View Is That the Family of Subsets Is in Fact a Multiset of the Set of All K-Subsets of X and One May Well Have a Multiple Occurence (Or Repetition) of a Block

THE 3−DESIGN PROBLEM

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for

the Degree Doctor of Philosophy in the Graduate

School of the Ohio State University

Niranjan Balachandran, M. Stat(India)

*****

The Ohio State University 2008

Dissertation Committee: Approved by Professor Neil Robertson, Advisor

Professor Akos´ Seress Advisor Professor Stephen Milne Graduate Program in Mathematics

ABSTRACT

The theory of Combinatorial designs is one of the oldest and richest disciplines in

Combinatorics and has wide ranging applications in as diverse ﬁelds as Cryptography,

Optics, Discrete Tomography, data structures and computer algorithms, hardware design, Interconnection networks, VLSI testing, Astronomical Imaging, and Neutron

Spectroscopy and also contributes to other disciplines of Mathematics such as The theory of Unimodular lattices, Coding Theory, Computational Group theory, and

Discrete and Combinatorial Geometry.

A t−(v, k, λ) design is a pair (X, B), where X is a set of size v and B is a collection of subsets of X of size k each such that every t-subset of X is contained in precisely

λ members of B.A t − (v, k, λ) design is also denoted Sλ(t, k, v). If λ = 1 then it is called a t-Steiner design and is denoted by S(t, k, v).

The problem of characterizing all triples (v, k, λ) for which a t − (v, k, λ) design exists is one of the fundamental problems in the theory of Combinatorial designs.

Clearly, these parameters cannot be chosen independent of one another since there are certain necessary conditions that are to be met in order that a design exists.

These are known as the arithmetic conditions or the ‘admissibility conditions’.

While the admissibility conditions are necessary, they are also not suﬃcient; there exist several cases of parameters that satisfy the admissibility conditions and yet no

ii design with these parameters exists. However, if the point set is large, then it is conjectured that the admissibility conditions would be suﬃcient as well. This is known as the ‘v-large existence conjecture’ or the ‘asymptotic existence’ conjecture.

The ‘asymptotic existence’ conjecture has been proved for t = 2 by Wilson, following the work of several including R.C. Bose, Marshall Hall, Jr., Haim Hanani, and Dijen

Ray-Chaudhuri.

This dissertation studies the ‘asymptotic existence’ conjecture in the speciﬁc case t = 3 with the primary goal of constructing new families of 3-designs . More speciﬁ- cally, this dissertation includes the following:

1. Firstly, by considering the action of the group PSL(2, q) on the ﬁnite projective

line and the orbits of the action of this group to construct simple 3-designs.

While the case q ≡ 3 (mod 4) is 3-homogeneous (so that orbits of any ‘base’

block’ would yield designs), the case q ≡ 1 (mod 4) does not work the same way.

We however overcome some of these issues by considering appropriate unions

of orbits to produce new inﬁnite families of 3-designs with PSL(2, q) acting

as a group of automorphisms. We also prove that our constructions actually

produce an abundance of simple 3-designs for any block size if q is suﬃciently

large. We also construct a large set of Divisible designs as an application of our

constructions.

2. We generalize the notion of a Candelabra system to more general structures,

called Rooted Forest Set systems and prove a few general results on combina-

torial constructions for these general set structures. Then, we specialize to the

iii case of k = 6 and extend a theorem of Hanani to produce several new inﬁnite

families of Steiner 3-designs with block size 6.

3. Finally, we consider Candelabra systems and prove that a related incidence ma-

trix has full row rank over Q. This leads to interesting possibilities for λ large theorems for Candelabra systems. While a λ-large theorem for Candelabra sys-

tems do not directly yield any Steiner 3-design (in fact, even simple 3 designs),

it allows for constructions of new Steiner 3-designs on large sets following the

methods of Block spreading.

iv To U. Koteswara Rao(1948-2008) - the ﬁrst real mathematician I knew personally.

v ACKNOWLEDGMENTS

No one who achieves success does so without acknowledging the help

of others. The wise and conﬁdent acknowledge this help with gratitude. Alfred North Whitehead.

It is very diﬃcult to make a complete list of all the people I am indebted to, since that would ﬁll more pages than I can imagine.

First and foremost goes my gratitude towards Professor Dijen Ray-Chaudhuri.

He has been more than a mentor to me. Apart from imparting to me a great deal of knowledge on a subject in which he is indeed a renowned expert, he taught me to think like a design theorist which is more important in the long run. He was kind even towards some of my silliest ideas and always had an encouraging word, especially when I felt I was getting nowhere.

My thanks to the members of my thesis committee cannot be underemphasized.

I am indeed grateful to Dr Neil Robertson, Dr. Akos´ Seress, and Dr. Stephen Milne for all their help.

I am also extremely thankful to Dr. Arasu for several useful mathematical discussions, his general advice, and great food at his place!

vi My interest in math came about would take me back to the time I was in elementary school. However, some of the teachers I have had, were instrumental in my desire to make a career out of being a mathematician. Indeed, one of my earliest math teachers Mr U. Koteswara Rao, made a mathematician look cool!

I also owe a lot of my mathematical training to my alma mater, The Indian

Statistical Institute, India. My 5 year stint at ISI, along with the summer camps at

TIFR, Mumbai, led me to believe that a career in mathematics was what I had to pursue.

In such a situation there are always several others who deserve more than a mere thank-you. My parents have always been extremely supportive of anything I did; my mom has spent several an occasion listening to me describing the ‘work I do everyday’, in an appreciative tone when, I am sure, she had no idea what I was talking about! My sisters Neeraja and Sindu have always been among my best friends, always encouraging, especially when I found myself in a slump. Among my close friends I particularly thank Rajeshpavan, Harish, Prasenjit, and Gopal for being a great help any time and every time!

Lastly, and by no means least, my gratitude to my wife Anupama. Anu has been a great support, ever helpful and cheerful, and has shown a lot of patience towards me and my irregular life style. I am inclined to think my fortunes have changed for the better ever since she became a part of my life.

vii VITA

December 24, 1977 ...... Born - Vellore, Tamil Nadu, India

1996-1999 ...... B. Stat(Hons), Indian Statistical Institute, Kolkata, India

1999-2001 ...... M. Stat., Indian Statistical Institute, Bangalore, India

2001-present ...... Graduate Teaching Associate, The Ohio State University

PUBLICATIONS

1. “Simple 3-designs and PSL(2, q) with q ≡ 1 (mod 4) ”, with Dijen Ray-Chaudhuri, Designs, Codes and Cryptography, 44(2007), 263-274.

2. “Graphs with restricted valency and matching number”, with Niraj Khare, submitted to Discrete Math (preprint available in arXiv.org :math/0611842v1), 2006.

3. “New inﬁnite families of Candelabra Systems with block size 6 and stem size 2 ”, submitted, 2007.

4. “ A λ-large theorem for Candelabra Systems”, in preparation, 2007.

viii FIELDS OF STUDY

Major Field: Mathematics

Specialization: Combinatorial Design theory, Finite Geometry

ix TABLE OF CONTENTS

Abstract ...... ii

Dedication ...... v

Acknowledgments ...... vi

Vita ...... viii

CHAPTER PAGE

1 The 3-design problem: General theory and results ...... 1

1.1 What is a Design? ...... 1 1.2 Steiner 2-designs and Finite Geometry ...... 5 1.3 Problems in Design Theory ...... 11 1.4 The problem of existence ...... 12 1.5 3-designs ...... 15 1.6 Algebraic constructions for 3-designs ...... 17 1.7 Combinatorial constructions for 3-designs ...... 19 1.8 λ-large theorems ...... 23

2 Algebraic Constructions ...... 27

2.1 Classical 3-transitive groups ...... 28 2.2 The Groups PSL(2, q), q odd...... 38 2.3 The case q ≡ 3 (mod 4) ...... 43 2.4 The case q ≡ 1 (mod 4) ...... 46 2.5 Large sets of 3-DDs ...... 62 2.6 Concluding remarks ...... 65

3 Combinatorial Constructions ...... 67

3.1 Introduction ...... 67 3.2 Candelabra systems and Rooted Forest Set Systems(RFSS) 68

x 3.3 General constructions ...... 74 3.4 New constructions for k = 6 from Candelabra systems . . 82 3.5 New constructions for k = 6: The Proofs ...... 85 3.6 Concrete instances of the theorems ...... 92 3.7 Steiner designs with block size 5 ...... 94 3.8 Concluding Remarks ...... 95

4 ‘λ-large’ theorems ...... 97

4.1 Preliminaries ...... 98 4.2 Proof of the Theorem ...... 101 4.3 Concluding remarks ...... 109

Bibliography ...... 111

xi CHAPTER 1

THE 3-DESIGN PROBLEM: GENERAL THEORY AND

RESULTS

If you would hit the mark, you must aim a little above it. Henry Wadsworth Longfellow.

1.1 What is a Design?

The notion of a Combinatorial design can be traced back to the Rev. T. P. Kirkman

(1806-1895) who ﬁrst posed a puzzle in Lady’s and Gentleman’s Diary in 1845 which has since then come to be known as the ‘15 Schoolgirl Problem’:

“Fifteen young ladies of a school walk out three abreast for seven days in succession: it is required to arrange them daily so that no two shall walk abreast more than once..”

Trial-and-error methods would quickly convince anyone that a solution to this problem is by no means trivial. Apparently, no one sent a solution to this puzzle posed by

Kirkman, who incidentally found all the ‘diﬀerent possible’ solutions to this problem.

1 One suggested solution is the following (see[27]):

Sun : ABC DEF GHI JKL MNO

Mon : ADH BEK CIO F LN GJM

T ue : AEM BHN CGK DIL F JO

W ed : AF I BLO CHJ DKM EGN

T hu : AGL BDJ CF M EHO IKN

F ri : AJN BIM CEL DOG F HK

Sat : AKO BF G CDN EIJ HLM

Independent of Kirkman, Jacob Steiner was interested in (what are now known as

Steiner Triple Systems) collections of subsets of size 3 each of a set X such that any two distinct elements of X occur together in precisely one member of the collection.

Though the solution to this problem involves some non-trivial constructions, it was still regarded as more of an ingenious puzzle rather than any serious mathematical endeavor.

By the middle of the 1930’s, as statistical methods and applications became increas- ingly relevant, the design of statistical experiments became an important branch of study among statisticians. Typically, one is interested in comparing v diﬀerent objects of interest (called treatments) by ‘testing’ them against b diﬀerent subjects (called blocks). A complete design would require each block being subject to all v treatments and this is, cost-wise, a bad experimental design, even for relatively small values of v and b.

2 To economize, an ‘incomplete’ design can be designed in which each subject is tested against k < v treatments in such a way that all pairs of treatments are equally tested among the subjects. The reasoning behind balancing the occurrences of pairs of treatments is that it aids in the analysis of variance.

In statistical parlance, this is known as a Balanced Incomplete Block design(BIBD).

In mathematical terms, a BIBD is simply what is called a 2-design. More generally, a t-design is deﬁned as follows.

An incidence structure Π is an ordered triple Π := (P, L, I) where P ∩ L = ∅, and

I ⊂ P ×L. The elements of P are called points and the elements of L, lines or blocks.

If an ordered pair (p, l) ∈ I, we say the point p lies on the line l (or that the line l contains the point p). For a block l let

(l) := {p | (p, l) ∈ I}

(p) := {l | (p, l) ∈ I}

Given non-negative integers t, v, k, λ, we say that an incidence structure Π = (P, L, I) is a t − (v, k, λ) design if and only if

1. |P| = n,

2. For each l ∈ L, we have |(l)| = k,

3. For each T ⊂ P of size t, we have |{l|T ⊂ (l), l ∈ L}| = λ.

If λ = 1, then the design is called a Steiner Design or a Steiner Set System.

3 Note that the deﬁnition of a t-design does not insist that the family of subsets is actually a set of subsets; indeed, the general point of view is that the family of subsets is in fact a multiset of the set of all k-subsets of X and one may well have a multiple occurence (or repetition) of a block.

In terms of incidence structures, we say that an incidence structure Π = (P, L, I) is simple if for l 6= l0, l, l0 ∈ L, we have (l) 6= (l0).

Thus, a simple t − (v, k, λ) design may thus be identiﬁed with the pair D := (P, L).

It is usually customary to write X, B to denote the point set, and the set of blocks, respectively.

Number of blocks in a BIBD: Let us count the number of blocks in a BIBD,

(X, B). In order to do that, we count the number of ordered triples (x, y, B) with x, y ∈ X, x 6= y, B ∈ B, x, y ∈ B. Clearly then,

X X k |{(x, y, B)|x, y ∈ X,B ∈ B, x, y ∈ B}| = b = , x,y 2 x,y∈X,x6=y B∈B by summing in two diﬀerent ways. Here, bx,y denotes the number of blocks containing the elements x and y. By the deﬁnition of a 2−(v, k, λ) design, it follows that bx,y = λ. Hence, k v v · |B| = · λ ⇒ |B| = λ 2 . 2 2 k 2 One can count the number of blocks in a t − (v, k, λ) design in precisely the same manner by counting the number of pairs (S,B) where S ⊂ X, |S| = t, B ∈ B,S ⊂ B, to obtain v |B| = λ t . k t

4 Returning to Kirkman’s Schoolgirl problem, since it is also required that no two girls walk abreast more than once, it follows that for any pair x 6= y, we have λx,y ≤ 1 where λx,y is the number of triples containing x, y. Hence the number of triples is at

15·14 most 3·2 = 35. But there are in fact 35 triples, each corresponding to a line (row) of girls walking abreast on some day. Thus in this terminology, Kirkman was basically interested in a 2 − (15, 3, 1) design (with an additional property1) Incidentally,

Kirkman later posed the same question as did Steiner, and also solved the problem of constructing Steiner triples in 1846, six years before Steiner posed the same question himself; Steiner’s name has however stuck on.

1.2 Steiner 2-designs and Finite Geometry

Steiner 2-designs also make a feature in Finite geometry, principally through the notion of a Finite Projective Plane. A Projective plane can be regarded as an incidence structure (X, B, I) where X is a set (in this case, a ﬁnite one, called the set of points),

B, a collection of subsets of X(called the set of lines) and I : X × B → {0, 1} the

‘incidence’ relation which further satisﬁes the properties:

1. Any two distinct points together lie on exactly one line.

2. Any two distinct lines intersect in a unique point.

1This involves the concept of what is called a resolvable design. One can pose the same kind of problem posed by Kirkman, the so-called Generalized Kirkman Schoolgirl Problem, on sets of size 6v + 3 which was fully resolved by Ray-Chaudhuri and Wilson [30].

5 3. There exists a quadrangle, i.e., there are four points, no three of which lie on

some line.

One can show, by very simple counting arguments, that |X| = |B| = n2 + n + 1 for some natural number n, every point lies on n + 1 lines, and every line contains n + 1 points. Hence a ﬁnite projective plane is simply a Steiner 2-design with parameters

2 − (n2 + n + 1, n + 1, 1). The integer n is called the order of the projective plane.

Some examples of ﬁnite projective planes of prime power order have been known for some time now. The standard way to construct a projective plane of prime power order is as follows:

3 Let V := Fq be a 3-dimensional vector space over the ﬁnite ﬁeld Fq. Let X be the set of all one-dimensional vector subspaces of V, let B denote the set of all two-dimensional vector subspaces of V, and let I denote the incidence relation

x ∈ X,L ∈ B, (x I L) if and only if x ⊂ L.

It is easily shown that the three conditions above are satisﬁed. Further, |X| =

|B| = q2 + q + 1 and for any x ∈ X, |x| = q + 1, so that we do indeed have a

2 − (q2 + q + 1, q + 1, 1) design.

However, these are not the only possible constructions; in fact for q a prime power but not a prime, several ‘inequivalent’ projective planes are known. To make this more precise, we make a deﬁnition; we say two designs (X, B) and (Y, C) are isomorphic

(or equivalent) if there exists a bijection

f : X → Y such that B ∈ B ⇔ f(B) ∈ C.

6 For q a prime power but not a prime, there exist projective planes that are not isomorphic to the example above.

For q prime, the only known projective planes are isomorphic to the construction mentioned above. It is conjectured that for prime q, these projective planes are the only possible ones.

Another interesting problem about ﬁnite projective planes is the question of existence of projective planes of order n when n is not a prime power. As of today, no such projective planes are known to exist. It is indeed a very longstanding conjecture in Finite Geometry that ﬁnite projective planes exist only for prime power orders.

Many partial results conﬁrming the belief in the conjecture have been proved but the problem still is wide open.

Though these conjectures concern ﬁnite sets and an exhaustive search is possible in principle, it is not practical even for small n. For instance, the question of the existence of a projective plane of order 10 was settled in the negative in 1989 by Lam et al (see [22]) using some ideas from Coding theory and a long series of exhaustive computer searches for certain set conﬁgurations within a hypothetical projective plane of order 10. The computation used the services of a super computer utilizing over

2000 hours of computing time. The next open case (n = 12) is too cumbersome to be dealt with even with present-day computing resources! So it is fallacious to think that these questions yield to brute-force calculations, simply because the magnitude of the problem is deceptively large.

7 Another class of Steiner 2-designs closely related to projective planes are the Affine planes. Affine planes arise as incidence structures involving lines and points, as before but with the following modifications: Any two lines intersect in at most one point2.

A projective plane of order n is hence, a kind of ‘geometric completion’ of an aﬃne plane of order n.

It is easy to show that if one deletes a line from a projective plane and all the points incident to it, the resulting structure is an aﬃne plane. Hence an aﬃne plane is a

Steiner 2 − (n2, n, 1) design. An affine plane consists of several ‘parallel classes’ of lines, i.e., subsets of lines, no two of which intersect non-trivially. One can construct in canonical fashion, a projective plane from a given affine plane of order n by appending another set of points called the ‘points at infinity’ in bijective correspondence with the parallel classes of lines of the affine plane, and adding to the set of lines, another line called ‘the line at infinity’, consisting of all the ‘points at infinity’. By a suitable incidence structure which extends the incidence structure of the given affine plane, one may show that this structure is indeed a projective plane of order n. Since this construction is reversible, the existence of a projective plane and the existence of an affine plane of the same order are equivalent notions.

2This deﬁnes what is called a ﬁnite geometry.

8 Another generalization of the notion of a projective plane leads to the deﬁnition of a Symmetric Design. A Symmetric Design3 is a 2 − (v, k, λ) design with the same number of blocks as points. In fact the following criteria are equivalent( see [10]):

1. The number of blocks equals the number of points.

2. Any two distinct blocks have the same number of common points.

3. Any two distinct blocks have λ common points.

The theory of Symmetric Designs is very rich with several very interesting and deep results; one may look at [2] for more details. We give here one instance of a deep result which has implications for the existence problem for projective planes: The

Bruck-Ryser-Chowla theorem.

The Bruck-Ryser-Chowla theorem gives a non-trivial number theoretic relation between the design parameters of a symmetric 2-design:

Theorem 1.2.1. (Bruck-Ryser-Chowla): If v, k, λ are positive integers such that

λ(v − 1) = k(k − 1) and there exists a 2 − (v, k, λ) symmetric design, then it is necessary that

1. If v is even then k − λ is a perfect square.

2. If v is odd then the equation

2 2 v−1 2 z = (k − λ)x + (−1) 2 λy

3These are not necessarily geometries. In fact, counting the number of blocks tells us that a symmetric design is a geometry if and only if it is a projective plane.

9 has a solution (x, y, z) 6= (0, 0, 0) in integers.

In particular, when v = n2 + n + 1, k = n + 1, λ = 1, then the Bruck-Ryser-Chowla theorem takes the form

Theorem 1.2.2. If a projective plane of order n exists and n ≡ 2 (mod 4), then the equation X2 + Y 2 = n has a solution in integers.

As of today, this is still one of the best known results addressing the question of non-existence of projective planes.

The Bruck-Ryser-Chowla theorem invokes the Minkowski-Hasse local-global principle for integral quadratic forms over Q. For a direct proof of the latter theorem, see [20].

However, we shall not get into this territory any further, since the present thesis and its goals deal with a diﬀerent problem altogether. The intent behind this digression was to give a glimpse of the kind of results one encounters while typically studying

Steiner 2-designs and projective planes.

We make one final remark about finite projective planes before we move on. A finite projective plane is not a ‘real configuration’, i.e., it cannot be realized by a finite set of points and lines in R2. This is so because the corresponding problem - The Sylvester-Gallai Problem - admits only the degenerate solution. Simply stated, it says:

Given a ﬁnite set of points in R2 such that for any two points there is a third point in the set that lies on the line joining the two, all the points are then collinear.

10 Thus the notion of a ﬁnite projective plane is in some sense, fundamentally combinatorial.

1.3 Problems in Design Theory

The fundamental problems in Design Theory are essentially the following:

1. Problem of Existence,

2. Problem of Construction,

3. Problem of Symmetries,

4. Problem of Enumeration.

The problem of existence deals with determining all tuples (t, v, k, λ) for which a t−(v, k, λ) design exists. The problem of construction asks for (if possible eﬃcient) a constructive solution to the existence problem. The problem of Symmetries involves constructions for designs with large automorphism groups and the problem of enumeration involves enumerating (at least asymptotically) all non-isomorphic designs with a ﬁxed set of parameters.

