Combinatorial Design Theory [expository notes]

101 Pages·2010·0.513 MB·English

by  Chris Godsil

Combinatorial Design Theory1
Chris Godsil
(cid:13)c 2010
1version: April 9, 2010
ii
Preface
This course is an introduction to combinatorial design theory.
iii
iv
Contents
Preface iii
1 Block Designs 1
1.1 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2.1 Fano Plane . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2.2 Trivial Cases . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.3 Complements . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.4 Another difference set construction . . . . . . . . . . . . . 2
1.2.5 Affine Planes . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Relations between Parameters . . . . . . . . . . . . . . . . . . . . 2
2 Symmetric Designs 5
2.1 Incidence Matrices of Designs . . . . . . . . . . . . . . . . . . . . 5
2.2 Constructing Symmetric Designs . . . . . . . . . . . . . . . . . . 6
2.3 Two Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Bilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.5 Cancellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.6 The Bruck-Ryser-Chowla Theorem . . . . . . . . . . . . . . . . . 11
2.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Hadamard Matrices 17
3.1 A Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 The Kronecker Product . . . . . . . . . . . . . . . . . . . . . . . 19
3.4 Symmetric and Regular Hadamard Matrices . . . . . . . . . . . . 21
3.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.5 Conference Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.6 Type-II Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4 Planes 27
4.1 Projective Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 Affine Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3 M¨obius Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
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vi CONTENTS
5 Orthogonal Arrays 33
5.1 Latin Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.3 Affine Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.4 Partial Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.5 Strongly Regular Graphs. . . . . . . . . . . . . . . . . . . . . . . 37
6 Block Intersections 39
6.1 Quasi-Symmetric Designs . . . . . . . . . . . . . . . . . . . . . . 39
6.2 Triangle-free Strongly Regular Graphs . . . . . . . . . . . . . . . 40
6.3 Resolvable Designs . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6.4 Designs with Maximal Width . . . . . . . . . . . . . . . . . . . . 44
7 t-Designs 47
7.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
7.2 Extending Fisher’s Inequality . . . . . . . . . . . . . . . . . . . . 48
7.3 Intersection Triangles . . . . . . . . . . . . . . . . . . . . . . . . 49
7.4 Complements and Incidence Matrices. . . . . . . . . . . . . . . . 50
7.5 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
7.6 Gegenbauer Polynomials . . . . . . . . . . . . . . . . . . . . . . . 52
7.7 A Positive Semidefinite Matrix . . . . . . . . . . . . . . . . . . . 53
8 Witt Designs 55
8.1 Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
8.2 Perfect Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
8.3 The Binary Golay Code . . . . . . . . . . . . . . . . . . . . . . . 57
8.4 The Plane of Order Four . . . . . . . . . . . . . . . . . . . . . . . 58
8.5 The Code of PG(2,4) . . . . . . . . . . . . . . . . . . . . . . . . 59
9 Groups and Matrices 61
9.1 Group Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
9.2 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . 62
9.3 Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
10 Mutually Unbiased Bases 67
10.1 Complex Lines and Angles . . . . . . . . . . . . . . . . . . . . . . 67
10.2 Equiangular Lines . . . . . . . . . . . . . . . . . . . . . . . . . . 68
10.3 Reality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
10.4 Difference Sets and Equiangular Lines . . . . . . . . . . . . . . . 70
10.5 MUB’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
10.6 Real MUB’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
10.7 Affine Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
10.8 Products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
11 Association Schemes 77
11.1 Intersection Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 77
CONTENTS vii
12 Incidence Stuctures 79
12.1 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
12.2 Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
12.3 Matrices and Maps . . . . . . . . . . . . . . . . . . . . . . . . . . 81
13 Matrix Theory 83
13.1 The Kronecker Product . . . . . . . . . . . . . . . . . . . . . . . 83
13.2 Normal Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
13.3 Positive Semidefinite Matrices . . . . . . . . . . . . . . . . . . . . 85
14 Finite Fields 87
14.1 Arithmetic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
14.2 Automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
14.3 Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
viii CONTENTS
Chapter 1
Block Designs
1.1 Definitions
We use the following definition for this lecture. (For the official definition, see
Corollary ??.) A block design D consists of a point set V and a set B of blocks
where each block is a k-subset of V such that:
(a) There is a constant r such that each point lies in exactly r blocks.
(b) There is a constant λ such that each pair of distinct points lies in exactly λ
blocks.
In fact (b) implies (a), which we leave as an exercise. What we call a block
design might also be called a 2-design.
1.2 Examples
We offer some simple examples of designs.
1.2.1 Fano Plane
Here we have V =Z and the blocks are as follows:
7
{0,1,3}
{1,2,4}
{2,3,5}
{3,4,6}
{4,5,0}
{5,6,1}
{6,0,2}
1
2 CHAPTER 1. BLOCK DESIGNS
This is a block design with parameters (v,b,r,k,λ)=(7,7,3,3,1).
Ingeneralbisverylargeanditmaybeinconvenientorimpossibletopresent
thedesignbylistingitsblocks. TheFanoplaneisanexampleofadifferenceset
construction. AdifferencesetS inanabeliangroupGisasubsetofGwiththe
property that each non-zero element of G appears the same number of times
as a difference of two elements of S. Here α = {0,1,3} is a difference set for
G=Z . If G is an abelian group and S ⊆G then the set
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S+g ={x+g |x∈S}
is called a translate of S. In our example, the design consists of all translates
of α. The Fano plane is the projective plane of order two.
1.2.2 Trivial Cases
A design is trivial if k ∈ {0,1,v−1,v}. These are valid designs, although not
so interesting. For a non-trivial design, 2 ≤ k ≤ v−2. The complete design
consists of allk-subsets of a set of size v.
1.2.3 Complements
If we take the complement in Z of each block of the Fano plane we get a
7
design on 7 points with block size 4. This holds in general, i.e. if we take the
complement of each block in a design we obtain another design.
1.2.4 Another difference set construction
LetV =Z ,thenα={0,2,3,4,8}isadifferencesetandthesetofalltranslates
11
of α is a 2-design with parameters (v,b,r,k,λ)=(11,11,5,5,2). A design with
b = v is called a symmetric design. We see later on that k = r in symmetric
designs.
1.2.5 Affine Planes
Let V be a vector space and let B be the set of lines (a line is a coset of a
1-dimensional subspace). This is a 2-design with λ=1. If we take V to be the
2-dimensional vector space over Z we get a 2-(9,3,1) design with b = 12 and
3
r =4. So
(cid:26)(cid:18) (cid:19) (cid:27)
a
V = |a,b∈Z .
b
This is an affine plane of order three.
1.3 Relations between Parameters
Theparametersv,b,r,k,λarenotindependent. Defineaflag tobeanordered
pair (v,α) where v ∈V and α ∈B. Then by counting with respect to the first