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Combinatorial Designs
and Their Applications
(組合設計及其應用)
應用數學系 傅恆霖
Combinatorial Design?
• Combinatorial design theory is the part of
combinatorial mathematics that deals with the
existence, construction and properties of systems
of finite sets whose arrangements satisfy
generalized concepts of balance and/or
symmetry. These concepts are not made precise
so that a wide range of objects can be thought of
as being under the same umbrella. At times this
might involve the numerical sizes of set
intersections as in block designs, while at other
times it could involve the spatial arrangement of
entries in an array as in Sudoku grids.
System of Sets
• A design defined on X is a collection
of subsets of X denoted by Ɓ.
• If all the subsets are of the same
cardinality, then it is called a block design.
The Fano plane
Seven points
Three points on each line
Every two points define a line
Seven lines
Three lines through each point
Every two lines meet at a point
4
The Fano plane as a set
system
0
{0,1,4}, {0,2,5}, {0,3,6}, {1,2,6},
{4,2,3}, {4,5,6}, {1,3,5}
1
3
2
4
5
6
5
Round robin tournament
Directed edge between every
pair of vertices
X  Y means X beats Y
{(1,2),(1,4),(2,4),(3,1),(3,2),(4,3)}
6
Doubles tournament
• Each game: a, b v
c, d
• Tournament has
many games
• Tournament usually
has structure (e.g.
everyone plays in
the same number of
games)
7
Whist tournament
every pair of players partner once and
oppose twice. Tournament is played in
rounds.
Example: Whist
Table 1with 8 players
Table 2
Round 1
∞
0
v
4
5
1
3
v
2
6
Round 2
∞
1
v
5
6
2
4
v
3
0
Round 3
∞
2
v
6
0
3
5
v
4
1
Round 4
∞
3
v
0
1
4
6
v
5
2
Round 5
∞
4
v
1
2
5
0
v
6
3
Round 6
∞
5
v
2
3
6
1
v
0
4
Round 7
∞
6
v
3
4
0
2
v
1
5
8
Research Strategies
• Use theoretical techniques to prove
that a given design exists (or doesn’t
exist) for certain sizes.
• Use experimental techniques to
prove that a given design exists (or
doesn’t exist) for certain sizes.
9
Quiz
Problem:
I have at most two favor numbers in
my pocket and the number are in
{1,2,3,…,63}. Can you find the
numbers by asking (me) as few queries
(問題) as possible?
Group Testing
• Applications!
A Latin square of order n is an n×n
array based on an n-set S such that each
element of S occurs exactly once in each
row and each column.
We can take S   n .
A latin square of order 3
Array Presentations
• We can use L = [li,j]nxn to represent a
latin square of order n.
• For convenience, li,j is read as the
(i,j)-entry of the latin square L.
• Two latin squares of the same order
are distinct if there is an ordered pair
(i,j) such that their corresponding
entries are not the same.
How many?
• Let L(n) denote the number of
distinct latin squares of order n.
• L(1) = 1
• L(2) = 2
• L(3) = 12
• L(4) = 576
• …
• L(5) = 161280
• L(6) = 812851200
• L(7) = 61479419904000
• L(8) = 108776032459082956800
• …
• L(11) =
776966836171770144107444346734
230682311065600000
• What’s next?
Sudoku
• Sudoku, or Su Doku, is a Japanese
word (or phrase) meaning something
like Number Place.
• There are about 5.525x1027 latin
squares of order 9 and 6.671x1021
valid Sudoku grids.
• Note here that a Sudoku grid is a
latin square with special properties.
Sudoku puzzle
1
3
4
1
2
3
4
4
3
1
18
A latin square of order n
defines a quasigroup on 3
elements.
<S,*> is a quasigroup if <S,*>is a groupoid
a*x  b
and 
have unique solution  a, b  S .

y * a  b
How much do you know?
• A quasigroup is not a group due to
the “associative law”.
• It is not difficult to prove that an
associative quasigroup is a group.
• Equivalently, if a quasigroup is also a
semigroup, then it is in fact a group!
• Group?
36 officers (Euler
1779)
----Euler 的困惑
Latin square
1 2


2
1


1 2 3


2
3
1


 3 1 2


Orthogonal Latin squares
1 2 3


 3 1 2
2 3 1


15:40:58
1 2 3


2 3 1
 3 1 2


21
1

2
3

4
2
3
4
1
4
1
2
3
3

4
1

2
 11 22 33 


 32 13 21
 23 31 12 


Two latin squares L  [lij ] and M  [mij ] of order n
are orthogonal if { (lij , mij ) 0  i, j  n  1}   2n 。
Two orthogonal latin squares of order 4
L1 , L2 ,, Lk are mutually orthogonal latin squares
of order n (MOLS(n)), if Li  L j for 1  i  j  k 。
MOLS(4)
Theorem Let n be a prime power and n≠2 .
Then there exists n-1 latin squares (best
possible!) of order n which is a collection of
mutually orthogonal latin squares.
1