Three of these problems - the problem of existence, the problem of construction, and the problem of symmetries - are closely related, at least in practice. The last problem, the problem of enumeration, appears to be too diﬃcult a problem to tackle, even for the case t = 2. There is some evidence that the number of non-isomorphic designs is

‘large’ (see for instance, [40] where Wilson gives an estimate for the number of Steiner

11 triple systems, i.e., 2 − (v, 3, 1) designs). But the asymptotic estimates are not still good enough and might need subtler techniques for estimation. Also, as mentioned in a preceding section, a brute-force technique is just not practicable since the universal v (t) set - in this case, a set of k-subsets of size k from the set of all possible sets of size k (t) v k - is one among v k diﬀerent choices, and this number grows enormously even t / t for small values of v and k.

1.4 The problem of existence

Suppose we are given a tuple (t, v, k, λ). Let X be a set of size v,(X, B) be a t − (v, k, λ) design and let S be a ﬁxed set of size i(0 ≤ i ≤ t). Counting the number of pairs

(T,B)|S ⊂ T ⊂ B|T | = t, B ∈ B in two diﬀerent ways (as in a preceding section), it follows that the number of blocks v−i (t−i) of the design containing S equals λ k−i . Hence, we have (t−i)

Proposition 1.4.1. If a t − (v, k, λ) design exists, then the following arithmetic conditions hold:

v − i k − i λ ≡ 0 (mod ), i = 0, 1, . . . , t − 1. t − i t − i

Thus, the existence of a t − (v, k, λ) design necessitates the validity of the aforementioned arithmetic conditions. A tuple (t, v, k, λ) is called admissible if the arithmetic

12 conditions hold and is called feasible if a t − (v, k, λ) design exists, so that a feasible tuple is necessarily admissible.

The converse is far from true, i.e., admissibility of a tuple does not guarantee feasibility. Indeed, the Bruck-Ryser-Chowla theorem implies that a projective plane of order 6 does not exist.

However, there is sufficient cause to believe in the validity of the ‘v-large’ conjecture which has been popularized most notably by Dijen Ray-Chaudhuri. The conjecture states that for fixed k, t and λ, if v satisfies the arithmetic conditions and v is ‘sufficiently large’, then the admissible quadruple (t, v, k, λ) is also feasible. This conjecture is sometimes also referred to as the ‘asymptotic-existence’ conjecture.

For t = 1 we have the the following

Theorem 1.4.2. Suppose (v, k, λ) is a triple satisfying k|λv with v ≥ k. Then there exists a 1 − (v, k, λ) design.

Proof: The equivalence of admissibility and feasibility of parameter triples (v, k, λ) can be seen by inducting on v as follows. More precisely, To see that, ﬁrst note that

λv the feasibility of the tuple (1, v, k, λ) implies that the number of blocks is b = k . The simplest case is when k|v. In this case we pick λ copies of a partition of the set V into k-subsets. The general case is reduced to this simple case easily; let d = (λ, k).

If d = 1, then since k|λv, it follows that k|v. Suppose then that d > 1 and that k = dk1, λ = dλ1; this implies that (k1, λ1) = 1 and that k1|v. Let v = mk1. Let

V := {V1, V2 ..., Vm}, with |Vi| = k1 for all i, Vi ∩ Vj = ∅ if i 6= j.

13 The given conditions imply that mλ = db, so that there exists a 1-design (M, B0) with parameters 1 − (m, d, λ). Note that by the assumption, since v ≥ k, we also have m ≥ d. Now for each B ∈ B0, let

[ B := {AB|B ∈ B0}, where AB := Vi. i∈B

Then (V, B) is a 1 − (v, k, λ) design.

The construction described above involves repeated blocks, i.e., multiple copies of some blocks. One might avoid repeated blocks for smaller values of λ by recourse to

Barany´ai’ theorem (see [2]) if necessary. We will not go into that for now.

For t = 2, the problem is a lot more involved. The validity of the asymptotic-existence conjecture was proved by R.M. Wilson([37],[38],[41]) building upon the work of many including Hanani, R.C Bose, Marshall Hall, and Dijen Ray-Chaudhuri.

For t ≥ 3, the problem is wide open with very little known in general. It was in fact not known for a long time if even simple t-designs exist for all t till it was settled in the aﬃrmative by Tierlinck [32]. However the designs constructed there (which are done so, by using the t-transitive action of the Symmetric group on t symbols St and an ingenious induction argument) are too large, i.e., the values of λ are too large.

Tierlinck made an attempt to reduce the values of λ in a subsequent paper [33] but the values of λ are still too large.

We shall concern ourselves with the case t = 3 in the rest of this dissertation.

One other reason to believe in the conjecture is the validity of the Erd¨os-Hanani conjecture. By a t-cover of a set X, we mean a pair (X, C) where C is a family of

14 k-subsets of a set X of size v such that every t-subset of X is contained in at least one member of C. Let

C(v, k, t) = min |C| C where the minimum is over all t-covers of a set X of size v. Erdösand Hanani conjectured that C(v, k, t) lim v = 1, v→∞ (t) k (t) for fixed k, t. This was proved in the affirmative by Rödl(via a probabilistic technique of the “nibble”[28]). This roughly says that for v ‘sufficiently large’, one can get as close as possible to a Steiner t-design.

We make one last remark in this section regarding techniques for attacking the problem of existence. The cases t = 1, 2 have been settled in the aﬃrmative by basically constructing (concretely or inductively), a design with the appropriate parameters.

The case t = 3 (resp. (t ≥ 3)) is much harder because the algebraic machinery available here is very limited and propositions analogous to theorems corresponding to the t = 2 case, that much harder to prove. R¨odl’s proof of the Erd¨os-Hanani conjecture (and simpler proofs by several others) essentially uses the second moment method in a rather clever way. To the best of our knowledge, that has been the most notable application of probabilistic methods in the context of design theory.

1.5 3-designs

The only instance of a complete solution to the asymptotic existence conjecture for

3-designs, is for the case of Steiner Quadruple systems (t = 3, k = 4, λ = 1): If k = 4

15 and v ≥ k then admissibility and feasibility are equivalent ([15]). But for larger k, even k = 5, the result is far from complete or satisfactory. In the case k = 5, although there are constructions for inﬁnitely many Steiner 3-designs, a proof of the asymptotic conjecture has been elusive. We shall come back to this point.

The fact is that it is quite a task to even construct inﬁnite families of 3-designs.

The proof of Wilson’s result of the asymptotic existence conjecture is essentially an elaborate construction. By the method of difference sets and difference families, one can obtain several infinite families of 2-designs for each k. All these constructions bear a distinct algebraic flavor in the sense that the underlying set upon which the design is constructed has a nice algebraic structure (Groups, fields, rings, etc).

One may then use other (combinatorial) methods to obtain design on larger sets by a variety of constructions, all of which essentially bear a distinct combinatorial nature.

Finally if v is ‘suﬃciently large’, then v succumbs to one or another of the various algebraic and combinatorial constructions. In fact, that is the essence of Wilson’s proof.

To attempt something similar for 3-designs, the ﬁrst task at hand would be to obtain a few inﬁnite family of 3-designs that can be constructed by exploiting some algebraic structure of an underlying set. Then we seek combinatorial methods that may be employed to expand an existing 3-design into a 3-design on a larger set. We describe these in the following sections.

16 1.6 Algebraic constructions for 3-designs

In general, since constructions of designs with block size k for arbitrary k has been a very diﬃcult problem as such, it seems a natural and good idea to require that a certain ﬁxed (big) group acts as a group of automorphisms for the desired design.

This basically reduces the magnitude of the problem and enables us to search for such designs in more reasonable computation time. The ﬁrst formulation of this idea on paper was due to Kramer and Mesner [21]. In that paper, they demonstrate several examples of t designs4. However, all their examples are concrete ones which culminate in a computer or computer-like brute-force search, albeit on a set of much smaller scale.

If however, we wish to construct an infinite family of 3-designs, we need an infinite family of groups to use this technique effectively.

An attempt to describe all possible algebraic tools that may be employed to construct

3-designs is an exercise in futility, so we restrict our attention to transitive (multiply- transitive) actions of groups in this thesis. One such interesting family of groups is

PGL(2, q), for q a prime power.

Returning to an earlier point, the fact that there are inﬁnitely many 3-designs (in fact, Steiner designs) with block size 5 is part of a more general result. There are in fact inﬁnitely many Steiner 3-designs with block size q + 1 with q being a prime power.

4Quite a few of their designs also have higher values of t.

17 Theorem 1.6.1. There exist Steiner 3-designs 3 − (qn + 1, q + 1, 1) whenever q is a prime power (When n = 2 they are referred to as Spherical Geometries).

The aforementioned result can be obtained through an algebraic construction which uses the sharply 3-transitive action of the group G =PGL(2, qn) on the ‘projective

n line of order q ’(Xq := Fq ∪ {∞}). All these designs admit as an automorphism

(q2−1)(q2−q) 2 group, the group PGL(2, q) which is a group of size q−1 = q(q − 1). Remark: While there exist several groups that are 3-transitive (for instance the permutation groups Sn), it is also necessary that the groups are not ‘too large’ since that would imply that λ grows large as well. We will see this in more detail in the next chapter.

Another closely related family of sharply 3-transitive groups are the ‘twisted’ PGL(2, q2) with q being a prime power. This group consists of two classes of mappings on the

finite projective line involving the finite field conjugate of any element in Fq. We will formally define this group in the next chapter. One can construct Steiner 3-designs admitting the twisted PGL(2, q2) as an automorphism group along the same lines as the theorem stated above. However, in terms of parameters, these designs are still

3 − (q2 + 1, q + 1, 1).

The following theorem which was proved by Zassenhaus in 1938 describes the set of all sharply 3-transitive groups and the corresponding actions; the aforementioned groups are the only such families.

Theorem 1.6.2. (Zassenhaus) Any sharply 3-transitive group is isomorphic to

18 1. PGL(2, q), q is a prime power.

2. the twisted PGL(2, q2), q is an odd prime power.

Since sharply 3-transitive actions are restricted to these families, we relax our requirement a little bit. In fact, we only need that the group PGL(2, q) acts transitively on subsets of size 3.

Hence if we relax our restriction so that we only require 3-homogeneity, i.e., transitivity on subsets of size three, we have another family of groups. The group PSL(2, q) acts 3-homogeneously on the ﬁnite projective line of order q if q ≡ 3 (mod 4). One may then use this action to obtain some other 3-designs. We shall see some of these results in the forthcoming chapter. Some of these results are summarized in the paper of Cameron et al (see [8]).

The action of PSL(2, q) on the ﬁnite projective line of order q is however, not 3- homogeneous if q ≡ 1 (mod 4). But it is not too far from being 3-homogeneous as well. We will deal with this case in some detail in the next chapter, and demonstrate constructions of designs using these groups. We shall also discuss some shortcomings of these techniques.

1.7 Combinatorial constructions for 3-designs

Combinatorial constructions basically produce designs on larger sets by building upon certain combinatorial structures as ‘ingredients’, and a method or ‘recipe’. We shall see the details in a subsequent chapter. In much of what follows, the constructions concern Steiner 3-designs.

19 One of the earliest such theorems for 3-designs was proven by Hanani.

Theorem 1.7.1. Suppose there exists a Steiner 3 − (v + 1, q + 1, 1) design where q is a prime power. Then there exists a Steiner 3 − (vq + 1, q + 1, 1) design.

As it turns out, this is a particular case of a more general theorem which bears such a combinatorial ﬂavor:

Theorem 1.7.2. (Product theorem[24]): If Steiner designs 3 − (a + 1, q + 1, 1) and

3 − (b + 1, q + 1, 1) exist with q being a prime power, then there exists a Steiner design

3 − (ab + 1, q + 1, 1).

Yet another result (due to Blanchard[3] and further generalized by Moh´acsy and

Ray-Chaudhuri) uses a technique called ‘block spreading’ to give an interesting combinatorial ‘mock-product’ theorem.

Theorem 1.7.3. ([3],[25]) : Let q be a prime power and a, a positive integer. Suppose there exists a Steiner design S(3, q + 1, a + 1). Then there exists an integer d0 = d0(q, v, a) such that for any positive integer v satisfying

v − 1 ≡ 0 (mod q − 1),

v(v − 1) ≡ 0 (mod q(q − 1)),

v(v2 − 1) ≡ 0 (mod q(q2 − 1)),

d there is a Steiner design S(3, q + 1, va + 1) whenever d ≥ d0.

Note that the validity of the ‘v-large’ conjecture yields the above theorem as a simple application of the product theorem (hence the name, ’mock-product’). Thus, these mock product theorems lend further support to the validity of the ‘v-large’ conjecture.

20 What uniﬁes all these results apart from the nature of these theorems is that they essentially deal with certain combinatorial objects which are not 3-designs but are closely related to Steiner designs.

One of the most systematic lines of approach to the problem of constructing Steiner 3- designs involves Candelabra Systems. Candelabra systems were ﬁrst studied by A.

Hartman ([19]) for block size 4 and then by Hedvig Moh´acsy in her PhD thesis (PhD,

Ohio State University, 2002) for an arbitrary block size ([24]) where some general constructions are also indicated. We formally state the deﬁnition of a Candelabra system (in its most general form as it appears in [24]).

Deﬁnition 1.7.4. Let v, λ, t are positive integers and K a set of positive integers. A

Candelabra system (or t-CS) or simply CS of order v, index λ and block sizes in K is a quadruple (X,S, Γ, A) satisfying:

1. X is a subset of v elements(also called points),

2. S is a subset of X of size s( called the stem of the Candelabra),

3. Γ := {G1,G2 ...} is a partition of X \S into non-empty subsets (called groups),

4. A is a family of subsets of X whose cardinalities are elements of K,

5. Every t-subset T ⊂ X with |T ∩ (S ∪ Gi)| < t for all i is contained in precisely λ blocks of A and any other t-subset of X is not contained in any element of

A more general class of structures which generalize Candelabra systems are Rooted

Forest Set Systems (RFSS); we shall deﬁne these objects in chapter 3 where we

21 also discuss a fundamental construction theorem. The fundamental construction for candelabra systems is a particular instance of this more general construction.

However, we must remark here that concrete instances of these structures are hard to come by. Moh´acsy and Ray-Chaudhuri construct a few Candelabra systems in

[24] but they all have the property that the stem size equals one. Indeed, one of the harder problems is the construction of an RFSS with a root set of size 0 or 2.

The general underlying philosophy is the following: Anytime a new CS (RFSS) is constructed (discovered), it spawns oﬀ an inﬁnite family of similar structures. Usually, this leads to new families of Steiner 3-designs.

A diﬀerent stream of such combinatorial constructions again traces its way back to

Hanani:

Theorem 1.7.5. (Hanani[17]) If a Steiner 3-design 3−(v +1, 6, 1) exists, then there exists a Steiner design 3 − (4v + 2, 6, 1).

This construction, of course is very speciﬁcally for the case k = 6.

One new construction for Steiner designs that we shall see in chapter 3 yields new Can- delabra systems with stem size 2 and block size 6 by generalizing Hanani’s theorem as a theorem for Candelabra systems. This result allows us to construct consequently, several inﬁnite families of Steiner 3-designs with block size 6. We shall also discuss some of the limitations of these methods and address the need for constructions of a

‘diﬀerent’ type.

22 1.8 λ-large theorems

An alternate formulation of the design problem is as follows:

Consider the incidence matrix of the set of t-subsets and k-subsets of a ﬁnite set X.

v v More precisely, consider the matrix A of order t × k whose rows are indexed by the t-subsets of X (a set of size v) and the columns, indexed by the k-subsets of X, with

AT,B = 1 if T ⊂ B,

= 0 otherwise.

A simple design with parameters t − (v, k, λ) is simply a 0 − 1 valued solution x to the system of linear equations

Ax = λ1, where the vector 1 is the vector with all its entries being one.

If one seeks solutions x, whose entries are non-negative integers, one may interpret the solution combinatorially as a design with repeated blocks.

This reformulation of the problem looks for any fundamental obstructions to the existence of a design, leading naturally to the question of the rank of this matrix.

This and other such questions were pursued in a concrete manner in a paper by

Wilson (see [40]) and also independently by Jurkat and Graver[14].

The main result in both papers (though Jurkat and Graver look more concretely at the module structure than Wilson does in [40]) proves that the incidence matrix

23 has full (row) rank. One consequence of importance is the following theorem due to

Wilson.

Theorem 1.8.1. (see [39]) Let t, k, v be given integers with t ≤ k ≤ v. Then there is an integer n ∈ N such that a t − (v, k, λ) design exists whenever the conditions v − i k − i λ ≡ 0 (mod ) (1.1) t − i t − i λ ≥ n (1.2) are fulﬁlled.

In all the discussions above, by the rank, we mean the rank over Q. One could also consider the p-rank of A, where p is a ﬁnite prime. For a good survey listing most known results for p-ranks for 2 − (v, k, λ) designs and their Smith normal forms, see the paper [42] by Qing Xiang. The p-ranks can be used to verify that a design is not isomorphic to another.

Remark: Note that for p large enough, the p-rank of any 0 − 1 matrix is the same as its rank over Q. Indeed, suppose A is a matrix of order m × n with m ≤ n, all of whose entries are 0 or 1, and that its rank over Q is full, i.e., r(A) = m. Then AAT has non-zero integral determinant. If p > det(AAT ), the p-rank of A is full since otherwise, det(AAT ) ≡ 0 (mod p) which is a contradiction. In general we may restrict to the sub-matrix of A consisting of its full set of independent rows(columns).

Note that the p-rank of A cannot exceed its rational (real) rank.

A corresponding result for transversal designs (equivalently, Orthogonal Arrays) is due to Ray-Chaudhuri and Singhi ([29]).

24 Theorem 1.8.2. There exists a function F (t, v, k) such that for any quadruple of integers (v, t, k, λ) with λ ≥ F (t, k, v), there exists an Orthogonal array At(v, k, t).

In the context of 3-designs, Blanchard proved the following theorem.

Theorem 1.8.3. (Blanchard, [4]) There are functions v0(k) and λ0(k, v) such that for any Steiner 2-design S(2, k − 1, v) with v ≥ v0(k) and any λ ≥ λ0(k, v) satisfying

v + 1 k λ ≡ 0 (mod ), 3 3 there exists an extension of λ copies of the Steiner 2-design, i.e., there exists a 3- design (X, B) with parameters 3 − (v + 1, k, λ) and a point x0 ∈ X which admits

λ · S(2, k − 1, v) as a derived design at x0.

Blanchard’s theorem essentially proves a corresponding ‘λ-large theorem’ for Cande- labra systems, though he did not recognize it in those terms. This is our starting point in the last chapter as we seek a λ-large theorem for uniform Candelabra systems with stem size one.

Though interesting from an algebraic point of view, a λ-large theorem is not immediately interesting to a design theorist or a ﬁnite geometer, since it is more interesting combinatorially to construct designs with small values of λ. A design with repeated blocks does not have any interesting analogue in Coding theory. In statistical terms too, a block design with a large value of λ usually means that a certain experiment is repeated (repeated block) is a sign of a more expensive experimental design.

However, the following combinatorial tool, called ‘block-spreading’ which ﬁrst ap- peared in a paper due to Wilson [41], and was later generalized for partial transversal

25 designs by J. Blanchard,[5] demonstrates an interesting utility of designs with large values of λ. the construction of Steiner 2-designs on large sets is the following theorem of Wilson.

Theorem 1.8.4. (Block Spreading) Given u and t ≥ 2, let q be a prime power satisfying q ≥ q0(t, u). Suppose there exists a partial design Pq(t, k, u, T ) for some family of t-subsets T , then for any d ≥ |T |, there exists a partial design

d P(t, k, u · q , T ⊗ Iqd )

where the elements of T ⊗ In are subsets of the form {(ai, x)|x ∈ In} where {ai, i = 1 . . . k} ∈ T .

It can be shown (as did Moh´acsy [24]) that the same proof can be modiﬁed for an arbitrary λ as well, so that it need not be prime as in Blanchard’s theorem.

An important aspect of this theorem is that it allows one to start with a partial design with large λ (in fact, the initial partial design could even have repeated blocks), use block spreading, and obtain another partial design with λ = 1.

We shall see some other consequences of these results in the ﬁnal chapter.

The rest of the thesis is structured as follows. We deal with algebraic constructions in chapter 2, with combinatorial constructions in chapter 3 and with ‘λ-large’ theorems and related material in chapter 4. Each chapter starts with the basic results known and lead to new results. Many of the known results have also been re-proven; care has been taken to give due credit to the original authors in all cases. In case a proof follows the same argument as that from another source, we indicate the source.

26 CHAPTER 2

ALGEBRAIC CONSTRUCTIONS

It’s really hard to design products by focus groups. A lot of times,

people don’t know what they want until you show it to them. Steve Jobs, CEO, Apple Inc., BusinessWeek, 1998.

Outline of the chapter:

The chapter is structured as follows. We ﬁrst describe the groups PGL(2, q), T PGL(2, q2), and PSL(2, q), and prove a few basic facts. After brieﬂy considering the actions of the groups PGL(2, q) and T PGL(2, q2), we turn our attention to the groups PSL(2, q) with q odd and consider the two cases (q ≡ 3 (mod 4) and q ≡ 1 (mod 4)) separately.

When I started my work for the case q ≡ 3 (mod 4), I discovered that many of my ideas and more were already present in [8]. So, I will not prove any result here but only give the motivation for approaching the problem this particular way. In a sense, it is a very natural progression of ideas.

