 4 I 0 1 0 2 5 38 I 0 6 1 4 27 I 0 6 2 8 4 1 1 0 6 1 4 2 7 1 0 6 2 8 4 1 1 0 6 1 3 21 I 0 6 3 4 4 2 1 0 6 1 3 4 1 1 0 6 1 4 4 2 1 0 6 1 3 41 I 0 6 1 4 4 2 1 0 6 1 3 21 I 0 6 3 4 4 7 1 0 f, 1 4 2 7 I 0 6 2 8 4 1 1 0 6 1 4 2 7 1 0 6 3 8 4 1 1 0 6 1 3 2 1 1 0 0 3 4 4 2 1 0 6 1 7 4 1 1 0 6 1 4 4 2 1 0 6 1 3 4 1 1 0 6 1 4 4 2 1 0 6 1 3 2 1 1 0 6 3 U 4 2 I r) 6 1 5 2 2 1 0 F 1 3 40 I 0 6 ‘ 5 22 I 0 6331101 0 6 1 5 2 2 1 0 6 3 3 4 0 1 0 6 1 5 2 2 1 0 6 3 7 4 3 1 O l O 2 1 3 U I 0 1 0 2 5 3 9 1 0 7 0 2 1 3 4 1 0 19 3 5 39 I 0 1 0 -71 34 I 0 1 0 2 5 3 2 I 0 1 0 2 1 3 4 1 0 1 0 2 5 3* I 0 7 1 6 3 1 1 0 7 2 5 4 0 1 0 7 1 6 3 1 ) 0 7 2 5 4 0 1 0 7 1 b 3 1 1 0 7 2 5 4 0 1 0 7 1 6 3 1 1 4 7 2s 40 I 0 6 7 4 2 4 1


o

21311 27111 2121) 21311 210 I 1 21511 21611 21711 218 1 1 21911

0 0 0 0 0 0 0 0 0

22311

C

22111 22211 22311

0 0 C 0

22411

1 a z g 1 4 2 3 1 4 2 0 1 4 2 0 1 4 2 0 1 4 2 0 1 4 3 0 ! 4 3 6 1 4 36 1 4 3 6 1 U 3 6 1 5 1 2 1 5 1 2 1 5 7 2 1 5 1 2

i

o

1 1 4 1 1 1 1

0

1 ) 1 1 1 1 1

0 0

0 0 0 0

0 0 0 0 0 0 0

2 9 3 6 1 2 9 3 6 1 2 3 3 6 1 2 1 5 4 2 1 2 1542 1 2 1 5 4 3 1 2 1 5 4 2 1 7’21291 2 ? 1 29 I 2 2 2 3 0 1

2 2 2 3 0 1 2 2 2 2

2 2 2 2

3 0 3 3

2 2 3 3

8 9 1 1

1 1 1 1

o 0 0 0 0 0 0 0 0

3 0 0 0

0 0

5 5 5 5 5 5 5 5 5 5 5 3 3 3 3

241 1 7 2 9 2 6 3 7 2 6 3 7 1 7 2 8 1728 2 6 3 7 2 6 3 7 1 ? 2 3 1 2 27 3 1 4 2 7 1 4 2 1 7 3 0 1 7 3 0 1 7 3 0 1 7 3 3

1 1 1 1

I 1 1 1

I 1 1 1 1 1 1

o

~ , 3 i u i 1

0 0 0 0

6 5 6 6 6

0 0

0 0 0 0 0 0

0 0

1 4 2 4 3 1 4 1 1 4 2 U 3 1 U1 1 4 2

1 1 I

I ~ 1

h 3 1 u l 1 6 1 5 6 16 6 1 5 6 1 6 6 1 5 6 1 6 b 1 5 6 1 6

3 9 1 40 I 3 9 1 4 0 1 3 9 ) 4 0 1 3 9 1 U O 1

248

M.J. Colbourn, R.A. Marhon 4211

0

4111 4 4 1 1

0 0

4 5 1 1

0

4 6 1 1 4711

0 0 0

4 8 1 1

4311 50 1 1 5 1 1 1 5 2 1 ) 5311 5411 5511

5611 5711 5911 5 7 1 1

6011 6111 6211

0 0

o

0

0 0 0 0 0 0 0 0 0 0

6 3 1 1

o

6411

0

6511 6611

0

6711 6 8 1 1 6911 7011 7111 ’211 7311 74 1 1

7511 7611 7711 7811 7 9 J l S O 1 1

8111 8 2 1 1 8311 5411 8 S J I R 6 j l 8711 A811

8911 9011

9111 0211 9311 9411 9511 q 6 i l

971) 9311 ’3911

10011 10llI

0 0 0 0

o

0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

’

3 1 1 1

1 7 1 1 1

1 3 1 7 1 3 1 7 1 3 1 7 3 1 7 1 3 1 7 1 3 1 7 1 3 l7

’

1 1 1 1 1 1

I

0 3 0 0

0 0 0 0 9

1 1 7 1

o

3 1 7 1 1 3 1 3 1 1 3 1 9 1

3 0 0 0 0 0 0 0 0 0 0

i 7

1 1 q I

1 7 1 1 3 l 3 1 1 3 1 1 3 1 1 ? l 1 3 1 1 3 7 1 3 1 1 7 1 1 3 1 1 3 1 1 3 1 1 3 2 1 7 3

’

9 q 9 7 7 s 3 0 q 9 7 9 0 3 3

1 I 1 1 1 I 1 1 I 1 1 1 1 1 1

? ? ? I

1

1 3 ? 1 3 2 1 1 2 1 3 2 1 3 2 1 3 2

7 3 3 3 3 5

I 1 1 1

1 3 Z C . I

1 1 1 1 1 1 1

’

1 1 1 1 1

1 1 1 1

1 1

1 1 1 1

3 2 3 3 3 3 3

2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 ? C I 2 5 1 3 2 5 1 3 2 5 1 3 2 5 1 3 2 5 1 3 2 5 1 ? 2 5 I 72c.I 3 3 1 1 3 3 1 1 3 3 1 1 3 3 1 1 3 T ’ I 1 3 ’ 1 3 3 1 1 3 3 1 1 3 3 1 1

? 9 3 4 2 7 3 4 2 9 3 4 1 0 2 5 , 1 0 3 5 1 0 2 5 1 0 2 5 3 1 4 6 3 1 4 6

1 ) 1 ( 1 1 1 )

4 3 1 ~ 6 1

4 3 4 1 4 1 4 1 4 1 U 1 4 1 4 1 U 1

1 1 1 1 1 1 1 1 1

4 3 3 1 3 3 3 3 3

6 2 7 2 2 7 2 2 ?

1 1 1 1 ) 1 1 1 I

4 2 4 4 5 1

4 2 4 4 5 1

a24u51

0

4 2 U U 5 j 4 2 4 4 5 1 4 3 4 4 5 1

0 0 0

J 0

o 0 0 3 9 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 3

3 0

0 0 0

6 2 0 3 7 1 6 2 1 3 8 1 6 2 1 3 8 1

0 0

5 1 2 3 4 )

0 0 0 0

I

o

0

I 1

4 4 4 4 1 4 4 4 4

o

0 0 0 0 0 0 0 0 0 0 0

o

0 3 0 4 2 4 4 5 1 0 4 2 4 4 5 1 0 4 1 1 1 6 1 0 ~ 1 1 1 6 10 4 1 1 1 6 1 0 4 1 1 1 6 1 0 4 4 0 4 5 1 0 4 4 0 4 5 I 0 4 4 0 4 5 1 0 4 U O U 5 I 0 4 9 1 6 1 0 0 9 1 6 ) 0 4 9 1 6 ) 0 0 9 1 6 1 0 4 9 1 6 1 0 4 3 1 6 1 0 4 9 1 6 1 0 9 1 6 1

U

3

4 4 0 4 7 1

0

U U O U 7 j

0 0 0

4 4 0 4 7 1 U U O U 7 I

4 4 4 4 4 U 4 U 4 4 4 4 4

4 4 4 4 1 1 1 1 1 1 1 1 1

0 0 0 0 1 1 1 2 2 2 2 6 6

4 a 4 4 1 1 1 1 1 1 1 3 7

7 7 7 7 6 6 6 ? 9 7 9 3 3

1 I 1 1 1 I 1 I 1 1 1 1 1

3 0 0

0 0 0

0 0 3 0 0 0 0

5 5 5 5 5

1 2 2 1

2 3 3 2

3 4 4 3

4 5 5 4

( 1 1 1

1,714 I 5 2 3 4 5 1

5 5 5 5 5 5 5

2 1 1 1 1 3 3

3 5 5 7 7 3 3

4 7 2 2 2 4 4

5 7 7 7 7 0 0

1 1 1 1 1 1 1

5 3 0 4 2 1

5 5 5 5 5 5

3 1 1 1 1 3

3 5 5 7 7 0

4 2 2 2 2 4

2 7 7 7 7 0

1 1 1 1 1 1

5 7 0 4 0 (

5 5 6 ~ 6 6 6 5 6 6 6 6 6 6 0

3 0 4 2 1 3 0 4 2 1 1 4 3 1 1 1 ~ 3 2 7 4 4 1 7 7 U U I 1 4 3 1 ) 1 4 31 I 2 7 4 4 ) 2 7 4 4 1 1 7 3 8 1 1 7 3 9 1 2 3 3 7 1 2 0 3 ’ 1 2 0 4 1 1

6 2 0 U l I

0 6 6 5 6

2 2 1 1 2

1 1 7 7 3

3 3 3 3 3

8 S 8 9 7

1 1 1 1 1

6 2 0 3 7 )

6 2 0 U 1 1 6 2 9 4 1 1 6 2 1 3 8 1 6 3 ’ 3 8 1

6 6 6 5 5 5 5 S 5

2 0 3 5 2 0 1 5 2 3 3 8 1 4 3 4 1 4 3 4 2 3 4 3 73113 ? l U 7 1 1 4 3

1 1 1 1 1 1 1 I 1

0

0 0 0 0 0 0

0 0

o

0 0 0 0 0 0 0 0 0 0 0

o

0 0 0 0 0 3 1o 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0

0 0 0

7 7 7 5 8 R 8 9 8 q 8 6 6 6 6 6 b 6

1 9 4 3 1 6 4 1 9 4 7 1 9 2 S 3 2 4 1 l 9 2 q 3 2 4 1 1 9 2 A 7 2 4 1 i 9 2 s 3 2 4 1 1 4 2 3 1 5 4 4 1 4 4 3 1 5 4 4 l U U l l 5 4 4 1 4 2 3

1 0 ~ 1 I 1 1 1 1

I l 1 1 1 1 1 I 1 1

6 7 5 4 4 1

6 6 6 6 6 6 6 6 9 10 9 9 9 3 9 3 8 R 8 R 8 8 8 R 8

1 4 2 3 1 3 5 4 4 1 1 ~ ~ 1 5 4 U I 1 4 4 3 1 1 5 4 4 1 1 0 2 3 1 3 5 U U I 1 9 3 7 1 2 4 4 2 1 1 9 3 7 1 2 4 4 2 1 1 9 3 7 ) 2 4 42 I 1 9 3 7 ) 2 4 4 2 1 1 8 3 7 1 ? 3 4 2 ( 1 8 4 1 1 1 9 4 2 ( 1 8 3 7 1 2 3 4 2 1 1 8 4 1 1 1 9 4 2 I 1 8 3 7 1

S ? 3 4 ? I 8 1 8 4 1 1

8 1 9 4 2 ) 8 1 8 3 7 1 A 2 3 4 2 1 8 1 8 4 1 ) 8131121 8 1 8 2 7 1 R 3 3 4 2 1 8 3 3 4 2 1 6 1 6 4 1 1 6 1 7 4 2 1 6 1 6 4 1 1 6 1 7 4 2 1 7 1 5 2 5 1 7 3 4 4 4 )

3

1

On cyclic Sfeiner 2-designs 10211 1331) 104 1 1

0 0 0

io511

o

1061) 10711 lOSIJ 139 1 1 110 1 1 111 1 1 11211 11311 11411 1 l S I I 11611 11711 11811 11911 12011 12111 12211 123 1 1 lZ411 125II 12611 12711 12811 127 1 1 13011 13111 13211 133 I I 134(11 13511 13611 137 1 1 138 I I 13911 14011 141 1 1 1421( 14311 144 1 1 14511 14611

