Error-correcting pooling designs associated with some distance-regular graphs

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Discrete Applied Mathematics

journal homepage:www.elsevier.com/locate/dam

Error-correcting pooling designs associated with some distance-regular

graphs

Yujuan Bai

a,b

, Tayuan Huang

c

, Kaishun Wang

a,∗ a_{Sch. Math. Sci. & Lab. Math. Com. Sys., Beijing Normal University, Beijing, 100875, China} b_{Beijing DongFangDeCai School, Beijing, 100026, China}

c_{Department of Applied Mathematics, National Chiao-Tung University, Hsinchu, Taiwan}

a r t i c l e i n f o

Article history:

Received 26 January 2008

Received in revised form 3 March 2009 Accepted 20 April 2009

Available online 28 May 2009

Keywords:

Pooling designs

se_{-disjunct matrices} Distance-regular graphs

a b s t r a c t

Motivated by the pooling designs over the incidence matrices of matchings with various sizes of the complete graph K2nconsidered by Ngo and Du [Ngo and Du, Discrete Math.

243 (2003) 167–170], two families of pooling designs over the incidence matrices of t-cliques (resp. strongly t-cliques) with various sizes of the Johnson graph J(n,t)(resp. the Grassmann graph Jq(n,t)) are considered. Their performances as pooling designs are

better than those given by Ngo and Du. Moreover, pooling designs associated with other special distance-regular graphs are also considered.

© 2009 Elsevier B.V. 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 that 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 a 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 contradictory, 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. (See [3]).

A group testing algorithm is error tolerant if it can detect some errors in test outcomes. A mathematical model of error-tolerance designs is a de-disjunct matrix. A binary matrix M is said to be de-disjunct if, given any d

+

1 columns of M with one designated, there are e

+

1 rows with a 1 in the designated column and 0 in each of the other d columns. A de_{-disjunct matrix} with e

=

0 is said to be d-disjunct. Macula [12] proposed a novel way of constructing d-disjunct matrices by the containment relation of subsets in a finite set, while in [13] he constructed de_{-disjunct matrices for certain values of e. Ngo and Du [14]} extended the construction to some geometric structures, such as simplicial complexes, and some graph properties, such as matchings. Huang and Weng [9] generalized the constructions to pooling spaces, while they proved that a d2e_-disjunct

matrix is e-error-correcting in [10].

Du and Ngo [15] pointed out that the subject of pooling designs is a young and interesting field with deep connections to coding theory and design theory, and they strongly believe that the theory of association schemes – in particular distance

∗_{Corresponding author.}

E-mail addresses:[email protected],[email protected](K. Wang). 0166-218X/$ – see front matter©2009 Elsevier B.V. All rights reserved.

regular graphs – should play an important role in improving pooling designs. For more information about pooling designs, see [2,6–8].

LetΓ

=

(

X

,

R

)

be a connected graph of diameter D, and let

∂(

x

,

y

)

denote the distance of the vertices x and y.Γis said to be distance-regular whenever for all non-negative integers h

,

i

,

j, and for any two vertices x and y at distance h, the number

ph_i,j

= |{

z

∈

X

|

∂(

x

,

z

) =

i

, ∂(

z

,

y

) =

j

}|

is independent of the choice of x and y. For more information, the reader may consult [1].

For any positive integer n we shall use

[

n

]

to denote the set

{

1

,

2

, . . . ,

n

}

. Also, given any set X and any vector space V over a finite field Fq,



X k



denotes the collection of all k-subsets of X , and

[

V

k

]

denotes the collection of all k-subspaces of V . The Johnson graph J

(

n

,

t

)

is defined on



[n] t



such that two vertices A and B are adjacent if and only if

|

A

∩

B

| =

t

−

1. Similarly, the Grassmann graph Jq

(

n

,

t

)

is defined on

h

Fnq

t

i

such that two vertices A and B are adjacent if and only if dim

(

A

∩

B

) =

t

−

1. Johnson graphs and Grassmann graphs are two families of well-known distance-regular graphs.

LetΓ

=

(

X

,

R

)

be a connected graph. An l-subset∆of X is said to be a t-clique ofΓ with size l if any two distinct vertices in∆are at distance t. 1-clique is the clique in traditional use. A strongly t-clique of Jq

(

n

,

t

)

with size l is a subfamily

{

A1

,

A2

, . . . ,

Al

} ⊆

h

Fnq

t

i

such that dim

(

A1

+

A2

+ · · · +

Al

) =

tl. Note that an l-matching on K2nis a 2-clique of J

(

n

,

2

)

with

size l. Hence a t-clique of J

(

n

,

t

)

with size l is a generalization of an l-matching.