27 2.1 Classical 3-transitive groups

Let G be a group that acts on a set Ω. Then the group is said to act t-transitively on Ω if for a given pair of t-tuples of pairwise distinct elements (xi), (yi), i = 1 . . . , t, there exists an element g ∈ G such that g(xi) = yi. Ω Now the action of G on Ω induces in a very natural way, an action of G on t , the set of t-subsets of Ω:

g(T ) := {g(x)|x ∈ T }.

Ω The group G is said to act t-homogenously on Ω if G acts transitively on t . In this thesis, we focus on the classical groups PGL(2, q) and PSL(2, q), for q, an odd prime power. We restrict to q odd (for PSL(2, q)) since for q even, PSL(2, q)=PGL(2, q).

One of the ﬁrst known instances of 3-designs occur as Steiner 3-designs with parameters 3− (q2 + 1, q + 1, 1), with q being a prime power. This construction is inspired by the collection of lines and circles (including the circle through inﬁnity) in the Complex

Projective line CP1: Any three points determine a unique circle in CP1.

To understand this more precisely, let Xq := Fq ∪ {∞} denote the ﬁnite projective line of order q. The group PGL(2, q) consists of mappings of the form:

ax + b f := x → cx + d with , a, b, c, d ∈ Fq, ad − bc 6= 0 with the group operation being composition of mappings.

Remark: In the description of a map f ∈ PGL(2, q) above, the choice of a, b, c, d ∈ Fq

28 is certainly not unique. In fact, the description above implies that we have a group homomorphism   a b   A : PGL(2, q) → GL(2, q), f → Af , where Af :=   . c d where GL(2, q) is the group of invertible matrices of order 2 over Fq. ax+b To compute the kernel of this homomorphism, suppose cx+d = x for all x ∈ Xq. ∗ Clearly, it follows that a = d 6= 0, b = c = 0, so that ker(A) ' Fq. Since this homomorphism is clearly onto, we have

|GL(2, q)| (q2 − 1)(q2 − q) |PGL(2, q)| = = = q(q2 − 1). |ker(A)| q − 1

Before we proceed further, we make a few observations.

1. Suppose we have g1, g2 ∈ PGL(2, q) with representatives A1,A2 ∈ GL(2, q),

then a representative for g1g2 is A1A2 ∈ GL(2, q).

2. For an element g ∈ PGL(2, q) and any representative Ag ∈ GL(2, q), the quan-

∗ 2 tity ∆(Ag) = ad − bc is unique up to an element of (Fq) .

The following proposition is easily veriﬁed.

Proposition 2.1.1. The action of the group PGL(2, Fq) is sharply three transitive, i.e., given a triple of three pairwise distinct elements (a, b, c) with a, b, c ∈ Xq and another triple of distinct elements (α, β, γ) with α, β, γ ∈ Xq, there exists a unique f ∈ PGL(2, q) such that f(α) = a, f(β) = b, f(γ) = c.

29 Proof: Suppose {α, β, γ} ⊂ Fq. Let us ﬁrst consider the special case a = 0, b = 1, c = ∞. Let β − γ x − α f (x) := . α,β,γ β − α x − γ

Then clearly, fα,β,γ(α) = 0, fα,β,γ(β) = 1, fα,β,γ(γ) = ∞. Moreover, it is easy to see that fα,β,γ ∈ PGL(2, q). The uniqueness of fα,β,γ is clear since any element f 6= 1 of

PGL(2, q) can ﬁx utmost 2 elements of Xq.

If ∞ ∈ {α, β, γ}, then fα,β,γ is described as follows.

β−γ 1. α = ∞: The element fα,β,γ(x) := x−γ .

x−α 2. β = ∞: The element fα,β,γ(x) := x−γ .

x−α 3. γ = ∞: The element fα,β,γ(x) := β−α .

−1 In the general case, the mapping fa,b,c ◦ fα,β,γ does the job.

We now describe another class of groups, the ‘twisted’ PGL(2, q2), denoted T P GL(2, q2).

q Let x := x ; note that (x) = x for all x ∈ Fq2 . x is called the ﬁnite ﬁeld conjugate of x. √ ∗ In more concrete terms, if we write Fq =< θ >, so that Fq2 = {a + b θ|a, b ∈ Fq}, √ √ we have x = a + b θ, ⇒ x = a − b θ. In particular, if x = x then x ∈ Fq.

2 The group T P GL(2, q ) is the set of mappings M := M1 ∪ M2, where

ax+b ∗ 2 •M1 := {x → cx+d , if ad − bc 6= 0, and ∆(f) := ad − bc ∈ (Fq2 ) },

ax+b ∗ 2 •M2 := {x → cx+d , if ∆(f) := ad − bc∈ / (Fq2 ) }.

30 2 2 Thus M1 = T PGL(2, q ) ∩ PGL(2, q ) as sets.

Proposition 2.1.2. T P GL(2, q2) is a group of order q2(q4 − 1).

Proof: Let g1, g2 ∈ M. For sake of simplicity, we abuse notation and write ∆(gi) to

denote ∆(Agi ), where Agi is a (ﬁxed) representative for gi, i = 1, 2. We also denote f(x) := ax+b for f ∈ M . cx+d 2

If g1, g2 ∈ M1, then the fact that g1g2 ∈ M1, follows from the observation ∆(g1g2) =

∆(g1)∆(g2) and the observation made earlier regarding ∆(g).

2 Suppose g1, g2 ∈ M2. Since ∆(g1g2) = ∆(g1)∆(g2), and ∆(gi) ∈/ Fq2 for i = 1, 2, it easily follows that g1g2 ∈ M1.

If g1 ∈ M1, g2 ∈ M2, then it is again easy to verify that g1g2 ∈ M2. Thus, the set M is finite, closed under composition of maps, and satisfies the cancellation laws (i.e., g1g = g2g ⇒ g1 = g2 and gg1 = gg2 ⇒ g1 = g2) The validity of the cancellation laws follows since M is a subset of the group of all bijective maps defined on Xq. Hence, it is necessarily a group.

Finally, note that the correspondence f → f gives a bijection between M2 and

2 2 4 PGL(2, q ) \M1, so that |M| = q (q − 1).

Remark: One can ﬁnd a less descriptive, yet more insightful deﬁnition of T PGL(2, q2) in [7]. We refrain from that line of approach since this description is more useful for calculation purposes. The utility of the other point of view will be explained later.

We prove the next fact that was stated in the previous chapter.

Proposition 2.1.3. The group T P GL(2, q2) acts sharply 3-transitively on the ﬁnite projective line of order q, where q is a prime power.

31 Proof: Consider, as before, any triple of distinct elements (α, β, γ) in X. It suﬃces to show that there is a unique element f ∈ T PGL(2, q2) such that f(α) = 0, f(β) = 1, and f(γ) = ∞.

2 If f := fα,β,γ ∈ PGL(2, q ), where fα,β,γ is the map deﬁned in the proof of the previous proposition satisﬁes f ∈ M1, we are through.

2 2 If not, ∆(f) ∈/ (Fq2 ) . Then fα,β,γ ∈ PGL(2, q ) \M1, and fα,β,γ(α) = 0, fα,β,γ(β) =

1, fα,β,γ(γ) = ∞, since the elements 0, 1, ∞ satisfy x = x. Hence if we write f (x) = ax+b , the map f 0(x) := (a)x+b satisﬁes f 0(α) = 0, f 0(β) = α,β,γ cx+d (c)x+d 0 0 2 1, f (γ) = ∞. Moreover, ∆(f ) = ∆(fα,β,γ ∈/ Fq2 . We have just shown that T PGL(2, q2) acts transitively on the set of all ordered triples in Xq2 . To show sharp 3-transitivity, i.e.,the uniqueness of the map, note that for a group G acting transitively on a set Ω, we have

|G| |Ω| = , |Stab(x)| for any element x ∈ Ω.

In our case, Ω is the set of all pairwise distinct ordered triples of Xq2 , and G = T PGL(2, q2). Since |Ω| = (q2 + 1)q2(q2 − 1) = |G|, it follows that the action is sharp.

The next proposition proves that these two groups are non-isomorphic.

Proposition 2.1.4. T PGL(2, q2) 6' PGL(2, q2).

Proof: We give an elementary proof of this fact. As before, we continue with the same abuse of notation. For each element g ∈ T P GL(2, q2),we ﬁx a representative

32 2 ax+b Ag ∈ GL(2, q ) and by ∆(g) we mean ∆(Ag).If g ∈ M2, and g(x) = cx+d , then a a b 2 representative for g is the element c d ∈ GL(2, q ). We shall prove that T PGL(2, q2) has fewer involutions than PGL(2, q2). Note that

2 2 since M1 = T PGL(2, q ) ∩ PGL(2, q ) as sets, it is enough to show that there are

2 fewer involutions in M2 than in PGL(2, q ) \M1.

2 2 Suppose g ∈ PGL(2, q ) \M1 satisﬁes g = 1, where 1 denotes the identity mapping

2 on Xq2 . By a preceding remark, if A := Ag is a representative for g in GL(2, q ), then A2 is a representative for g2. Since g2 = 1, we have:

a2 + bc = d2 + bc ⇒ a2 = d2, (2.1)

ab + bd = 0, (2.2)

ac + cd = 0, (2.3) where   a b   g A =   is a representative for . c d

Hence we have a + d = 0 or b = c = 0. Suppose a + d = 0, and say b = 1. Then we have     a b a 1     A :=   =   c d c −a

2 2 2 is an involution in PGL(2, q )\M1 if and only if −(a +bc), or equivalently, a +bc is not a square in Fq2 . For each non-square α, any choice of a ∈ Fq2 gives a unique value

2 q2−1 2 2 of c so that a +c = α. Hence there are at least 2 ·q involutions in PGL(2, q )\M1.

2 On the other hand, suppose g ∈ M2 satisﬁes g = 1. Let as before, A := Ag be

33 a representative for g in GL(2, q2). It follows that (a straightforward calculation) a representative for g2 is AA, where     a b a b     A =   , A =   . c d c d

Since g2 = 1, we have:

aa + bc = dd + bc, (2.4)

b a + b d = 0, (2.5)

c a + c d = 0. (2.6)

The second and third equations yield b = c = 0 or c = ηb and c = ηb so that η ∈ Fq.

In either case, we have c = ηb where η ∈ Fq. This reduces the ﬁrst equation to aa = dd. If b 6= 0, the second equation can be rewritten as

a ab + bd = 0 ⇒ d = − b, b and consequently, we have

ax + b ξx + 1 a g(x) := = , where ξ = . a ηx − ξ b ηbx − ( b )b

But then ∆(g) = −(ξξ +η) ∈ Fq since ξξ ∈ Fq which contradicts the assumption that g ∈ M2.

ax 2 If b = c = 0, then g(x) = d = ξx, for some ξ∈ / Fq2 . q2−1 The number of non-square elements in Fq2 = 2 , so the number of involutions in q2−1 M2 is utmost 2 . This proves the claim and completes the proof.

34 Note that the subset M1 is a subgroup since it is the intersection of the groups PGL(2, q2) and T PGL(2, q2). This is the group PSL(2, q2).

Formally, we deﬁne the group PSL(2, q) as the set of all mappings on the ﬁnite projective line of the form ax + b x → cx + d

∗ with a, b, c, d ∈ Fq, ad − bc being a square in Fq.

Remark: The deﬁnition for T PGL(2, q2) in [7] is given in more universal terms ,i.e.,

T PGL(2, q2) is deﬁned thus: The outer automorphism group of PSL(2, q2) contains

PSL(2, q2) < 1, a, b, c >, where {1, a, b, c} is a Klein group of order 4 with PSL(2, q2) <

2 1, a > being isomorphic to PGL(2, q ) and b is the conjugation map on Fq2 . Define T PGL(2, q2) := PSL(2, q2) < 1, c >. This definition is not quite explicit . One reason for giving the elementary definition for T PGL(2, q2) is that it is easier to make calculations explicitly.

The next proposition establishes the construction of designs using the action of these groups.

Proposition 2.1.5. : Suppose G is a t-transitive group acting on a set Ω. Let

Ω B0 ⊂ Ω. Consider the induced action of G on . If O is the orbit of B0 for this |B0| |B0| ( t )|G| action, (Ω, O) is a t− |Ω|, |B0|, |Ω| design, , where G0 := {g ∈ G|g(B0) = B0} ( t )|G0| is the set stabilizer of B0.

Proof: Consider any distinct sets T1,T2 of size t. We shall show that the number of blocks containing T1 is the same as the number of blocks containing T2.

35 Let Bi := {B ∈ O|Ti ⊂ B}, i = 1, 2. Order the elements of T1 and T2 i.e., write

Ti = (xi,1, xi,2, . . . , xi,t) for i = 1, 2. Since G is t-transitive, there exists g0 ∈ G such that g(x1,j) = x2,j for all 1 ≤ j ≤ t.

Note also that every block of O is simply g(B0) for g ∈ G. Hence, X |B1| = 1,

g∈G:T1⊂g(B0) X |B2| = 1.

g∈G:T2⊂g(B0) But we can rewrite the second sum as follows: X X X X |B2| = 1 = 1 = 1 = 1 = |B1|, g∈G:T ⊂g(B ) g∈G:g−1(T )⊂B −1 −1 g∈G:g−1(T )⊂B 2 0 2 0 g0g∈G:g g0 (T2)⊂B0 1 0 and that establishes the fact that (Ω, O) is a t-design for some λ.

To calculate λ note that G acts as a group of automorphisms of the design, i.e., if B is a block, then so is g(B) for any g ∈ G. Hence the number of blocks is |G| . But |G0| v (t) since a t − (v, k, λ design has λ k blocks, the result follows from equating these two (t) expressions and solving for λ.

As a consequence let us construct a few 3-designs. Let q be a prime power and let

2 G:=PGL(2, q ), Ω := Xq2 and B0 := Fq ∪ {∞}.

2 The orbit of B0 in the ﬁnite projective line of order q yields a 3-design. In order to

36 Hence, it follows that λ = 1 and the design is a Steiner design.

The same proof works if we have G = T PGL(2, q) if q is an odd square, with Ω,B0 being the same. In fact, the same idea (with Ω and B0 being the same) works for G = PGL(2, qn) acting on the ﬁnite projective line of order qn. Thus we have the following (classical) result:

Theorem 2.1.6. (Classical Steiner designs): There exist Steiner designs with parameters 3 − (qn + 1, q + 1, 1), where q is a prime power.

Since PGL(2, q) and T PGL(2, q) have the same size, the parameters of the designs from these actions are the same.

We make a few remarks before we move to the next section.

1. As stated in the ﬁrst chapter, these two groups are the only groups which are

sharply 3-transitive. The proof of this fact invokes a fair bit of ﬁnite group

theory and Frobenius’ theorem, which digress from the questions we would like

to focus upon. The upshot of this theorem is that we need to look beyond

sharply 3-transitive groups to construct 3-designs with diﬀerent parameters.

2. There are other groups that act 3-transitively (though not sharply) - for instance

Sn, the symmetric group on n symbols. These groups grow very quickly in size as n increases, so λ for such designs increases.

3. One might also have a union of disjoint orbits (rather than one single orbit) to

37 form t-designs; λ however keeps increasing as one considers designs with several

orbits.

4. If a group acts transitively on subsets of size t, then orbits for that action yield

designs as well. Indeed, the proof that (X, O) is a design only needed the fact

that there exists g0 ∈ G such that g0(T2) = T1. The same holds if the induced action of G on subsets of size t is transitive.

The final remark has some further ramifications. A group is said to act t-homogenously on Ω if if acts transitively on the set of t-subsets of Ω. In the next section we look closely at the action of PSL(2, q) with q an odd prime power, on the finite projective line of order q and investigate its consequences, vis-á-visconstructions of simple designs.

The case q ≡ 3 (mod 4) was studied in some detail by Cameron et al [8]. We shall give a brief outline of the ideas involved there. The case q ≡ 1 (mod 4) presents some further problems. Some of these problems can be circumvented to yield several simple 3-designs.

2.2 The Groups PSL(2, q), q odd

We now turn our attention to another class of classical groups: The Special Projective

Linear groups PSL(2, q).

38 In the rest of this chapter, we let X := Fq ∪ {∞}. We recall that the group PSL(2, q) consists of all mappings of the form:

ax + b x → cx + d

∗ with a, b, c, d ∈ Fq, ad − bc being a square in Fq. We insist that q is odd since if q = 2n, then PSL(2, q)=PGL(2, q). This follows

n q immediately since any element of Fq for q = 2 satisﬁes x = x and is hence necessarily a square.

Proposition 2.2.1. PSL(2, q) acts transitively on subsets of size 3 of X, if and only if q ≡ 3 (mod 4).

Proof: For subsets T1, T2 of X, we say T1 ∼ T2 if and only if there is an element g ∈

PSL(2, q) such that g ·T1 = T2. We start with a simple observation which is quite straightforward to verify; any subset T of X of size 3 satisﬁes

T ∼ {0, ∞, 1} or T ∼ {0, ∞, θ}

∗ where < θ >= Fq, i.e., θ is a primitive root of unity in Fq and for two subsets T1, T2 of order 3.

We note here that {0, ∞, 1} 6∼ {0, ∞, θ} if q ≡ 1 (mod 4) and that accounts for the non-homogeneous action for the case q ≡ 1 (mod 4). Indeed, if we write T0 =

{0, ∞, 1}, then the only maps g ∈ PSL(2, q) such that g(T0) = T0 are

1. g1(x) := x,

x 2. g2(x) := x−1 ,

39 3. g3(x) := 1 − x and the corresponding maps 1 , i = 1, 2, 3. gi θ Thus the only maps g ∈ PGL(2, q) satisfying g(T0) = θT0 are the maps θgi, , i = gi 1 1, 2, 3. Since q ≡ 1 (mod 4), g ∈ PSL(2, q) if and only if g ∈ PSL(2, q), and since gi ∈PSL(2, q), it follows that

{0, ∞, 1} 6∼ {0, ∞, θ}.

The proposition above suggests that we deal with each case separately, which is what we shall do in the next couple of sections.

But before we embark upon each case separately, it is useful to note some general structural results about PSL(2, q). We begin with the following proposition which is a useful step towards understanding the structure of orbits of k-subsets of X induced by the action of PSL(2, q).

Proposition 2.2.2. : Let k ∈ N and consider the action of G := PSL(2, q) on the set of k-subsets of X. If Γ is an orbit for this action then so is θΓ, where θ is a primitive

2 root of unity in Fq. Moreover, θ Γ = Γ.

Proof: Suppose Γ := GB0 = {gB0 : g ∈ G}, where gB0 := {gb : b ∈ B0} for a subset

B0 ⊂ X of order k. We have

n ay + b o Γ = G(B ) = : y ∈ B : ad − bc ∈ ( ∗)2 0 cy + d 0 Fq

40 so that we have

θΓ := {θB : B ∈ Γ} n ay + b o = θ : y ∈ B : ad − bc ∈ ( ∗)2 cy + d 0 Fq n aθy + bθ o = : y ∈ B : ad − bc ∈ ( ∗)2 cy + d 0 Fq

na(yθ) + bθ ∗ 2o = c : y ∈ B0 : ad − bc ∈ (Fq) θ (yθ) + d n ay + b o = : y ∈ θB : ad − bc ∈ ( ∗)2 = G(θB ) cy + d 0 Fq 0

ax+bθ ax+b since the map fθ := x → c ∈ G if and only if the map f := x → ∈ G. θ x+d cx+d

An alternate proof: Let gθ := x → θx ∈ PGL(2, q). Then, θΓ = gθΓ = gθGB0 =

−1 gθGgθ B0 = GgθB0 = GθB0, since G is a subgroup of PGL(2, q) of index 2, and is therefore normal.

As a consequence we have the following corollary, which also appears in [34]. But,

ﬁrst, we make a deﬁnition. For a set Y , by the term Set system of Y , we mean a multiset of subsets of Y . Hence in a set system, one could have several copies of a set.

Corollary 2.2.3. : Let λ∆(T ) denote the number of subsets of the set system ∆ containing the set T . If Γ is an orbit for the action of G on k-subsets of X, we have

λΓ({0, ∞, 1}) = λθΓ({0, ∞, θ}), (2.7)

λθΓ({0, ∞, 1}) = λΓ({0, ∞, θ}). (2.8)

Consequently, if B := Γ ∪ θΓ, then λB({0, ∞, 1}) = λB({0, ∞, θ}).

41 Proof: For the set system ∆, let A∆ := {A ∈ ∆ : 0, ∞, 1 ∈ A} and let B∆ := {B ∈

∆ : 0, ∞, θ ∈ B}. From the previous proposition, it is clear that A ∈ AΓ if and only if θA ∈ BθΓ, so the ﬁrst statement is clear. For the second part, note that B ∈ Γ holds if and only if θB ∈ θΓ. But θB ∈ θΓ if and only if θ−2(θB) = θ−1B ∈ θΓ since

−2 the map x → θ x is an element of PSL(2, q). This implies that B ∈ BΓ if and only

−1 if θ B ∈ AθΓ and that completes the proof.

The last statement follows by adding equations (1) and (2).

In the course of the proof of the results in the later sections, we shall need a few facts

(theorems) of a group-theoretic nature. We list references for convenience.