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

14711 14811

lUSIJ 15011 15111 15211 15311

15U 11 155 1 1 15611 15711

158II 15s 1 1 160 1 1 1 h l I I

C

0 0 0 0 C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

o

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 3 3 1 1 1 3 3 1 1 1 3 31 I 1 3 3 1 1 1 3 3 1 1 1 7 3 1 1 1 3 7 1 1 1 3 31 I 1 3 3 1 I 1 3 3 1 1 1 3 7 1 1 1 3 3 3 1 1 3 3 3 1 1 3 3 3 1 1 3 3 3 1 1 3 3 7 1 1 3 3 3 ) 1 3 3 3 1 1 3 3 3 1 1 3 3 3 1 1 3 3 3 1 1 3 7 3 1 1 3 3 7 1 1 3 3 4 1 1 3 7 4 1 1 3 3 4 1 1

1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

3 3 4 1 3 34 I 7 3 4 1

3 3 U 3 3 4 43 3 4 7 3 4 3 3 4 7 3 43 2 43 3 4 3 3 4 7 3 4c 3 4 5 3 4 5 3 45 4 1 2 4 1 2 4 1 2 4 1 2 4 1 2 4 1 2 U 1 7 4 1 2 4 7 0 4 20 4 23 4 2 3 U Z O 4 2 0 4 20 4 20 U 2 1

I 1

0 0

0

o

0 0 0 0 0 0 0 0 3 0 0 0 3 0 0 0 0 0 3 0 0 0 0 0 0 0 0

I

0

1 1 1

0

I I

0 0 9 0 3 0 0 0

1 1

I 1 1

I 1 1 1 1 1 1 I 1 1

I I 1 I 1

1 1 ]

0 0

o

0

0 0 0 0

4 4 4 4 4 4

1 6 1 6 73 2 3 2 3 2 3

3 3 1 3 3 1 40

1

4 0 1 4 0 1 4 0 1

0 0 0 0 0 9

4 3 7 4 4 1

0

4 3744 1 4 3 7 4 4 1 4 3 7 4 4 I

0

6 1 6 1 6 1 1 1 1 ) 1 1 1 1 2 ) 2 1 2 1 3 1 5 1 5 1 5 )

0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 3 3

q U 5 1

I)

4 4 0 4 5 1

4 4 4 4 4 4 4 4 4 4 3 4 4 4 4

1 1 1 1 1 1 1 1 1 1 4 3 1 4 3 3 4 3 1 4 3 2 5 4 2 5 4 2 5 4 254 1 0 4 7 4 9 4

4 9 4 5 4 11 47 4 1 1 4 7 4 1 1 4 7 U 1 1 4 7 4 2 2 2’) 4 2 2 2 9 4 2 2 2 7 4 7 2 2 9 U 2 7 34 4 2 7 34 4 2 7 3 4 $ 7 7 3 4 4 16 37 I r 1 6 3 7 u 1 6 3 7 4 19 40 3 3 3 0 2 2

1

3

I

0 0 3 0 0

1 1 I

I 1 1 1

1

I 1 1

I 1 1

I 1

9 3 3 1

7 2 9 2 3 4 2 2 4

3 0 1 3 0 1

0 5 I 4 5 J

0

2 2 4 4 5 1

0 9 0 0 0 0 0 0 0 0

274451 2 1 4 3 1 j 2 14 23 I 2 14 7 3 I 211,231 2 3 1 4 3 1 2 3 1 4 0 ) 2 31 40 1 2 31 40 I 2 1 6 2 U J

3

0 0 0 0

0 0 0 0 0 0

o

0

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

5 1 4 5 1 4 5 11 b i i

249 4 6 1 4 6 1 43 I u 3 l

0 0 0

o

0 5 1 4 4 6 1 0 5 1 4 3 4 1 0 5 1 4 3 4 1 0 5 2343 I 0 5 2 3 4 3 1 0 6 2 3 3 9 1 0 6 1 4 3 5 1 0 f i 1 4 3 5 1 0 6 2 3 4 4 1 0 5 1 1 2 3 1 0 5 1 1 2 3 1 0 5 3 4 4 6 1 0 5 3 3 4 6 1 0 5 1 1 2 3 1 0 5 1 , 2 3 1 0 5 3 4 4 6 1 0 5 34U6 I 0 6 2 3 4 4 1 0 6 2 3 3 0 1 0 620101 0 6 2 9 3 8 1 0 ~ 2 8 3 8 ]0 5 20 39 I 0 6 2 0 3 0 1 0 6 2 q 3 9 1 0 6 2 f 3 3 3 1 0 5 16 2 4 I 0 5 1 6 2 4 1 0 5 3 3 4 1 1 0 5 3 3 4 1 1 0 5 If, 24 1 0 5 16 2 4 I 0 5 3 3 4 1 1 0 5 3 3 4 1 1 0 5 22 28 I 0 5 2 2 2 8 1 0 5 2 7 3 S ( 0 5 27 3 5 I 0 ~ 2 3 3 ~n 5 2 3 3 4 1 0 5 2 1 3 7 1 0 5 2 3 3 7 1 0 5 2 0 3 4 1 0 5 2 3 3 4 1 0 5 2 3 3 7 1 0 5 2 3 3 7 1 0 5 1 1 3 5 1 0 5 11 35 I 0 5 22 46 J 0 5 2 2 4 6 1 0 5 1 1 3 5 1 0 5 ‘ 1 3 5 1 0 5 2 2 46 I 0 ‘5 2 2 40 I 0 5 1 3 2 3 1 0 5 1 9 U h I

7 7 7 7 7 7 6 6 6 b 8 9 9 9 7 7 7 7 7 7 7

1 5 2 5 1 3 4 4 4 1 1 5 25 1 3 4 4 4 1 1 5 2 5 1 3 4 4 4 1 1 6 I r l l 1 7 4 2 I 1 6 4 1 I 1 7 4 2 I 3 3 4 2 1 2 4 3 4 1 2 7 3 7 1 2 7 3 7 1 1 5 4 3 1 1 6 4 4 1 1 5 4 3 1 1 6 4 4 1 1 5 4 3 1 1 6 4 U 1 1 5 4 3 1 7 1 6 1111 I 9 2 7 3 7 1 8 2 3 3 5 1 A 2 5 3 7 1 9 2 3 3 5 1 8 2 5 3 7 1 R 2 3 35 I 8 2 5 3 7 1 A 2 3 3 5 1 8 2 5 3 7 1 6 2 0 37 I 6 2 1 3 8 ) 6 2 0 3 7 ) 6 2 1 3 8 1 6 2 0 37 I 6 2 1 38 I 6 2 0 3 7 1 6 2 1 3 8 1 9 20 34 I 9 2 7 4 1 1 9 2 7 4 1 1 9 ? 7 41 I 16 1 6 3 3 1 6 2 5 4 7 ) 6 1 6 1 3 1 6 2 5 4 2 1 6 1 6 3 3 1

6 6 6 7 7

2 1 2 1

7

15 3 4 1 5 3 4 15 34 1 5

7 7 7 7 7 6

5 6 5 5

4 3 4 2

2 7 7 5

1 1 1 1

34 44

I

25 I 4 4 1 2 5 1 4 4 1 25 I 44

I

2 5 1

250

M.J. Colbourn, R.A. Mathon

25 1

O n cyclic Sreiner 2-designs

7 e s i g n s , v = 2 1 t!tr3ugh 6 5

C y c l i c 7t.einer ?(?,b,v)

v = 21 1

11

u

1

0 0

1 3 3 1 4 5 j 1 3 3 1 4 5 1

14 I t

I

I

0

I

v = 41

v = 61

v = 65

1 1 1 2 1 1

0

0

4 1 0 1 9 5 7 1 u 1 0 1 3 5 7 1

0 D

5 1 b U 1 4 R 1 5 2 2 ? 7 5 4 1

I

I

,

f

r

U I

Pi

I

P.

.4

,

c.

r

Y

n

252

x

x

x

X

Y

x x x x

x

C

I

r

M.J. Colboum, R.A. Mathon

X

x

X

x

I

X

x

X

Y

Y

I

c

x x

V K X C

x

x

I I I

N I

7

a a

01

Y

x Y c ’ P Y V X

On cydic Sreiner 2-designs

II

lr

I1

N

r

t

c

m m r c

rn:

253


Annals of Discrete vathematics 7 (1980) 255-264 @ North-Holland Publishing Company

NON-ISOMORPHIC REVERSE STEINER TRIPLE SYSTEMS OF ORDER 19 R.H.F. DENNISTON Deparrmenf of Mafhematics, Uniuersiry of Leicesfer, Leicesfer LEI 7RH, England

1. Introduction It has been known since 1917 that, up to isomorphism, there are just 80 Steiner triple systems of order 15. The next admissible order for such systems is 19; but the corresponding problem is too big to be feasible, and it may not be without interest if a restricted form of it can be solved. One severe restriction is to require a system to have what we may call a “reversal”, namely an involutory automorphism with only one fixed point. Triple systems with reversals have in fact been given the name of reverse Steiner triple systems, and have recently been the subject of various papers, beginning with [ 13 and [4].The present paper establishes that the maximum number of non-isomorphic reverse systems of order 19 is 184.

2. Uniqueness of reversal

Theorem 1. If a Steiner triple system of order u has two different reversals, then u is divisible by 3 . Proof. Let r and s be the fixed points of different reversals p and a. Then points x and y correspond in p if and only if rxy is a line; so s must be distinct from r, since otherwise a would be the same as p. Let R be the group of automorphisms generated by p and a: one orbit under R consists of r, s, and the point r collinear with r and s. Let x be any other point. Then a p x and papx are in line with r ; using a, we find that px and a p p x are in line with t ; using p, we find that x is in line with s and papapx, and therefore coincides with a p p p x . So we see that, in the orbit to which x belongs, there are just six points. That is, the group R partitions the set of u points into orbits, each of cardinality three or six, and t h e Theorem is proved. We may observe that apa is a reversal with t as fixed point, and that R is naturally isomorphic with the group of permutations of { r , s, t}. An affine space of order three and any number of dimensions has, of course, as many reversals as it has points. I have found it easy to construct, on 33 points, a system with at least 255

256

R.H.F. Dennisron

three reversals; and I conjecture that such a system exists whenever u is congruent to 3 or 9 modulo 24.

3. Projections and coverings Let S be a reverse Steiner triple system of order 19. We suppose that S has its points so named that its reversal is the permutation

Then we may construct a design D on nine varieties 1 , 2 , . . . , 9 , by taking in turn each pair of corresponding lines (not through 0) of S, and simply dropping the primes. If, for instance, 1’2‘3’, 1”2”3“, 1’2”4‘, l”2’4” are lines of S, then 123 and 124 are blocks of D. We naturally call D “the projection” of S ; we see at once that, given any two of the nine varieties, we can find just two distinct blocks of D to which they belong. So the projection D is a balanced incomplete block design without repeated blocks, and its parameters are given by v=9,

b=24,

k=3, r=8,

h=2.