2
3

4
1

2
3

4
5

2
3
4
3
4
5
4
5
5
1
1
2
1
2
3
15:40:58
5

1
2

3
4 
2
1
4
3
3
4
1
2
4

3
2

1
1

3
5

2
4

2
5

5 1 2
2 3 4

4 5 1
1 2 3 
4
1
3
5
3
1

3
4

2
2
4
3
1
4
24
3
1
2
4
1

4
2

5
3

4

2
1

3
1

4
2

3
2
3
1
4
2
3
4
5
1
2
3
1
4
2
5
3
4
5
1
3
2
4
1
4

1
3

2
5

3
1

4
2 
1

5
4

3
2

2
3
4
1
2
3
5
4
1
5
2
1
3
4
5
5

4
3

2
1 
An Updated Result
 2  MOLS (n) for n  2, 6;
 3  MOLS (n) for n  2, 3, 6, 10;
 4  MOLS (n) for n  2, 3, 4, 6, 10, 14, 18, 22;
NEXT!
15:40:58
25
f :  2n   2n , f ((i, j ))  (lij , mij )
利用L⊥M我們可定義函數f，f為  2n 的一個
Permutation，或者說f為由  2n 對映至  2n 的1-1,
onto 函數。
例：f((1,1))=(0,3)
f((2,3))=(1,0)
(＊) 令α為的一個排列，β也是排列，
則 L  M  L  M  ，
其中 L  [ (lij )]， L  [lij ] 。
由上述的結果，我們可以發現當L⊥M時，與L
垂直的n階拉丁方陣至少有n！個。所以，要判斷
key是由哪兩個方陣所形成並不容易！
Cryptosystem from MOLS(n)
1. 使用n階拉丁方陣。
2. 有 n 2 個 Messages (Plaintexts)。
3. 以n階拉丁方陣建構一個圖(OLSG)，G，則有
|E(G)| 個keys。
(最早使用k個MOLS(n)，則有   個keys.)
k
2
討論：
1.當n相當大時，MOLS(n)的個數也會很大。
2.如果只考慮MOLS(n)，key space較小。
3.Orthogonal mate。不容易找！(全部找出來！)
分散模式解 (Sharing Scheme)
在近代有很多重大的決定，為了確保決策過程
沒有暇疵，通常會採用由多個人都同意的情況下才
執行；例如開金庫，發射核彈…。所以，建立一個
系統使得較小的人數無法開啟是有它的必要性。
A critical set C in a latin square
L  [lij ] is
a set
C  {( i, j; k ) | i, j , k   n & k  lij } (partial latin square)
with the following two properties：
1. L is the only latin square of order n which
has symbol k in (i,j)-cell for each i, j; k  C；
and
2. no proper subset of c has property (1).
A critical set
例
上面圈出的三個位置，任兩個都會形成一個臨界
集(Critical Set)。少了，或多了都不是臨界；然而
多了(3個)也可以繼續填成唯一的拉丁方陣。
Critical Set for Sudoku
• If we expect the solution of a Sudoku
puzzle is unique, then the partial
latin square shown must have
contain a “critical” set in the sense of
satisfying the requirements of a
Sudoku game.
• Sometimes, we did find more than
one solution for some game.
Fact 1
A critical set C of a latin square L provides
minimal infos from which L can be reconstructed.
Fact 2
Deciding whether a partial latin square is a
critical set is NP-complete. (From completion
point of view.)
Fact 3
Denote the minimum size of a critical set of order
n2
n by M(n). M (m) 
[D. Curran & G.H.J.van Rees,
4
Cong. Numer. 1979]
Critical sets, n=5
分散模式解
1.Key → L (拉丁方陣)
n public
2.選一個集合它是L中多個臨界集的聯集：S。
3.把其中 t’≦|S|。(可以容許高階者持有多一些
(i,j ; k), 甚至一個Critical Set！)
4.足夠多的人即可得到“Key“。
A (s,t)-secret sharing scheme is a system
where k pieces of information called shares
or shadows of a secret key K are distributed
so that each participant has a share such
that:
1. the key K can be reconstructed from
knowledge of any t or more shares; and
2. the key K can not be reconstructed from
knowledge of fewer than t shares.
臨界集的選擇有很多！
↑
對於臨集的了解不多。
↓
增加破解難度
More Applications
• Coding Theory
• Group Testing
• Experimental Designs
• More to be introduced!