Theorem 2.2.4. (see [12] [18],(pf 285-286)): The subgroups of PSL(2, q)(q = pn) are as follows:

q±1 1. Cyclic subgroups Cd, d| 2 ,

q±1 2. Dihedral subgroups of order 2d, d| 2 ,

3. A4 (size 12)

2 4. S4 when q ≡ 1 (mod 16) (size 24),

2 5. A5 when q ≡ 1 (mod 5). (size 60),

6. Subgroups PSL(2, pm) with m|n,

7. Subgroups PGL(2, pm) with 2m|n,

42 8. The elementary abelian groups of order pm, m ≤ n,

9. A semidirect product of the elementary abelian group of order pm and the cyclic

q−1 m group of order d where d| 2 and d|(p − 1).

In fact, the theorem in the references cited also count the number of subgroups of each type.

Proposition 2.2.5. Suppose G, H are isomorphic subgroups of PSL(2, q) of types

(1)-(5) in the theorem above. Then G and H are conjugate in PGL(2, q).

2.3 The case q ≡ 3 (mod 4)

We shall only discuss the main ideas involved in this case. Most of the calculations involved are a little tedious, although very straightforward conceptually. We shall only state the relevant results ([8]) and not provide any proofs.

As seen in the previous section, the case q ≡ 3 (mod 4) implies that the action of

PSL(2, q) on the projective line Xq is 3-homogeneous. As remarked at the end of a preceding section, it follows that for any block B of size k,(Xq, OB) is a 3-design, where OB is the orbit of B in the induced action of PSL(2, q) on subsets of size k. One might also have a union of several orbits; this gives a design with more blocks, and subsequently, a larger value of λ.

To determine all possible λ that result by forming designs as described above, we need to calculate the size of the stabilizer for an arbitrary block B. Since the stabilizer of a block is a subgroup of PSL(2, q), it suﬃces to determine all possible blocks that

43 admit a given subgroup G as its stabilizer. Since all the subgroups of PSL(2, q) are known, this can be carried out in principle.

In mathematical terms, if G ⊂ PSL(2, q) is a subgroup, then we are interested in the set

BG := {B ⊂ Xq|G is the stabilizer of B} .

However, it is a little diﬃcult to describe all the blocks that admit a given group as its stabilizer. Since we are more interested in classifying all possible values of λ such that there is a simple design (no repeated blocks) by taking unions of orbits, it suﬃces to calculate |BG|.

In practice, it is often easier to work the other way around, i.e., given a group G, it is easier to count the number of subsets that are ﬁxed by G, i.e., suppose

X f (G) : = |{B ∈ q |g(B) = B for all g ∈ G}|. k k X g (G) : = |{B ∈ q |G is the stabilizer of B}|, k k then it is easier to calculate fk(G). Furthermore, we have

X fk(G) = gk(H). G⊆H⊆PSL(2,q)

To solve for gk(G) we simply use the principle of M¨obiusinversion:

X gk(G) = fk(H)µ(G, H), G⊆H⊆PSL(2,q) where µ(·, ·) is the M¨obiusfunction on the lattice of subgroups of PSL(2, q).

Note that if G is the stabilizer of a block B, then for any other block B0 in the orbit

0 OG of B, the stabilizer of B is a conjugate of G.

44 We now wish to count the number of k-orbits that admit as its stabilizer, a group isomorphic to G. Counting the number of pairs (O,B) where B is a block in the orbit O which has a stabilizer isomorphic to G, we have

X X |PSL(2, q)| X q(q2 − 1) q(q2 − 1) |O| = = = N · , |Stab(O)| 2|G| G 2|G| O O O where NG denotes the number of orbits whose stabilizer is isomorphic to G. On the other hand, this sum equals gk(B)ψ(G), where ψ(G) is the number of subgroups of PSL(2, q) that are isomorphic to G. Equating these, we have

2ψ(G)g (B)|G| N = k . G q(q2 − 1)

By calculating the M¨obius function for PSL(2, q), and fk(G) for each G (This is easier said than done, though!), one may calculate λ in principle. The authors in [8] give a list of all possible designs (possible values of λ) that can obtained in this process by a union of orbits with each orbit having as its stabilizer, a group of types (1)-(5) in the list of subgroups of PSL(2, q) as listed in the previous section. The authors restrict to these subgroups because any two subgroups of these types are isomorphic if and only if they are conjugate in PGL(2, q) so that it is easier to count the number of subgroups of G isomorphic to G.

Theorem 2.3.1. (Cameron et al,[8]): Let 3 ≤ k ≤ q − 2 and k 6≡ 0, 1 (mod p)

n k (here q = p ). Then there exist 3 − (q + 1, k, 3λ 3 ) designs with automorphism group PSL(2, q) if and only if

a2 a3 a4 a5 X id X jd λ = a + + + + + + 1 4 12 24 60 d 2d q±1 q±1 d>1,d| 2 d>2,d| 2

45 2gk(1) where a1 ≤ q(q2−1) , a2 ≤ gk(D4)/3, a3 ≤ gk(A4), a4 ≤ 2gk(S4), a5 ≤ 2gk(A5), id ≤

dgk(Cd) q±1 , jd ≤ gk(D2d) with ai, id, jd being non-negative integers.

Before we move on, we make a few remarks.

1. It must be noted that though the above theorem of Cameron et al is very

comprehensive, it is rather diﬃcult to establish a more arithmetic formula for

λ. Hence, in practice, one still has to calculate orbit stabilizers for chosen

blocks.

2. One can also do a similar calculation for the groups PGL(2, q) as in [9]. Some

more concrete and arithmetically easier answers are available in some speciﬁc

cases (see [35],[36]).

The case q ≡ 1 (mod 4) is different since the action is not 3-homogenous. Our attention in this case however, is not so much a general set up ála Cameron et al; we seek to construct simple 3-designs on the finite projective line that admit PSL(2, q) as a group of automorphisms. Often we only deal with two orbits. The details follow in the next section.

2.4 The case q ≡ 1 (mod 4)

Since we are dealing with the case q ≡ 1 (mod 4), we have q2 ≡ 1 (mod 16) if and only if q ≡ 1 (mod 8), and q2 ≡ 1 (mod 5) if and only if q ≡ 1, 9 (mod 20).

We are now ready to state our ﬁrst theorem.

46 Theorem 2.4.1. : Let q ≡ 13 (mod 16) or q ≡ 1 (mod 16) be a prime with q > 121.

q−1 (k−1)(k−2) Let k = 4 . Then there exists a non-trivial simple 3 − (q + 1, k, 2 ) design. Further, if q ≡ 13 (mod 16) then this value of λ is minimal, i.e., for any simple

(k−1)(k−2) 3 − (q + 1, k, λ) design, 2 divides λ.

Proof : Note that q ≡ 1 (mod 4) holds, by the hypothesis. Consider the projective

∗ line X := Fq ∪ {∞} and the group G := PSL(2, q) acting on X. Let θ ∈ Fq be a primitive root of unity and let B := {1, θ4, θ8, θ12, . . . , θ(q−5)}. Let Γ denote the orbit,

X Γ := GB of the induced action of G on k , the set of k-subsets of X. Consider the set system B := Γ ∪ θΓ. From corollary 2 it follows that (X, B) is a 3-design with parameters 3 − (q + 1, k, λ0) for some λ0. To calculate λ0, note that,

|B| = 2|Γ| (2.9) |PSL(2, q)| = 2 (2.10) |GB| (q2 − 1)q (q2 − 1)q = 2 = . (2.11) 2|GB| |GB| (q + 1)q(q − 1) On the other hand, |B| = λ since (X, B) is a 3-design (simple or 0 k(k − 1)(k − 2) not). Comparing the two expressions, we get

k(k − 1)(k − 2) λ0 = . |GB|

Since q ≡ 13 (mod 16) or 1 (mod 16), it follows that k ≡ 0, −1 (mod 4). Now note

4 1 that for the block B as before, the following maps f(x) := θ x, g(x) := x are both −1 members of GB. Since gf = f g it follows that D2k, the dihedral group of size 2k, is

47 (k−1)(k−2) contained in GB. Hence |GB| is divisible by 2k and λ0| 2 . By the hypothesis, (k−1)(k−2) 2 is odd and that implies that Γ 6= θΓ. Therefore, the design is simple. To complete our calculation of λ0, we make use of the list of subgroups of G. Since q is prime, there are no nontrivial subgroups of types 6 and 7 from the list in the theorem. Since D2k ⊂ GB, the group GB cannot be of types 1 or 8. By size considerations (from the hypothesis on q), types 3, 4 and 5 are also ruled out. Finally, for type 9, note that such a subgroup of G has size, a multiple of q. But we have

k(k−1)(k−2) k(k−1)(k−2) |GB|| 2 , so that we have q| 2 and that is a contradiction since q is prime and q > k.

Hence it follows that GB is a dihedral group containing D2k. But again, from the list of subgroups of PSL(2, q), the dihedral subgroups of G are the groups D2d where d|2k + 1 or d|2k, since q = 4k + 1. Since D2k ⊂ GB, we must have GB = D2k or

(k−1)(k−2) GB = D4k. The later is ruled out since 2 is odd. Therefore, we have GB = D2k (k−1)(k−2) and λ0 = 2 .

Now, for any 3 − (q + 1, k, λ) design, we have the arithmetic conditions

• k(k − 1)(k − 2) | λ(q + 1)q(q − 1),

• (k − 1)(k − 2) | λq(q − 1),

• (k − 2) | λ(q − 1).

The hypothesis implies that k is odd so that (k, k−2) = 1. Also, from the assumptions on q, we have (k − 1, q − 1) = (k − 1, 4k) = 2. It is now a simple check to see that

48 (k−1)(k−2) 2 |λ. Since the designs constructed achieve this value of λ, the designs are minimal, as claimed.

And last but not the least, it is quite simple to see that this is not the trivial design either. Our next theorem is a companion theorem to the one above.

Theorem 2.4.2. Let q ≡ 5 (mod 16) or q ≡ 9 (mod 16) be a prime with q > 121.

q−1 (k+1)(k+2) Let k = 4 . Then there exists a non-trivial simple 3−(q +1, k +2, 2 ) design. Further, if q ≡ 1 (mod 12) and q ≡ 5 (mod 16) then this value of λ is minimal, i.e.,

(k+1)(k+2) for any simple 3 − (q + 1, k + 2, λ) design, 2 divides λ.

Proof: As in the proof of the previous theorem, we deﬁne the block B0 as follows:

4 8 12 (q−5) B0 := {0, ∞} ∪ {1, θ , θ , θ , . . . , θ }, B := Γ ∪ θΓ, where Γ := GB.

∗ where as before, Fq =< θ >, i.e., θ is a primitive root of unity. We note that the exact same proof (including the maps f, g) of the previous theorem works in this case

(k+1)(k+2) as well. Since q ≡ 5 (mod 16) or q ≡ 9 (mod 16), it follows that 2 is odd, so that (X, B) is indeed a simple 3 − (q + 1, k + 2, λ) design for some λ. Again, the same

(k+1)(k+2) proof as that of the previous theorem proves that GB = D2k and λ = 2 .

To prove the last assertion on the minimality of these designs under some further arithmetic conditions, note that by the hypothesis on q, it follows that

1. k is odd,

2. (4k+2, k+2) = 1, since if d = (4k+2, k+2), then d|4(k+2)−(4k+2) = 6. Since

49 k is odd (so that k + 2 is odd as well), it follows that d|3. But the hypothesis

on q implies that k ≡ 0 (mod 3) and hence k + 2 ≡ 2 (mod 3).

(k+1)(k+2) The same argument as in the proof of the previous theorem proves that 2 |λ for any 3 − (q + 1, k + 2, λ) design which proves minimality.

Remark: It is possible to improve upon the result stated in the theorem above by ruling out possibilities other than D2k for GB with some more detailed calculation, for smaller values of q. We shall however not get into those details now.

We now show another application of the same idea to obtain another inﬁnite family of simple 3-designs. This family has a ﬁxed block size of 7.

Theorem 2.4.3. : Let q ≡ 1 (mod 20) be a prime. Then there exists a non-trivial simple 3-design 3 − (q + 1, 7, 21).

∗ Proof: Suppose q = 20r + 1. As before, let θ be a primitive root of unity in Fq. Let

4r 8r 12r 16r B0 = {0, ∞} ∪ {1, θ , θ , θ , θ }.

Again, as in the previous theorem, consider the set system B := Γ∪θΓ where Γ := GB0 is the orbit of B0 under the action of G := PSL(2, q). Again (X, B) is a 3-design 3 − (q + 1, 7, λ) with λ = 7·6·5 . |GB0 | 1 Now note that the map f := x → x is an element of PSL(2, q) when q ≡ 1 (mod 4) and that f(B0) = B0, so that 2||GB0 |(since f ◦ f = I, where I is the identity map on X). Then clearly, λ is odd which implies that Γ 6= θΓ. So, the design is simple as before. It remains to check that λ = 21. Note also that h := x → θ4rx also

50 2 5 stabilizes B0 and hence H :=< f, h >⊂ GB0 . Since f = 1, h = 1, and as before,

−1 fh(x) = h f(x), so that H ' D10. It follows that λ|21. We now make use of the following simple proposition; this can be found in [?], for instance, so we omit the proof.

Proposition 2.4.4. : Let g be an element of G(= PSL(2, q)) of order m. Then the number of ﬁxed elements χ(g) by g, in X(= Fq ∪ {∞}), is given by

1. χ(g) = 1 if m = p, q = pn for some n.

q−1 2. χ(g) = 2 if m| 2 .

q+1 3. χ(g) = 0 if m| 2 .

Now if the subgroup GB0 contains an element of order 3, then it has to ﬁx exactly one element of B0. But since q is a prime and q > 3, the proposition above gives us a

contradiction. Next, suppose that GB0 contains an element g of order 7. Since the set

B0 is partitioned as a union of cycles for the action of the cyclic subgroup generated by g with each cycle size being a multiple of 7, no proper subset of B0 is invariant under the action of g.

In this case, 2, 5, 7||GB0 |, so that GB0 is a non-abelian group (since it contains H) whose size is at least 70. From the list of subgroups of G (see the theorem stated in

the preliminary section), it follows that GB0 is a dihedral group and hence g centralizes h, i.e., gh = hg.

Consequently, g(0) = g(h(0)) = h(g(0)) ,so, g(0) is fixed by h. Since g has no fixed points, we have g(0) = ∞. By the same token, we get g(∞) = 0. Hence g fixes the subset {0, ∞} of B0 which is a contradiction.

51 Consequently, we have GB0 = H ' D10 and that completes the proof. .

Remark:An important point in the preceding proof is this: In fact taking a block

B0 satisfying B0 = −B0, and k ≡ 3 (mod 4) always gives us a simple design by this technique, with the only restriction on q being it is prime. However, calculation of λ depends on the choice of B0.

Remark: The union Γ ∪ θΓ in fact turns out to be simply an orbit for the action of the action of PGL(2, q). The reason we are more interested in dealing with the action of PSL(2, q) is due to the fact that for q prime, the lattice of subgroups of

PSL(2, q) is well known and hence one could, as in [8], calculate the M¨obius function for PSL(2, q) in less messy a fashion than for PGL(2, q). In principle, one could list the subgroups of PGL(2, q) using the fact that there is an embedding PSL(2, q) <

PGL(2, q) < PSL(2, q2) and use the M¨obiusfunction from PSL(2, q) to calculate the

M¨obiusfunction on the lattice of subgroups of PGL(2, q) as well.

Another aspect of our point of view is that we are speciﬁcally interested in the nature of the action of PSL(2, q) on the orbits of PGL(2, q). We shall,shortly, state this more precisely .

Note that if for some B, we have Γ = θΓ, where Γ is the orbit of B, then, from the preceding discussions, (X, Γ) would be a simple 3-design. Hence we can always obtain a simple design by either simply taking an orbit Γ or taking the union Γ ∪ θΓ.

∗ It is possible that Γ = θΓ for a non-trivial block B (B = Fq is the trivial case). We indicate two such instances here.

52 ∗ • k = 4: If 2 is a primitive root of unity in Fq, we could take θ = 2. Consider the x−1 block B = {0, 1, 2, ∞}. The map f(x) := 4 x satisﬁes f(0) = ∞, f(∞) = 4, f(1) = 0 and f(2) = 2, so that f(B) = θB and it is clear that f ∈ PSL(2, q).

• k = 5: If q ≡ 5 (mod 8), consider the block B0 = {0, θ, ∞, θα, θβ}, where

i 2 ∗ α−1 x α = i−1 with i = −1 in Fq, and β = α . Then the map f(x) := β x−θ satisﬁes, f(0) = 0, f(θ) = ∞, f(∞) = β, f(θα) = 1 and f(θβ) = α so that

−1 −1 ∗ f(B) = θ B (note that θ also is a primitive root of unity in Fq). Again, it is easy to see that f ∈ G.The fact that f ∈ G follows from the assumption that

∗ i (as above) does not have a square root in Fq.

We shall say a little more on this in the last section.

The arguments in the preceding theorems can be viewed from a more general per- spective as follows. Fix k ≥ 4 and consider the induced action of PSL(2, q) on the set

X X of k-subsets of X, namely, k . The set k is partitioned into orbits Γi, i = 1, . . . , r for some r, so that we have X [ = Γ . k i i

Let O = {Γ1, Γ2,..., Γr}. Then the content of proposition 1 simply implies that the

∗ group Fq acts on O as ∗ g · Γ = gΓ, for g ∈ Fq, Γ ∈ O.

Thus we can write [ O = Oi i∈I

53 ∗ where each Oi is an orbit of O under the action of Fq. Furthermore, since the map

2 x → θ x is an element of PSL(2, q), it follows that |Oi| = 1 or 2. Hence each orbit for the action described above gives us a simple 3-design with

PSL(2, q) acting as a group of automorphisms.

We now work to understand this viewpoint, a little more concretely. The next proposition is a starting step of sorts; it can also be viewed as an independent result.

Proposition 2.4.5. : Let q ≡ 1 (mod 8) be an odd prime. Let B0 := {0, ∞, 1, −1}

∗ and Γ = GB0. Then O := O(Γ), the orbit of Γ under the action of Fq as described above has size 2. Consequently, there exist simple 3-designs 3 − (q + 1, 4, 3) for q ≡ 1

(mod 8) admitting PSL(2, q) as an automorphism group.

Proof: Proposition 1 yields that (X, B) is a 3 − (q + 1, 4, λ)-design (simple or not) for some λ, where B = Γ ∪ θΓ, and θ is a primitive root of unity. From the proof of theorem 3, we have λ = 4·3·2 . Consider the maps f(x) := −x, g(x) := 1 and |GB0 | x 1+x h(x) := − 1−x in G = PSL(2, q). The claim that f, g ∈ G follows from the fact that ∗ −1 is a square in Fq and that h ∈ G follows, if 2 is a square in Fq which is indeed the case, by the hypothesis. It is straightforward to verify that f ﬁxes the subset

{1, −1} and fixes 0 and ∞, g fixes the set {0, ∞} and fixes 1, −1 and finally, h fixes the sets {1, ∞} and {0, −1} but fixes no point. Hence G :=< f, g, h > fixes B0 so

that G ⊂ GB0 . Also, it is simple to see that |G| = 8.

Note that in particular, the group G acts transitively on B0.

Now, if GB0 has an element of order 3, then by conjugation with an element of G, if necessary, we may assume that it has one such element which ﬁxes ∞ since any

54 element of order 3 ﬁxes exactly one element of B0. Taking its square, if necessary, we may assume that it acts on B0 as (1 0 − 1)(∞). But g := x → x − 1 is the unique element of G satisfying g(1) = 0, g(0) = −1, g(∞) = ∞ and g(−1) = −2 6= 1 since q is prime and q ≥ 4.

This contradicts the assumption of the existence of an element of GB0 of order 3. Finally since λ is odd, it follows that |O| = 2 and hence Γ 6= θΓ and so (X, B) is a simple design. . Our next result is an ‘existential’ result for simple 3-designs inasmuch as the exact value of λ is not speciﬁed.

Theorem 2.4.6. : For any k ﬁxed, k ≥ 4, and q, a prime satisfying q ≡ 1 (mod 4) and q being suﬃciently large, i.e., for q ≥ q0(k) for some integer q0(k), there exists a set B0 ⊂ X of size k such that the orbits Γ and θΓ are distinct, where, Γ := G(B0), where G = PSL(2, q). Consequently, there exist simple 3 − (q + 1, k, λ) designs for all

‘suﬃciently large’ q.

Remark: The preceding proposition considers the case k = 4 and q ≡ 1 (mod 8).

In that case, we did exhibit a concrete instance of a set B0 and were also able to compute λ.

Proof: We show that for each k ≥ 4, there exists B0 ⊂ X, such that there is no element g ∈ G satisfying, gB0 = θB0, where as always, θB0 := {θx : x ∈ B0}. Let us ﬁrst consider the case k = 4. Note that by the observation in the Preliminaries section, we know that {0, ∞, 1} and {0, ∞, θ} are G-inequivalent, i.e., there exists no g ∈ G such that g {0, 1, ∞} = {0, θ, ∞}. Consider the sets By := {0, 1, ∞, y} with

55 ∗ ∗ y ∈ Fq \{1}. We shall prove that for some η ∈ Fq \{1}, the sets Bη and θBη are G-inequivalent.

Suppose there exists g ∈ G such that gBy = θBy. For the sake of notation we write

g(0) = α, (2.12)

g(1) = β, (2.13)

g(∞) = γ, and (2.14)

g(y) = δ, (2.15)

where {α, β, γ, δ} = θBy. Note that if δ = θy, then g maps the set {0, 1, ∞} into {0, θ, ∞} which, by the observation above is impossible. Hence, θy 6= δ.