The situation would be complicated if a system could have more than one reversal, and so more than one projection: but Theorem 1 reassures us that this will never happen with a system of order 19. We can therefore see that the projections of two isomorphic reverse systems will be isomorphic ( 9 , 2 4 , 3 , 8 , 2 ) designs. So,if we are to solve the problem of non-isomorphic reverse systems, we should begin with the problem of non-isomorphic (9,24,3,8,2)-designs without repeated blocks. After that, we should consider the problem of setting up “a covering” for a given design D - namely, a reverse system S which has D as its projection. We shall see that there is a design which has no covering, but that other designs can be covered in 213 or 214 ways. The appropriate point of view will be that the varieties of D are numbered 1 , 2 , . . . , 9 in a fixed way, and that the labelling 0, l‘, . . . .9“ of the points of the covering S is required to be consistent with this; also that different arrangements into triads of the symbols 0. l‘, . . . , 9 ” are to be regarded as “distinct” coverings. Suppose, then, that S is a covering of 0,and that we interchange the two symbols 1’ and 1” wherever they occur in S. This will give a covering of D, distinct from S according to the convention just made, but isomorphic with S as a Steiner system. We might say that we have “re-primed” the variety 1 ; and we could go o n to re-prime other varieties of D. However, the result of re-priming all nine varieties would be a reverse system which, by definition, coincided exactly with S, On the other hand, we shall certainly get a covering distinct from S if we re-prime any proper subset of ( 1 , . . . ,9}. To prove this, we have only to consider

Non-isomorphic rewrse Steiner triple systems of order 19

251

the effect on the four lines 1’2’3’, 1”2”3”, 1’24‘, l”2‘4“ of any re-priming that includes 1 but not 2-and to remember that any two varieties of D will be covered by four lines in essentially this way. It follows that, by taking S as equivalent to any covering obtained from S by re-priming, we partition the set of all coverings of D into equivalence classes (“prime classes”, shall we say), each of cardinality 2’ exactly. Two coverings of D, if they are in the same prime class, are isomorphic as reverse Steiner systems. Now let p be an automorphism of the given design D. Then we can transform any covering S into an isomorphic reverse Steiner system, p*S say, by making the obvious permutation of the points: if, for instance, 1 4 2 in p, then 1 ’ 4 2 ‘ and 1”+2” in p * , while 0 is fixed. Moreover, p* preceded by the re-priming of 1, and p* followed by the re-priming of 2, will in this instance have the same effect: we can assert that p will permute the prime classes in a well-defined way. If p happens to fix the prime class of a given covering S, we can get an automorphism of S if we follow p* with a suitable re-priming: and conversely. We shall see that, when a design D has coverings, a non-trivial automorphism of D never fixes all t h e prime classes, though it may fix a quarter of them. The conclusion is that the automorphism group of a design D will act as a group of permutations on the set of prime classes of coverings of D. If the prime class of a covering S is in the same orbit as that of a covering S*, then the reverse Steiner systems S and S* are isomorphic. Conversely, we remarked above that isomorphic systems have isomorphic projections: if we think concretely of two isomorphic coverings of the same design, we see that their prime classes are in the same orbit. Or, if we are given two isomorphic reverse systems that are specified in some other way, we can label their points so that they will have the same design as projection - and then, over that design, they will be in the same prime class (or, at least, in two prime classes in the same orbit). We can now see how the investigation should go. First, we exhibit a maximal set of non-isomorphic (9,24,3,8,2)-designs without repeated blocks, each design having its automorphism group determined. Then, for each of these designs, we find the number of prime classes, and sort these into orbits under the automorphism group (by looking for cases where an automorphism can fix a class). The total number of orbits will be the required number of non-isomorphic reverse Steiner systems.

4. A set of designs A solution to the problem of non-isomorphic designs, obtained by means of a computer, has been published by Gibbons [2]. I had independently solved the problem by hand, and have now checked that my solution agrees with Gibbons’. I will write out the designs as I found them, with Gibbons’ Roman numbers in brackets.

258

R. H.F. Denniston

D,: 123 124 134 156 159 167 178 189 234 256 259 267 (I) 278 289 357 358 368 369 379 457 458 468 469 479 We do not need the automorphism group of D , (which is of order 80). D,: 123 129 134 145 156 167 178 189 234 246 257 258 (VII) 267 289 357 358 368 369 379 459 468 478 479 569 Group of order 2 generated by (12) (39) (48) (5) (67).

D,:

123 129 134 145 156 167 178 189 234 246 258 259 (IX) 267 278 357 358 368 369 379 457 468 479 489 569 Group is trivial. D,:

123 129 134 145 156 167 178 189 235 246 247 257 (HI) 268 289 348 359 367 369 378 458 469 479 568 579 Group of order 2 generated by (18) (2) (36)(45) (7) (9).

D,: 123 129 134 145 156 167 178 189 235 246 247 257 (IV) 268 289 348 358 367 369 379 459 469 478 568 579 Group is trivial. Dh: 123 129 134 145 156 167 178 189 234 246 257 259 (V) 268 278 357 358 368 369 379 458 467 479 489 569 Group is trivial. D,: 123 129 134 145 156 167 178 189 235 246 247 257 (VI) 268 289 348 358 367 369 379 459 468 479 569 578 Group of order 2 generated by (14) (28) (3) (5) (69) (7).

D,: 124 127 136 139 146 157 158 189 235 238 245 268 (X) 269 279 347 349 356 378 458 467 489 569 579 678 Group of order 6 generated by (123) (456) (789) and (14) (26) (35) (7) (89). D,: 126 127 134 139 145 158 168 179 237 (XI) 256 289 348 356 359 367 467 469 478 Group of order 8 generated by (1 2345678) (9).

238 245 249 578 579 689

Ill,,: 124 127 136 138 145 159 169 178 235 238 249 256 (XII) 267 289 346 349 357 379 457 468 478 568 589 679 Group of order 6 generated by (123456) (789). D l l : 123 126 139 147 148 156 157 189 234 247 258 259 (VIII) 268 279 345 358 367 369 378 456 469 489 579 678 Group of order 6 generated by ( I 23456) (789).

Non-isomorphic reverse Steiner triple systems of order 19

259

D12: 123 125 134 148 157 168 169 179 236 247 249 259 (XIII) 268 278 349 357 358 367 389 456 458 467 569 789 Group of order 18 generated by (123) (456) (789) and (174) (235689). D13: 126 129 137 139 147 148 156 158 237 238 246 248 (11) 257 259 346 349 356 358 457 459 678 679 689 789 Group of order 8 generated by (1234) (5) (67) (89) and (1) (24) (3) (5) (68) (79). 5. Construction of coverings

The problem of finding coverings, for a given design D, will be easier to handle if we set up some kind of arithmetic for it. Let us think again of the example at the beginning of Section 3, where 123 and 124 are blocks of D. We may express the assertion that each of the pairs { l’,2’) and { l”,2”) is collinear with a point that projects onto 3, while {l’,2”) and 11”. 2’) are collinear with points that project onto 4, by writing [l, 2; 31 = + 1,

[ 1,2; 41 = -1.

The symbol [p, q ; r ] is defined only when pqr is a block of the given design D. The following equations are satisfied when the symbols in them are defined:

[P, 4 ; rl = [q, p; rl. [ p , q ; r l . [p, q ; s I = -1

-

(1) ( r f s),

[P, 4 ; r l . [r, p; ql Eq, 7; PI= + I .

(2) (3)

In fact, we can see the truth of (3) by imagining that we have tossed three coins, and want to take one away so as to leave a head and a tail; there will be either two ways of doing this, or none. Equations (l),(2), (3) are not only necessary, but sufficient: if we have written down symbols [p, q; r ] , three for each block of D, that satisfy these equations, we have effectively set up a covering of D. A topologist might visualise the blocks of D as two-dimensional simplexes of a pseudomanifold, and might arrange a covering by setting up a one-dimensional cochain. However, a dual interpretation can be made in terms of graph theory. Let us consider the blocks of D as the vertices of a graph, two vertices being adjacent when the blocks have a pair of points in common [3]. Eleven of our designs will then have connected graphs; one exception is DI3,where the four vertices 789, 689, 679, 678 make up one component, and the other twenty make up a second. The graph of D, has three components: one consists of ten vertices 156, 167, 178, 189, 195, 256, 267, 278, 289, 295; another has ten vertices, and the third has four. The choice of a covering €or D will now correspond to the standard process of converting the graph into a directed graph (will “orientation” serve as a name for

260

R.H.F. Dennisron

this process?). Taking once more the example symbolized by the equations [ 1.2; 31 = + I and [ 1.2; 41 = - 1, we may say that the edge which represents {1,2} has been oriented, the head end being at the vertex which represents the block 123, and the tail end at 124. Equation (3) now tells us that, of the three edges through any given vertex, either all have their head ends there, or one has a head end and two have tail ends. A simple counting argument then shows that, in any component of such a directed graph, exactly a quarter of the vertices have the former property (of being at the head ends of all their edges). But this means that a graph. if ten of its vertices make up a component, cannot possibly be oriented according to t h e rule. We conclude that there is no reverse system that has D , as its projection. As it turns out. each of the other twelve designs has some coverings, and accordingly has a graph for which there is, shall we say, at least one “admissible orientation”. This fact, which we shall establish in the next section, does not seem easy to infer from the graph theory. However, it does seem obvious what happens when G. the graph of a design D, has been provided with two admissible orientations A , and AZ. Let C be the set of edges of G which we have to reverse in changing from A , to A,. Then at any vertex, since the number of head ends is odd for A , , and likewise for A,. there must be an even number of edges that belong to C.That is, C will be a cycle of G. Conversely, given any cycle C of G, and one admissible orientation A , , we can change from A , to another admissible orientation by reversing all the edges of C. It follows that a given graph with first Betri number v, i f it has any admissible orientations at all, will have exactly 2” of them To find v, we subtract 24, the number of vertices of any one of our graphs, from 36, the number of edges, and add the number of components: so v is 13 for each of the designs from D, to D,, inclusive, and 14 for DI3.According to section 3. the respective numbers of prime classes are 2’ and 2‘, for the coverings of these designs.

6. Construction of coverings: an effective procedure It remains to be seen how the symbols [ p , q ; r ] can be used to handle these large sets of coverings. Let us take the design D, by way of example. We may choose to say that a covering of D, is “standardised” when the lines through 7’ are 7’7”0, 7’1’2”. 7’2’9”, 7‘9’5”. 7’5’1’’, 7’4’6, 7‘6’8”, 7’8‘3’’, 7’3‘4“. It is in fact easy to see that, in any prime class of coverings of D,, there is one and only one standardised covering. For such a covering of D,, we see that [ 1 , 2 ; 71 = - 1; and it follows, by equation (2) of section 5 , that [ l , 2; 41 = + I . Likewise, [2,9;61 = + 1,

[ 5 , 9 ;61 = + 1,

[ 1 , 5 ; 81 = + 1,


26 1

Let us now give names to five (suitably chosen) of the other symbols; it turns out that a convenient set is a = [2,8; 61,

fi = [8, 9; 41,

6 = [3,9; 41,

= [l,9; 81.

y = [2,5; 31,

Then we should be able to find all the other symbols in terms of these. First, we use Eq. (2) again:

[2,8;3]=-a,

[8,9; 1]=-/3,

[3,9; 1]=-6,

[1,9;5]=-~.

[2,5;4]=-y,

Now, using Eq. (3) as well, we deduce from the equations [3,8; 21 = +1 and [2,8; 3]= -a that [2,3; 81 = -a, whence again [2,3; 51 = a. Likewise, we find successively that

[2,6; 81 = a, [ 5 , 6; 91 = a, P - 6 ; 51 = Y, [4,8;9]=-p6, [ 5 , 8 ; 11= B E , [2,4; 5]=-y&, [ 1,6; 41 = - Y ~ E ,

[2,6; 9]= - a ; [5,6;3]=-a, [3,6; 1]=-7, [4,8;5]=@; [5,8;4]=--P~; [2,4; 1 ] = ~ 6 ~ [ 1,6; 31 = y& ;

[6,9; 2]= -a, [3,5;2]=ay, [4,9;3]=6, [1,8;9]=-/3~, [4,5; 8]=-&, ;[1,4;2]=~6~, [ 1,3; 61 = - 6 ~ ,

[6,9; 5]= a ; [3,5;6]=-ay; [4,9; 8]= -6; [1,8;5]=@, [4,5; 2 ] = 6 ~ ; [1,4;6]=-yb; [ 1,3; 91 = SE.

In the course of this arithmetic, we have ensured that condition (3) is satisfied for all but one of the 24 blocks of D,. We now observe that

[3,9; 1]=-6,

[1,9;3]=-~,

[1,3;9]=6~;

so condition (3) is satisfied for the last block, and there is no doubt that the system exists. Each of the symbols a,p, y, S, E can take the values +1 and -1 independently. So we have effectively constructed 2’ coverings of D8,all of them standardised as above. No two of these are in the same prime class, and we can obtain from them, by re-priming, the complete set of 213 “distinct” coverings of D,. It turns out that the same algorithm works for each of the designs D,, . . . , DI3,except that D,, with its higher Betti number needs six Greek letters.