A class of pooling designs over the incidence matrices of matchings with various sizes of the complete graph K2n is

considered by Ngo and Du [14]. In this paper, we try to generalize Ngo and Du’s construction. The rest of the article is organized as follows. In Section2, we review some results on pooling designs over Johnson graphs and Grassmann graphs, and then compute one important parameter for these pooling designs. (SeeTheorems 2.9and2.10). In Section3, with an interpretation of matchings as 2-cliques of the Johnson graph J

(

n

,

2

)

, the pooling designs by Ngo and Du are generalized to the incidence matrices of t-cliques with various sizes of the Johnson graph J

(

n

,

t

)

and strongly t-cliques with various sizes of the Grassmann graph Jq

(

n

,

t

)

, respectively. We show that our pooling designs have the same capability of error-detecting and error-correcting as Ngo and Du’s. However, the test-to-item ratio of ours is much smaller. In Section4, we construct pooling designs associated with some special distance-regular graphs.

2. Disjunctness over Johnson graphs and Grassmann graphs

For a binary matrix M of order N

×

T , let B

(

D

)

denote the Boolean sum of those columns indexed by elements of D

⊆ [

T

]

, and let dH

(

B

(

D

),

B

(

D0

))

denote the Hamming distance between B

(

D

)

and B

(

D0

)

whenever D and D0are two distinct subsets of

[

T

]

.

Let

es

=

min |D|=|D0|=s

dH

(

B

(

D

),

B

(

D0

)).

The larger the parameter es, the better its capacity of error correcting.

In this section, we first review some results on pooling designs over Johnson graphs and Grassmann graphs, and then compute the parameter esfor those se-disjunct matrices.

2.1. Some known results

D’yachkov et al. [5] proposed the concept of fully se-disjunct matrices. An se-disjunct matrix is fully se-disjunct if it is not de0_{-disjunct whenever d}

_>

_{s or e}0

_>

_{e. D’yachkov et al. [4] gave the lower bounds of e}

sfor a fully se-disjunct matrix.

Proposition 2.1 ([4, Lemma 3.4]). Let M be a fully se_{-disjunct matrix. Then e}

s

≥

2

(

e

+

1

)

.

Macula [12] constructed d-disjunct matrices using the containment relation in a structure. D’ yachkov et al. [5] discussed the error-correcting property of Macula’s construction.

Definition 2.1 ([12]). For positive integers d

<

k

<

n, let J

(

n

,

d

,

k

)

be the binary matrix with row-indexed (resp. column-indexed) by



[n] d

 

resp

.



[n] k



such that M

(

A

,

B

) =

1 if and only if A

⊆

B and 0 otherwise.

Theorem 2.2 ([5, Proposition 2]). Suppose 1

≤

s

≤

d

<

k

<

n and e

=

e

(

s

) =



k−s k−d



−

1. Then J

(

n

,

d

,

k

)

is fully se-disjunct. Ngo and Du [14] gave a q-analogue of Macula’s construction. The error-correcting property of Ngo and Du’s construction was discussed in [5,4], respectively.

Definition 2.2 ([14]). For positive integers d

<

k

<

n, let Jq

(

n

,

d

,

k

)

be the binary matrix with row-indexed (resp. column-indexed) by

h

Fnq d

i 

resp

.

h

Fnq k

i

such that M

(

A

,

B

) =

1 if A

⊆

B and 0 otherwise. Let Fqbe the finite field with q elements, where q is a prime power. Let Fn

qbe the n-dimensional vector space over Fq. For a positive integer n, the Gaussian binomial coefficients with basis q is defined by

h

n i

i

q

=























i−1

Y

j=0 n

−

j i

−

j

,

if q

=

1

,

i−1

Y

j=0 qn

₋

_qj qi

₋

_qj

,

if q

6=

1

.

Naturally,



n 0



q

=

1 and



n i



q

=

0 if i

>

n. In the case q

=

1, we shall write n i

instead of



n i



1for convenience.

In the rest of this paper, for positive integers d

<

k and k

−

d

≥

2r, we always assume that

pq

(

r

) =











h

k d

i

q

−

h

k−r d

i

q

h

k−r d

i

q

−

h

k−2r d

i

q











and eq

(

s

,

r

) =



k d



q

−



k

−

r d



q

−

(

s

−

1

)



k

−

r d



q

−



k

−

2r d



q

!

−

1

.