Before we proceed, we introduce some further notation here; ±i denote the elements

∗ 2 ∗ in Fq satisfying i = −1 (such elements exist in Fq since q ≡ 1 (mod 4)). Similarly, √ ∗ x = ± 3 are the elements (such elements exist if and only if q ≡ 1 (mod 3)) of Fq satisfying x2 = 3.

The following table presents a list of all the (revised in light of this observation) possibilities as we vary over the ordered quadruple, (α, β, γ, δ).

56 (α, β, γ, δ) g g ∈ PGL(2, q) iﬀ y = g ∈ PSL(2, q) iﬀ

(θy, θ, ∞, 0) −θ(x − y) 2 Never in PSL(2, q) √ θ(1−y) 1±i 3 (θy, θ, 0, ∞) x−y 2 Never in PSL(2, q)

x−1 ∗ 2 (θy, 0, θ, ∞) θ x−y −1 y − 1 ∈/ (Fq)

x−y ∗ 2 (θy, ∞, θ, 0) θ x−1 g ∈ PGL(2, q) y − 1 ∈/ (Fq) √ θ(x−1) 1±i 3 (θy, 0, ∞, θ) y−1 2 Never in PSL(2, q)

θ(y−1) 1 ∗ 2 (θy, ∞, 0, θ) x−1 2 y − 1 ∈/ (Fq) (0, θy, ∞, θ) (θy)x ±1 Never in PSL(2, q) √ x 1±i 3 ∗ 2 (0, θy, θ, ∞) θ x−y 2 y∈ / (Fq)

θy ∗ 2 (∞, θy, 0, θ) x g ∈ PGL(2, q) y∈ / (Fq)

x−y 1 ∗ 2 (∞, θy, θ, 0) θ x 2 y∈ / (Fq)

−θy ∗ 2 (θ, θy, 0, ∞) x−y 2 y∈ / (Fq) √ −θ(x−y) 1±i 3 ∗ 2 (θ, θy, ∞, 0) y 2 y∈ / (Fq) √ x 1±i 3 ∗ 2 (0, ∞, θy, θ) θy x−1 2 y∈ / (Fq)

x 1 (0, θ, θy, ∞) θy x−y 2 Never in PSL(2, q)

θy(x−1) ∗ 2 (∞, 0, θy, θ) x 2 y∈ / (Fq) √ θy(x−y) 1±i 3 (∞, θ, θy, 0) x 2 Never in PSL(2, q)

x−1 ∗ 2 (θ, 0, θy, ∞) θy x−y g ∈ PGL(2, q) y(y − 1) ∈/ (Fq)

x−y ∗ 2 (θ, ∞, θy, 0) θy x−1 −1 y(y − 1) ∈/ (Fq) The table above is to be read as follows: for instance, the third row in the table tells us that corresponding to the quadruple (α, β, γ, δ) = (θy, 0, θ, ∞), there exists g ∈ PGL(2, q) satisfying g(0) = θy, g(1) = 0, g(∞) = θ, g(y) = ∞ if and only if

57 x − 1 y = −1 and g(x) := θ . Further, this element of PGL(2, q) is also an element x − y ∗ 2 of G if and only if y − 1 = −2 ∈/ (Fq) . Similarly, row 4 of the table corresponds to x − y the case where the map g(x) := θ is always an element of PGL(2, q) but is an x − 1 ∗ 2 element of G if and only if (y − 1) ∈/ (Fq) .

∗ 2 Note that if q is a prime satisfying q ≡ 1 (mod 8), we have 2 ∈ (Fq) , so that the case B := {1, 0, ∞, −1} (row 3 of the table) corresponds to the statement of the preceding theorem.

∗ In order that Bη and θBη are G-inequivalent, we need η ∈ Fq such that all the conditions in the last column of the table are violated. By an inspection of the table,

∗ we conclude that if there exists η ∈ Fq \{1} such that η and (η −1) are both non-zero squares in Fq, then all the conditions of the last column are simultaneously violated, so that the set Bη cannot be mapped by an element of G to the set θBη.

To see that, we ﬁrst prove a very simple lemma.

Lemma 2.4.7. : Suppose n > 1. Then a set A ⊂ {1, 2,..., 4n} satisfying

1. |A| = 2n,

2. x ∈ A if and only if 4n + 1 − x ∈ A,

3. 1, 4 ∈ A, contains two consecutive elements.

Proof of the lemma: By condition 2, the set A contains precisely n elements among {1, 2,..., 2n}. If 2n ∈ A, then 2n + 1 ∈ A by condition 2 and we are through.

58 Similarly, if {2, 3, 5} ∩ A 6= ∅, then we are through since 1, 4 ∈ A. Hence, suppose

2, 3, 5, 2n∈ / A. In particular, n ≥ 3. If n ≥ 4, then since the set {6,..., 2n − 1} can be partitioned into n − 3 disjoint pairs (6, 7), (8, 9) ..., (2n − 2, 2n − 1), and

|A∩{6,..., 2n−1}| = n−2 by the observation above, the lemma follows by the pigeon- hole principle. For n = 3, since |A ∩ {1, 2 ..., 6}| = 3 and 2, 3, 5 ∈/ A, 6 = 2 · 3 ∈ A but then 7 ∈ A by condition 2 and we are through. ∗ Now, for q > 5 prime and q ≡ 1 (mod 4), we can represent Fq by the set of residues

∗ congruent modulo q, so that we may write Fq := {1, 2, . . . , q − 1}. Since q ≡ 1

∗ 2 (mod 4), A := (Fq) satisﬁes the conditions 1,2 and 3 of the lemma above. Hence

∗ 2 ∗ 2 there exists η ∈ (Fq) such that η − 1 ∈ (Fq) . By the observation made before the lemma, the proof of theorem 8 for the case k = 4 is complete.

Suppose now that k ≥ 4. By the proof of the part for k = 4, we know that there

∗ 2 exists an element η ∈ (Fq) such that Bη and θBη are G-inequivalent. Fix one such η. In order to prove the theorem for k ≥ 4, we prove the following stronger statement: For q, a ‘suﬃciently large’ prime (we shall see a more precise meaning of this in the course of the proof) satisfying q ≡ 1 (mod 4), there exists a set B0 ⊂ X of size k, containing the set Bη = {0, 1, ∞, η} such that no g ∈ PSL(2, q) satisﬁes g(0) = θα, g(1) = θβ, g(η) = θγ and g(∞) = θδ for all 4-tuples of distinct elements

(α, β, γ, δ) in B0. Clearly, this stronger statement proves our theorem since for such a B0, it must necessarily follow that B0 and θB0 are G-inequivalent. We prove this by induction on k. The case k = 4 has already been settled. Hence let

0 k > 4. Suppose by the induction hypothesis that we have a set B0 of size k − 1 that contains Bη and satisﬁes the conditions of the stronger statement.

59 Let (α0, β0, γ0) be an ordered triple of the elements of Bη and (α, β, γ), an ordered

0 triple of the elements of B0. Since PGL(2, q) acts sharply 3-transitively on X, there exists a unique element g ∈ PGL(2, q) such that g(α0) = θα, g(β0) = θβ and g(γ0) =

θγ. Hence there is at most one element δ = δ(α0,β0,γ0)(α, β, γ) in X such that g(δ0) =

θδ, where {δ0} = Bη \{α0, β0, γ0}. Let

0 Y(α0,β0,γ0) := {δ(α0,β0,γ0)(α, β, γ): α, β, γ ∈ B0 and α, β, γ are distinct},

0 [ Y := B0 ∪ Y(α0,β0,γ0),

(α0,β0,γ0) where the union is over all the ordered triples (α0, β0, γ0) of distinct elements of Bη.

0 ∗ 0 Now, if Y := Fq \Y 6= ∅, there exists an element δ ∈ Y , which by deﬁnition, satisﬁes

∗ the following: For all 4-tuples (α1, β1, γ1, δ1) with {α1, β1, γ1, δ1} = {α, β, γ, δ } where

0 α, β, γ ∈ B0 and distinct, there exists no element g ∈ PSL(2, q) satisfying g(0) =

θα1, g(1) = θβ1, g(η) = θγ1 and g(∞) = θδ1. Indeed, if not, then for some α, β and γ

0 ∗ in B0, we have δ ∈ Y(α0,β0,γ0) where (α0, β0, γ0) is a suitable permutation of (α, β, γ) and that is a contradiction.

0 0 Set B0 := B0 ∪ {δ0}. Since B0 satisﬁes the inductive hypothesis, it now follows that

B0 fulﬁlls the conditions of the stronger statement as a consequence of the preceding statement.

3 0 3 Finally, note that since |Y | = O(k ), we have Y 6= ∅ if q > q0(k) = ck for some suitably large constant c (independent of k). That completes the induction and the proof of theorem 8.

The theorem proven above also holds if q is a prime power. Indeed, the induction

60 step never explicitly needs the fact that q is prime. Thus to prove the same statement for prime powers, it suﬃces to prove the same for the case k = 4. This can be proven in exactly the same lines as in the proof above. In fact, if q = pn, for a prime p,

∗ the proof in the theorem above guarantees the existence of η ∈ Fp satisfying the conditions required in the proof, if p ≡ 1 (mod 4). If p ≡ 3 (mod 4), then since

2 n ∗ 2 q ≡ 1 (mod 4), it follows that q = (p ) for some n. Hence Fp2 ⊂ Fq, and Fp ⊂ Fp2 ,

∗ any choice of η ∈ Fp, η 6= 1 satisﬁes the requirements of the proposition.

Remark: One can actually count exactly the number of choices for η with a little more work. If p ≡ 3 (mod 4) is a prime, then consider the number of solutions n of

2 2 a the equation X − Y = 1 in Fp. If ( p ) denotes the Legendre symbol, then the fact 2 a that the equation X = a has 1 + ( p ) solutions gives us,

X a b n = 1 + 1 + p p (a,b): a−b=1 X a b ab = 1 + + + p p p (a,b): a−b=1 X ab = p + , p (a,b): a−b=1 since the other two sums are zero. In order to compute the sum, let

X ab S = u p (a,b): a−b=u

61 ∗ for u ∈ Fp. Then X ab S = u p (a,b): a−b=u X ab = p (a,b): a/u−b/u=1 X au · bu = = S . p 1 (a,b): a−b=1

X X X a(u − a) X a X (u − a) Moreover, S = = = 0, so we u p p p u∈Fp u∈Fp a∈Fp a∈Fp u∈Fp have (p − 1)S1 + S0 = 0. But

2 X −1 a S = = −(p − 1). 0 p p a∈Fp 2 2 Hence, S1 = p−1. But this counts the number of solutions of the equation X −Y = 1 which includes the trivial solutions (±1, 0). Also since if (x, y) is a solution, so are

∗ 2 (±x, ±y), we conclude that the number of choices of η ∈ Fp such that η, η − 1 ∈ (Fp) p−3 is 4 > 0 if p > 3. p−5 If p ≡ 1 (mod 4), then a similar calculation yields 4 choices for η.

The calculation performed above is in fact standard number theory fare and is one of the many techniques to compute Gaussian sums over ﬁnite ﬁelds.

2.5 Large sets of 3-DDs

We consider an application of the ideas discussed in the preceding section and demonstrate an instance whereby one can obtain a large set of Divisible Designs. By a 3 -

Divisible Design or 3 − DD, we mean a triple (Y, Γ, B) satisfying:

62 1. Y is a set of size v,

2. Γ = {G1,G2,...} is a partition of Y into non-empty subsets (called groups or point classes),

3. B is a family of subsets of Y (called blocks), each of cardinality k such that each

block intersects any group in fewer than two points,

4. Each 3-subset of points with each point from a diﬀerent group is contained in

a unique block.

In the general deﬁnition of a Divisible design, the groups need not be equal in size; however in our constructions, that shall be the case.

By a Large set of 3 − DDs on a set Y admitting a partition Γ into groups, we mean a partition (B1, B2,..., Br) of the set

B := {B ⊂ Y : |B| = k, |B ∩ Gi| ≤ 1 for all i}

where, for each i, (Y, Γ, Bi) is a 3 − DD. We now show a construction for a Large set of DDs with block size k, for any k ≥ 4 and q + 1 groups, where q is a prime power.

Theorem 2.5.1. : Let q be a prime power. Then there exists λ such that for all

q+1 d ≥ 3 , there exists a Large set of 3 − DDs with q + 1 groups and ﬁxed group size of λd.

63 Proof: We ﬁrst consider the case q ≡ 3 (mod 4). Consider the induced action of

PSL(2, q) on the set of k subsets of X := Fq ∪ {∞}. As seen in section 3, r X [ = O , k i i=1 where each Oi is an orbit of this induced action and as mentioned in the introduction,

(X, Oi) is a 3 − (q + 1, k, λi) for some λi. Consider the set of designs (X, Ai), where

lcm(λ1,λ2,...,λr) ni = and Ai := ni · Oi denotes the design where each block of Oi is λi repeated ni times. Writing n = lcm(λ1, λ2, . . . , λr), we have, by the choice of the ni’s, the conclusion that each (X, Ai) is a design 3 − (q + 1, k, n). q+1 Now for any d ≥ 3 , consider the set Y := X × Ind , where In denotes the set {1, 2, . . . , n}, and the family of 3-subsets n o T := {(x, i1), (y, i2), (z, i3)}, x, y, z ∈ X, x, y, z distinct and i1, i2, i3 ∈ Ind .

By the method of ‘block spreading’ (see [23]), there exists a partial design (X×Ind , Bi) such that every element of T is contained in exactly one block of Bi. Moreover since each member of Bi has size k and contains at most one element from each set

Ix := {x} × Ind , it follows that (X × Ind , {Ix : x ∈ X}, Bi) is a 3 − DD with block Sr size k. Moreover, since (X, Oi) is a 3-design for each i, it follows that i=1 Bi = B, as claimed.

The case of q ≡ 1 (mod 4) is entirely similar; of course, here each (X, Oi) is not necessarily a 3-design, but, as seen in section 3, the set of orbits O := {Oi : 1 ≤ i ≤ r}

∗ ∗ is partitioned into orbits for the action of Fq on O and each orbit for the action of Fq gives us a simple design. The same procedure as above gives us the desired result.

64 Remark: The condition on d arises as one of the conditions of the statement in

[26]. Roughly speaking, the blocks of Bi project onto the blocks of Ai and hence the name, ‘Block Spreading’. We will return to the notion of Block spreading in the last chapter.

2.6 Concluding remarks

We now turn our attention to a couple of questions from the preceding sections.

Denniston([11]) ﬁrst used the 3-homogeneous action of PSL(2, q) to obtain, in fact,

Steiner 5-designs on q + 1 points for some values of q (q = 27, 47, 83) which was emulated by others ([13], [?]) to obtain similar results. It is indeed desirous to see if one can suitably also obtain Steiner 5-designs for q ≡ 1 (mod 4).

In general, the quest for Steiner 3-designs admitting ‘large’ automorphism groups has one possible approach; this is in the spirit of [11].

All sharply 2-transitive groups are known; these are precisely affine groups of trans- formations over near-fields (This is due to Zassenhaus who classified all near-fields).

If a design (X, B) admits a sharply 2-transitive group G as a group of automorphisms and for some x0, y0 ∈ X, the collection

Bx0,y0 := {B \{x0, y0}|x0, y0 ∈ B ∈ B}

is a 1-design on X \{x0, y0}, then the design (X, B) is in fact a Steiner 3-design. It is reasonable enough to expect that such constructions for Steiner 3-designs may yield fruitful results. Concrete instances of (in fact Steiner 5-designs) are due to

Denniston([11]), Mills([23]) and many others.

65 Coming back to an earlier point, the occurrence of blocks B0 for which the corresponding orbit Γ satisﬁes Γ = θΓ is certainly a non-trivial possibility. As seen earlier, the orbits of these blocks give a 3-design and we have already seen instances with k = 4, 5. Ad-hoc methods allow for constructions of such blocks in many of these situations as well. For instance, suppose q ≡ 1 (mod 40) or q ≡ 9 (mod 40). Then,

∗ q ≡ ±1 (mod 5), so that 5 is a square in Fq. √ 3+ 5 2 Let α := 2 (so that α satisﬁes the equation α − 3α + 1 = 0) and β := α − 1. α−1 β Note that α = α−2 = β−1 . Consider the block {0, ∞, θ, α, β} and the map

x − θ g := x → α ∈ PGL(2, q). x

It is easy to see that g satisﬁes g(θ) = 0, g(0) = ∞, g(∞) = α, g(θα) = β and g(θβ) =

∗ 1. Note that if α is a non-square in Fq then g ∈ G(= PSL(2, q)). One could also start √ 3− 5 with α := 2 to obtain the same conclusion.

It seems likely that the same holds for arbitrary values of k, i.e., one can always construct a block B0 of any ﬁxed size k for which Γ = θΓ holds, where Γ = G(B0). It is worth characterizing the set,

X B := B ∈ : Γ = θΓ, Γ = G(B) and Stab(B) = G G k for small subgroups G ⊂ G. A computationally simple answer to some of these questions might lead to constructions of more 3-designs.

66 CHAPTER 3

COMBINATORIAL CONSTRUCTIONS

Necessity is the mother of invention. Old saying.

3.1 Introduction

The question of the existence of Steiner 3-designs has been completely resolved only for the case k = 4 (see [15]): There exists a Steiner design S(3, 4, v) if and only if v ≡ 2 (mod 6) or v ≡ 4 (mod 6), v ≥ 4.

For t ≥ 3, k ≥ 5, the problem is wide open. Even the task of constructing new Steiner designs has proven to be extraordinarily diﬃcult.

One important technique for constructing new families of Steiner 3-designs starts with

Steiner designs 3−(qn +1, q +1, 1), where q is a prime power. As seen in the previous chapter, these designs can be constructed by considering the 3-transitive action of the group PGL(2, qn) on the ﬁnite projective line. Building upon these designs, one can obtain Steiner 3-designs on larger sets of points by what are essentially combinatorial constructions.

Most of these combinatorial constructions essentially involve Candelabra systems.

Candelabra systems were ﬁrst studied by A. Hartman ([19]) for block size 4 and then

67 by Dijen Ray-Chaudhuri and Mohácsyfor an arbitrary block size ([24]). Though they define t-Candelabra systems, we shall restrict our attention to the case t = 3. All the concrete realizations of the various construction theorems of Hartman, Mohácsy et al have stem size (We shall formally define this in the following section when we make the formal definitions) one. In the rest of this chapter, the term Candelabra system shall mean a 3-Candelabra system.

The rest of this chapter is arranged as follows. We briefly recall Candelabra systems, then define a more general set structure called Rooted Forest Set System, and prove a generalized ‘fundamental construction theorem’. Then we consider Candelabra systems and Lattice Candelabra systems in particular and obtain new infinite families of Steiner 3-designs with block size 6. Finally, we shall comment on some of the methods employed and address the need for ‘new’ constructions.

3.2 Candelabra systems and Rooted Forest Set Systems(RFSS)

We start with a few deﬁnitions. We recall the deﬁnition for Candelabra systems restricting our attention to the case t = 3 rather than an arbitrary t.

Deﬁnition 3.2.1. (Candelabra system, [24]): Let v, k be positive integers. A

Candelabra system (or CS) of order v and block size k is a quadruple (X,S, Γ, A) satisfying:

• X is a subset of v elements (also called points),

• S is a subset of X of size s (called the stem of the Candelabra),

68 • Γ := {G1,G2 ...} is a partition of X\S into non-empty subsets (called groups),

•A is a family of k-subsets of X,

• Every subset T ⊂ X of size 3 satisfying |T ∩(S∪Gi)| < 3 for each i, is contained in precisely one block of A and any other 3-subset of X is not contained in any

block.

By the group type of a CS (X,S, Γ, A), we mean the list (|G|,G ∈ Γ: |S|) (the stem size is separated by a colon). If a CS has ni groups of size gi and has |S| = s, then

n1 n2 we notate the group type as (g1 g2 ... : s). If all the groups are of the same size, then the Candelabra system is called uniform.

A straightforward relation between Steiner 3-designs and Candelabra systems is as follows. Suppose there exists a Steiner 3-design on v + 1 points with block size k.

Then there exists a Candelabra system on v + 1 points with group type (1v : 1) with block size k. This is clear: Suppose (X, B) is a Steiner 3-design. Let x0 ∈ X be a

ﬁxed point. Consider the partition Γ := {{x} : x ∈ X, x 6= x0} of the set X \{x0}.

v Then (X, {x0}, Γ, B) is a Candelabra system with block size k and group type (1 : 1).

A less trivial connection between Steiner 3-designs and Candelabra systems comes through the following proposition.

Proposition 3.2.2. : Let k be a positive integer, k ≥ 2. Suppose there exists a

Steiner design S(3, k + 1, k2 + 1). Then there exists a CS of group type (kk : 1) with block size k + 1. Conversely, if there exists a CS of group type (kk : 1) with block size k + 1, there exists a Steiner design S(3, k + 1, k2 + 1).

69 Proof: Suppose (X, B) is the given Steiner 3-design, where |X| = k2 +1. Fix a point

∞ ∈ X. The derived design at the point ∞ is an aﬃne plane of order k and hence admits a parallel class of blocks, B1,B2,...,Bk. Then clearly, Γ := {B1,B2,...,Bk} is a partition of the set X \ {∞}. Let

A := B\{B1 ∪ {∞},B2 ∪ {∞},...,Bk ∪ {∞}}.

