Two error-correcting pooling designs from symplectic spaces over a finite field

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Linear Algebra and its Applications

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / l a a

Two error-correcting pooling designs from symplectic

spaces over a finite field

Zengti Li

a

, Tayuan Huang

b

, Suogang Gao

a,∗

a_{Mathematics and Information College, Hebei Normal University, Shijiazhuang, 050016, PR China} b_{Department of Applied Mathematics, Chiao-Tung University, Hsinchu 30050, Taiwan}

A R T I C L E I N F O A B S T R A C T

Article history:

Received 11 July 2008 Accepted 20 April 2010 Available online 20 May 2010 Submitted by R.A. Brualdi

Keywords:

Pooling design Disjunct matrix Symplectic space DNA

In this paper, we construct two classes of t×n, se_-disjunct

ma-trix with subspaces in a symplectic spaceF(q2ν)and prove that the

ratio efficiency t/n of two constructions are smaller than that of D’yachkov et al. (2005) [2].

© 2010 Elsevier Inc. All rights reserved.

1. Introduction

The basic problem of group testing is to identify the set of defective items in a large population of items. Suppose we have n items to be tested and that there are at most d defective items among them. Each test (or pool) is (or contains) a subset of items. We assume some testing mechanism exists which if applied to an arbitrary subset of the population gives a negative outcome if the subset contains no positive and positive outcome otherwise. Objectives of group testing vary from minimizing the number of tests, limiting number of pools, limiting pool sizes to tolerating a few errors. It is conceivable that these objectives are often contradicting, thus testing strategies are application dependent. A group testing algorithm is non-adaptive if all tests must be specified without knowing the outcomes of other tests. A non-adaptive testing algorithm is useful in many areas such as DNA library screening [1,7].

A group testing algorithm is error tolerant if it can detect some errors in test outcomes. A mathe-matical model of error-tolerance designs is an se-disjunct matrix.

A binary matrix M is said to be se-disjunct if given any s

+

1 columns of M with one designated, there are e rows with a 1 in the designated column and 0 in each of the other s columns. An s1-disjunct

∗_{Corresponding author.}

E-mail address:[email protected](S. Gao).

0024-3795/$ - see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2010.04.040

matrix is said to be s-disjunct. In [3], D’yachkov et al. proposed the concept of fully se-disjunct matrices. An se-disjunct matrix is fully se-disjunct if it is not de-disjunct whenever d

>

s or e

>

e.

Macula [5] proposed a novel way of constructing s-disjunct matrices using the containment relation in a structure.

Huang and Weng [4] gave a comprehensive treatment of construction of d-disjunct matrices by using of pooling spaces, which is a significant and important addition to the general theory.

Ngo and Du [6] extended the construction to some geometric structures, such as simplicial com-plexes, and some graph properties, such as matchings.

D’yachkov et al. [2] claimed that the “containment matrix” method has opened a new door for constructing s-disjunct matrices from many mathematical structures.

In this paper, we construct two classes se-disjunct matrix with subspaces in a symplectic space

F

(2ν)

q and exhibit their disjunct properties. Given some fixed items, our goal is to detect the positive items. For a pooling design, the less the number of tests is, the better the pooling design is. In order to discuss easily in the following, we give a new definition. We call the ratio between the number of tests and the number of detected items test efficiency, that is the ratio between the number of rows and the number of columns in the se-disjunct matrix, i.e., t

/

n. We will give some discussions on the

ratio t

/

n and compare them with others, such as in [2]. 2. Symplectic space Let K

=

 0 I(ν)

−

I(ν) 0 

.