Theorem 2.3 ([5, Proposition 4],[4, Theorem 4.4 and Corollary 4.5]). Let q be a prime power. Suppose k

−

d

≥

2 and e

=

eq

(

s

,

1

)

. (i) If s

∈ [

pq

(

1

)]

, then Jq

(

n

,

d

,

k

)

is se-disjunct.

(ii) If s

∈ [

q

+

1

]

, then Jq

(

n

,

d

,

k

)

is fully se-disjunct.

Based on J

(

n

,

d

,

k

)

, Macula [13] proposed another family of se_{-disjunct matrices. D’yachkov et al. [5] also discussed their} error-correcting property.

Definition 2.3. (i) A familyK

⊆



[n] k



is called an

{

r

,

r

+

1

, . . . ,

k

}

-clique of J

(

n

,

k

)

if

|

K

∩

K0

| ≤

k

−

r for any two distinct K

,

K0

_∈

_K. (ii) A familyF

⊆

h

Fnq k

i

is called an

{

r

,

r

+

1

, . . . ,

k

}

-clique of Jq

(

n

,

k

)

if dim

(

K

∩

K0

) ≤

k

−

r for any two distinct K

,

K0

∈

F.

Definition 2.4 ([13]). For positive integers d

<

k

<

n, letK

⊆



[n] k



. Suppose J

(

n

,

d

,

K

)

denotes the binary matrix with row-indexed (resp. column-indexed) by



[n] d



(resp.K) such that M

(

A

,

B

) =

1 if A

⊆

B and 0 otherwise.

Theorem 2.4 ([5, Proposition 3], [13, Theorem 2]). LetKbe an

{

r

,

r

+

1

, . . . ,

k

}

-clique of J

(

n

,

k

)

. (i) Let d

≥

1 with 1

+

r

k−d

≤

r and

α

d

=

min

(

r

d

_,

_k

₋

_d

₎

_{. Then J}

₍

_n

_,

_d

_,

_K

₎

_{is d}αd−1_-disjunct.

(ii) J

(

n

,

d

,

K

)

is se_{-disjunct where s}

_{∈ [}

_p

1

(

r

)]

and e

=

e1

(

s

,

r

)

.

As the q-analogue of J

(

n

,

d

,

K

)

, we propose the following definition.

Definition 2.5. For positive integers d

<

k

<

n, letF

⊆

h

Fnq

k

i

. Suppose Jq

(

n

,

d

,

F

)

denotes the binary matrix with row-indexed (resp. column-row-indexed) by

h

Fnq

d

i

(resp.F) such that M

(

A

,

B

) =

1 if A

⊆

B and 0 otherwise. Similar toTheorem 2.4, we have the following result.

Corollary 2.5. LetF be an

{

r

,

r

+

1

, . . . ,

k

}

-clique of Jq

(

n

,

k

)

. Then Jq

(

n

,

d

,

F

)

is se-disjunct where s

∈ [

pq

(

r

)]

and e

=

eq

(

s

,

r

)

. Let X

= {

x

=

(

x1

,

x2

, . . . ,

xn

) |

xi

∈

F

}

where F

= {

0

,

1

, . . . ,

q

}

, and let Pidenote the set of elements with weight i of X . For positive integers 1

≤

d

≤

k

≤

n, define H

(

n

,

q

,

d

,

k

)

to be the binary matrix with row-indexed (resp. column-indexed) by Pd(resp. Pk) such that M

(

x

,

y

) =

1 if xi

=

0 or xi

=

yiand xi

6=

0.

D’yachkov et al. [5] proposed the above matrix and discussed its disjunctness.

Theorem 2.6 ([5]). For 1

≤

s

≤

d

≤

k

≤

n, H

(

n

,

q

,

d

,

k

)

is fully se_{-disjunct, where e}

₌



k−s k−d



Let V be the vector space of dimension n

+

_{r over Fq}, and let W be a fixed r-subspace of V . Let Pi

= {

A

|

A

∈

h

_V

i

i

and A

∩

W

=

0

}

. For positive integers 1

≤

d

≤

k

≤

n, define Bq

(

n

,

r

,

d

,

k

)

to be the binary matrix with row-indexed (resp. column-indexed) by Pd(resp. Pk) such that M

(

A

,

B

) =

1 if A

⊆

B.

Huang and Weng [9] proved that Bq

(

n

,

r

,

d

,

k

)

is a de-disjunct matrix for some e. Similar toTheorem 2.3, we may obtain the following results.

Theorem 2.7. Let q be a prime power. Suppose k

−

d

≥

2 and e

=

eq

(

s

,

1

)

. (i) If s

∈ [

pq

(

1

)]

, then Bq

(

n

,

r

,

d

,

k

)

is se-disjunct.