7. Effect of an automorphism

Our next task, according to the programme at the end of Section 3, is to determine how the automorphism group of a design acts on the set of prime classes of its coverings. In particular, we should consider how often a non-trivial automorphism will leave a prime class fixed.

262

R.H.F. Denniston

By way of example. suppose that C , a prime class of coverings of D8,is fixed by the automorphism (123) (456) (789). In C there is one system, S say, which is "standardised" as in Section 5. According to Section 3, we are concerned with the "obvious" transformation 1'+2', 1"+2", 2'+3', . . . , determined by our automorphism. This will take S to a system, 7-say, which by hypothesis is also in the class C. Let us choose one of the two re-primings that take T back to S ; and let us write nq= -1 if the re-priming interchanges q' and q", but nq= +1 if it does not. Now the standardised system S will certainly have 7'9'5" as one of its lines; this will correspond to a line 8'7'6 in T, and the re-priming will take this to one or the other of two lines 7'6'8" and 7"6"8' that belong to S. So definitely n,r6 = - 1. In terms of the symbols a. 0, y , 6, E that belong to S, and of its numbers [p, q ; r], we can work out necessary and sufficient conditions for the re-primed system to be a standardised one: [ 9 . 3 ; 11 = -6, [9, 1: 81 = E , [9,2: 6]= + I , [9.6; 51 = a, [9.4; 31 = 6, [ 9 , 5 ;71-1, [9,7:2]=-1, [9.8; 41 = p,

[7,1: 2]= + l ; [7,2;9]=+1; [7,3; 41 = + 1; [7 , 4 ;6] = + 1; [7,5; 1 ] = + 1 ; [7,6; 81 = + 1; [7,8; 3]= + I ; [7,9; 51 = + 1;

These equations determine the set of numbers 7rq (to within multiplication of the whole set through by -1). But we are assuming, not merely that the re-primed system is standardised, but that it actually coincides with S. So we need five more conditions:

This is as far as we need to go, because the standardised covering of D, is uniquely determined by its symbols a, p, y, 6, €.'The only restrictions on S arc thc last five, which in fact reduce to two independent conditions a = ps = E .

The conclusion is that (123) (456) (789) fixes just a quarter of t h e prime classes of coverings of D, (namely those whose standardised representatives satisfy these two conditions). The same arithmetic, applied to the involution (14) (26) (35) (7) (89) of D,, leads to an impossible condition a = -a. So this automorphism does not fix any


263

prime class. It turns out, likewise, that no prime class is fixed by any involution of any standard design (nor, consequently, by any automorphism of period 4 or 8).

8. Counting the systems Accordingly, we see in the case of D9that the automorphism (12345678) (9) must permute the 32 prime classes in four orbits of length 8; and this means that, up to automorphism, D9is the projection of only four reverse systems. Likewise, each of the designs D,, D4, D,, having an automorphism group of order two, is the projection of 16 systems. For D13, there are 64 prime classes, permuted by the eight automorphisms in eight orbits of length 8; so D,, is the projection of eight reverse systems. Going back to D,, we see that its automorphism group has two generators, one of order three fixing eight of the 32 prime classes, and one of order two fixing none. It follows that there are four orbits of length 2 and four of length 6. And therefore a maximal set of non-isomorphic coverings of D, will consist of eight reverse systems, four having automorphism groups of order 6 (which turn out to be cyclic), and four of order 2. The designs D,, and D,,, each of which has an automorphism group of order 6, give results analogous to those for D,. Each of these has eight nonisomorphic coverings, of which four have cyclic automorphism groups of order 6 . The design D,,, with a group of order 18, has two prime classes which are fixed by all its automorphisms of period 3 (but interchanged by all its involutions). Then there are six classes that are fixed by (123) (456) (789), but the automorphism group is transitive on this set of six; and likewise for (147) (258) (369). Finally, there is a set of 18 prime classes on which the group is sharply transitive. The conclusion is that D,, has four non-isomorphic coverings, with automorphism groups of orders 18, 6, 6, 2. These groups turn out to be Abelian, even though the group for D,, is not. To summarise, we have 32 non-isomorphic coverings for each of the designs that have trivial automorphism groups, namely D,, D,, and D,. We have 16 for each of D,, D4, D,; eight for each of D,, Dlo,D I 1DI3; , four for each of D, and D,,: but none for D,.

Theorem 3. Up to isomorphism, there are just 184 reuerse Steiner triple systems of order 19. One of these has an automorphism group of order 18 (Abelian, not cyclic), and fourteen have (cyclic) groups of order 6. For each of rhe others, the unique reversal and the identity are the only automorphisms. We may observe that the system with the group of order 18 is the one that was discovered by Rosa [4].

264

R.H.F. Denniston

References [ I ] J . Doyen, A note on reverse Steiner triple systems, Discrete Math. 1 (1972) 315-319. [2] P.B. Gibbons, Computing techniques for the construction and analysis of block designs, Ph.D. thesis, University of Toronto 1976 (=Technical Report no. 92, Department of Computer Science, University of Toronto, May 1976). [3] E.S. Kramer, Indecomposable triple systems, Discrete Math. 8 (1974) 173-180. [ I ]A. Rosa, On reverse Steiner triple systems, Discrete Math. 2 (1972) 61-71.


A N ENUMERATION METHOD FOR NOMSOMORPHIC

COMBINATORIAL DESIGNS L.P. PETRENJUK and A.J. PETRENJUK Department of Mathematics, Institufe of Agricultural Engineering, Kirovograd, USSR

In this paper we analyse the substance of an approach to the problem of the constructive enumeration of nonisomorphic combinatorial designs. This approach consists in constructing, by means of transformations described in Section 3, of a large set of designs with a given parameter set and their subsequent pairwise comparison for isomorphism. Distinguishing between nonisomorphic designs can be done by using invariants; some of them are described in Section 2. These transformations and invariants in combination with computers form a convenient machinery when searching for new nonisomorphic designs and trying to obtain estimates for their number.

1. Basic definitions and problems Let E be a set of cardinality r, P ( E ) be the collection of all subsets of E, and P,(E) the collection of all subsets of E with cardinality r, r s n . Define on P ( E ) a function n(P) taking on values in the set Z' of integers. To this function corresponds a collection B of subsets of E in which the subset (block) occurs n(P) times. Such a collection will be called a finite incidence system (FIS) based on E, and the function n(P) will be called the block multiplicity function of this system. The value n(P) will be called the multiplicity of the block p in the system B. A finite incidence system defines o n the set P ( E ) another function

called the weight function of the system B; its value p ( X ) is called the weight of the set X in the system B. In order to indicate that n(P) and p ( X ) is the block multiplicity function and the weight function of a FIS B, they are denoted by n, ( p ) and pB ( X ) , respectively. The function n(P) determines p ( X ) uniquely; however, given a function p ( X ) , the corresponding function n(P) (and consequently, the corresponding incidence system) may not exist. Necessary and sufficient conditions for its existence (which are quite complicated) were obtained in [2]. 265

266

L.P. Perreniuk, A.J. Pewenjuk

A problem arises here whose special cases have been studied extensively. Let Q c P ( E ) , and let a function p * ( X ) be defined on Q, with values in Z'. Under what conditions does there exist a FIS B such that its block multiplicity function n ( X ) equals zero provided I X l $ K , and the weight function on Q coincides with p * ( X ) ? Here K c { l , 2 , . . . , n } . If such a FIS exists, we will call it a ( K , p*, Q, E)-design. The formulated problem will be called the existence problem for a (K, p * , Q, E)-design. The case when Q = P , ( E ) is of greatest interest. It is natural to call a (K, p*, Q, €)-design with such a Q an I-weighted design [3]. If, in addition, p * ( X )= A =constant then we have an I-balanced design [3]. 1-balanced designs were considered first by Hanani [4] who denoted them by P ( K , I, A, n ) . If ( K ( = 1, i.e. the design consists of blocks of size k then an 1-weighted ( K , p * , 0, E)-design is called a tactical configuration and is denoted by C ( k , I, A, n). Thus, a tactical configuration C ( k , I, A, n ) is a collection of blocks P I , . . . , pb (some of them may be the same subsets of the set E) such that each block contains exactly k elements and every 1-subset of E is contained in exactly A blocks. Tactical configurations with 1 = 2 are called balanced incomplete block designs. Tactical configurations with 1 = k - 1, A = 1 are called Steiner systems; if k = 3 we have Steiner triple systems (STS), if k = 4 we have Steiner quadruple systems (SQS) etc. Tactical configurations C(2, 1, A, n ) are nothing else than undirected regular graphs of degree A. Other generalizations of tactical configurations which can be studied by methods described here were defined and investigated in [l], Two FISs B and B based on sets E and E, respectively, are said to be isomorphic if there exists a 1-1 correspondence 4 : E-E such that for each block P in B, @$ = 6 implies n,@) = ne(P). Isomorphic FISs differ only by names of elements, thus the distinction between them is unessential. Isomorphism between FISs is an equivalence relation. The enumeration problem for FISs is the following: determine (or, at least, find bounds for) the total number of equivalence classes under this relation. If one is interested not in the number of equivalence classes but in a list of their representatives (one from each class), the corresponding problem is that of a constructive enumeration of nonisomorphic FISs. A natural way to solve the problem of constructive enumeration is in the following: (1) one constructs a large set of ( K , p * , 0, E)-designs, and then (2) one shows that among them there is a certain number of pairwise nonisomorphic ones. As a result, one obtains a lower bound for the number of nonisomorphic designs of this kind. This is how a lower bound for the number of nonisomorphic SQS of order 16 wa5 obtained in [19], and the number of all nonisomorphic STS of order 19 with a

An enumeration method for nonisomorphic combinatorid designs

261

head was obtained in [13]. It is not an exaggeration to say that almost all known bounds for the number of nonisomorphic FISs of various kinds have been obtained by using this kind of an approach.

2. Methods of distinguishing nonisomorphic designs A description of several methods of constructing FISs with given properties can be found in the literature (see e.g., [20]). Often a tactical configuration can be constructed in several ways. How does one determine in such a case whether the obtained designs are isomorphic or not? This problem of distinguishing and identifying is meant to be solved (and in many cases is successfully solved) by invariants. The notion of an invariant is widely used in mathematics. In general it can be defined as follows [ 5 ] . Let D and R be sets, and let c and p be equivalence relations defined on D and R, respectively. If there is a 1-1 mapping f : D + R such that dlud, implies f(d,)pf(d,) for any dl, d,E D, then f is said t o be an invariant in the set D. It follows from this definition that if for the elements d l , d, E D the relation f(dl)pf(d2) does not hold then d,cd, cannot hold either. This principle forms the basis for distinguishing through invariants. Naturally, in order to justify actually using an invariant f to distinguish elements of D with respect to an equivalence c it is necessary that certain conditions be satisfied. The simplest of them are formulated in [ 5 ] . The essence of these conditions is that the complexity of computing f(d,) and f(d2)and of testing their equivalence under p should be much smaller than the complexity of a direct testing of the equivalence of d , and d, under u. To distinguish nonisomorphic FISs it is preferable to use invariants constructed according to the following principle. One determines a class of fragments of FISs, i.e. such a collection of parts of FISs which is mapped by isomorphisms into corresponding collections of parts of the images. Elements of the basic set E, blocks, collections of blocks, collections of elements, subsystems etc. can all serve as fragments. Denote by M ( B ) a collection of fragments of a given kind of a certain FIS B. With each fragment 4 E M ( B ) we associate a characteristic c ( 4 ) which must have the property that if under an isomorphism of two FISs B and B, the image of 4 E M ( B ) is 6 E M ( B ) , then c(4)= E(6).Here c and E are characteristics of the same kind defined in M ( B ) and M ( B ) , respectively. A characteristic of a fragment reflects either properties of its internal structure or those of its position in FIS, in particular, among other fragments. A specification of a set of fragments by their characteristics represents the mentioned invariant. In this case D is the collection of all FISs, R the set of the mentioned specifications, c the isomorphism of FISs and p the equality of specifications.