The symplectic group of degree 2

ν

over

F

q, denoted by Sp(2ν)

(

F

q

)

, consists of all 2

ν ×

2

ν

matrix T over

F

qsatisfying TKT

=

K. The vector space

F

q(2ν)together with the right multiplication action of Sp2ν

(

F

q

)

is called the 2

ν

-dimensional symplectic space over

F

q[8]. Let P be an m-dimensional subspace of

F

(q2ν), denote also by P an m

×

2

ν

matrix of rank m whose rows span the subspace P and call the matrix P a matrix representation of the subspace P. An m-dimensional subspace P is said to be of type

(

m, r

)

, if PKPis of rank 2r. In particular, subspaces of type

(

m, 0

)

are called m-dimensional totally isotropic

subspaces. The subspaces of type

(

m, r

)

exist if and only if 2r m 

ν +

r. The subspace of type

(

m, r

)

, which contains subspaces of type

(

m1, r

)

, exists if and only if 2r m1 m 

ν +

r. It is known that the

number of subspaces of type

(

m, r

)

, denoted by N

(

m, r

;

2

ν)

, is given by

N

(

m, r

;

2

ν) =

q2r(ν+r−m) _ν i=ν+r−m+1

(

q 2i

₋

1

)

r i=1

(

q2i

−

1

)

m−2r i=1

(

qi

−

1

)

.

(1)

Let N

(

m1, r

;

m, r

;

2

ν)

denote the number of subspaces of type

(

m1, r

)

contained in a given subspace of type

(

m, r

)

. It is known that

N

(

m1, r

;

m, r

;

2

ν) =

q2r(m−m1) m−2r i=m−m1+1

(

q i

₋

₁

₎

m1−2r i=1

(

qi

−

1

)

.

(2)

Let N

(

m1, r

;

m, r

;

2

ν)

denote the number of subspaces of type

(

m, r

)

containing a given subspace of type

(

m1, r

)

. It is known that

N

(

m1, r

;

m, r

;

2

ν) =

_ν+r−m1 i=1

(

q2i

−

1

)

_ν+r−m i=1

(

q2i

−

1

)

m−m1 i=1

(

qi

−

1

)

.

(3)

Lemma 2.1. Let

F

(_q2ν)denote the 2

ν

-dimensional symplectic space over a finite field

F

qwith 2r m0 i  m

ν +

r

.

Fix an

(

m0, r

)

-subspace W0of

F

_q(2ν), and an

(

m, r

)

-subspace W of

F

(_q2ν)such that W0

⊂

W

.

Then the number of

(

i, r

)

-subspace A of

F

(_q2ν), where W0

⊂

A

⊂

W , is N

(

i

−

m0, 0

;

m

−

m0, 0

;

2

(ν +

Proof. Since the symplectic group Sp2ν

(

F

q

)

acts transitively on each set of subspaces of the same type, we may assume that W has the matrix representation of the form

W

=

⎛ ⎜ ⎜ ⎝ r m0−2r ν+r−m0 r m0−2r ν+r−m0 I 0 0 0 0 0 0 0 0 I 0 0 0 I 0 0 0 0 0 0 W1 0 0 W2 ⎞ ⎟ ⎟ ⎠ r r m0−2r m−m0 , where

(

W1, W2

)

is an

(

m

−

m0, 0

)

-subspace of

F

2(ν+r−m0)

q . By (2), the number of

(

i, r

)

-subspace A, where W0

⊂

A

⊂

W , is N

(

i

−

m0, 0

;

m

−

m0, 0

;

2

(ν +

r

−

m0

))

.



3. Construction I

Definition 3.1. For 2r d0

<

d

<

k

ν +

r, assume that P0 is a fixed

(

d0, r

)

-space of

F

(q2ν)

.

Let M be a binary matrix whose columns (rows) indexed by all

(

k, r

)

-spaces containing P0 (

(

d, r

)

-spaces

containing P0) in

F

(q2ν) such that M

(

A, B

) =

1 if A

⊆

B and 0 otherwise. This matrix is denoted by

M1

(ν

, d, k

)

.

Theorem 3.1. Suppose 2r d0

<

d

<

k

ν +

r and set b

=

q(q k−d0−1₋₁₎ qk−d₋₁

.