(ii) If s

∈ [

q

+

1

]

, then Bq

(

n

,

r

,

d

,

k

)

is fully se-disjunct. 2.2. Parameter esfor error tolerance

The complement Mcof a binary matrix M is the matrix that results when one interchanges the 0’s and 1’s in M. LetK

be any subset of



[n] k



. Macula [13] considered the matrix J∗

₍

_n

_,

_d

_,

_K

₎

_{as that which results by row augmenting the matrix} J

(

n

,

d

,

K

)

with Jc

₍

_n

_,

₁

_,

_K

₎

_{. He claimed that e}

1

≥

4 for J∗

(

n

,

d

,

K

)

. Hwang [11] gave a proof.

Theorem 2.8 ([11, Theorem 2]). Given J∗

₍

_n

_,

_d

_,

_K

₎

_{with k}

₋

_d

_≥

_{3. Then e}

1

≥

4.

In the rest of this subsection, we shall compute the parameter esfor J

(

n

,

d

,

k

)

and Jq

(

n

,

d

,

k

)

, respectively. We begin with an example.

Example 2.1. Given a matrix J

(

n

,

d

,

k

)

with 1

≤

d

<

k

<

n

.

For s

∈ [

d

]

, let D0

= {b

1

,

b

2

, . . . ,

s[

−

1

,

k\

+

1

}

and D0₀

= {b

1

,

b

2

, . . . ,

s[

−

1

,

b

k

}

where

b

i

= [

k

+

1

] − {

i

}

and i

∈ [

k

+

1

]

. Then











R

|

R

∈

 [

k

]

d



,

R

6⊆

b

1

,

b

2

, . . . ,

s[

−

1

,

b

k









=



k

−

s k

−

d



.

By the symmetry, dH

(

B

(

D0

),

B

(

D00

)) =

2



k−s k−d



.

Theorem 2.9. Given a matrix J

(

n

,

d

,

k

)

with 1

≤

s

≤

d

<

k

<

n

.

Then es

=

2



k−s k−d



.

Proof. The upper bound for esis derived fromExample 2.1. ByTheorem 2.2andProposition 2.1, it is also a lower bound for es. Hence the desired result follows. 

Similar to the case for Johnson graphs, we consider the following example. 

Example 2.2. Given a matrix Jq

(

n

,

k

,

d

)

with k

−

d

≥

2

.

For each i

∈ [

k

+

1

]

, let eibe the row vector of V whose i-th coordinate is 1 and all other coordinates are 0. Suppose Fq

= {

a1

,

a2

, . . . ,

aq

}

and s

≤

q

+

1.

For i

∈ [

s

+

1

]

, let

D0

= {

C1

, . . . ,

Cs−1

,

Cs

}

and D00

= {

C1

, . . . ,

Cs−1

,

Cs+1

}

,

where

Cs

= h

e1

,

e2

, . . . ,

ek

i

,

Cs+1

= h

e2

,

e3

, . . . ,

ek

,

ek+1

i

,

Ci

= h

e1

+

aie2

,

e3

, . . . ,

ek

,

ek+1

i

.

By the principle of inclusion and exclusion,











R

|

R

∈



Cs d



,

R

6⊆

C1

, . . . ,

Cs−1

,

Cs+1









=



k d



q

−



s 1





k

−

1 d



q

+



k

−

2 d



q s

X

j=2

(−

1

)

j



s j



=

eq

(

s

,

1

) +

1

.

By the symmetry, dH

(

B

(

D0

),

B

(

D00

)) =

2eq

(

s

,

1

) +

2

.

Theorem 2.10. Given a matrix Jq

(

n

,

d

,

k

)

with k

−

d

≥

2

.

If s

∈ [

q

+

1

]

,

then es

=

2eq

(

s

,

1

) +

2.

Proof. The upper bound for esis derived fromExample 2.2. ByTheorem 2.3andProposition 2.1, it is also a lower bound. Therefore the desired result follows. 

3. Disjunctness over matchings on K_2mand its extensions

For positive integers d

<

k

≤

m, let M be a binary matrix with row-indexed (resp. column-indexed) by d-matchings (resp. k-matchings) of K2msuch that M

(

A

,

B

) =

1 if A

⊆

B and 0 otherwise. This matrix is denoted by M

(

2m

,

d

,

k

)

. In [14],

Ngo and Du proposed the matrix and discussed its disjunctness.

Theorem 3.1 ([14, Theorem 11,Corollary 12]). Let 1

≤

d

<

k

≤

m. Then (i) M

(

2m

,

d

,

k

)

is a d-disjunct matrix.