268

L.P. Petrenjuk, A.J. Pelrenjuk

If for two FISs the values of an invariant are computed and they are different then the two FISs are nonisomorphic. If the invariants are equal then, in general, the problem of distinguishing and identifying remains unsolved, and different methods must be employed (for instance, using a different invariant). Invariant f is complete in D if from f(d,)pf(d,) always follows d,ad2. The opposite end of this ideal situation represents a triuial invariant which maps all elements of D into the same equivalence class of p in R. Let D contain exactly d equivalence classes of u,and let the invariant f take on values representing exactly r equivalence classes of p in R. The sensitivity a ( f )of the invariant f is the ratio r/d. This quantity characterizes (in a first approximation) t h e distinguishing capability of the invariant f. The sensitivity of a complete invariant is maximal and equals 1 while the sensitivity of a trivial invariant is minimal. Let us describe now an invariant used for distinguishing STSs. The main notion w e will use is that of interlacing of two elements in an STS; this notion is encountered i n Cole [6] and Cummings [7]. Let A, be an STS of order n. For each element x of the basic set E and some element y E E form the following set of pairs of elements of E:

Let now x E E, y E E (x # y). The undirected graph r,, whose vertex set is the set E and whose edge set is I7: U I7: is said t o be the graph of interlucing of elements x, v in the system A,. It follows from the properties of STS that the graph I ; , is a union of disjoint cycles o f even lengths, with length of each cycle no less than 4. The symbol Cs',!, . . . , SLY) where si and ri are natural numbers, 2 s s 1< . . .< s,,,, r,s, = $ ( r ~ 3 ) , determines the type of inferlacing of elements x, y in A, if the graph r,, contains exactly ri cycles of length 2si. Clearly, the number of distinct types of interlacing for every order of STS does not exceed the number of partitions of i ( n - 3) into parts not less than 2. For example, when n = 19, the types of interlacing are: TI = (2"). T , = (2,, 41, T3= (2, 32),Z'; = ( 2 , 6 ) , T , = (3,5), T(, = 14,4), T , = ( 8 ) . For any n. the types of interlacings can be ordered lexicographically provided one treats the symbol S ' as a string of r letters s. Number the types of interlacing in lexicographic order: T , , Tz,. . . , T,,. Then to each element x E E we may assign a p-dimensional vector

where 1, is t h e number of elements of E having type of interlacing T, ( j = 1.2, . . . , p ) with x. Obviously, t , + t 2 + - . . + tp = n - 1. We call ( f , , t2, . . . , t,,) the vector-index of X in the STSA,. This vector-index plays the r61e of t h e characteristic, and the elements of the basic set E are fragments. After the vector-indices of all elements of E in the STSA, have been

269

A n enumeration method for nonisomorphic combinatorid designs

computed, we may assign to this STS a table

T(A,,)=

. ........ ... . ... ...t(k) @)

where in each row the number lh, ih > O , is the number of elements of E having (tih)',. . . , t:)) as their vector-index in the STS A,,. For the sake of uniqueness, the rows of the table are in increasing lexicographic order. Observe that I , + * + lk =

n. The obtained table represents a specification of elements of the set E by their vector-indices. We will call it invariant table (or T-table) of the system A,,. Below are two STS of order 19 obtained by the Skolem method [8], and their T-tables. One can see that the values of the invariant are distinct, consequently these STSs are nonisomorphic. STS No. 1 0 0 0 0 0 0

1 2 3 4 6 7

0 9 0 11 0 14 1 2 1 3 1 4

5 8 10 18 17 16 12 13 15 6 9 11

1 1 1 1 1 2 2 2 2 2 2 2

7 8 10 12 15 3 4 5 9 11 13 16

18 17 13 14 16 7 10 12 18 14 15

3 3 3 3 3 3 4 4 4 4 4

17

5

4 5 6 12 14 17 5

8 11 13 15 16 18 9 6 12 7 14 13 16 15 17 6 10

5 5

5 5 6 6 6 6 7 7 7 8

7 8 14 16 7 8 9 15 8 9 10 9

13 15 17 18 11 14 16 18 12 15 17 13

8 8 9 9 10 10 11 12 13

10 11 10 11 11 12 12 13 14

16 18 14 17 15 18 16 17 18

T-table of STS No. 1 10 0 0 6 2 0

101 191

The T-table is a complete invariant in the set of STS of order 13 and 15. For orders n 3 19 as of yet there is no information on the sensitivity of the T-table as an invariant. Other invariants for distinguishing STSs are described in [9]. A description of how to use invariants to obtain information about the internal structure of STS is also contained in [9]. To distinguish between SQSs, one could use an invariant of the following type. Let an SQS (denote it by P) be based on the set E = (1, . . . ,n}. For each a E E denote by aP/aa the STS consisting of all triples (x, y, z) of elements of the set E \{a} such that (a, x, y, Z ) E P. The system aP/aa is called the derived STS of P with respect to a.

L.P. Pefrenjuk, A.J. Petrenjuk

270 STS No. 2

~~

0 0 0 0 0

1 2 3 4 5 0 6 0 13 0 14 0 15 1 2 1 3 I 4

9 7 10 8 11 12 17 16 18 10 % 11

1 1 1

1 1 2 2 2 2 2 2 1

5 7 12 15 16 3 4 5 8 12 13 17

6 13 14 17 18 11 6 9 14 16 15 18

3 3 3 3 3 3 4 4

4 5 6 12 14 15 5 7 4 12 4 13 4 I6 5 8

9 7 18 13 17 16 10 18 15 14 17 18

5 5 5 6

6 6 6 6 7 7 7 7

12 13 14 7 8

17 16 15 15 13 9 16 10 14 11 17 8 16 9 14 10 17 1 1 12

8 8 8 9 9 9 10 10 11

9 10 11 10 I1 12 11 13 14

17 12 15 15

13 18 16 18 18

T-table of STS No. 2 I 8 I 1 2 2 11 3 10 4 6 5 6 6 0

6 1 0 2 3 2 I)

1 1 2 1 2 2 2 1 3 2 2 3 0 3

1 1 1 0 0 0 0

0 0 0 0 0 0 0

3 3 3 3 3 3 I

-

Consider the collection of tables

If the table 7‘’’ oxcurs among them a , times, T”’ a2 times etc. then the system P is assigned a symbol-specification

d ( P ) = u , T ” + a , T Z ’ +. . where the ”symbolic” summation is commutative. It is easy to see that d ( P ) is an invariant in the set of SQSs. W e will call it the derivative invariant (or derivant) of the system P. O n request o f one of the authors, S.G. Buzdugan has written a program to obtain derivants of SQSs of order 16. H e found 41 SQSs of order 16 having distinct invariants. This suggests good distinguishing possibilities of the derivant*. The following generalization of this invariant seems interesting. Let A be a tactical configuration C(k. I, A, n j based on E and let X c E, = t where 1 d t < 1. Denote by dA/dX the FIS based on E \ X and determined by the multiplicity function o f blocks

1x1

r t ( P ) = n*iP

u X).

* Editor’s romrnenf. The invariant d ( P ) does not distinguish the two nanisornorphic transitive SOS of order 16 based on PG(3.2) (see. e.g. P.B. Gibbons, R.A. Mathon, D.G. Corneil, Steiner quadruple \wtern\ o n 16 symbols. Proc. 6th S.-E. Conference Comhinatoria. Crraph Theory and Computing, H O C ~Raton. 1075. pp. 345-365).

An enumeration method for nonisomorphic cornbinatonal designs

271

The system dA/dX is a tactical configuration C(k - r, 1 - t, A, n - r) and is called derivative of A with respect to X. The system aP/aa considered above is a special case of such a derivative for 1 x1= 1. Let further I be an invariant on the set of tactical configurations C(k- r, 1 f, A, n - t ) which takes on values from a finite set { I , , . .. , I N } . Consider the collection of configurations {dA/dX,X c E, 1 x1 = 1). If for a1 among them the invariant I takes on the value I , for a2 the value I , etc. then the "symbolic" sum d,(A)=alIl+(~212+* * * +aNZN

is an invariant in the set of tactical configurations C(k, 1, A, n). Obviously a1+ . . .+a, =(Y). The invariant d,(A) will be called the (f, I)-derivant of the system A. The derivant d(P) is thus a (1,t)-derivant of the SQS P. The following invariant penetrates even more deeply into the structure of a tactical configuration. For each S, S c E, IS(= s, O < s < t, let there exist & ( S ) sets X, X c E, X =) S, 1 x1= r such that dA/dX = Zj ( j = 1,2,. ... N ) . Denote 4 ( S )= I?==, pj(S)Zi.If the subsets of S are considered fragments and 4 ( S ) their characteristics then the specification of the set P,(E) by 4 ( S ) is an invariant in the set of tactical configurations C(k, I, A, n ) . Call it (t, I),-derivant of the configuration A, and denote it by d:(A). The (f, I),-derivant can be represented by a table

K I p:"

dh(A)=

p:". . .

.................. p y ...

pi"'

where

E,

is the number of sets S, S c E, IS1 = s such that

4 ( s ) = p : " I , + p ~ ) ~ 2 " " + p ~ ) I N , Y = 1 , 2 , . .. ,w. The rows of the table are in increasing lexicographic order. Let us describe how to construct another invariant. Let B be a FIS based on E, and let b be the number of its blocks. For each of the (!) unordered collections of I blocks form the intersection vector (xl,x2.. . . . x), where is the number of pairs of blocks p, p' in the considered collection such that Ip n p'( = i ( i = 1,2, . . . . n). Form a table

where xy) + x y ' + . . . + x(+')= (,I, ' h , + . . + h, = (p), and h, is the number of collections of t blocks having as their intersection vector (x'i"', . . . ,x!,")), The rows of the table are in increasintg lexicographic order.

L.P. Petrenjuk, A.J. Petrenjuk

272

This table is also an isomorphism invariant in the set of FISs. Call it the rr -invariant.

Clearly, due to the complexity of computations, computing invariants like the ones described above without an aid of a computer is not feasible. Invariants based on the same principle are described and have been used successfully to distinguish nonisomorphic objects in [ 5 , 9 , 11-18].

3. Construction of weighted HSs by using mutually balanced collections of blocks Two FISs A and B both based on the set E are mutually balanced on

Q, Q = P(E) if PA

(x)= p B (x)

for all X E 0. Using mutually balanced FISs one can construct from a given (K, p * , Q, E)-design other (K, p*, Q, E)-designs that are, in general, not isomorphic t o the design one starts with. In order to show this let us introduce some relations and operations on FISs. We will write A s B provided n A ( P ) S n B ( P for ) all PEP(E). The sum A + B of two FISs based on E is a FIS with the block multiplicity function n A t L ) ( 6 )= nA (0) + nB (0). The difference A - B is defined to be the FIS whose block multiplicity function is

The intersection of two FISs A and B is defined to be the FIS A n B with the block multiplicity function nAnB

(PI = min (nA( P ) , nB(P))-

Theorem 1. Let H be a ( K , p*. 0, E)-design, A and B be FISs based on E and mutually balanced on Q, and A c H . Then (H - A ) + B is a ( K , p*, Q, El-design.

the summation here is over the set

(0 : 0 2 X).

Theorem 2. Let H and G be two (K,p*, Q, E)-designs. There exist FISs A and B based on E and mutually balanced on Q such that ( H - A ) + € ?= G.