Then M1

(ν

, d, k

)

is s e_-disjunct for 1 d  b and e

=

qk−dN

(

d

−

d0

−

1, 0

;

k

−

d0

−

1, 0

;

2

(ν +

r

−

d0

))

−(

s

−

1

)

qk−d−1N

(

d

−

d0

−

1, 0

;

k

−

d0

−

2, 0

;

2

(ν +

r

−

d0

)).

Proof. Let C, C1,

. . .

, Csbe s

+

1 distinct columns of M1

(ν

, d, k

)

. To obtain the maximum number of

subspaces of type

(

d, r

)

which contain P0in C

∩

s  i=1 Ci

=

s  i=1

(

C

∩

Ci

)

,

we may assume that each C

∩

Ci

(

1 i  s

)

is a subspace of type

(

k

−

1, r

)

.

Then each C

∩

Cicovers N

(

d

−

d0, 0

;

k

−

d0

−

1, 0

;

2

(ν +

r

−

d0

))

subspaces of type

(

d, r

)

con-taining P0from Lemma 2.1. However, the coverage of each pair of Ciand Cjoverlaps at a subspaces of type

(

k

−

2, r

)

containing P0, where 1 i, j  s. Therefore, from Lemma 2.1 only C1 covers the

full N

(

d

−

d0, 0

;

k

−

d0

−

1, 0

;

2

(ν +

r

−

d0

))

subspaces of type

(

d, r

)

containing P0, while each of

C2,

. . .

, Cs can cover a maximum of N

(

d

−

d0, 0

;

k

−

d0

−

1, 0

;

2

(ν +

r

−

d0

)) −

N

(

d

−

d0, 0

;

k

−

d0

−

2, 0

;

2

(ν +

r

−

d0

))

subspaces of type

(

d, r

)

not covered by C1. By (2), the subspaces of type

(

d, r

)

of C not covered by C1, C2,

. . .

, Csis at least

e

=

N

(

d

−

d0, 0

;

k

−

d0, 0

;

2

(ν +

r

−

d0

)) −

N

(

d

−

d0, 0

;

k

−

d0

−

1, 0

;

2

(ν +

r

−

d0

))

− (

s

−

1

)(

N

(

d

−

d0, 0

;

k

−

d0

−

1, 0

;

2

(ν +

r

−

d0

))

−

N

(

d

−

d0, 0

;

k

−

d0

−

2, 0

;

2

(ν +

r

−

d0

)))

=

qk−dN

(

d

−

d0

−

1, 0

;

k

−

d0

−

1, 0

;

2

(ν +

r

−

d0

))

− (

s

−

1

)

qk−d−1N

(

d

−

d0

−

1, 0

;

k

−

d0

−

2, 0

;

2

(ν +

r

−

d0

)).

Since by (2)N(d−d0−1,0;k−d0−1,0;2(ν+r−d0)) N(d−d0−1,0;k−d0−2,0;2(ν+r−d0))

=

qk−d0−1−1 qk−d₋₁ and e

>

0, we obtain s

<

q

(

q k−d0−1

−

₁

)

qk−d

₋

₁

+

1

.

Set

b

=

q

(

q

k−d0−1

−

₁

)

qk−d

₋

₁

.

Then 1 s  b.



Corollary 3.2. Suppose that 2r d0

<

d

<

k

ν +

r and 1 s  min{b, q

+

1

}.

Then M1

(ν

, d, k

)

is not se+1-disjunct, where b and e are as in Theorem3.1

.

Proof. Let C be a

(

k, r

)

-space containing P0, and E be a fixed

(

k

−

2, r

)

-space containing P0and

con-tained in C. By Lemma2.1, we obtain the number of

(

k

−

1, r

)

-spaces containing E and contained in C is

N

(

1, 0

;

2, 0

;

2

(ν +

r

−

k

+

2

)) =

q

+

1

.

