關於組合設計理論及其相關應用的研究(I)

行政院國家科學委員會專題研究計畫 期中進度報告

關於組合設計理論及其相關應用的研究(1/3)

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中

華

民

國 92 年 5 月 30 日

A Study of Dual Pairs of Set System

s

Tayuan Huang and Jiahsian Wang

Abstract: Each (0,1) matrix is associated with a dual pair of set systems. Some classical combinatorial structures including 2-designs, symmetric designs,

quasi-symmetric designs are represented in this way. In terms of union or Boolean sums, some xondition were posed over a dual pair of set systems, so that it provides a way for pooling designs, i.e., non-adaptive group testing.

1. Preliminary

For a given (0,1)-matrix M of order t n× , two set systems with ground sets {1, 2,K, }n and {1, 2,K, }t respectively, can be associated with rows as well as the columns of M naturally. Some combinatorial structures can be defined over these

set systems by posing some conditions over those rows and columns of M . For

example, the inner products among pairs of rows and pairs of columns were used in posing those conditions for 2-designs, symmetric designs and quasi-symmetric designs. On the other hand, Boolean sums among columns were used to define for disjunction, which lead to the notions of d -disjunct, d -separability and used for

non-adaptive group-testing and pooling designs as well.

Let M be a

( )

0,1 -matrix of order t×n, let T be the subset of _i

{

1,2,K,n

}

with the i -th row of M as its characteristic vector, and C be the subset of _j

{

1,2,K,n

}

with the j -th row of M as its characteristic vector. We consider the

following properties over the family C =

{

C₁,C₂,K,C_n

}

⊆2[ ]t and their dual families T=

{

T₁,T₂,K,T_t

}

, where

( )

U

D i Ci D C = _∈ for

[ ]

    ∈ d n D : (i) _{i j} 1 C_i d C C ∩ < ∀i≠ j,

(ii) C ⊄C

( )

D if j∉D,

[ ]

    ∈ d n D , (iii) C

( ) ( )

D ≠C D′ wherever

[ ]

    ∈ ′ d n D D, are distinct, i.e., the d -term union are pairwise distinct.

Definition Let M be a

( )

0,1 -matrix of order t×n and let C be the subset of _j

{

1,2,K,n

}

with the j -th row of M as its characteristic vector. Then the matrix M

is called d -disjunct if C=

{

C1,K,Cn

}

⊆2[ ]t satisfying the condition that

U

i D i

j C

C ⊄ _∈ .

2. d-disjunct

A d -disjunct matrix M of order t×n, M provide a strategy for a

non-adaptive group testing which can identify to d defects. Thus form the above

discussion we have sufficient conditions for disjunctness of matrices : If

∑

= ∩ > d j i i i C Cj C ₁ 0 0 , since

(

)

0 0 0 0 1 1 1 i d j i i d j i i d j i i C C C C C C C ∩

U

_{= j} =

U

₌ ∩ _j ≤

∑

₌ ∩ _j < ,

U

d j i i C j C ₁ 0 ⊄ = , then M is called d -disjunct.

In particular, let T be the test consisting of _i

{

j M

( )

i,j =1

}

, and

(

)

t

t x x x

r = 1, 2,K, be the corresponding outcome vector. Suppose D⊆

[ ]

n is the set of d defects, then x_j =1 if and only of j∈C

( )

D . This group testing reports support

( )

r as the set C

( )

D , consequent -ly, j∈

[ ]

n is a defect if

( )

r C

( )

D C_j ⊆support = .

Thus form the sufficient conditions for disjunctness of matrices, if M is a

( )

0,1 - matrix of order t×n such that JM =kJ and the off-diagonal entries of

M

(

)

≤ ∩

_U

d₌ j i i Aj A ₁ d A A d k j i ∩ ij ≤ ≤

∑

=1 α . It follows that

U

d j i i Aj A ⊄ ₌₁ . This shows that M is a d -disjunct matrix of order t×n. Therefore, an interesting problem in extremal set theory is the following: Find

[ ]

[ ]t

k t F ⊆2     ⊆ with F large

as possible such that A∩B ≤1 whenever A,B∈F are different.