A n enumeration method for nonisomorphic combinatorial designs

273

Proof. It is easy to see that A = H - G and B = G - H are just those FISs whose existence is asserted in the statement of the theorem. The transition from the FIS H to ( H - A ) + B performed in Theorem 1 is called a substitution in H of the collection of blocks A by a collection of blocks B that is mutually balanced with A on Q. The essence of the two theorems is in that we can obtain from a ( K , p*, Q, E)-design H any other (K, p*, Q, E)-design by an appropriate substitution. Observe that to achieve this end it suffices to consider only pairs of mutually balanced collections of blocks A, B such that A r l B = 9. A special case of the substitution of blocks is the d-transformation of graphs described in [101. For STS the notion of substitution of collections was introduced in [9], and in general form this notion appeared in [18]. The question arises, how to find pairs A, B of mutually balanced collections of blocks on Q such that one of these collections is contained in H? We will suggest a construction method for such pairs of collections for the case of tactical configurations C(k, I, 1, n ) . Let H be a configuration based on E and let x E E, y E E, x f y . Denote by aH/axI, the FIS obtained as a result of deleting from aH/dx of all blocks containing y . Clearly, dH/dx), and aH/ay, 1 are mutually balanced collections of blocks on P[-,(E). Consider a bipartite graph r,, whose vertices are the blocks of these collections, and edges are joining two blocks from different collections if they have an ( 1 - 1)-subset in common. Suppose this graph has m ( m5 1) components. Consider a subgraph r of the graph r,, which is a union of several components. Denote by M and N, respectively, the collection of all those blocks from aH/ax(, and aH/ayl,, respectively, that are vertices of the graph Clearly, the collection of blocks

r.

is contained in H , and the collection

N = { p : P = f l ’ U { y } , p ’ ~ M } U { p: P = P ’ U { X } , ~ ’ E N } is mutually balanced with on Pl(E). ( H - fi)+ N is a tactical configuration different from H. Let us call a substitution of collection of blocks performed according to the way described above an X-substitution. Observe that to determine in H a collection M and to construct the collection 13 that is mutually balanced with it depends on the choice of elements x , y and on the choice of the components that determine the graph r. In the degenerate case when r,, [email protected] described collections are undefined. Consider an example. Let H denote the STS No. 1 considered earlier. Write in columns the sets of pairs aH/a3 and dH/a5.

L.P. Petrenjuk, A.J. Petrenjuk

274

a~ i a 3

awas

5

11

3

11

12 1s 2 7 6 13

0 4 8 2 7 6

1 9 15 12 13 10

17 18

Below the horizontal line are the sets of pairs aH/a3Is and aH/d5)3.Between the columns is the diagram of the graph r3.5. If we take now for r the larger component of the graph r3.5 then the described method yields the following mutually balanced collections of triples: -

M 0 1 3 3 2 3

N

3 10 3 9 4 8 12 15 3 7 6 13

0

1

4 5 2 5

5 9 8 1s 5 12 7 13 6 10

5

5

0 5 10 1 5 9 4 5 8 5 12 15 2 5 7 5 6 13 0 1 3 3 4 9 3 8 15 2 3 12 3 7 13 3 6 10

Thus, we defined a collection of transformations each of which transforms the tactical configuration C ( k ,I, 1, n ) into a tactical configuration based on the same set and having the same parameter set but having its block structure different from that of the original configuration. If we take into account that on tactical configurations obtained from H we can perform X-substitutions of collections of blocks as well then it will become clear that this construction method can produce a large family of tactical configurations C(k,I, 1, n). Is it possible for any two classes of isomorphic configurations C(k, I, 1, n) to form a finite sequence of X-substitutions of mutually balanced collections of

An enumeration method for nonisomorphic combinatonal designs

275

blocks that transforms representative of one of these classes in some representative of the other class? This is the completeness problem of the constructed family of transformations. It can be formulated as follows. Two tactical configurations C(k, I, 1, n ) are reachable if there exists a sequence of X-substitutions which effects a transformation of one of them into a configuration isomorphic to the other one. Reachability is an equivalence on the set of configurations C(k, I, 1, n ) and the question is, what is the number r = r ( k , I, 1, n) of its equivalence classes. If there is only one such class then knowing a single tactical configuration C(k, I, 1, n ) one can obtain through X-substitutions a family containing representatives of all isomorphism classes of the set of tactical configurations having these given parameters. If r > 1 then for such a construction we need a basis consisting of r pairwise unreachable tactical configurations C ( k ,I, 1, n). One has r ( 3 , 2 , 1, n ) = 1 for n = 3 , 7 , 9 , 13 (the corresponding X-substitution in the case n = 13 can be found in [6]) and this leads one to the following conjecture: r(k, I, 1, n ) = 1 for all parameter sets k , I, n such that the corresponding configuration exists. On the other hand, it should be observed that in a similar problem on H-transformations of 1-factorizations of order n (see [ll]) already for n = 8 and 10 the number of corresponding reachability classes equals 2, and for n = 12 it is not less than 6 [12].

References [l] P, Hell and A. Rosa, Graph decompositions, handcuffed prisoners and balanced P-designs, Discrete Math. 2 (1972) 2 2 9 2 5 2 . [2] A.J. Petrenjuk, Kriterii realizujemosti funkcii kak vesovoi funkcii koneEnoi sistemy incidencii, Vestnik Mosk. Univ. Ser. Mat. Meh. 1971, No. 4, pp. 16-20. [3] A.J. Petrenjuk, 0 suEestvovanii vzveiennych koneEnyh incidentnyh struktur, Doklady AN SSSR 193 (1970) 535-536. [4] H. Hanani, On some tactical configurations, Canad. J. Math. 15 (1963) 702-722. [5] A.J. Petrenjuk, Primenenie invariantov v kombinatornych issledovaniach, in: Voprosy Kibernetiki, Trudy Seminara po Kombinatornoi matematike, Moskva, Sov. Radio 1973, pp. 129-136. [6] F.N. Cole, The triad systems of thirteen letters, Trans. Amer. Math. Soc. 14 (1913) 1-5. [7] L.D. Cummings, On a method of comparison for triple systems, Trans. Amer. Math. Soc. 15 (1914) 311-327. [8] T. Skolem, Some remarks on the triple systems of Steiner, Math. Scand. 6 (1958) 273-280. [9] A.J. Petrenjuk, Priznaki neizomorfnosti sistem trojek Stejnera, Ukr. Mat. 2. 24 (1972) 772-780; English translation: Tests for nonisomorphic Steiner triple systems, Ukr. Mat. Zh. 24 (1972) 620-626. [lo] S. Hakimi, O n realizability of a set of integers as degrees of the vertices of graph, J. SIAM 10 (1962) 496-506. [ll] V.V. Voznjak and A.J. Petrenjuk, Ob odnom algoritme peretislenija sistem grupp par, in: Kombinatornyj Analiz, Vyp. 2 pp. 38-41, [ 121 L.P. Petrenjuk and A.J. Petrenjuk, 0 perecislenii soverknnyh 1-faktorizacii polnyh grafov (to appear). [ 131 I.P. Neporoinev and A.J. Petrenjuk, Konstruktivnoje peretislenije sistem grupp par i oglavlennyje sistemy trojek Stejners, I. Kombinatornyj Analiz 2 (1972) 17-37; 11. 3 (1974) 28-42; 111, t o appear.

276

L.P. Pefrenjuk, A.J. Petrenjuk

[ 141 L.P. Petrenjuk and A.J. Petrenjuk, 0 konstruktivnom perefislenii 12-ver5innyh kubikskih yafov, Kornbinatornyj Analiz 3 (1974) 72-82. [ 151 L.P Petrenjuk and A.J. Petrenjuk, 0 perefislenii desiativerginnyh odnorodnyh grafov stepeni 4, in: Voprosy Kibernetiki. Trudy Serninara po Kombinatornoi matematike, Moskva, Sov. Radio 1975 1161 L.P. Petrenjuk and A.J. Petrenjuk, Postrojenie nekotoryh klassov kubiteskih grafov i neizornorfnost' Kirkmanovyh sistem trojek, Kombinatornyj Analiz 4 ( 1976) 73-77. [17] Kh.J. Kurek and A.J. Petrenjuk, 0 pokrytii grafov zviozdami, in: Teorija Grafov. Kiev (1977) 145-156. [ 181 A.J. Petrenjuk. lssledovanija v teorii konefnyh sistem incidentnostei, Cand. Diss., Moskva, 1971. [191 C.C. Lindner and A. Rosa, There are at least 31,021 nonisomorphic Steiner quadruple systems of order 16. Utilitas Math. 10 (1976) 61-64. [20] J Doyen and A. Rosa. A bibliography and survey of Steiner systems. Bollet. Unione Mat. Ital. (4) 7 (1973) 392-419.


ON CYCLIC STEINER SYSTEMS S(3,4,20) K.T. PHELPS School of Mathematics, Georgia Institute of Technology Atlanta. G A 30332, USA A Steiner quadruple system of order n is said to be cyclic if it has an n-cycle as an automorphism. In this paper we enumerate all cyclic Steiner quadruple systems of order 20. As a necessary prelude t o this enumeration, a number of results are established, including a new characterization of the existence problem in terms of hypergraphs. It is also shown that the necessary conditions for an S-cyclic SQS (n) is that n = 2,4,10 or 20 mod 24, and that there does exist an infinite class of cyclic Steiner quadruple systems, thus answering a question posed by Lindner and Rosa [16].

1. Introduction A Steiner system S(3,4, n), more often called a Steiner quadruple system of

order n (briefly SQS (n)) exists for all n = 2 or 4 mod 16. In general a Steiner system of order n is said to be cyclic if it has an automorphism consisting of a single n-cycle. A cyclic Steiner System S(t, k, n), consisting of an n-set P and a collection B, of k-element subsets of P, can thus be assumed to have as its n-set Z,,, the residues m o d n and (Z,,,+>, the integers mod n under addition as a subgroup of its automorphism group. The existence of cyclic Steiner triple systems was settled by R. Peltesohn [17]. For Steiner quadruple systems, this question is unresolved. In a previous computer investigation, Guregovii and Rosa [12] have enumerated all cyclic SQS (n)for n s 16: they established that no cyclic SQS (n)exists for n = 8, 14 or 16 and exactly one exists for n = 2, 4 and 10. Others have established the existence of cyclic SQS (n) for various values of n (e.g. [8, 161). Until recently (Phelps [19]), the only known cyclic S Q S ( n ) were S-cyclic (for a definition of S-cyclic see Section 3 below). Jain [14] constructed the first S-cyclic SQS(20) and showed that it was the unique S-cyclic system. Later Phelps [19] constructed another cyclic SQS (20). More recently Griggs and Grannell [ll] and Cho [3] have constructed other examples of cyclic SQS (20). With these diverse results in mind it appeared that a complete enumeration of cyclic SQS (20) would be especially warranted. A primary purpose of this paper then is to establish that there are exactly 29 nonisomorphic cyclic SQS (20), thus extending the results of Guregovii and Rosa r121. A complete enumeration of cylic SQS(20) given limited resources, did not appear feasible at first. To solve this enumeration problem, a better understanding 277

K.T. Phelps

218

of the general existence problem had t o be developed. This mathematical basis along with a proper characterization of this problem, enabled the author to enumerate all cyclic SQS(20) with only a modest utilization of computer resources (less than an hour of computer time on a CDC Cyber 70). Thus it is felt that the approach used, that is the algorithmic methods and the general results upon which they are based is almost as interesting as the results that were obtained. In the next section, the general existence problem for cyclic SQS(N) is discussed. Some new results are presented. Having established the mathematical background, we then proceed t o discuss the methods used and finally the results that were obtained. For more on cyclic quadruple systems the reader is referred to an excellent survey article by Lindner and Rosa [16].

2. Background Let Pr(Z,,)be the collection of all r-element subsets of Z,, = {O, 1 , 2, . . . , n - 1). To each triple {i. j , k} in P3(Zn). i < j < k, we can associate a difference triple ( j - i, k - j , i - k) with the differences taken mod n. Two difference triples are equivalent if and only if one is a cyclic re-ordering of the other. Under the action o f (Z,,,+), two triples of CPJZ,) will be in the same orbit if and only if the associated difference triples are equivalent. For example, all distinct difference triples mod 20 are listed in Table B.l.Note that for n = 2 or 4 mod 6 every orbit in 9,(Z,) is full (i.e. of size n). Table 6 . I . Difference triple mod 20 I

1

6

I

11

1 1 h i i 1 ’

41 46

‘,

i

j

’

’-

1”’-

4

‘I

s

(?

4.) 47

--

1 1

.’

1 7

/

1.’

.’ .’

.’

1 i

G,

,’

J 1

lh

‘i

ti

I I

s

11

’

. )

.’ .’ .’

.)

5 6

4 4

4 4

c

5

1

2 3 5 1 3

/

I,
I

I I

3 3

I i

I0

/

;3

6

1

.J

4 67

4 5 Y 4 I0 4 !I i) / h ? 10

3

2

t, ?

.’