For 1 s  min{b, q

+

1

}

, we choose s distinct

(

k

−

1, r

)

-subspaces containing E and contained in

C, denote these subspaces by Qi

(

1 i  s

)

. For each Qi, we choose a

(

k, r

)

-subspace Cisuch that

C

∩

Ci

=

Qi, where 1 i  s. Hence each pair of Ciand Cjoverlaps at the same

(

k

−

2, r

)

-subspace E, where 1 i, j  s. By Theorem3.1, it follows that the corollary hold.



Corollary 3.3. Suppose that d

=

d0

+

1 and 1 s  q

.

Then M1

(ν

, d, k

)

is se-disjunct, but it is not se+1 -disjunct, where e

=

qk−d0−2

(

_q

−

_s

+

₁

).

Proof. Setting d

=

d0

+

1 in the e formula of Theorem3.1, we obtain e

=

qk−d0−2

(

_q

−

_s

+

₁

).

The second statement follows directly from Corollary3.2.



The following theorem tells us how to choose k so that the test to item ratio is minimized. Theorem 3.4. For 2r m0

<

m

ν +

r, the sequence N

(

m0, r

;

m, r

;

2

ν)

is unimodal and gets its peak at m

=

2ν+2r+m0 3 or m

=

2ν+2r+m0 3

+

1

.

Proof. For 2r m0 m1

<

m2

ν +

r, by (3), we have

N(m0,r;m1,r;2ν) N(m0,r;m2,r;2ν)

=

m2−m0 i=m1−m0+1

(

q i

₋

₁

₎

_ν+r−m1 i=ν+r−m2+1

(

q 2i

₋

₁

₎

=

m2−m1−1 i=0

(

qm1−m0+1+i

−

1

)

m2−m1−1 i=0

(

q2(ν+r−m2+1+i)

−

1

)

(4)

=

m2−m1−1 i=0 qm1−m0+1+i

−

₁ q2(ν+r−m2+1+i)

−

₁

.

If2ν+2r+m0 3

+

1 m1

<

m2

ν +

r, then2ν+2r₃+m0

<

m1. It implies that

2m1

+

m2

>

3m1

>

2

ν +

2r

+

m0

.

(5)

Since i m2

−

m1

−

1, by (5) we have

m1

+

2m2

>

2

ν +

2r

+

m0

+

1

+ (

m2

−

m1

−

1

)

 2

ν +

2r

+

m0

+

1

+

i

.

So

q2(ν+r−m2+1+i)

−

₁

<

_qm1−m0+1+i

−

₁

.

Therefore, qm1−m0+1+i

−

₁ q2(ν+r−m2+1+i)

−

₁

>

1

.

From (4) we have N

(

m0, r

;

m2, r

;

2

ν) <

N

(

m0, r

;

m1, r

;

2

ν).

If 2r m0 m1

<

m2  2ν+2r+m0 3 , then m22ν+2r₃+m0. Thus m1

+

2m2

<

3m2 2

ν +

2r

+

m0

<

2

ν +

2r

+

m0

+

1

+

i

.

It follows that m1

−

m0

+

1

+

i

<

2

ν +

2r

−

2m2

+

2

+

2i

=

2

(ν +

r

−

m2

+

1

+

i

).

So qm1−m0+1+i

−

₁

<

_q2(ν+r−m2+1+i)

−

_1, and hence qm1−m0+1+i

−

₁ q2(ν+r−m2+1+i)

−

₁

<

1

.

From (4) we have N

(

m0, r

;

m2, r

;

2

ν) >

N

(

m0, r

;

m1, r

;

2

ν).