It is clear that (X, {∞}, Γ, A) is a CS with block size k + 1 and group type (kk : 1).

The converse is equally simple and we omit the details. We now recall the definition of another very important class of combinatorial structures which are crucial in the construction of Steiner designs, namely, Group Divisible designs, and Transversal Designs. We had already defined Divisible designs in the preceding chapter. Let us once more recall the definition.

Deﬁnition 3.2.3. (Group Divisible design or t-GDD): Let v, k be positive integers. A t-Group Divisible Design(or t-GDD) is a triple (X, Γ, A) satisfying

• X is a set of size v(elements are called points),

• Γ = {G1,G2,...} is a partition of X into non-empty subsets (called groups),

•A is a family of subsets of X(called blocks), each of cardinality k such that,

each block intersects any group in fewer than two points,

• Each t-set of points with each point from a diﬀerent group is contained in exactly

one block.

70 A t-Group Divisible design with block size k and l groups, each of size k, is called a t-Transversal Design (denoted TD(t, k, l)). Analogously, we deﬁne Lattice Group

Divisible designs (LGDD) and Lattice Transversal designs (LTD):

Deﬁnition 3.2.4. (Lattice Group Divisible design or LGDD): Let t, k, n be positive integers with 1 ≤ t ≤ k. Let (X, Γ, A) be a uniform t-Group Divisible design

n of group type (m ) constructed over the set X := Im × In, where Γ = {Ri : i =

1, 2 . . . , n}, where Ri := Im × {i}. The triple (X, Γ, A) is called a Lattice Group

Divisible design (or t−LGDD) if for any block A ∈ A and any set Cj := {j}×In, j =

1, 2 . . . , m, either |A∩Cj| < t or A ⊆ Cj. A Lattice Group Divisible design with block size k and k groups of size m is called a Lattice Transversal design and is denoted

LTD(t, k, m).

Group divisible designs and transversal designs play a very important role in the construction of Candelabra systems. The following theorem, called the fundamental construction for Candelabra systems, appears in [24]; this is essentially a generalization of a construction of Hartman (see [19]) for Steiner Quadruple systems.

Theorem 3.2.5. (The fundamental construction for Candelabra systems,[24]):

Let (X,S, Γ, A) be a Candelabra system with S = {s0} (called the ‘ingredient CS) and let K be a set of positive integers. Let ω : X → N ∪ {0} be a function(called a weight X function). There exists a CS with block sizes in K and group type ( ω(x)|G ∈ Γ: s) x∈G if the following conditions hold:

• For every block A of the CS not containing s0, there exists a 3 − GDD with block sizes in K and group type (ω(x)|x ∈ A, ω(x) 6= 0),

71 • For each block of the ‘ingredient’ candelabra system containing s0, there is a CS

with block sizes in K and group type (ω(x)|x ∈ A, x 6= s0, ω(x) 6= 0 : s).

We now generalize the notion of a Candelabra system to consider more hierarchical structures. Let F be a Rooted Forest, i.e., a forest with each connected component having a distinguished vertex called the root of the component. Let X 6= ∅ , |X| < ∞.

Consider a surjective map τ : X → V (F ). We call the sets τ −1(v), v ∈ V (F ) the groups of X induced by F. Alternately we could start with a partition of X into non-empty sets Xv, v ∈ V (F ) and consider a labelling of F by the sets Xv.

Deﬁnition 3.2.6. A t-Rooted Forest Set System(RFSS) on F w.r.t X is a triple denoted (t, F ) − (X, τ, A) where A ⊂ P(X) is a collection of subsets satisfying the following:

1. For any maximal path from a leaf of F to a root(the root of its connected com-

ponent) P , no t-subset of [ τ −1(w) w∈V (P ) (which shall henceforth be called subsets along a maximal path) is contained in

any member of A.

2. Any t-subset U of X which is not a subset along a maximal path is contained

in a unique member A(U) of A.

When we impose an additional restriction that |A| ∈ K for some K ⊂ N for each A ∈ A, we shall call it a K- RFSS.

Examples

72 1. A Candelabra System is an instance of a Rooted Tree Set System(RTSS)(If

the forest is in fact a tree, we shall call the set system by this name): Consider

a t-CS System (X,S, Γ, A). Let F be a star of degree |Γ|(i.e. a tree with a single

vertex of degree |Γ|, v0 and all other vertices of degree one). Consider the map τ : X → V (F ) as follows: Fix a bijection between Γ and the vertices of F of

degree one; call it γ. Deﬁne

τ(x) = γ(G) if x∈ / S and x ∈ G ∈ Γ

= v0 if x ∈ S.

Then it clearly follows from the deﬁnitions that (X, τ, A) is an RTSS on F .

2. (t=2) A Group Divisible Design is an instance of an RFSS. Consider a GDD

(X, Γ, A) and let F be a forest on |Γ| vertices with no edges. As before ﬁx a

bijection γ :Γ → V (F ) and deﬁne τ(x) = γ(G), if x ∈ G ∈ Γ.

As before, (X, τ, A) is an RFSS.

For t > 2, note that a GDD is not a t-RFSS.

Remark: In all discussions that ensue, one may assume that no non-root vertex of any tree in the forest has degree 2 because in that case by contracting a vertex v of degree 2, one can form another RF SS T/v (T being the tree containing v) through the map τ 0 : X → V (T ) \{v}

τ 0(x) = τ(x), if τ(x) 6= v

= τ(w), if τ(x) = v

73 where w is the farther of the two neighbors of v in T from the root of its component.

Conversely for a tree T , one can always add a vertex v0 of degree 2(a path of length 2 between 2 adjacent vertices v, w of T ) and consider any non-empty partition A∪B of

τ(w), where again, w is further away from the root of the corresponding component.

Then we have the new map

τ 0 : X → V (T ) \{v} deﬁned as

τ 0(x) = τ(x) if τ 0(x) 6= w

= w if x ∈ A

= v0 if x ∈ B and again (X, τ 0, A) is an RFSS.

Remark: The role of the map τ is only to induce the partition on X so as remarked earlier, one could think of τ as a partition {Xv}v∈V (F ) of X with each Xv 6= ∅ (the sets

Xv shall be called vertex-sets). We shall follow one convention or the other depending on convenience.

3.3 General constructions

Since the deﬁnition very clearly generalizes the notion of Candelabra Systems, the existence of an RFSS generally assures the existence of Candelabra Systems. We specialize to the context of t = 3, though much of this will carry over for arbitrary t.

Suppose we have an RTSS (3,T ) − (X, τ, A)(T being a rooted tree) on a set X. Let

74 w be a vertex of T adjacent to a unique non-leaf vertex of T . Consider a new tree

Tw deﬁned as follows:

V (Tw) := V (T ) \{w},

[ 0 E(Tw) := E(T ) \ {{w, v}|v is a leaf of T } {{v, w }|v is a leaf of T },

0 where w is the(unique) non-leaf neighbor of w in T . We deﬁne an RTSS on Tw by deﬁning

w Xv := Xv for v ∈ V (Tw),

w A := {A \ Xw| A ∈ A}.

Let the corresponding projection be denoted τ w.

It is now an easy veriﬁcation to check that (X, τ w, Aw) is indeed an RTSS. Note that the blocks in Aw have sizes in K ∪ (K − 1) where K − 1 := {k − 1|k ∈ K}. We shall refer to this procedure as puncturing the RTSS at Xw.

We now describe a recursive construction for an RTSS in terms of existence of Can- delabra Set Systems. To describe the recurrence we shall introduce some more no- tations. Consider a tree T on which we wish to describe an RTSS. By Sn we denote a star with n leaves. Now each neighbor of the root of T is again the root of another tree Tv, v ∈ N(v). Making a planar drawing of T with neighbors of the root v0 being v1, v2, ..vn, for some n from left to right, we recursively denote this tree as

T = Sn(Tv1 ,Tv2 , ...Tvn ).

Proposition 3.3.1. : Suppose the set X admits a partition n G X = Xi i=0

75 with Xi 6= ∅ ∀ i ∈ {0, 1, 2, ..., n} and suppose there exists a Candelabra Set System

(X, Γ,X0, Av0 ) with Γ = {X1,X2, ...Xn}. Now for each i,if there exists an RTSS

−1 (3,Tvi ) − (Xi ∪ X0, τi, Avi ) with τi (vi) ) X0 ∀ i, then there exists an RTSS on T w.r.t X where T = Sn(Tv1 ,Tv2 , ..., Tvn ).

Proof: The proof is indeed quite straightforward. We shall write Ti instead of Tvi

and Ai instead of Avi for simplicity. Deﬁne the partition τ on X by:

Xv0 = X0,

−1 Xv = τi (v), if v ∈ V (Ti), v 6= vi

−1 Xvi = τi (vi) \ X0

By the assumptions, each of these sets Xv 6= ∅. Let the block collection be A = n [ Ai. We claim that (X, τ, A) is the required RTSS. Consider any 3-element set i=0 U = {x, y, z} ⊂ X. If U is contained along a maximal path in T , then it lies along a maximal path in Ti or U ∩ X0 6= ∅. In either case, no element of A contains U. On the other hand, suppose U is not contained along any maximal path of T. Then we have the following possibilities:

1. U ∩ X0 = ∅: Then U ⊂ Xi for some i ∈ {1, 2, .., n} and so there exists a unique

A ∈ Ai s.t. U ⊂ A.

2. U ∩ X0 6= ∅: Suppose U ∩ X0 = {x}(note that |X0 ∩ U| ≤ 1). Then we again have the following cases:

76 (a) y, z ∈ Xi for some i: In this case we have a unique A ∈ Ai s.t. U ⊂ A(since

we have an RTSS on Ti w.r.t. Xi ∪ X0).

(b) y ∈ Xi, z ∈ Xj, i 6= j. Then by the existence of the Candelabra Set System

(X, Γ,X0, A0), there exists a unique block A ∈ A0 s.t. U ⊂ A.

Remark: If we further have |A| ∈ K for some K ⊂ N in each of the set systems in the hypothesis then the same is true of the resulting RTSS as well.

To make a similar proposition for the construction of RFSSs in general, we need

‘ingredient’ RFSSs. To that end, consider a forest with n vertices labeled 1, 2, .., n n G and no edges(the empty forest). Suppose we have a set X and a partition τ, X = Xi i=1 and Xi 6= ∅ ∀ i. An RFSS on this forest w.r.t.X implies the existence of a collection A such that every set U ⊂ X, U ( Xi, i ∈ {1, 2, ..., n} of size 3 is contained in a unique element A(U) ∈ A. The subsets of X of size 3 of the aforementioned type contain 0, 1 or 2 elements in every Xi. We shall need constructions for an RFSSs on an empty forest(forest with no edges) on n vertices. Note that constructions for RFSSs on the empty forest is equivalent to constructing a Steiner design with a collection of blocks that also partition the point set.

Some constructions are already present in literature. Since another way of viewing

RFSSs on an empty forest is the same as the existence of a Candelabra Set System with an empty stem, we have the following:

Proposition 3.3.2. (Mills): There exists an RFSS on an empty forest of n nodes with each vertex-set of size 6 and uniform block size 4.

77 Hanani’s constructions for Steiner Quadruple Systems also gives the following result, though it is not stated explicitly:

Proposition 3.3.3. (Hanani): There exists an RFSS on an empty forest of 3n + 2 nodes with vertex-sets of size 4 and uniform block size 4.

The preceding proposition occurs as a consequence of one of Hanani’s constructions

(the ‘doubling’ construction) in [16].

Another construction of RFSSs on an empty forest(with varying block size) comes from the construction of Candelabra Set Systems. For a Candelabra Set System

(X, Γ,S, A), we obtain a corresponding RFSS by puncturing the stem S.

Another important observation: Suppose we have an RFSS on an empty forest which admits a 2-GDD as a sub-design. Then one obtains a Candelabra Set System with root being a singleton set as follows: To each subset of the sub-2-design add an additional point(call it ∞). The resulting collection of blocks is easily seen to be a

Candelabra Set System. More generally, if the RFSS admits s disjoint 2 − GDDs, then one can adjoin a root set of size s and obtain a Candelabra Set System with a root set of size s. However the blocks of this Candelabra Set System are not all of the same size.

This simple observation in fact leads to a replication construction for RFSSs on an empty forest(This is a similar proposition to the fundamental construction for

Candelabra Systems).

Theorem 3.3.4. Suppose (X, {Xi}, B) is an RFSS on the empty forest F with vertex

78 set V which contains a 2-GDD, A. For a map ω : V → N, suppose the following hold:

1. For each B ∈ B \ A there exists a 3 − GDD with block sizes from K for some

K ⊂ N and group type (ω(x)|x ∈ B, ω(x) 6= 0).

2. For each block A ∈ A there is an RFSS on an empty forest with vertex-sets of

size(s) (ω(x)|x ∈ A) with block sizes in K. Then there exists an RFSS on the X empty forest F and with vertex-sets of size ( ω(x)) and with block sizes in

x∈Vi K.

Proof: For each x ∈ X, let Ω(x) be disjoint sets, |Ω(x)| = ω(x), and let X∗ =

[ ∗ [ Ω(x),Xi = Ω(x). For each block B ∈ B \ A, there exists a 3-Group Divisible x∈X x∈Xi design (3-GDD) [ ( Ω(x), {Ω(x)|x ∈ B}, BB) x∈B [ with block sizes in K. For each block A ∈ A, there exists an RFSS w.r.t Ω(x) with x∈A ∗ ∗ ∗ vertex-sets being Ω(x), x ∈ A and block collection AA. We claim that (X , {Xi }, B ),

∗ [ [ ∗ ∗ ∗ where B = BB ∪ AA, is the required RFSS. Consider any set {x , y , z }. B∈B\A A∈A To make things notationally simpler, we say that the projection of x∗ is x if x∗ ∈ Ω(x).

Suppose the projections of x∗, y∗, z∗ are x, y, z respectively. We have the following cases:

∗ ∗ ∗ ∗ ∗ ∗ ∗ 1. x, y, z ∈ Xi for some i: Suppose {x , y , z } ⊂ B for some B ∈ B . If B ∈ BB for some B, then {x, y, z} ∈ B, and that is a contradiction. So no subset of size

3 which lies along a maximal path is contained in any of the blocks.

79 2. x ∈ Vi, y ∈ Vj, z ∈ Vk, i 6= j 6= k 6= i: By assumption, there exists a unique block B ∈ B such that {x, y, z} ⊂ B. Since we have a 3 − GDD on the set

S ∗ ∗ ∗ ∗ ∗ w∈B Ω(w) there exists a B ∈ BB such that {x , y , z } ⊂ B . Suppose if possible that A∗,B∗ both contain {x∗, y∗, z∗}. Since every block is a member of

BB for some block B ∈ B, it follows that {x, y, z} is contained in two distinct

blocks A, B, since in each collection BB every triple is contained in atmost one block. And that is a contradiction.

3. x, y ∈ Vi, z ∈ Vj, i 6= j: Again we have two subcases:

• x 6= y: Again by assumption there exists a unique block B ∈ B such that

∗ {x, y, z} ⊂ B. As before there again exists a block B ∈ BB such that {x∗, y∗, z∗} ⊂ B∗. The uniqueness part for B∗ is similar.

• x = y: Since there exists a sub-collection of blocks of the RFSS which is

a 2 − GDD, there exists a unique A ∈ A such that {x, z} ∈ A. Again S since we have an RFSS on the set w∈A Ω(w), so there exists a unique

∗ ∗ ∗ ∗ ∗ A ∈ AA such that {x , y , z } ⊂ A . Finally as for the uniqueness of the block A∗, suppose if possible A∗ 6= B∗ and x∗, y∗, z∗ are contained in both

∗ ∗ these sets. Clearly, as before, {x, z} ⊂ A ∩ B, where A ∈ BA,B ∈ BB(so A 6= B as well). Since the blocks in A form a 2 − GDD, B/∈ A. Then by

assumption, the block collection BB forms a 3−GDD and so in particular, |B∗ ∩ Ω(w)| ≤ 1 for all w ∈ B. Hence {x∗, y∗, z∗} cannot be contained in

B∗ and we are through.

80 One may analogously deﬁne RTSSs and RFSSs with a replication factor. We say that an RFSS has replication λ if each t-subset not along a maximal path is contained in precisely λ blocks. One can get more RFSSs under this broader deﬁnition. However there may be repeated blocks.

Proposition 3.3.5. Given positive integers k, then for some λ > 0 there exist RFSSs on the empty forest with n = 2k nodes and block size 2k for all blocks with each vertex- set of the same size .

Proof: Let the vertex-sets be the disjoint sets |Xi| = m, i = 1, 2, .., n. Consider

(2) the set collection B = {B||B ∩ Xi| ∈ {0, 2}}. For 3-sets U = {x, y, z} with x, y ∈ Xi for some i and z ∈ Xj, j 6= i, the number of sets B,U ⊂ B is λ2 = n − 2mk−2 (m − 1) . Similarly if U = {x , x , x } with x ∈ X , x ∈ X , x ∈ k − 2 2 i j k i i j j k n − 3mk−3 X , i 6= j 6= k, then the number of sets B,U ⊂ B is λ = (m − 1)3 . k 1 k − 3 2 Now n−2mk−2 (m − 1) (n − 2)m k−2 2 = > 1, 3n−3mk−3 (m − 1)(k − 2) (m − 1) k−3 2 so the number of sets of the latter type are covered fewer times than the sets of the former type by the blocks of B(2). We know that for m >> 0, there exist 3 − TD(2k, m), (X, B1) (Due to Blanchard [6]). Let λ0 = λ2 − λ1. Let B =

(2) S B λ0B1(where by λB we mean λ disjoint copies of every member of B). Then

2k (X, {Xi}i=1, B) is the required RFSS.

Combinatorial structures with large replication parameters λ, are not combinatorially as interesting as the corresponding structures with λ = 1. However, they may be used

81 as ‘ingredients’ to construct combinatorial structures with λ = 1. We will discuss some of these issues later.

In the rest of this chapter, we shall divert our attention towards concrete techniques for constructions of Steiner designs, speciﬁcally for k = 6.

3.4 New constructions for k = 6 from Candelabra systems

Before we state our theorems, we recall a few other deﬁnitions.

Closely related objects to CS systems are Lattice Candelabra systems. These are basically uniform Candelabra systems with some additional structure on the point set and the block collection. However, these have stem size either zero or one.

Deﬁnition 3.4.1. (Lattice Candelabra system or LCS,[24]): Let k, n, m be positive integers. Let (X,S, Γ, A) be a uniform candelabra system of group type (mn :

|S|) over the set X = Im × In ∪ S with S = ∅ or S = {∞}. (X,S, Γ, A) is called a

Lattice Candelabra system (or LCS) if for any block A ∈ A, either |A ∩ (Cj ∪

S)| < 3 or A ⊆ Cj ∪ S, where Cj = {j} × In.

Note that ‘trivial’ Lattice Candelabra systems can be constructed from Steiner 3- designs in a straightforward fashion. If (X, B) is a Steiner 3-design, with |X| = v + 1 and block size k, then let x0 ∈ X be the stem, and consider the partition Γ :=

{{x} : x ∈ X, x 6= x0}. It is clear that (X, {x0}, Γ, B) is a Lattice Candelabra system with group type (1v : 1). Note that in this construction, we have m = 1, n = v.

Though trivial, this observation is handy, as we shall see.

However, these are not the only instances of LCS. The following proposition gives us

82 LCS with block sizes q + 1, where q is a prime power. This proposition also appears in [24].

Proposition 3.4.2. (Existence of nontrivial LCS): Suppose that for a positive integer k, there exists a Steiner design S(3, k + 1, k2 + 1). Then there exists a 3 − LCS with block size k + 1 and group type (kk : 1).

Proof: It suﬃces to construct a Candelabra system (X,S, Γ, A) with group type

(kk : 1) and block size k +1 such that S is a singleton set, and the Candelabra system admits a collection of k blocks B1,B2,...,Bk ∈ A satisfying Bi ∩ Bj = S for any i 6= j, 1 ≤ i, j ≤ k.

To do that, note that there exists a 3 − CS (X,S, Γ, A) with group type (kk : 1), stem S of size one and block size k + 1 as seen earlier. Observe that the Steiner

3-design S(3, k + 1, k2 + 1) has as its derived design, an aﬃne plane of order k. The partition Γ in the CS corresponds to a parallel class of blocks in the derived design.

Now choose a second parallel class of blocks, Γ0 of the derived design. The blocks in the CS corresponding to Γ0 clearly satisfy the requirement above and that completes the proof. .

Since for each prime power q, there exists a Steiner design S(3, q + 1, q2 + 1), there exist a LCS with group type (qq : 1) and block size q + 1, whenever q is a prime power.

Our starting point towards new Steiner design constructions involves the construction

83 of new Candelabra systems with stem size 2 for the case k = 6. Having done so, we generalize the following theorem of Hanani:

Theorem 3.4.3. (‘4v + 2-theorem’, [15]): Suppose there exists a Steiner design 3 −

(v + 1, 6, 1). Then there exists a Steiner design 3 − (4v + 2, 6, 1).

This enables us to obtain a new ‘mock-product’ theorem which gives us several inﬁnite families of Steiner 3-designs with block size 6. Finally, we say a few words on attempts to attack the existence problem for Steiner 3-designs in general.