The following two theorems show that the relation between the family

{

C1,C2,

C = K,C_n

}

and its dual family when they satisfied the above properties. Theorem If C =

{

C₁,C₂,K,C_n

}

⊆2[ ]t satisfying the condition that C

( ) ( )

D ≠C D′

for distinct

[ ]

    ∈ ′ d n D

D, , then the dual family T =

{

T₁,T₂,K,T_t

}

⊆2[ ]n satisfying the following condition that for each x∈

{ }

0,1t, there exist

[ ]

    ∈ d n D such that 0 = ∩Ti D if and only if x_i =0. Theorem Let

[ ]

    ∈ d n

D , if C=

{

C₁,C₂,K,C_n

}

⊆2[ ]t satisfying the condition that

U

i D i

j C

C ⊄ _∈ , then the dual family T =

{

T₁,T₂,K,T_t

}

⊆2[ ]n satisfying the following condition that _{( )}T

[ ]

n D

D C

j∉ i = −

U

, and vice verse.

3. Some connections with combinatorial designs

A family

{

T₁,T₂,K,T_t

}

⊆2[ ]n is called a d-complete design if ( )DT

[ ]

n D C i∉ i = −

U

where

[ ]

    ∈ d n

D . Note that if i∉C

( )

D then M

( )

i,α =0 for all α∈D, i.e. α∉T_i for all α∈D. Hence T_i ∩D=∅, i.e. T_i ⊆

[ ]

n −D. For example,

1.

{

C1,C2,K,C_b

}

is a 2-

(

v,k,1

)

design, then Mv×b is

(

k−1

)

-disjunct, and

hence

{

T1,T2,K,T_v

}

is

(

k−1

)

-complete.

2. If

{

C1,C2,K,C_b

}

is an affine resolvable 2-

(

v,k,λ

)

design, then

v k C Ci j 2 ≤ ∩ , hence M_v×b is       −1 v k

{

T1,T2,K,Tv

}

is       −1 v k -complete.

3. If

{

C1,C2,K,C_b

}

is any affine resolvable incomplete block design with k

C_i = and C_i ∩C_j ≤q₁or q₂ then M is t -disjunct,

{

}

     = 2 , 1 max q q k

t , and hence

{

T₁,T₂,K,T_v

}

is t -complete.

Theorem If M is the point-block incidence matrix of a 2-

(

v,k,1

)

design and

[ ]

    ∈ d n

D , then D=

[ ]

n −

U

{

T_i for each i with M

( )

i,D =0

}

.

Example :

A (16, 4, 1)-BIBD is presented by

{1 2 3 4, 5 6 7 8, 9 10 11 12, 13 14 15 16, 1 5 9 13, 2 8 10 15, 3 6 11 16, 4 7 12 14, 1 6 10 14, 2 7 9 16, 3 5 12 15, 4 8 11 13, 1 7 11 15, 2 6 12 13, 3 8 9 14, 4 5 10 16, 1 8 12 16, 2 5 11 14, 3 7 10 13, 4 6 9 15}.

The blocks of the dual incidence structure are as follows:

A1 = {1,5,9,13,17} A2 = {1,6,10,14,18} A3 = {1,7,11,15,19} A4 = {1,8,12,16,20}

A5 = {2,5,11,16,18} A6 = {2,7,9,14,20} A7 = {2,8,10,13,19} A8 = {2,6,12,15,17}

A9 = {3,5,10,15,20} A10 = {3,6,9,16,19} A11 = {3,7,12,13,18} A12 = {3,8,11,14,17}

A13 = {4,5,12,14,19} A14 = {4,8,9,15,18} A15 = {4,6,11,13,20}

A16 = {4,7,10,16,17}.

Suppose we obtain the following result vector:

R(U) = (0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1).

When we carry out the algorithm identify with input R(U), we compute the following:

j M 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 3 0 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 4 0 1 1 1 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 6 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 10 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 12 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 15 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 (Note that boxed entries are used to indicate when a “1” is changed to a “0”.) The positive set U is thus U = {10, 18}.

4 Some Classes Based on Ranked Posets

A few classes of non-adaptive pooling designs were proposed by Du and Ng, Macula and Weng and Huang . The first one is guaranteed to be d-error-detecting and

thus

 

d 2 -error-correcting, where d, a positive integer, is the maximum number of

defectives (or positives). Hence, the number of errors which can be detected grows linearly with the number of positives. Also, this construction induces a construction of a binary code with maximum Hamming distance of at least 2d+2. The second design is the q-analogue of a known construction on d-disjunct matrices.