4. Discussions of test efficiency for construction I

Identifying most positive items with least tests is one of our goals. Therefore, discussing how to make the ratio t

/

n smaller is significative. In our matrix,

t

/

n

=

N 

₍

_{d0, r}

_;

_{d, r}

_;

₂

_ν)

N

(

d0, r

;

k, r

;

2

ν)

=

k−d0 i=d−d0+1

(

q i

₋

₁

₎

_ν+r−d i=ν+r−k+1

(

q2i

−

1

)

.

We first will explain several facts on the ratio:

(1) Parameter d0

(ν

, r

)

only appears in the numerator (denominator). It is easy to show that the

larger the d0,

ν

and r are, the smaller the ratio is.

(2) Noting that the increasing speed of q2i

−

1 is larger than qi

−

1, so the smaller the d and k are, the smaller the ratio is.

In [2], D’yachkov et al. constructed with subspaces of GF

(

q

)

, where q is a prime power, each of the columns(rows) is labeled by an k

(

d

)

-dimensional space, mij

=

1 if and only if the label of row i is contained in the label of column j. In order to compare with t

/

n, we should take the dimension of the

space of GF

(

q

)

to be 2

(ν +

r

−

d0

)

. Assume that the test efficiency of [2] is t1

/

n1. Then t1

/

n1

=

_2(ν+r−d 0) d  q _2(ν+r−d 0) k  q

=

k i=d+1

(

q i

₋

₁

₎

2(ν+r−d0)−d i=2(ν+r−d0)−k+1

(

q i

−

₁

)

.

Theorem 4.1. If 2d0

>

k

−

1, then t

/

n

<

qd0(d−k)t1

/

n1, wherek−₂1

<

d0

<

d

<

k

.

Proof t n

/

t1 n1

=

k−d0 i=d−d0+1

(

q i

₋

₁

₎

_ν+r−d i=ν+r−k+1

(

q2i

−

1

)

 k i=d+1

(

qi

−

1

)

2(ν+r−d0)−d i=2(ν+r−d0)−k+1

(

q i

₋

₁

₎

=

k−d−1 i=0

(

qd−d0+1+i

−

1

)

k−d−1 i=0

(

q2(ν+r−k+1+i)

−

1

)

 k−d−1 i=0

(

qd+1+i

−

1

)

k−d−1 i=0

(

q2(ν+r−d0)−k+1+i

−

1

)

=

k−_d−1 i=0 qd−d0+1+i

−

₁ qd+1+i

₋

₁ k−_d−1 i=0 q2(ν+r−d0)−k+1+i

−

₁ q2(ν+r−k+1+i)

₋

₁

<

k−_d−1 i=0 qd−d0+1+i qd+1+i k−_d−1 i=0 q2(ν+r−d0)−k+1+i

−

₁ q2(ν+r−k+1+i)

−

₁

=

k−d−1 i=0 q2(ν+r)−k+1+i−2d0

−

₁ qd0

(

_q2(ν+r)−k+1+2i−(k−1)

−

₁

)

.

Since 2d0

>

k

−

1, we have q 2(ν+r)−k+1+i−_2d0−1 q2(ν+r)−k+1+2i−(k−1)₋₁

<

1. Therefore, t

/

n

<

qd0(d−k)_t1

/

_n1, wherek−₂1

<

d0

<

d

<

k.



5. Construction II

Definition 5.1. For 2 2r  d

<

k

ν +

r, let M be a binary matrix whose columns (rows) indexed by

all subspaces of type

(

k, r

) ((

d, r

))

in

F

_q(2ν)such that M

(

A, B

) =

1 if A

⊆

B and 0 otherwise. This matrix

is denoted by M2

(ν

, d, k

)

.

Theorem 5.1. Suppose 4 2r

+

2 d

<

k

−

1

ν +

r

−

1

.

If 1 s  q2r, then M2

(ν

, d, k

)

is se-disjunct, where e

=

q(k−d−1)d+2r

.

Proof. Let C, C1,

. . .