182) 3 (187) 4

3 4 5

I I

1

;3 3 3

(192) (197) (202) (207) (212) (217)

1

’

5 Y 4 Y 4

Y

1 11 - ) 6 / 5 1 7 4 7 2 8 3 1 3 1 7 1 4 4. 2 1 14 3 J t 13 3 3 1 7 3 4 2 J 1 3 4 7 6 3 5 3 9

/

L0

5

I!

2

I .i

6

13

!I

1

1 I 1 1

:’

:’

‘I

LJ

10

.I

I

I I

J

14 1 Ll Y

3

2

n

6 2

:)

7

8 3 J 1 15 1 10 6 2 9 r, 3 4 8 4 1 12 4 6 7

5

:’

1 14 7 H 3 11 8 6 3 10 Y 4

3 4 4 4 5 6

1 J 1

4

3

4

1 1 1 1

A H

1

1

I I

3 3 3 3 3

4 4

I, C

.

I

I /

?

,

5 v

‘r‘

tl

7

h

!I

11 1

h 4 4 I.’ 5. 1 ?

4,

.!

4

4

-5

10

Y

LI

4

Y

3 4 9 4 3 5 6 6 , 3 3

.3 4

tl 6

4

6 7 1

4

?lO

4 4

4 111 5 2 9 1 2 1 2 1 1 1 2

411 4

:’

I .’

’p

.’ I I

;51

I

’

F

Table B.4. Edges of Hypergraph 1 2 312 1 7 817 2 4 12 22 2 9 17 27 3 5 22 32 3 10 27 37 4 6 32 42 4 38 48 55 5 9 40 44 6 7 50 55 6 53 54 55 7 9 20 56 7 41 48 54 8 10 19 20 8 31 39 48 9 36 37 5 0 2 1 1 13 22 1.0 1.1 30 56 I2 16 18 36 13 14 21 41 13 20 27 47 14 1.7 33 52 14 38 51. 5:3 1.5 18 30 43 15 3:’ 37 49 16 4 7 49 56 17 20 28 30 17 22 ?Y 47 1 ?1 2? 31 10 ?1 40 52 19 2 2 49 92 ? 3 :?6 ? 9 53

24 ?9 31 55 24 30 36 54 2 0 2 5 27 42 25 33 38 48 181) ?6 46 49 52 i n & ) i 31 32 41 171) 12 32 34 51 1 9 6 ) 2? 33 35 56 ?Ol) 34 .36 40 5.5 :!06) 35 37 39 44 : ’ 43 55 ? l l ) 11 4 :’16) 1 1 51 5? 5 7

(

2)

(

7) 12) 17) 22) 27)

(

( ( (

( (

32)

(

42)

(

(

47) 52) 57)

(

63)

(

67) 72) 77)

(

( (

37)

(

82)

(

87) Y2)

( (

’37) 102)

107) 1.12)

117) 12?)

127)

132) 137) 142) 147) 152) 157) 162) 167) 1.72)

177) 182)

187) 192) 197) 202) 207) 212) 217)

1 3 413 1 8 918 2 5 13 23 2 10 18 28 3 6 23 33 3 28 38 57 4 7 33 43 4 39 49 50 5 46 53 57 6 8 51 56 b 41 47 53 7 10 29 30 7 31 38 47 8 28 29 57 8 21 29 38 9 31 40 49 3 1 1 14 23 1 1 12 13 31 1.2 17 19 37 13 15 2” 42 13 28 48 56 14 18 34 40 1.4 39 42 46 15 1.9 39 44 16 1.7 50 57 16 42 48 54 17 37 39 56 18 127 29 56 2 21 23 32 1 1 22 23 41 2 1 23 24 50 23 27 30 54 24 26 32 56 24 37 52 54 25 28 29 43 10 26 27 50 26 43 49 53 2 31 33 42 13 32 35 52 2 8 i33 44 47 34 37 44 48 35 44 46 49 12 42 44 56 1 55 56 57

(

3)

(

8)

(

13) 18) 23)

(

( ( ( ( ( (

28)

33) 38) 43) 48) 53) 58) 63) 68) 73) 78)

1 1 2 2 3 3 4 5 5 6 6 7 7

4 9 6 19 7 29 8

6 47 9

31 ( 38 ( 21 ( 8 37 ( 8 11 ( 9 21 ( 83) 4 11 ( 88) 12 13 ( 93) 12 18 ( 9 8 ) 13 16 (1.03) 13 29 (108) 14 19 (113) 14 32 ( 1 1 8 ) 15 46 (123) 16 18 (1:!8) 16 32 (133) 17 46 (138) 18 36 (143) 3 21 (148) 1 2 22 (153) 22 23 (158) 23 28 (163) 24 27 (168) 14 38 (175) 25 29 (178) 19 26 (183) 26 33 (188) 3 31 (193; 19 3 2 (198) 2Y 35 ( 2 0 3 ) ‘0 35 ( 2 0 8 ) I 41 (213) 21 45 (

514 10 19 14 24 29 57 24 34 39 55 34 44 41 50 54 55 30 52 37 46 39 57 28 37 38 55 19 28 30 39 15 24 15 33 20 38 23 43 49 51 35 49 36 40 54 56 :?O 5 l 38 53 48 51 38 51 24 33 24 42 25 51 52 53 30 33 43 47 38 44 28 51 .39 54 34 43 44 46 34 37 36 41 42 50 44 57

( (

4) 9)

1 1 2 2 3 3 4 5 5 6 6 7 7 8 9

14) 19) 24) ( 29) ( 34) ( 39) ( 44) ( 49) ( 54) ( 59) ( 64) ( 69) ( 74) 79) 9 ( 84) 5 ( 89) 12 ( 94) 12 ( 99) 13 (104) 14 (109) 14 (114) 15 (119) 15 (124) 16 (129) 16 (134) 17 (139) 18 (144) 4 (149) 13 (154) 23 (159) 23 (164) 24 (169) 24 (174) 25 (179) 26 (184) 27 (189) 4 (1Y4) 20 (199) 31 (204) 19 (209) 2 (214) 1 ( ( (

5 615 10 20 57 7 15 25 20 30 55 8 25 35 30 40 50 10 36 46 7 42 51 48 50 53 10 39 40 21 27 36 47 48 55 1 1 18 27 46 47 50 18 19 57 1.1 20 29 1 1 16 25 14 16 34 19 39 56 17 24 44 15 31 50 20 36 53 16 41 55 47 51 54 19 29 52 22 28 46 42 49 54 32 40 53 21 25 34 22 25 43 24 27 40 29 43 46 28 34 39 33 37 39 46 52 53 28 29 52 36 39 52 31 35 44 32 34 36 34 35 57 35 37 42 41 43 51 50 51 55

5)

10) 15) 20 ) 25 ) 30) 35) 40) 45) 50 ) 55) 60 ) 65) 70) 75)

1 2

2 3 3 4

4 5 5

6 7 7 8

8 9 ( 80) 1 6 ( 85) ( 90) 12 ( 95) 12 (100) 13 (105) 14 (110) 14 (115) 15 (120) 15 (125) 16 (130) 10 (135) 17 (140) 18 (145) 5 150) 14 155) 23 160) 33 165) 24 170) 15 175) 25 180) 26 185) 27 190) 1 1 195) 21 200) s4 ?05) 28 210) 3 215) 2

6 716 3 1 1 21 8 16 26 4 21 31 9 26 36 5 31 41 37 47 57 8 43 52 41 46 49 48 49 57 8 55 57 50 53 54 9 10 57 41 49 53 27 2a 55 1 1 12 21 1 1 17 26 15 17 35 20 40 51 19 26 46 16 32 51 37 54 56 17 42 56 42 47 48 20 38 40 17 18 55 32 39 54 22 30 48 21 26 35 72 26 44 25 38 49 30 33 36 2Y 35 48 26 41 57 4.4 47 48 30 37 40 33 40 54 32 33 50 33 34 55 35 39 47 35 38 43 41 44 5:’ 50 52 56

284

K.T.Phelps

Lemma 3.3. There exists a cyclic SQS (3' other than the units mod n.

+ l ) , for all k > 0 having automorphisms

Proof. PGL (2, q ) always has a cyclic subgroup of order q + 1 (Huppert [13, p. 1873). There exists a S(3,4,3' + 1) having PGL (2, 3 k ) as its automorphism group C2, 161. As for isomorphisms, the question remains open. Among the cyclic SQS (20) the only isomorphisms are those induced by the group of units mod20. Both these questions will be discussed further in conjunction with the presentation of the enumeration results.

4. The enumeration of cyclic SQS (20) As was previously mentioned, enumerating cyclic SQS ( n ) is equivalent to finding all 1-factors or perfect matchings in the associated hypergraph. More specifically consider the (valid) orbits of quadruples of 2,". To each orbit we can assign a subset of X, the set of all differences triples mod 20. Letting 8 denote the collection of these subsets of X we have that H(Z,,) = ( X , 8)is a hypergraph. If we find a l-factor in ( X , %), then the union of the corresponding orbits will be a cyclic SQS (20) since every difference triple is in exactly one edge and hence every triple must occur in exactly one quadruple. The units of Z,,,, induce automorphisms of the hypergraph H(Z,,). These automorphisms acting on the edges of H ( Z 2 J can be used to simplify the enumeration search. For example having found all l-factors that contain a particular edge, then one can eliminate that edge and all edges in its orbit from further consideration. Also having found a particular partial l-factor, the subgroup of these automorphisms that fixes this partial 1-factor can be used in a similar fashion to simplify the remaining search. Every cyclic SQS (20) must contain a short orbit of size 5 and hence, by Lemma 7,. 1, must contain the orbit having ( 5 , 5 , 5 , S) as its difference quadruple. That is, it must contain the base hlock {1,6, 11, 16). Thus the difference triple ( 5 , S , 10) is accounted for and hence this vertex (45) and all edges incident to it can be eliminated from the hypergraph H(Z,,J. In what follows, it will always be assumed, that we are working with this reduced hypergraph. Besides the short orbit of size 5, a cyclic SQS (20) can contain 0.2, or 4 short orbits of size 10. From Lemma 2.2 we know that these short orbits have these diKerence quadruples:

( 1 , 9, 1,9), (3.4,3.7).

(2, tA2.8).

(4,6,4.6).

In attacking this problem then we divide it into 3 cases, I. 11 and 111 in which we enumerate all cyclic SQS (20) having 0 , 2 and 4 orbits of size 10 respectively. In Case 11, there are (j) possible choices of 2 such short orbits. However this

On cyclic Steiner systems S(3.4,20)

285

collection of six choices is reduced to 4 possible subcases (namely A, B, C, D) by using the automorphlsms induced by the units of Z20. Furthermore every 1-factor must contain an edge which covers vertex 12 (i.e. difference triple #12). The degree of 12 in this hypergraph is at most 15 (depending upon the case). However, these edges fall into only 6 orbits, having edges numbered ( l ) , (‘ll), (88), (89), (91) and (93) as representatives (see Table B.4). Thus the other 9 edges can be eliminated without any loss. Moreover having enumerated all 1-factors containing one of these edges we can eliminate the remaining edges in its orbit from further consideration. The enumeration, then, consisted of a series of computer runs in which the program found all 1-factors in a suitably reduced hypergraph. Finding all perfect matchings in a k-uniform hypergraph (k >2) is in general “hard” problem (Garey and Johnson [9]).However, properties of the hypergraph are such that there is a simple algorithm that is rather efficient. First we describe the algorithm and then discuss why it is efficient. The algorithm is a simple backtrack search using many known techniques (see Gibbons [lo]). Assuming that (X, 8) is the uniform hypergraph in question, we have that at level i of the search F ( i ) contains the partial 1-factor; T(i), T ( i )G X , contains the vertices covered by the edges in F(i)and S ( i ) , S ( i ) s 8,contains the edges of the sub-hypergraph induced by the vertices of X\T(i). Let u ( i ) be a vertex of minimum degree in (X\T(i), S ( i ) ) . If the degree of u(i) is zero then this partial 1-factor can not be completed so we backtrack. If the degree of u ( i ) is nonzero we choose an edge, E E S ( i ) such that u ( i )E E. Then F(i + 1)= F ( i )U { E } , T ( i + l ) = T ( i ) U E and S ( i + l ) = { E r I E r ~ S (and i ) E’nE=@}.Since we know exactly how many edges are needed we backtrack if IS(i + 1)1C k - IF(i + 1)1, where k is the number of edges in a 1-factor. An outline of the algorithm follows: Initialize: F(O), S(O), T(O),k, i = 0. Do while i 3 0 Next-Level: Compute u(i). If degree of u ( i ) < O then go to Back-Track. Next-Edge: Find E c S ( i ) such that u(i)EE. If no such edge exists then go to Back-Track. Set F ( i + l ) = F ( i ) U { E } ; T ( i + l ) = T(i)UE. Compute S(i + 1). i=i+l. If ( S ( i ) (< k - IF(i)l then go to Back-Track. If i < k then go to Next-Level. Print F( i). Back-Track i = i - 1. S(i) = S(i)\F(i + 1). Go to Next-Edge. End; End;