The main theorems that we shall prove are the following:

Theorem 3.4.4. (CS theorem): There exists a Candelabra system with 5 groups with

4m−4 group size 3 and stem size 2, and block size 6, for m ≥ 2.

Having obtained this, we prove a lemma,

Lemma 3.4.5. (Steiner design lemma): Suppose there exist Steiner designs S(3, k, l+

2),S(3, k, v + 1), a CS with group type (lk−1 : 2) and a TD(3, k, l), then a Steiner design S(3, k, vl + 2) exists. which leads us to new families of Steiner designs with block size 6.

Theorem 3.4.6. Let k ≥ 2 be an integer and suppose v is a positive integer satisfying

v ≡ 1 (mod 20) or (3.1)

v ≡ 5 (mod 20). (3.2)

Then, there exist integers m0, v0, and d0(v) such that whenever m ≥ m0, v ≥ v0 and d ≥ d0(v), there exist

84 k d 4m−4 • Steiner designs S(3, 6, lv(4 · 5 + 1) + 2) where l = 3 .

• Steiner designs S(3, 6, lv · 5m + 2)

for m ≥ m0(v) for some ﬁxed integer m0(v).

3.5 New constructions for k = 6: The Proofs

We begin with the proof of the Steiner design lemma. By Iv we shall denote the set

Iv := {1, 2, . . . , v}. Proof of Steiner design lemma: We need to construct a Steiner design S(3, k, vl+

2) which we shall do so on the point set X := Il × Iv ∪ {∞1, ∞2} of size lv + 2. Since there exists a Steiner design S(3, k, v + 1), there also exists a CS (Y, {∞}, C) with

v group type (1 : 1) on the point set Y := Iv ∪ {∞}, |Y | = v + 1 by the observation following the deﬁnition of a Candelabra system in the preceding section.

Let C∞ := {B ∈ C|∞ ∈ B}. Consider the weight function ω deﬁned by ω(∞) :=

2, ω(i) := l for 1 ≤ i ≤ v. For each block B ∈ C∞, let YB := Il ×(B\{∞})∪{∞1, ∞2}

k−1 and let (YB, BB) be a CS with group type (l : 2) (this exists by the hypothesis).

Similarly, for each block B ∈ C \ C∞ let (YB, CB) be a TD(3, k, l) (which again exists, by the assumption).

Let [ [ B = ( BB) ∪ ( CB).

B∈C∞ B∈C\C∞ By the fundamental construction for Candelabra systems, it follows that (X, B) is a

v CS with group type (l : 2). For each 1 ≤ i ≤ v, let (Xi, Bi) be a Steiner 3-design

85 S(3, k, l + 2) on the set Xi := Il × {i} ∪ ({∞1, ∞2}). Such a Steiner design again exists by the assumption of the existence of a Steiner design S(3, k, l + 2).

Finally let ! [ A := B ∪ Bi .

i∈Iv We claim that the pair (X, A) gives us a Steiner design S(3, k, lv + 2).

Let x, y, z be distinct elements of X. Suppose, ﬁrst that x, y, z are all distinct from

∞1, ∞2. If not all x, y and z lie in a subset of the form Il × {a} for some a ∈ Iv, then there exists a unique block B ∈ B such that x, y, z ∈ B. If x, y, z ∈ Il × {a} for some a ∈ Iv, then there exists a unique block B ∈ Ba such that x, y, z ∈ B.

If {x, y, z} ∩ {∞1, ∞2}= 6 ∅, two cases arise:

1. x = ∞1, ∞2 6= y, z: Suppose y ∈ Il × {a} and z ∈ Il × {b} with a 6= b, a, b ∈ Iv then there exists a unique block (in the candelabra system) B ∈ B such that

∞1, y, z ∈ B. If a = b, then there exists a unique block B ∈ Ba such that

∞1, y, z ∈ B.

2. x = ∞1, y = ∞2: In this case if z ∈ Il × {a} for some a ∈ Iv, there exists a

unique block B ∈ Ba such that x, y, z ∈ B.

Observe that the collections B, Bi, i ∈ Iv are mutually disjoint, since any block in Bi is contained in Il × {i}, and any block of B intersects any set of the form {∞1, ∞2} ∪

(Il × {i}) , i ∈ Iv in fewer than 3 points. These observations establish the uniqueness of the block containing the set {x, y, z} and so, the proof of the lemma is complete.

86 In order to prove the CS theorem, we need a few other results.

Firstly, one can prove a ‘Fundamental theorem for LCS’ starting with appropriate

‘ingredient’ designs. Not surprisingly, the ingredient designs here, are LTDs and LCSs.

We merely state the result here. One may look up [24] for a proof and further details.

We mention in passing that the details are quite similar to that of the fundamental theorem of Candelabra systems.

Theorem 3.5.1. (Fundamental construction for LCS): Let n, a and b be positive integers and let K be a set of positive integers. Let (X, {∞}, Γ, A) be a 3 − LCS of group type (an : 1). For any given positive integer b, there exists a 3 − LCS of group type ((ab)n : 1) with block sizes from K, if the following holds:

1. For each block A of the 3-LCS not containing ∞, there is a 3-LGDD of group

type b|A| with block sizes from K.

2. For each block A of the 3-LCS containing ∞, there is a 3-LCS of group type

(b|A|−1 : 1) with block sizes from K.

Thus, this theorem requires the existence of LCS and LGDDs as ingredients in order to be applicable. Note that a Lattice Transversal t-design is equivalent to a Transversal design admitting a parallel class of blocks. Thus if one can exhibit a parallel class of blocks in a Transversal design, one has a Lattice Transversal design.

The following theorem ([24]) provides a nontrivial family of Lattice Transversal designs.

Theorem 3.5.2. (Existence of nontrivial LTDs): Suppose q is a prime power satisfying (q − 1, t − 1) > 1. Then one can obtain a transversal design TD(t, q + 1, q)

87 admitting a parallel class of blocks. Hence if q is an odd prime power, then there exists an LTD(3, q + 1, q).

Proof of the CS Theorem: To prove the theorem we ﬁrst make the following claim:

Claim: There exists a Lattice Candelabra system, CS(5l0 : 1) with block size 6,

4m−1 where l0 = 3 , m ≥ 2. We use the fundamental construction for LCSs. Firstly, note that there exists a

4m+2 Steiner design S(3, 6, l0 + 1) (note that l0 + 1 = 3 ). This follows by repeated application of the product theorem (theorem 1) and the existence of a Steiner design

3−(22, 6, 1). This 3-design occurs as a two point derivation of the well known Steiner

5-design on 24 points. Hence there exists a 3-LCS with group type (1l0 : 1).

By the proposition on the existence of non-trivial LCS, there exists a 3-LCS with group type (55 : 1). Again, by the existence of nontrivial LTDs, there exists a

LTD(3, 6, 5)., Let a = 1, b = 5,K = {6}, s = 1, so that the conditions of the fundamental construction for LCS are satisﬁed. Thus we have a Lattice Candelabra system

5 with group type (l0 : 1) and block size 6.

Now, let X := I5 × Il0 , Γ = {I5 × {i}, i ∈ Il0 } , R = {{i} × Il0 , i ∈ I5}, and let

5 (X, {∞}, Γ, B) be a Lattice Candelabra system with group type (l0 : 1) and block size 6. Let us denote the elements of Γ by Gi, i ∈ Il0 for brevity. Let

Bi = {B ∈ B|B ⊂ {∞} ∪ {i} × Il0 } , i ∈ I5.

Let ! [ [ A = B\ Bi ({∞} ∪ Gi).

i∈I5 i∈Il0

88 Then (X, {∞}, R, A) is a Lattice Candelabra system with group type (5l0 : 1) with block size 6.

To see that this is indeed the case, we only need to show that this is a Candelabra

system; since ({∞} ∪ Gi), i ∈ Il0 is a block, it follows trivially that it is also a Lattice Candelabra system.

Consider any set T = {x, y, z} of size 3 in X. Suppose T ⊂ {∞} ∪ {i} × Il0 for some i ∈ I5, and that if possible we have T ⊂ A for some A ∈ A. Clearly, A ∈ B\ S B , so A ∈ B. But since (X, Γ, {∞}, B) is a LCS, we have A ∈ B and that i∈I5 i i is a contradiction.

We now consider two cases again:

1. ∞ ∈/ T : In this case, if T ⊂ Gi for some i, then T ⊂ {∞} ∪ Gi and this is the

unique block of A containing T . Suppose T 6⊂ Gi. Then there exists a unique block A ∈ B \ S B such that T ⊂ A. i∈I5 i

2. T = {∞, x, y}: Suppose x, y ∈ Gi for some i. Then again, T ⊂ {∞} ∪ Gi and

this is the unique block of A containing T . If x ∈ Gi, y ∈ Gj, i 6= j, then again there exists a unique block A ∈ B \ S B such that T ⊂ A. i∈I5 i

That completes the proof of the claim.

Before we complete the proof, we make one other observation. Let (X, B) be the

Steiner 3-design 3 − (22, 6, 1) which exists, as noted earlier.

For any two distinct points x, y ∈ X, let Bx,y = {B ∈ B|x, y ∈ B}. Since (X, B) is a

Steiner design, Bx,y induces a partition Π on the set X \{x, y} into 5 groups of size

89 4 each. Then (X, {x, y}, Π, B\Bx,y) is clearly a Candelabra system with group type (45 : 2) and block size 6.

We ﬁnally complete the proof as follows: Let (X, {∞}, Γ, B) be a Candelabra system

5 4m−1−1 with group type (l0 : 1), with l0 = 3 and the element ∞ being the sole element of the stem. Consider the weight function ω : X → N deﬁned by ω(∞) := 2, ω(x) := 4 for x ∈ X, x 6= ∞. Since there exist a CS(45 : 2) with block size 6 and a TD(3, 6, 4)

(this is due to Brouwer though he did not publish it; see [15]) the conditions for the fundamental theorem for Candelabra systems are satisﬁed and so there exists a

5 4m−4 CS(l : 2) with block size 6, where l = 3 .

A couple of remarks are in order.

1. Assmus and Key in [1] also construct Steiner 3-designs with the parameters

3 − (l0 + 1, 6, 1) though their construction uses the points and lines of the underlying PG(m − 1, 4) in a more geometric manner.

2. Note that we use the deﬁnite article since such a design is indeed unique. In

fact, the uniqueness of this design was established ﬁrst by Witt in 1938.

Before we turn to the proof of the theorem , we recall an important result due to Blanchard (see [6]) which states that transversal designs of strength 3 exist for

‘suﬃciently large’ m, for a ﬁxed block size k. More precisely, Blanchard proves in [6], the following result:

Theorem 3.5.3. (Blanchard’s theorem for transversal designs [?]): Let k be a given

90 positive integer. There exists an integer m0 := m0(k) such that for all integers m ≥ m0, there exists a Transversal design TD(3, k, m).

This is an extension of the well known Chowla-Erd¨os-Straus theorem for transversal designs of strength two.

Notation-wise, by writing m 0, we mean that m is an element of the set {n ∈ N : n ≥ m0} for some absolute constant m0. Thus, Blanchard’s result ensures that for

4m−4 m 0, there always exists a transversal design TD(3, 6, l) with l = 3 . 4m−4 Proof of Theorem: For l = 3 , note that there exists a Steiner 3-design S(3, 6, l+ 4m−4 4m+2 4m+2 2), since l + 2 = 3 + 2 = 3 and a Steiner design S(3, 6, 3 ) exists for all m ≥ 2.

The CS theorem yields a Candelabra system with group type (l5 : 2) and block size

6. By Hanani’s ‘4v + 2-theorem’, the Steiner designs S(3, 6, 5k + 1), d ≥ 2, yield the

Steiner design S(3, 6, 4 · 5k + 1) + 1). Repeated application of the product theorem gives us Steiner designs S(3, 6, (4 · 5k + 1)d + 1) for all k, d ≥ 2. The mock-product theorem (theorem 1) implies that for each ﬁxed u 0, and u satisfying the conditions

(1), (2) of the statement of theorem 13, there exists d0(u) such that for d ≥ d0(u),

k d a Steiner design S(3, 6, u · (4 · 5 + 1) + 1), for all d ≥ d0(v), m ≥ 2 exists. Also note that by Blanchard’s theorem, there exists a TD(3, 6, l) if l ≥ L0 for some ﬁxed constant L0. Let m0 = blog4(3L0 + 4)c.

4m−4 k d For m > m0, l = 3 , k = 6, v = u · (4 · 5 + 1) , the hypotheses of the Steiner design lemma are satisﬁed, yielding a Steiner design S(3, 6, lv(4 · 5k + 1)d + 2) as desired.

91 For the second family of Steiner 3-designs, let v = u · 5m + 1 in the Steiner design lemma. Remark: The product theorem is used repeatedly in the last construction. However this construction presents us with certain shortcomings which will be addressed in the ﬁnal section.

3.6 Concrete instances of the theorems

The theorem proved in the previous section results in several new Steiner 3-designs with block size 6. We shall indicate a concrete instance of such an inﬁnite family here. The bounds in the result of Blanchard ([6]) are too large and often, one can obtain smaller bounds. The following proposition which was ﬁrst observed by Bush as a generalization of McNeish’s theorem for Transversal designs, gives us a concrete result in this context.

Proposition 3.6.1. : Suppose k is a positive integer and q is a prime power with k < q. Then there exists a TD(t, k, q). Consequently, if for some m, all the prime factors q of m satisfy q > k, then there exits a TD(t, k, m).

Proof: Consider the point set X = Fq × A, where A ⊂ Fq is some ﬁxed subset of Fq with |A| = k and let Γ = {G1,G2,...,Gk} := {Fq × {x}, x ∈ A} be a partition of X into groups. Consider the collection of blocks

B := Ba0,a1,...,at−1 : ai ∈ Fq where

Ba0,a1,...,at−1 := {(x, fa1,a2,...,at (x)), x ∈ A}

92 t−1 X i with the functions fa0,a1,...,at−1 deﬁned by fa0,a1,...,at−1 (x) = aix . i=0 t Clearly, |B| = q . Consider any set T of size t such that |T ∩ Gx| ≤ 1 for each x ∈ A.

Suppose T = {(xi, yi): xi ∈ A, xi 6= xj for i 6= j, yi ∈ Fq}. The system of equations

2 t−1 a0 + a1x1 + a2x1 + ··· at−1x1 = y1,

2 t−1 a0 + a1x2 + a2x2 + ··· at−1x2 = y2,

···

2 t−1 a0 + a1xt + a2xt + ··· at−1xt = yt has a unique solution (a0, a1, . . . , at−1) with ai ∈ Fq, 0 ≤ i ≤ t − 1 since the matrix   t−1 1 x1 ··· x1    t−1   1 x2 ··· x   2     ···      t−1 1 xt ··· xt is non-singular (Vandermonde type matrix). Hence there is a unique block B =

Ba0,a1,...,at−1 ∈ B such that T ⊂ B. The latter statement is a simple consequence of MacNeish’s ‘product theorem’ for transversal designs: If there exist Transversal designs TD(t, k, l),TD(t, k, m), there exists a Transversal design TD(t, k, lm).

An alternate way of writing this proof would be the following ‘recipe’: consider a

TD(t, q +1, q) which exists for prime power q. Now simply ‘puncture’ this transversal design at q + 1 − k points to obtain the desired TD(t, k, q).

93 45−4 Let l = 340 = 3 . All the theorems proved in the previous section hold for this particular value of l, except possibly the existence of a TD(3, 6, 340). Now the observation made above implies that we can construct a transversal design TD(3, 6, 17).

One can also construct a TD(3, 6, 5),TD(3, 6, 4) (see [15]). Since 340 = 4 · 5 · 17, one can also construct a TD(3, 6, 340) as a consequence of MacNeish’s product theorem for Transversal designs.

k d Thus there exist Steiner designs S(3, 6, 340v(4 · 5 + 1) + 2) with v ≥ v0 and v satisfying (1), (2) of the statement of Theorem.

3.7 Steiner designs with block size 5

Before we conclude, we demonstrate another application of the of the product (mock- product) theorem. Though the result here does not prove anything new per se, it brings to attention, an infinite family that was not noticed earlier by Mohácsy et al, thus emphasizing the fact that every time we have a new 3-design with block size q + 1, we can generate an infinite family of new Steiner designs from the product

(mock-product) theorem.

Using the 3-homogeneous action of the group PSL(2, 27), Denniston[11] constructed a Steiner 5-design S(3, 7, 28). A two point derived design yields a Steiner 3-design

S(3, 5, 26). A Steiner design with the same parameters was also constructed independently by Hanani[15]. We now produce an inﬁnite family

Proposition 3.7.1. (New Steiner 3-designs with block size 5): Suppose v 0 is a

94 ﬁxed integer satisfying

v ≡ 1, 4, 16, 25, 40, 40 (mod 60)

Then, there exists an integer m0(v) such that whenever m ≥ m0(v), there exist Steiner 3-designs, S(3, 5, v · 4m · 52d + 1), for d ≥ 1.

Proof: From Denniston’s construction [11] (or Hanani [15]), we have a Steiner

S(3, 5, 52 + 1). By the product theorem, we have the existence of Steiner designs

S(3, 5, 52d + 1) for all d ≥ 1.

The mock-product theorem stated in the introduction gives us designs S(3, 5, v·4m+1) where v satisﬁes the hypothesis, since the inversive geometric constructions yield

S(3, 4 + 1, 4m + 1). The product theorem applied once more to these two Steiner

3-designs ﬁnally yields the desired families of Steiner designs with block size 5.

3.8 Concluding Remarks

Before we conclude, we make a remark on the general problem for constructing Steiner

3-designs. All known combinatorial constructions for Steiner 3-designs generally involve two kinds of constructions:

1. Constructions of Transversal designs/ Group divisible designs,

2. Recursive constructions.

Most constructions of the ﬁrst type give Steiner designs with parameters of the type in [24]. Improvements in constructions involving the techniques of ‘block spreading’

(see [3],[24]) can lead to constructions of newer Steiner 3-designs (new parameters).

95 The harder problem seems to be in doing away with recursive constructions. While recursive constructions appear frequently in constructions of Steiner designs (for instance the fundamental construction for Candelabra systems is essentially a recursive construction), the new parameter values that result from these constructions increase in geometric progression while the necessary arithmetic conditions for the existence of Steiner 3-designs increase in arithmetic progression. What we seem to be missing fundamentally are what can be termed ‘adjunction theorems’ or ‘additive theorems’.

In computational terms, it appears that brute force cannot take us very far in our quest for new Steiner 3-designs. For instance, since the classical Witt designs have block size q + 1 where q is a prime power, all these combinatorial techniques again yield Steiner 3-designs with block size of the same type. It is indeed curious that there is no single known Steiner 3-design whose block size is not of the aforementioned type.

The smallest such integer is k = 7 and there is no known Steiner 3-design with block size 7. The smallest possible such set of parameters for which there potentially exists a Steiner 3-design would be an S(3, 7, 77). At the moment, it is not clear how one might construct Steiner designs with an arbitrary block size. Such designs are bound to bring in new techniques and ideas.

96 CHAPTER 4

‘λ-LARGE’ THEOREMS

The necessary conditions for t-designs are suﬃcient for something. Title of a paper of R.M.Wilson.

Outline of the chapter

As mentioned in the ﬁrst chapter of this thesis, the matrix formulation of the problem of existence of a t-design allows one to look for other ‘obstructions’ (if any) for the design by considering the rank of the incidence matrix. Wilson’s theorem in [39] addresses the corresponding algebraic problem for t-designs for general t leading to constructing t-designs with ‘suﬃciently large’ values of λ.

In this chapter, we restrict our attention to the case of t = 3 (though we will state some results in generality when needed) and consider the analogous problem for

Candelabra systems. The rest of the chapter is arranged as follows. We ﬁrst describe some basic known results in this direction and then state the results, with the proofs in the next section. In the ﬁnal section we shall say a few words on ‘block spreading’ and related results.

97 4.1 Preliminaries

We begin by recalling the following theorem of Wilson (see [39]):

Theorem 4.1.1. (see [39]) Let t, k, v be given integers with t ≤ k ≤ v. Then there is an integer n ∈ N such that a t − (v, k, λ) design exists whenever the conditions

v − i k − i λ ≡ 0 (mod ) (4.1) t − i t − i λ ≥ n (4.2) are fulﬁlled.

The following theorem due to Blanchard (see [4]) has a similar ﬂavor to it:

Theorem 4.1.2. There are functions v0(k) and λ0(k, v) such that for any Steiner

2-design S(2, k − 1, v) with v ≥ v0(k) and any λ ≥ λ0(k, v) satisfying

v + 1 k λ ≡ 0 (mod ), 3 3 there exists an extension of λ copies of the Steiner 2-design, i.e., there exists a 3- design (X, B) with parameters 3 − (v + 1, k, λ) and a point x0 ∈ X which admits

λ · S(2, k − 1, v) as a derived design at x0.

Suppose the Steiner design S(2, k − 1, v) admits a parallel class of blocks, B0 (in particular, k − 1|v). Then, Blanchard’s theorem above tells us a little more: There exists a CS with group type ((k − 1)n : 1) with replication parameter λ. Indeed, since the derived design at the point x0 is λ · S(2, k − 1, v), we may simply discard the collection of blocks λ · ({x0} ∪ B),B ∈ B0 from the collection of blocks of the 3-design

98 constructed in the theorem. Since, the general deﬁnition of a Candelabra system in

[24] also includes the possibility of a replication parameter λ > 1, we see that the resulting structure above clearly is a Candelabra system with replication parameter

λ.