, Csbe s

+

1 distinct columns of M2

(ν

, d, k

)

. To obtain the maximum number of

subspaces of type

(

d, r

)

in C

∩

s  i=1 Ci

=

s  i=1

(

C

∩

Ci

)

,

we may assume that each C

∩

Ciis a subspace of type

(

k

−

1, r

)

, where 1 i  s. By (2), the number of the subspaces of type

(

d, r

)

of C not covered by C1, C2,

. . .

, Csis at least

N

(

d, r

;

k, r

;

2

ν) −

sN

(

d, r

;

k

−

1, r

;

2

ν)

=

q2r(k−d) k₋2r i=k−d+1(q i₋₁₎ d−2r i₌1 (qi−1)

−

sq2r(k−d−1) k₋2r₋1 i=k−d (q i₋₁₎ d−2r i₌1 (qi−1)

=

q2r(k−d−1) k−2r−1 i=k−d+1(q i₋₁₎ d₋2r₍ qi₋₁₎

(

q k

₋

_q2r

₋

_s

₍

_qk−d

₋

₁

_)).

Since 2r

+

2 d

<

k

−

1, we obtain k₋2r₋1 i=k−d+1(q i₋₁₎ d₋2r i₌1 (qi−1)

=

d−2r−2 i=0

(

qi+k−d+1

−

1

)

d−2r−2 i=0

(

qi+1

−

1

)

1 qd−2r

−

₁

=

d−_2r−2 i=0 qi+k−d+1

−

1 qi+1

−

₁ 1 qd−2r

−

₁

=

d−2r−2 i=0 qk−d qi+1

−

_qk1−d qi+1

₋

₁ 1 qd−2r

₋

₁

>

q(d−2r−1)(k−d)−(d−2r)

.

Since 1 s  q2r_{, and 2r}

₊

_{2 d, we obtain}

qk

−

q2r

−

s

(

qk−d

−

1

)

 qk

₋

_q2r

₋

_q2r

₍

_qk−d

₋

₁

_{) =}

_qk−d+2r

₍

_qd−2r

₋

₁

₎

_{ q}k−d+2r

_.

Hence e

=

q(k−d−1)d+2r.



Theorem 5.2. Suppose 2 2r  d

< ν +

r

.

Let p

=

qd+_q1₋−₁q2r

−

1

.

If 1 s  p, then M2

(ν

, d, d

+

1

)

is fully se-disjunct, where e

=

p

−

s

.

Proof. By (2), we have N

(

d, r

;

d

+

1, r

;

2

ν) =

p

+

1. It follows that we can pick s

+

1 distinct sub-spaces C, C1,

. . .

, Csof type

(

d

+

1, r

)

such that C

∩

Ciand C

∩

Cjare two distinct subspaces of type

(

d, r

)

, where 1 i, j  s. By the principle of inclusion and exclusion, the number of subspaces of type

(

d, r

)

in C but not in each Ciis p

−

s

+

1, where 1 i  s. It follows that e  p

−

s. On the other hand, similar to the proof of Theorem5.4we obtain

e N

(

d, r

;

d

+

1, r

;

2

ν) −

s

−

1

=

p

−

s

.

Hence e

=

p

−

s.



The following theorem tells us how to choose k so that the test to item ratio is minimized. Theorem 5.3. For m goes from 2r to

ν +

r, the sequence N

(

m, r

;

2

ν)

is unimodal and gets its peak at m

=

 2ν+2r 3 or m

=

 2ν+2r 3

+

1

.

Proof. For 2r m1

<

m2

ν +

r, by (1), we have

N(m2,r;2ν) N(m1,r;2ν)

=

_ν+r−m1 i=ν+r−m2+1

(

q 2i

₋

₁

₎

m2−2r i=m1−2r+1

(

q 2r+i

₋

_q2r

₎

=

m2−m1−1 i=0

(

q 2(ν+r−m2+1+i)

−

₁

)

m2−m1−1 i=0

(

qm1+1+i

−

q2r

)

=

m2−m1−1 i=0 q2(ν+r−m2+1+i)

−

₁ qm1+1+i

−

_q2r

.

