Chapter 10 Combinatorial Designs
BIBD
Example
(a,b,c)
(b,c,f)
(a,b,d)
(b,d,e)
(a,c,e)
(b,e,f)
(a,d,f)
(c,d,e)
(a,e,f)
(c,d,f)
Here are 10 subsets of the 6 element set {a, b, c, d, e, f }.
BIBD
Definition
A balanced incomplete block design is a collection of k-subsets,
called blocks, of a v-set S, k < v, such that each pair of elements of S
occur together in exactly λ of the blocks.
Notation
(v, k, λ)-BIBD or (v, k, λ) design. Alternately, we could say we have a
(b, v, r, k, λ)-BIBD.
b is the number of blocks
r is the number of times each element appears in a block
Necessary Conditions
Theorem
In a (v, k, λ) design with b blocks, each element occurs in r blocks
such that
1
λ(v − 1) = r(k − 1)
2
bk = vr
Example
Existence
Show that no (11, 6, 2) design can exist.
Example
‘
Existence
Does a (43, 7, 1)-BIBD exist?
Example
Example
(7, 3, 1)-BIBD
Finite Projective Plane
Definition
A finite projective plane is a (n2 + n + 1, n + 1, 1) design.
Fisher’s Inequality
Theorem
Fisher’s Inequality: In any (v, k, λ) design, b ≥ v.
Definition
The incidence matrix of a (v, k, λ) design is a b × v matrix A = (aij )
defined by
1 if the ith block contains the jth element
aij =
0 otherwise
Theorem We Need
Theorem
If A is a (v, k, λ) design, then
A0 A = (r − λ)I + λJ
where A0 is the transpose of A, I is the v × v identity matrix and J is
the v × v matrix of all 1’s.
Proof of Fisher’s Inequality
Let A be the incidence matrix. We first show what A0 A is nonsingular
by showing that its determinant is non-zero. Now,
r λ ··· λ λ r ··· λ λ λ ··· λ 0
|A A| = .
.
.
.
.
.
.
.
. .
.
. ..
λ λ
.
r Subtract the first row from each of the other rows.
r
λ
···
λ
λ−r r−λ
·
·
·
0
λ−r
0
r
−
λ
0
= .
.
.
..
..
..
..
.
.
..
λ−r
0
r−λ
Proof of Fisher’s Inequality (cont.)
Now, add to the first column of the sum of all the other columns.
r + (v − 1)λ
λ
·
·
·
λ
0
r−λ
···
0 0
0
r−λ
0 = .
.
.
..
..
..
..
.
..
0
0
.
r−λ = [r + λ(v − 1)](r − λ)v−1
= [r + r(k − 1)](r − λ)v−1
= rk(r − λ)v−1
But, k < v, so by property (1) r > λ, so |A0 A| =
6 0, But, A0 A is a v × v
matrix, so the rank ρ of A0 A is ρ(A0 A) = v. Finally, since
ρ(A0 A) ≤ ρ(A) and since ρ(A) ≤ b (A has b rows), v ≤ ρ(A) ≤ b.
Existence
Example
Can a (16, 6, 1) design exist?
Complementary Designs
Definition
Let D be a (b, v, r, k, λ) design on a set S of v elements. Then the
complementary design D has as it’s blocks the complements S − B of
the blocks B in D.
Complementary Designs
Theorem
Suppose that D is a (b, v, r, k, λ) design. Then D is a
(b, v, b − r, v − k, b − 2r + λ) design provided that b − 2r + λ > 0.
Why must b − 2r + λ > 0?
Symmetric Designs
Corollary
If D is a symmetric (v, k, λ) design with v − 2k + λ > 0 then D is a
symmetric (v, v − k, v − 2k + λ) design.
Residual Designs
Definition
The (v − 1, v − k, r, k − λ, λ) design obtained from a symmetric
(v, k, λ) design by deleting all elements of one block is called a
residual design.
Example
The residual design created from a (7, 3, 1)-BIBD
(1,2,4) (2,3,5) (3,4,6) (4,5,7) (5,6,1) (6,7,2)
(7,1,3)
Affine Plane
Definition
A design with the parameters (n2 , n, 1) is called an affine plane of
order n.
If we can arrange the blocks into groups so that each group contains
each element exactly once, we say the design is resolvable.
Resolvability
Definition
A BIBD isresolvable
if the blocks can be arranged into v groups so
that the br = kv blocks of each group are disjoint and contain in
their union each element exactly once. The groups are called
resolution classes of parallel classes.
Kirkman(1850)
Fifteen young ladies in 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 twice.
(4,2,1) Design
Suppose we wanted a league schedule for 4 teams where each team
played each other team one time.
How many weeks do we need?
How many total games?
The Turning Trick
A (2n, 2, 1) design exists for all integers n ≥ 1. But this would get
tedious to develop for a large league unless we had a trick ...
If we wanted to construct a league schedule for 8 teams, what would
the parameters of the corresponding block design be?
The Turning Trick
1
u
7 u
u2
u
6
∞
u
u
5
u
3
u
4
The Turning Trick
1u
7
u
u2
6 u
u
∞
u
5
u3
u
4
The Turning Trick
1u
7
u
u2
6 u
u
∞
u
5
u3
u
4
Back to the (4,2,1) Design
We can ‘go backwards’ from the residual design idea to build finite
projective planes.
These constructions show that affine planes of order n exist iff finite
projective planes of order n exist. There is also a correspondence with
...
Latin Squares
Definition
A Latin square on n symbols is an n × n array such that each of the n
symbols occurs exactly once in each row and in each column. The
number n is called the order of the square.
Example