The theorem proved above is a crucial tool in the construction of new Steiner 3-designs in [3] and also [25]. It is therefore a question of suﬃcient interest to generalize this result in an appropriate context.

Theorems for existence of combinatorial structures with large values of λ and col- loquially called λ-large theorems. The earliest such results for t-designs are due to

Jurkat and Graver [14] and R.M. Wilson[39]. A similar result for Transversal designs (Orthogonal arrays) is due to Singhi and Ray-Chaudhuri [29]. A more recent paper due to Singhi[31] gives a uniﬁed approach towards these problems by deﬁning appropriate tags on subsets of the underlying set.

All these theorems essentially prove that a certain incidence matrix admits full (row) rank over Q, so it follows that there are no ‘linear constraints’ on the existence of the concerned structure.

We shall prove a similar theorem for Candelabra systems. We start with a couple of X deﬁnitions. As always, denotes the set of all k-subsets of X. k

Deﬁnition 4.1.3. Let X be a non-empty ﬁnite set. We say X admits a stem-partition

99 n [ (S, Γ) if Γ := {G1,G2,...,Gn},X = S ∪ Gi, where Gi ∩ Gj = ∅ if i 6= j. The i=1 subset S is called the stem of X and Gi, the groups of X.

Deﬁnition 4.1.4. Let X be a non-empty ﬁnite set admitting a stem-partition (S, Γ).

Let

X T = T ∈ : |T ∩ (S ∪ G )| < 3 for all i , 3 i X K = K ∈ : |K ∩ (S ∪ G )| < 3 for all i . k i

By the Incidence matrix of X relative to Γ and S we mean the matrix

AΓ,S := ((AT,K ))T ∈T ,K∈K , where

AT,K := 1 if T ⊂ K and

:= 0 otherwise.

When there is no confusion regarding the partition Γ and the stem S, we shall simply write A for the incidence matrix of X relative to Γ and S.

Our goal is the following

Theorem 4.1.5. Let X be a ﬁnite set admitting a stem-partition (S, Γ) with |S| =

1, |Gi| = |Gj| for all Gi,Gj ∈ Γ. Then the incidence matrix of X relative to Γ and S has full row rank over Q.

100 4.2 Proof of the Theorem

n [ In the rest of this section, we assume we have a stem-partition X = S ∪ Gi i=1 relative to the partition Γ := {Gi|1 ≤ i ≤ n} and the stem S. Further, we assume

|S| = 1, |Gi| = |Gj| = m for all i 6= j. Also we denote the incidence matrix A of X relative to the partition Γ and stem S by A = ((AT,K )). We start with a couple of simple observations.

Proposition 4.2.1. Suppose |Gi| = m for all i and |Γ| = n. Then n m n |T | = m3 + n · ((n − 1)m) + m2, (4.3) 3 2 2 n m n − 1 n |K| ≥ mk + n · mk−1 + mk−1. (4.4) k 2 k − 1 k − 1 In particular, if 3 < k < n − 3, we have |K| ≥ |T |.

Proof: The two statements above are fairly easy to prove. For each T ∈ T , since

|T ∩ (Gi ∪ S)| ≤ 2, we have the following three possibilities:

1. |T ∩ Gi| ≤ 1 for all i, T ∩ S = ∅: In this case, the number of such sets T is given

n 3 by 3 · m .

2. |T ∩ Gi| = 2 for some i: Note that in this case, T ∩ S = ∅, |T ∩ Gj| ≤ 1 for all m j 6= i. The number of such T is n · 2 · ((n − 1) · m).

3. T ∩ S = S: In this case, note that |T ∩ Gi| ≤ 1 for all i. The number of such

n 2 sets T is 2 · m and that completes the proof of (1).

The proof of 4.4 is similar; we only count those sets K ∈ K satisfying |K ∩ Gi| ≤ 1 for all i, K ∩ S = ∅; |K ∩ Gi| = 2 for some i, |K ∩ Gj| ≤ 1 for all j 6= i; K ∩ S = S.

101 The last part of the proposition follows since the expression on the right hand side of 4.4 is term-wise greater than the expression on 4.3. The preceding proposition, in particular implies that for 3 < k < n − 3, the incidence matrix of X relative to Γ and S has full rank.

In the rest of the section, by rank of a matrix, we shall mean its rational rank, i.e., rank over Q.

Lemma 4.2.2. Let K0 ⊂ K denote the collection of k-subsets K ⊂ X that satisfy one of the following:

1. |K ∩ Gi| ≤ 1 for all i, K ∩ S = ∅.

2. K ∩ S = S.

3. |K ∩ Gi| = 2 for some i, |K ∩ Gj| ≤ 1 for all j 6= i.

Let A0 denote the submatrix of A, the incidence matrix of X relative to Γ and S consisting of the columns indexed by the elements of K0. Then A0 is permutation- equivalent to a matrix of the form   AA0 A00      0 B B0      0 0 C for some 0-1 matrices A, A0, A00, B, B0, C.

102 1 2 3 Proof: Deﬁne T1, T2, T3 ⊂ T , and K0, K0, K0 ⊂ K0 as follows:

T1 := {T ∈ T | |T ∩ Gi| ≤ 1 for all i, T ∩ S = ∅}

T2 := {T ∈ T | T ∩ S = S}

T3 := {T ∈ T | |T ∩ Gi| = 2 for some i} and

(1) K0 := {K ∈ K0| |K ∩ Gi| ≤ 1 for all i, K ∩ S = ∅}

2 K0 := {K ∈ K0| K ∩ S = S}

3 K0 := {K ∈ K0| |K ∩ Gi| = 2 for some i}.

The lemma follows immediately from the following simple observations:

1 1. If T ∈ T2 ∪ T3,K ∈ K0 then AT,K = 0.

2 2. If T ∈ T3,K ∈ K0 then AT,K = 0.

Before the next lemma, we make another deﬁnition.

Deﬁnition 4.2.3. Suppose X is a nonempty set admitting a partition Γ, i.e., X := n [ Gi for nonempty sets Gi with Gi ∩ Gj = ∅ for i 6= j. Let A, B be two nonempty i=1 families of subsets of X such that each member A ∈ A (resp. B ∈ B) satisﬁes

|A ∩ Gi| ≤ 1 (resp. |B ∩ Gi| ≤ 1) for all i. By the Incidence matrix of X w.r.t.

X X (A, B), we mean a 0-1 valued matrix I := ((IA,B)) with its rows indexed by the members of A and columns indexed by the members of B, deﬁned as follows:

X IA,B := 1 if A ⊂ B and

:= 0 otherwise.

103 Lemma 4.2.4. Suppose X is a nonempty set admitting an equi-partition Γ, i.e., let n [ X = Gi, with Gi = m for each i. Let i=1 X T := T ∈ : |T ∩ G | ≤ 1 for all i , X t i X K := K ∈ : |K ∩ G | ≤ 1 for all i . X k i

X X If n ≥ k, we have r(I ) = |TX |, where r(I ) denotes the rank of the incidence matrix

X I , of X w.r.t. (TX , KX ).

Proof: Without loss of generality, we write X := Im × In where In := {1, 2, . . . , n}.

Let Y := Im × Ik, Γk := {Im × {i} : i ∈ Ik}. Firstly note that if k = n (so that the Divisible design is a Transversal design or

Orthogonal array), the result is a direct consequence of a ‘λ-large’ theorem due to

Ray-Chaudhuri and Singhi (Lemma 3.2 in [29]). For notational convenience let us denote the incidence matrix by IY in this case. Note that the notation is consistent.

Now suppose n > k. Consider the matrix

In := ((IT,K )) where

IT,K := 1 if T ⊂ K and

:= 0 otherwise,

for subsets T,K ⊂ In with |T | = t, |K| = k. If r(I) denotes the rank of I, Wilson’s n theorem in [39] implies that r(I) = . t

104 Now to conclude the proof of the lemma, observe that IX is permutation equivalent

Y Y to I ⊗ In, the Kronecker product of I and In. This observation completes the proof since n r(IX ) = r(IY ⊗ I ) = r(IY ) · r(I ) = mt = T . n n t X

Lemma 4.2.5.

n r(A) = m3, 3 m r(B) = n · ((n − 1)m 2 where r(A), r(B) denote the row ranks of the matrices A, B as in Lemma 4.2.2.

Proof: Note that the rows of A are indexed by the members of K1 and the columns of A by the members of T1. Since T ∈ T1 ⇒ |T ∩ Gi| ≤ 1,T ∩ S = ∅ and K ∈ K1 ⇒

|K ∩ Gi| ≤ 1,K ∩ S = ∅, it follows that the matrix A is the incidence matrix for a Group divisible design with n groups of size m each, of strength 3, and block size k. n By the preceding lemma, it follows that r(A) = |T | = m3. 1 3 In a similar manner, note that for the matrix B, the rows and columns are indexed by the member of T2 and K2 respectively. Since each set T ∈ T2,K ∈ K2 satisﬁes S ⊂ T,K, the matrix B is the same as the incidence matrix for the stem-partition for X \ S with an empty stem. Again, this is the incidence matrix for a Divisible design with n groups of size m each, strength 2 and block size k − 1. Once again, by

m the preceding lemma we have r(B) = |T2| = n 2 · ((n − 1)m, and that completes the proof.

105 We now turn to the matrix C of Lemma 4.2.2. By deﬁnition, the rows of C are

3 indexed by the members of T3 and the columns by the members of K0. Note that every element T ∈ T3 also satisﬁes |T ∩ Gi| = 2 for some i, |T ∩ Gj| ≤ 1 for all j 6= i.

3 Deﬁne the following subfamilies of T3, K0 as follows. For 1 ≤ i ≤ n, deﬁne

T3(i) := {T ∈ T3| |T ∩ Gi| = 2}

3 1 K0(i) := K ∈ K3| |K ∩ Gi| = 2, |K ∩ Gj| ≤ 1 for all j 6= i

Then it follows immediately that

3 For T ∈ T3(i),K ∈ K0(j) we have T ⊂ K ⇒ i = j.

3 Hence, arranging the elements of T3, K0 as T3 = T3(1) T3(2) ···T3(n) and K3 = 3 3 3 0 K0(1) K0(2) ···K0(n) respectively, we see that C again admits the form   C1 0 ... 0      0 C2 ... 0    C =  . . . .   . . . .      0 0 ... Cn

Thus in order to determine the rank of C, we need to calculate the ranks of Ci, i =

1 . . . , n. Since Gi = m for all i, it follows that r(Ci) = r(Cj), for all i ≤ i, j ≤ n.

106 Lemma 4.2.6. In the form above, r(C1) = |T3(1)|.

3 Proof: For notational convenience, let us write T := T3(1), K := K0(1), so that the rows and columns of C1 are indexed by the members of T and K respectively. Also, as before, we shall identify G1 := {1} × Im with the set Im.

We make a further simpliﬁcation; let us denote by Ti,j, Ki,j, the following subfamilies of T, K:

Ti,j := {T ∈ T| i, j ∈ T } ,

Ki,j := {K ∈ K| i, j ∈ K} .

Then, arranging the members (as a list) of T (resp. K) in, say, the lexicographic order of the two element subsets {i, j} of Im, we write T = T1,2 T1,3 ··· Tm−1,m , K = K1,2 K1,3 ··· Km−1.m.

so that once again the matrix C1 admits the form   C1,2 0 ... 0      0 C1,3 ... 0    C1 =  . . . .   . . . .      0 0 ... Cm−1,m with Ci,j denoting the incidence matrix for the family Ti,j in the family of k-subsets,

Ki,j. Hence, it suﬃces to calculate the rank of Ci,j.

107 Now note that every member T ∈ Ti,j and K ∈ Ki,j satisﬁes {i, j} ∈ T,K. Conse- quently, if we consider the pair (Y, ∆) where,

Y := X \ G1, ∆ := Γ \{G1}, and the families of subsets

Y T0 := (= Y ), 1 Y K0 := K ∈ : |K ∩ G | ≤ 1 for all G ∈ ∆ , k − 2 i i

Y 0 0 then the incidence matrix I of Y w.r.t (T , K ) is permutation-equivalent to Ci,j, so

Y 0 r(Ci,j) = r(I ). But by lemma, 4.2.4, it follows that r(Ci,j) = |T | = |Ti,j|. Hence, X X r(C1) = r(Ci,j) = |Ti,j| = |T|, Im Im {i,j}∈( 2 ) {i,j}∈( 2 ) as required.

Proof of Theorem 4.1.5: Note that by Lemma 4.2.2, r(A0) ≤ r(A) ≤ |T |, where

A, A0 are as in Lemma 4.2.2. Again by Lemma 4.2.2, we have r(A0) = r(A)+r(B)+ r(C), since A0 is a block partitioned matrix. By lemmas 4.2.5, and 4.2.6, it follows that r(A0) = |T |, so that r(A) = |T | and that completes the proof.

Remarks:

1. The proof used the fact that the partition Γ is an equi-partition, i.e., that

|Gi| = |Gj| for 1 ≤ i, j ≤ n in lemma 4.2.4.

108 2. Theorems that prove that a certain incidence matrix has full rank are usually

proved by establishing a corresponding ‘λ-large’ theorem ([14],[39],[29]) and

these two facts are usually equivalent. We however shall not prove a corre-

sponding λ-large theorem for Candelabra systems in this thesis.

3. The above proof can be easily modiﬁed for the case S = ∅. In general, one

can prove in similar fashion, a corresponding result for appropriately deﬁned

incidence structures for RFSS as well. We hope to have a general structural

result in the future.

4. Though Candelabra systems have been constructed for quite a few values of k

(the block size) and s (stem size), they predominantly have s = 1. There are a

few examples where this is not the case: our examples in the preceding chapter

construct Candelabra systems with s = 2. Mills has constructions for CS with

k = 4, 6. Mills and Hanani (as indicated in the preceding chapter) also outline

constructions for CS with s = 0. However for other values of s (note that the

deﬁnition does not force s ≤ 2), there are no known instances. A λ-large CS

theorem would construct CS systems with larger values of s as well, albeit with

λ being large.

4.3 Concluding remarks

A λ-large theorem for a class of combinatorial structures is in some sense equivalent to the fact a corresponding incidence matrix has full rank. Once a λ-large theorem has been proven, one can construct corresponding structures on ‘large sets’ with

109 λ = 1, by ’Block-Spreading’ methods. The block spreading theorems of Wilson[41]

(for t = 2, ﬁber size = qd for d ‘suﬃciently large), Blanchard [5] (for arbitrary t,

fiber size = qd), and Mohácsy and Ray-Chaudhuri[25] (arbitrary t, fiber size = λd) use a specific algebraic structure of each fiber of a partial divisible design (a vector space or direct sum of vector spaces) to obtain a corresponding theorem for another partial design with λ = 1. Again, the bounds on d are too large which reflect in the parameter values of the theorems of Blanchard [3], Mohácsy[25] and the new Steiner designs of the preceding chapter. Blanchard and Mohácsyuse strong restrictions on the ingredient partial designs of their constructions. It would be worthwhile obtaining block-spreading theorems with less stringent restrictions in order to see new Steiner

3-designs.

110 BIBLIOGRAPHY

[1] Assmus, E. F., and Key, J. D., On an inﬁnite class of Steiner systems with t = 3 and k = 6, Journal of Combinatorial theory A, 42(1986), 55-60.

[2] Beth, T. , Jungnickel D., and Lenz, H., Design Theory Volumes 1 and 2, En- cyclopedia of Mathematics and its Applications 78, second edition, Cambridge University Press, 1999.

[3] Blanchard, J. L., A construction for Steiner 3-designs, Journal of Combinatorial Theory A, 71(1995), 60-67.

[4] Blanchard, J. L., An extension theorem for Steiner systems, Discrete Mathemat- ics 141(1995), no. 1-3, 23-35.

[5] Blanchard, J. L., A construction for orthogonal arrays with strength t ≥ 3, Discrete Mathematics, 137(1995), no. 1-3, 35-44.

[6] Blanchard, J. L., The existence of orthogonal arrays of any strength with large order, unpublished manuscript.

[7] Cameron, P. J., Permutation Groups, Cambridge University Press, 1999.

[8] Cameron, P. J., Maimani, H. R., Omidi, G. R., and Tayfeh-Rezaie, B., 3-designs from PSL(2, q), Discrete Mathematics, 306(2006), vol. 23, 3063-3073.

[9] Cameron, P. J., Maimani, H. R., Omidi, G. R., and Tayfeh-Rezaie, B., 3-designs from PGL(2, q), Electronic Journal of Combinatorics, volume 13, 2006.

[10] Cameron, P. J., and van Lint, J. H., Designs, Graphs, Codes and their Links, Cambridge University Press, 1991.

[11] Denniston, R. H. F., Some new 5-designs, Bulletin of the London Mathematical Society, 8(1976), 263-267.

[12] Dickson L. E., Linear Groups with an Exposition of the Galois Field Theory, Dover Publications Inc., New York, 1958.

111 [13] Granell M. J., Griggs T. S., and Mathon R. A., Some Steiner 5-designs with 108 and 132 points, Journal of Combinatorial designs, 1(1993), 213-238.

[14] Graver, J., and Jurkat, W. B., The module structure of integral designs, Journal of Combinatorial Theory A, 15(1973), 75-90.

[15] Hanani, H., On quadruple systems, Canadian Journal of Mathematics, 12(1960),145- 157.

[16] Hanani, H., On some tactical conﬁgurations, Canadian Journal of Mathematics, 15(1963), 702-722.

[17] Hanani, H., A class of 3-designs, Journal of Combinatorial Theory A, 26(1979), 1-19.

[18] Huppert, B., Endliche Gruppen, Volume 1, Springer Verlag, Berlin, 1967.

[19] Hartman, A., The fundamental construction for 3-designs, Discrete Mathematics, 124(1994), no 1-3, 107-131.

[20] Hughes, D. R., and Piper, F. C., Projective Planes, Springer Verlag (Graduate Texts in Mathematics), 1982.

[21] Kramer, E. S., and Mesner, D. M., t−designs on hypergraphs, Discrete Mathe- matics, 15(1976), 247-274.

[22] Lam, C. W. H., Thiel, L. H., and Swiercz, S., The non-existence of ﬁnite projective planes of order 10, Canadian Journal of Mathematics, 41(1989), 1117-1123.

[23] Mills, W. H., A new 5-design, Ars Combinatoria 6(1972), 193-195.

[24] Moh´acsy, H., and Ray-Chaudhuri, D. K., Candelabra systems and designs, Jour- nal of Statistical Planning and Inference, 106(2002), 419-448.

[25] Moh´acsy, H., and Ray-Chaudhuri, D. K., A construction for inﬁnite families of Steiner 3-designs, Journal of Combinatorial Theory A, 94(2001) 127-141.

[26] Moh´acsy, H., and Ray-Chaudhuri, D. K., A construction for group divisible t- designs with strength t ≥ 2 and index unity, Journal of Statistical Planning and Inference, 109(2003) 167-177.

[27] http://mathworld.wolfram.com/KirkmansSchoolgirlProblem.html

112 [28] R¨odl,V., On a packing and covering problem, European Journal of Combina- torics, 6(1985), 69-78.

[29] Ray-Chaudhuri, D. K., and Singhi, N. M., On the existence and number of orthogonal arrays, Journal of Combinatorial Theory A, 47(1988), 28-36.

[30] Ray-Chaudhuri, D. K., and Wilson, R. M., Solution of Kirkman’s schoolgirl problem, Proceedings of Symposia in Pure Mathematics, 19(1971), 187-203.

[31] Singhi, N., Tags on subsets, Discrete Mathematics, 306(2006), no. 14, 1610-1623.

[32] Teirlinck, L., Non-trivial t−designs without repeated blocks exist for all t, Dis- crete Mathematics, 65(1987), 301-311.

[33] Teirlinck, L., Locally trivial t−designs and t-designs without repeated blocks, Discrete Mathematics, 77(1989), 345-356.

[34] Weixia Li, The existence of a simple 3 − (30, 7, 15) design and a simple 3 − (26, 12, 55) Design, unpublished manuscript.

[35] Weixia Li, and Hao Shen, Simple 3-designs of PSL(2, 2n) with block size 7, unpublished manuscript.

[36] Weixia Li, and Hao Shen, Simple 3-designs of PSL(2, 2n) with block size 6, unpublished manuscript.

[37] Wilson, R. M. An existence theory for pairwise balanced designs I. Composition theorems and morphisms, Journal of Combinatorial Theory A, 13(1972), 220- 245.

[38] Wilson, R. M., An existence theory for pairwise balanced designs II. The structure of PBD-closed sets and the existence conjecture, Journal of Combinatorial Theory A, 13(1972), 246-273.

[39] Wilson, R. M., The necessary conditions for a design are suﬃcient for something, Utilitas Mathematica, 4(1973), 207-215.

[40] Wilson, R. M., Non-isomorphic Steiner triple systems, Mathematische Zeitschrift, 135(1974), 303-313.

[41] Wilson, R. M., An existence theory for pairwise balanced designs III. Proof of the existence conjectures, Journal of Combinatorial Theory A, 18(1975), 71-79.

113 [42] Qing Xiang, Recent progress in algebraic design theory, Finite Fields and their Applications 2(2005), 622-653.

114