If2ν+₃2r

+

1 m1

<

m2

ν +

r, then2ν+₃2r

<

m1. It implies that

2m1

+

m2

>

3m1

>

2

ν +

2r

.

(6)

m1

+

2m2

>

2

ν +

2r

+

1

+ (

m2

−

m1

−

1

)

 2

ν +

2r

+

1

+

i

.

Thus m1

+

1

+

i

>

2

(ν +

r

−

m2

+

1

+

i

).

It follows that m1

+

i

−

2r 2

(ν +

r

−

m2

+

1

+

i

) −

2r

.

So q2(ν+r−m2+1+i)−2r_{ q}m1+i−2r_, and hence q2(ν+r−m2+1+i)−2r

−

1 q2r

<

q m1+i−2r

+ [(

_q

−

₁

)

_qm1+i−2r

−

₁

] =

_qm1+1+i−2r

−

₁

.

It follows that q2(ν+r−m2+1+i)−2r

−

1 q2r qm1+1+i−2r

−

₁

<

1

.

Therefore, q2(ν+r−m2+1+i)

−

₁ qm1+1+i

−

_q2r

<

1

.

From (4) we have N

(

m2, r

;

2

ν) <

N

(

m1, r

;

2

ν).

If 2r m1

<

m2  2ν+2r 3 , then m22ν+₃2r. Thus m1

+

2m2

<

3m2 2

ν +

2r

<

2

ν +

2r

+

1

+

i

.

It follows that m1

+

1

+

i

<

2

ν +

2r

−

2m2

+

2

+

2i

=

2

(ν +

r

−

m2

+

1

+

i

).

So qm1+1+i

−

_q2r

<

_q2(ν+r−m2+1+i)

−

_q2r

<

_q2(ν+r−m2+1+i)

−

₁

.

It follows that q2(ν+r−m2+1+i)

−

₁ qm1+1+i

−

_q2r

>

1

.

From (4) we have N

(

m2, r

;

2

ν) >

N

(

m1, r

;

2

ν).



Theorem 5.4. If d

=

2r, k

=

2r

+

1, then the test efficiency of construction II is smaller than that of [2].

Proof. If d

=

2r, k

=

2r

+

1, then the disjunct matrix of construction II is M2

(ν

, 2r, 2r

+

1

)

and the

disjunct matrix of [2] is M

(

n, 2r

+

1, 2r

)

. Let_nt be the test efficiency of M2

(ν

, 2r, 2r

+

1

)

and let_nt1 1 be the test efficiency of M

(

n, 2r

+

1, 2r

)

, respectively. Then

t n

=

N

(

d, r

;

2

ν)

N

(

k, r

;

2

ν)

=

N

(

2r, r

;

2

ν)

N

(

2r

+

1, r

;

2

ν)

=

q 2r(ν+r−2r)ν i=ν+r−2r+1

(

q 2i

₋

₁

₎

r i=1

(

q2i

−

1

)

2r−2r i=1

(

qi

−

1

)

·

r i=1

(

q 2i

₋

₁

₎

2r+1−2r i=1

(

q i

₋

₁

₎

q2r(ν+r−2r−1)ν i=ν+r−2r−1+1

(

q2i

−

1

)

=

q2r

(

q

−

1

)

q2(ν−r)

₋

₁

=

q2r+1

−

q2r q2ν−2r

₋

₁, and t1 n1

=

_2ν d  q _2ν k  q

=

k i=d+1

(

q i

₋

₁

₎

2ν−d i=2ν−k+1

(

qi

−

1

)

=

q2r+1

−

1 q2ν−2r

₋

₁

.

Therefore,_nt

<

t1 n1.