For 1≤d <k<n, let J n d k be the ( 1

(

, ,

)

0 )-matrix of order,

    ×     k n d n with rows

and columns be indexed by respectively we define the matrix J∗

(

n d k, ,

)

. Initially, we

define the by letting its represented by the members of [ ]n

d       and [ ]n k      

respectively in the following way: for D [ ]n d   ∈ _{ } and K [ ]n k   ∈ _{ } the

(

D,K

)

th entry of the matrix J n d k is 1 if and only if

(

, ,

)

D⊂ K. We then define J∗

(

n d k, ,

)

of order

the     ×     +     k n n d n

obtained by row augmenting the matrix J n d k with

(

, ,

)

(

n k

)

c , 1 , δ .

For q being a prime power, let F denote the Galois field GF(q) of q _q

elements. Let _      l Fm q

denote the set of all l-dimensional subspaces (l-subspaces for short)

of the m-dimensional vector space onF . For _q m≥k>d ≥1, Let M_q

(

m,k,d

)

be the 0,1-matrix whose rows (resp. columns) are indexed by elements of _

     d Fm q (resp.       k Fm q

matrixM_q

(

m,k,d

)

is 1 if and only if the i-th d-subspace is a subspace of the j-th

k-subspace of F ._qm

Given integers m≥k>d ≥1. An l-matching is a matching of size l (i.e. it has l

edges). Let M(m,k,d) be the (0,1)-matrix whose rows are indexed by the set of all d-matchings on K2_m, and whose columns are indexed by the set of all k-matchings on

m

K2 . All matchings are to be ordered lexicographically, i.e., M(m,k,d) is 1 if and only if

the i-th d-matching is contained in the j-th k-matching.

Theorem For m≥k>d ≥1,

1. M_q

(

m,k,d

)

is a d-disjunct matrix of order v×n with row weight

q d k d m       − − and column weight q d k       .

2. M(m,k,d) is a d-disjunct matrix of order v×n with row weight g(m-d,k-d) and

column weight     d k .

Let V be an n-dimensional space over a finite field of order q. Let M be the

incidence matrix with row entries indexed by the set of all 1-dimensional subspaces, columns entries indexed by the set of all k-dimensional subspaces of V, where a_i,j =1 if and only if the i-th row is a 1-dimensional subspace contained in the k-dimensional

subspace corresponding to the j-th column.

Theorem

1. Both J n d k and

(

, ,

)

J∗

(

n d k, ,

)

are d -disjunct; moreover

2. d_H

(

B_d

(

J∗

(

n d k, ,

)

)

)

≥4 if k−d ≥3.

Corollary Given integers m>d ≥1, the following holds:

1. M(m, k, d) is d-error-detecting and

 

d 2 -error-correcting；moreover

if known to be exactly d.

Theorem Let V be a vector space of dim n over a finite field GF(q), and let M be a

(0,1)-matrix row-indexed by _      d V , column-indexed by _      k V

such that M(A,B)=1 if

B

A⊆ , and 0 otherwise. For a subset D of _

     k V

, let L(D) be the Boolean sum of those

columns corresponding to those k-subspaces in D. Then

e d_H

(

L

( ) ( )

D LD

)

m D D k V D D m = ′ = ′ =       ⊆ ′ , min : , , . 2 , if , 0, otherwise. q k m m d d m  _× −  _≤   ₋  =_{  }   5. References

[1] D.J. Balding, W.J. Bruno, E. Knill, D.C. Torney, A comparative survey of non-adaptive pooling designs, in: Genetic Mapping and DNA Sequencing, MN,

1994, Springer, New York, 1996, pp. 133-154.

[2] T. Beth, D. Jungnickel, H. Lenz, Design Theory, Cambridge University Press,

Cambridge, 1986.

[3] T. Huang and K.S. Wang, Association Schemes and Group Testing with eror-tolerance, manuscript 2003.

[4] J.H. van Lint, R.M. Wilson, A Course in Combinatorics, Cambridge University

Press, Cambridge, 1992.

[5] A.J. Macula, Error-correcting nonadaptive group testing with d -disjunct e

matrices, Discrete Appl. Math. 80 (2-3) (1997) 217-222.

[6] A. Macula, A simple construction of d-disjunct matrices with certain constant weights, to appear, Discrete Math (1994).

[7] H.Q. Ngo, D.-Z. Du, A survey on combinatorial group testing algorithms with applications to DNA , DIMACS /DU2, Amer. Math. Soc., Providence, RI, 2000.

[8] Weng, and Huang, Pooling Spaces and non-adaptive pooling designs, Discrete Maths, to appear.