A
 D

 C
B
B
A
D
C
C
B
A
D

D
C 

B 
A
Latin Squares and League Schedules
Suppose a league schedule has been arranged for 2n teams in 2n − 1
rounds. Then, define a 2n × 2n array A = (aij ) by
aii = n, aij = k
i 6= j
where the ith and jth teams play in round k. Since each team plays
precisely one game per round, A is a Latin square.
Latin Squares and League Schedules
Example
Construct a Latin square of order 2n from a league schedule on 8
teams.
1u
7
u
T
T
T
T
T
"u2
"
"
T
T
"
6 u
" T
T "
T
u
T
T
T
T
∞
T
T
Tu
T
T 3
T
T
T
TT
T
u
Tu
5
4
MOLS
So just how many Latin squares are there of order n, up to labeling? Is
there just one of each order?
1
2
2
1
Order 2

1
 2
3
2
3
1

3
1 
2

1
 3
2
Order 3
2
1
3

3
2 
1
MOLS

1
 2

 3
4
2
1
4
3
3
4
1
2


4
3 

2 
1

1
 3

 4
2
1
 4

 2
3
2 3
4 1
3 2
1 4
Order 4

4
2 

1 
3
2
3
1
4
3
2
4
1

4
1 

3 
2
MOLS
Definition
Join (A, B) is the n × n array where the i, jth entry is (aij , bij ) where
aij ∈ A and bij ∈ B.
These two Latin squares are called mutually orthogonal. For short, we
say MOLS. For n ≥ 4, the MOLS are pairwise orthogonal.
MOLS
Definition
A complete set of MOLS of order n consists of n − 1 pairwise
orthogonal Latin squares.
Notation: Number of MOLS of order n is given by N(n).
MOLS
Definition
A complete set of MOLS of order n consists of n − 1 pairwise
orthogonal Latin squares.
Notation: Number of MOLS of order n is given by N(n).
Theorem
For all numbers n ≥ 3, N(n) ≥ 2, except for N(6) = 1.
MOLS
Theorem
N(n) ≥ 2 whenever n is odd, n ≥ 3.
MOLS Example
Example
Construct 2 MOLS of order 3.
MOLS Example
Example
Construct 2 MOLS of order 5.
Moore-MacNeish
Theorem
Moore-MacNeish: N(mn) ≥ min{N(m), N(n)}
Corollary
N(n) ≥ min{pαi i }-1
Latin Squares and Finite Projective Planes
Theorem
An affine plane of order n exists iff a finite projective plane of order n
iff n − 1 MOLS of order n exists.
Construction
Example
Construct an affine plane of order 4 from the three MOLS of order 4.
Initial Designs
Definition
A (cyclic) (v, k, λ) difference set (mod v) is a set D = {d1 , d2 , . . . , dk }
of distinct elements of Zv such that each non-zero d ∈ Zv can be
expressed in the form d = di − dj in precisely λ ways.
Back to the (7,3,1) Design
Example
{1, 2, 4} is a (7, 3, 1) difference set.
Justification
Why is it that the design we will obtain will be balanced?
Translates
Definition
If D = {d1 , d2 , . . . , dk } is a (v, k, λ) difference set mod v) then the set
D + A = {d1 + a, d2 + a, . . . , dk + A} is called a translate of D.
Theorem
If D = {d1 , d2 , . . . , dk } is a cyclic (v, k, λ) difference set then the
translates D, D + 1, . . . , D + (v − 1) are the blocks of a symmetric
(v, k, λ) design.
Translate Example
Example
We will illustrate this with our {1, 2, 4} difference set.
Another Example
Example
Verify that {1, 2, 4, 10} is a (13,4,1) difference set in Z13 .
Another Example
Example
Verify that {1, 3, 4, 5, 9}(mod 11) yields a (11,5,2) design.
Difference Sets in Groups Other Than Zv
Definition
A (v, k, λ) difference set in an additive abelian group G of order v is a
set D = {d1 , d2 , . . . , dk } of distinct elements of G such that each
non-zero element g ∈ G has exactly λ representations as g = di − dj .
How The Example Gives a Design
We can obtain the translates in the same manner, but how does this
give us a (16,6,2) design?
Difference Systems
Definition
Let D1 , . . . , Dt be sets of size k in an additive abelian group G of
order v such that the differences arising from the Di give each
non-zero element of G exactly λ times. The D1 , . . . , Dt are said to
form a (v, k, λ) difference system in G.
Note: the Di need not be disjoint.
Difference System Example
Example
Show that {1, 2, 5},{1, 3, 9} form a (13,3,1) difference system in Z13 .
Important: the differences are only taken within blocks.
Starters
Definition
A starter in an abelian group G of order 2n − 1 is a set of n − 1
unordered pairs {x1 , y1 }, . . . , {xn−1 , yn−1 } of elements of G such that
i. x1 , y1 , . . . , xn−1 , yn−1 are precisely all the non-zero elements of G
ii. ±(x1 − y1 ), . . . , ±(xn−1 − yn−1 ) are precisely the non-zero
elements of G
Example of a Starter
Example
The pairs {1, 2}, {4, 8}, {5, 10}, {9, 7}, {3, 6} form a starter in Z11 .
Whist Tournaments
Difference systems can also be used to construct whist tournaments.
Definition
A whist tournament, denoted Wh(4n), on 4n players is a schedule of
games involving two players against two others, such that:
(i) the games are arranged in 4n − 1 rounds, each of n games
(ii) each player plays in exactly one game each round
(iii) each player partners every other player exactly once
(iv) each player opposes every other player exactly twice
Example
Construct a Wh(4).
Example of Whist Tournament
Example
Verify that
∞,0 v 4,5
1,10 v 2,8
is the initial round of a cyclic Wh(12).
3,7 v 6,9