6. Conclusion

We construct two classes se-disjunct matrix with subspaces in symplectic space

F

(_q2ν). For a pooling design, the less the number of tests is, the better the pooling design is. Assume that the test efficiency in [2] is t1

/

n1. We prove that the test efficiency in construction I is less than qd0(d−k)t1

/

n1, where

k−1

2

<

d0

<

d

<

k, and that the test efficiency in construction II is less than t1

/

n1

−

q2r₋₂

q2ν−2r₋₁. From Theorem 4.4 of [2], the matrix of construction is sz-disjunct. To compare the error-correcting capability, we give two tables in the following. Take s

=

q, d0

=

2r

+

1, d

=

2r

+

2, k

=

4r

+

1,

ν =

20 and m

=

2

ν =

40, we have Table1from Theorem 3.1, 4.1 above and Theorem 4.4 in [2]; similarly, take

s

=

q, d

=

2r

+

1, k

=

2r

+

2,

ν =

100 and m

=

2

ν =

200, we have Table2from Theorem 5.1, 5.4 above and Theorem 4.4 in [2]. From Table1, we know that the error-correcting capability of construction [2] is better than that of ours on some values (for example,

(

s, r

) = (

3, 2

)

or

(

5, 4

)

). But in some cases (for example,

(

s, r

) = (

7, 5

)

or

(

17, 6

)

), the test efficiency of [2] is not good; whereas on these values (for example,

(

s, r

) = (

7, 5

)

or

(

17, 6

)

), the construction I above is feasible. For comparison of construction II with construction [2], their error-correcting capability is better than that of ours from Table2.

Table 1

Comparison of construction I with D’yachkov et al.

q=s 3 5 7 17 r 2 4 5 6 t n 1.9065×10−39 6.3166×10−84 5.2357×10−92 1.7941×10−95 t1 n1 6.3650×10− 28 _{2.0972×10−}15 _1.0701×1038 _9.0322×10175 e 9 15625 5764801 2.0160×1012 z 675782226 1.1140×1049 _2.2283×1091 _3.2890×10189 Table 2

Comparison of construction II with D’yachkov et al.

q=s 3 5 7 17 r 1 3 4 5 t n 6.0991×10−94 1.5693×10−131 1.9060×10−155 5.2883×10−221 t1 n1 8.8098×10− 94 _{1.9616×10−}131 _{2.2236×10−}155 _{5.6188×10−}221 e 9 15625 5764801 2.0160×1012 z 37 97651 47079201 3.6414×1013

Acknowledgements

The authors are grateful to the referees for their valuable suggestions and comments. This paper is supported by Natural Science Foundation of China (10971052); Natural Science Foundation of Hebei Province, China (No. A2008000128) and Natural Science Foundation of Langfang Normal University, China (LSZZ200901).

References

[1] D. Du, F.K. Hwang, Combinatorial Group Testing and Its Application, second ed., World Scientific, Singapore, 2000. [2] A.G. D’yachkov, F.K. Hwang, A.J. Macula, P.A. Vilenkin, C. Weng, A construction of pooling designs with some happy surprises,

J. Comput. Biol. 12 (2005) 1129–1136.

[3] A.G. D’yachkov, A.J. Macula, P.A. Vilenkin, Nonadaptive group and trivial two-stage group testing with error-correction

de-disjunct inclusion matrices, preprint.

[4] T. Huang, C. Weng, Pooling spaces and non-adaptive pooling designs, Discrete Math. 282 (2004) 163–169.

[5] A.J. Macula, A simple construction of d-disjunct matrices with certain constant weights, Discrete Math. 162 (1996) 311–312. [6] H. Ngo, D. Du, New constructions of non-adaptive and error-tolerance pooling designs, Discrete Math. 243 (2002) 161–170. [7] H. Ngo, D. Zu, A survey on combinatorial group testing algorithms with applications to DNA library screening, DIMACS Ser.

Discrete Math. Theoret. Comput. Sci. 55 (2000) 171–182.