K.T. Phelps

286

Using this simple algorithm the longest run on any one case was about a minute. Case I took about 4 minutes in total. T h e majority of the runs were completed in 10-25 seconds. Of course, this is partly due to the pre-run analysis, discussed above, which simplified the search. Others reasons for this efficiency lie in t h e nature of the hypergraph. Note, that in this algorithm the number of steps is dependent on the minimum degree at each level. That is if d ( i ) is the maximum degree of any vertex u ( i ) during the entire search then the number of steps is less then d ( i ) where k depends upon the case being considered. Looking at H(Z,,,) (see Table B.4) any vertex has degree between 14 and 17. Moreover any 2 vertices are contained in at most 2 edges. In fact the majority of pairs of vertices are contained in at most one edge. T h u s the minimum degree drops from one level to the next. In fact for the first couple of levels. t h e degree of almost every vertex drops and the minimum degree usually drops by at least 3. Hence the upper bound o n the number of steps, d(i). is relatively modest (e.g. 14.11 . S * 6 * ( 4 ! as ) compared to 14!). Furthermore backtracking (because the minimum degree at that level is zero) should often occur at a relatively low level (say 5-6). The foregoing discussion gives a rationale for choosing this algorithm based on the worst case and is not intended as a proof in any sense of the word. The above procedures will enumerate all non-isomorphic cyclic SQS (20). However further tests are needed to show that these designs are in fact nonisomorphic. One technique is to look at the derived triple systems of these designs. Using a procedure initially proposed by Cummings [5] and later developed by Petrenjuk [18], we determined that of the 29 cyclic SQS(20), 25 have nonisomorphic derived triple systems and hence must be nonisomorphic. A full description of this procedure can also be found in Lindner and Rosa [ l S ] or M. Colbourn [4]. For those cyclic SQS (20) that have isomorphic derived triple system. (i.e. I.A.l, I.A.3. and 1II.A. 1, III.A.2, III.B.2, III.B.3) another procedure (described by Gibbons [ 101) is used. We count the number of block triangles in the quadruple system as well as t h e number of such triangles that involve each particular block. A block triangle of a SQS (n),( P , B),is a triple of quadruples b,, bi,bk from B such that any 2 quadruples have 2 points i n common but n o point is common to all 3 blocks (i.e. hi nb, n b k =g). The total number of block triangles is different for each of these cyclic SQS (20) and hence they are nonisomorphic. Both of these procedures usually give the automorphism partitioning of the points of the designs. This will be considered later in our discussion of the results.

nF==,

nf=,

4. Results of tbe enumeration

The main results are that there are 29 nonisomorphic cyclic SQS (20). Of these I S have no short orbits of size 10: 6 have 4 short orbits of size 10 (4 of these

On cyclic Steiner systems S(3,4,20)

287

where originally constructed by Cho [3]). The remaining 8 have 2 short orbits of size 10 with each subcase A, B, C, D having at least one cyclic SQS (20). The only isomorphisms between cyclic SQS (20) are those corresponding to the units of 220. There are 6 cyclic SQS (20) having the unit 13 as an automorphism giving 12 distinct SQS (20); 4 designs have the unit 3 as an automorphism giving 8 distinct SQS(20); 4 designs have the unit 19 as an automorphism giving 16 distinct SQS (20) (one of these was discovered by Griggs and Grannell [ l l ] another by Jain [14]); only one SQS (20) has the unit 9, and no other units as its automorphism giving 4 distinct SQS (20). None of the cyclic SQS (20) has 11 as an automorphism. Hence there are 152 distinct SQS (20) having (Zzo,+) as its automorphism group, but only 29 nonisomorphic cyclic SQS (20). Let us consider the full automorphism group G, for these cyclic SQS (20) and look at the stabilizer of a point (e.g. Go). Then the derived triple system associated with this point, 0, must have Go as a subgroup of its automorphism group. In all but 4 instances the automorphism group of the derived triple system is precisely Go which as it turns out is always a subgroup of the group of units of Zz0. In particular, 14 of these cyclic SQS (20) have derived triple systems that are automorphism-free. For III.B.2, III.B.3, 1II.A.1 and III.A.2, the derived triple systems are all isomorphic and have an automorphism group of order 144 (e.g. Table B.5). In what follows any reference to derived triple systems can be assumed to refer to the derived triple system associated with the point zero. The permutation (2, 12) (4,14) (6,16) (8,18) is an isomorphism between the derived triple systems of III.A.1 and III.A.2, between those of III.B.2 and III.B.3 as well as between those of I.A.1 and I.A.3. An isomorphism from the derived triple systems of III.B.2 to that of III.A.1 is (2, 12) (3,7,5) (13, 17, 15) (4,8,9, 14, 18, 19). The generators of the automorphism group of the derived triple system of III.A.l along with the derived triple system itself is listed in Table B.5. The routine used to generate this automorphism group is based on the work of Druffel er al. [7] and Gibbons [lo]. One further remark; this derived triple systems has a large number of subsystems of order 7. Although the derived triple systems are isomorphic these quadruple systems have different automorphism groups. In two cases III.B.2 and III.B.3, Go, the subgroup of automorphisms fixing zero is precisely the automorphism group of the derived triple system and hence lGol= 144. In the other two cases III.A.1 and III.A.2, Go is merely the unit 13. This is easy to see. Consider III.A.l; the derived triple systems (see Table B.5) has a subgroup of order 8 that fixes 1. None of these are automorphisms of III.A.l. Thus there can be at most one automorphism mapping 1 to j (and fixing 0) for any j > l . Since the unit 13 is an automorphism, there are only 5 possible choices for j . Testing these possibilities out by hand, we see that the unit 13 is the only possible automorphism. Finally, in Table A we list the base blocks and difference quadruples for each of the 29 nonisomorphlc cyclic SQS(20). The block # in this listing refers to the

2 88

K.T.Phelps

Table B.5. Derived triple system of III.A.1 and generators of its automorphism group 1 1 1

3 8 4

4 5 6 7 19 6 3 19 2 3 4 16 15 1. 19 7 6 ’ 15 9 11

2 2 6

1

9

2

1

3

4

7 8 4 3 8 7 8 7 5 8 6 18

9 15 15 2 14 12

10 11 12 10 11 12 10 11 16 10 11 19 10 12 1 1 10 13 14

13 18 14 13 13 7

14 15 16 17 18 19 17 9 16 14 13 5 13 9 12 18 17 5 1 4 6 5 18 17 13 9 17 16 15 18 4 5 1 9 1 1 6 8 2

144

ORDER

PARTITION :

1

1

1

1

7 12 5 13 17 7 9 16 6 9 17 6 8 13 5 12 18

8 9 14 2 4 1 7 0 11 17 1 17 18 6 10 16 2 7 14

13 16 18 1 5 9 1 13 14 1 7 8 1 10 11 4 9 18

6 14 15

4

3 14 2 2 2

8 1 6 18 19 9 1 3 10 12 6 11

1

1

1

1

11 12 16 3 6 1 8 4 11 13 7 11 18 7 10 17 5 10 15 4 1 1 3 3

15 16 6 1 2 15 19 9 1 2 10 13

1 1 0 8 4 3 6 8 9

1

12 15 8 19 11 14 7 1 9 10 18 11 15

2 15 18 3 5 7 16 17 15’ 1 3 4 4 10 14

1

1 12 5 12 2 9 5

1

1

1

1

1

1

14 17 11 19 13 19 3 1 9 10 19 1 4 16

3 15 17 2 5 8 1 2 16 4 5 6 7 13 15

associated edge in the hypergraph (see Table B.3 and B.4). Otherwise the listing should be self explanatory.

5. Problems

There are a host of problems relating to cyclic S Q S ( n ) . For instance are the necessary conditions on n for an S-cyclic SQS (n)to exist, sufficient? Can one find necessary or sufficient conditions for a cyclic SQS ( n ) to have specific additional automorphisms such as particular subgroups of the units of Z,, or automorphisms different from the units of Z,,? Of course, there is the basic problem of determining the spectrum for cyclic SQS (n). Finally, the program was modified t o search for cyclic SQS(22). During a 3 minute run 7 nonisomorphic cyclic SQS(22) where found. Indications are that there will be many (100-200) such systems. However, a complete enumeration will have to wait until the necessary resources are available.*

References [ 11

'?

Yes )

Yes '? *?

? 2 10'4 .? 2 4 1

~ 4 1 [R17]. A.E. Brouwer ~ 3 1

Bibliography and survey of Steiner systems

V

S(t, k , li)

Existence

N(t, k, u)

References

349


Topics on Steiner Systems (Annals of Discrete Mathematics, Volume 7)

Topics on Steiner systems

The Steiner Tree Problem (Annals of Discrete Mathematics, Volume 53)

Topics on Perfect Graphs (Annals of Discrete Mathematics, Volume 21)

The Steiner Tree Problem (Annals of Discrete Mathematics, Volume 53)

Studies on Graphs and Discrete Programming (Annals of Discrete Mathematics)

Topics in Discrete Mathematics

Theories of Computational Complexity (Annals of Discrete Mathematics, Volume 35)

Euler Systems (Annals of Mathematics)

Theories of Computational Complexity (Annals of Discrete Mathematics, Volume 35)

Eulerian Graphs and Related Topics: Part 1, Volume 2 (Annals of Discrete Mathematics, Volume 50)

Eulerian Graphs and Related Topics: Part 1, Volume 1 (Annals of Discrete Mathematics, Volume 45)

Eulerian Graphs and Related Topics: Part 1, Volume 1 (Annals of Discrete Mathematics, Volume 45)

Lukasiewicz-Moisil Algebras (Annals of Discrete Mathematics, Volume 49)

Graph Theory and Applications (Annals of discrete mathematics, Volume 38)

Combinatorics '86 (Annals of Discrete Mathematics)

Convexity and Graph Theory (Annals of Discrete Mathematics, Volume 20)

Lukasiewicz-Moisil Algebras (Annals of Discrete Mathematics, Volume 49)

Threshold Graphs and Related Topics (Annals of Discrete Mathematics, Volume 56)

Threshold Graphs and Related Topics, Volume 56 (Annals of Discrete Mathematics)

Discrete Mathematics of Neural Networks: Selected Topics

Discrete mathematics of neural networks: selected topics

Optimization of Stochastic Systems: Topics in Discrete-time Systems, Volume 32 (Mathematics in Science and Engineering)

Notes on discrete mathematics

Discrete Mathematics of Neural Networks: Selected Topics (Monographs on Discrete Mathematics and Applications)

Discrete Optimization: Part 1: Symposium Proceedings (Annals of Discrete Mathematics, Volume 4)

Discrete Optimization: Part 2: Symposium Proceedings (Annals of Discrete Mathematics, Volume 5)

Algorithmic aspects of combinatorics (Annals of discrete mathematics 2)

Algorithmic aspects of combinatorics (Annals of discrete mathematics 2)

Fourth Czechoslovakian Symposium on Combinatorics, Graphs and Complexity (Annals of Discrete Mathematics, Volume 51)

Fourth Czechoslovakian Symposium on Combinatorics, Graphs and Complexity (Annals of Discrete Mathematics, Volume 51)

Topics on Steiner Systems (Annals of Discrete Mathematics, Volume 7)