(ii) M

(

2m

,

d

,

m

)

is d-error-detecting and

b

d

/

2

c

-error-correcting.

(iii) Moreover, if the number of positives is known to be exactly d, then M

(

2m

,

d

,

m

)

is

(

2d

+

1

)

-error-detecting and d-error-correcting.

With an interpretation of matchings as 2-cliques of Johnson graph J

(

n

,

2

)

, we shall generalize Ngo and Du’s designs to the incidence matrices of t-cliques with various sizes of the Johnson graph J

(

n

,

t

)

and strongly t-cliques with various sizes of the Grassmann graph Jq

(

n

,

t

)

, respectively. We show that our pooling designs have the same capability of error-detecting and error-correcting as Ngo and Du’s. However, the test-to-item ratio of ours is much smaller.

Definition 3.1. Given positive integers d

<

k and kt

≤

n

.

(i) Let J

(

n

,

t

,

d

,

k

)

be the binary matrix with row-indexed (resp. column-indexed) by t-cliques with size d (resp. k) of J

(

n

,

t

)

such that M

(

A

,

B

) =

1 if A

⊆

B and 0 otherwise.

(ii) Let Jq

(

n

,

t

,

d

,

k

)

be the binary matrix with row-indexed (resp. column-indexed) by strongly t-cliques with size d (resp. k) of Jq

(

n

,

t

)

such that M

(

A

,

B

) =

1 if A

⊆

B and 0 otherwise.

Since J

(

2m

,

2

,

d

,

k

) =

M

(

m

,

d

,

k

)

, J

(

n

,

t

,

d

,

k

)

is a generalization of M

(

m

,

d

,

k

)

.

Lemma 3.2. Let W be a given k-subspace of Fn

q. Then the number of d-subspaces of Fnqintersecting trivially with W is

h

n−k d

i

q qdk

_.

Proof. Let D

= {

A

|

A

∈

h

Fnq

d

i ,

A

∩

W

=

0

}

. Counting the set

{

(v

1

, v

2

, . . . , v

d

)| v

i

6∈ h

W

, v

1

, v

2

, . . . , v

i−1

i

,

i

∈ [

d

]}

in two ways, we have

(

qn

−

qk

)(

qn

−

qk+1

) · · · (

qn

−

qk+d−1

) = |

D

| ·

(

qd

−

1

)(

qd

−

q

) · · · (

qd

−

qd−1

).

Hence

|

D

| =

h

n−k d

i

q qdk_{as required. }

Lemma 3.3. (i) The number of t-cliques of J

(

n

,

t

)

with size l is u

(

n

,

t

,

l

) =



n

tl

 (

tl

)!/(

t

!

)

l_l

_!

_.

₍₁₎

(ii) The number of strongly t-cliques of Jq

(

n

,

t

)

with size l is

uq

(

n

,

t

,

l

) =

h

n t

i

q



n

−

t t



q

· · ·



n

−

(

l

−

1

)

t t



q

·

q t2l(l−1)/2 l

!

.

(2)

Proof. (i) Since every tl-subset of

[

n

]

forms₍(tl)!

t!)l_l!many t-cliques of J

(

n

,

t

)

with size l,

(

1

)

holds.

(ii) ByDefinition 2.3, uq

(

n

,

t

,

l

)

is the number of

{

A1

,

A2

, . . . ,

Al

} ⊆

h

Fnq

t

i

satisfying dim

(

A1

+

A2

+ · · · +

Al

) =

tl

.

Let N

(

n

,

t

,

l

)

be the number of ordered tuples

(

A1

,

A2

, . . . ,

Al

)

of t-subspaces of Fnqsuch that dim

(

A1

+

A2

+ · · ·

Al

) =

tl. Then uq

(

n

,

t

,

l

) =

N(n_l,!t,l)

.

Hence, if we want to get uq

(

n

,

t

,

l

)

, it suffices to compute N

(

n

,

t

,

l

)

. There are



n t



qways to choose A1, then



n−_tt



_qqt2ways to choose A2byLemma 3.2and so on. It follows that

N

(

n

,

t

,

l

) =

h

n t

i

q



n

−

t t



q qt·t



n

−

2t t



q q2t·t

· · ·



n

−

(

l

−

1

)

t t



q q(l−1)t·t

.

Hence(2)holds. 

Theorem 3.4. Let 1

≤

s

≤

d

<

k and kt

≤

n. Then J

(

n

,

t

,

d

,

k

)

is an se_{-disjunct matrix of order N}

_×

_{T with row weight} u

(

n

−

td

,

t

,

k

−

d

)

and column weight



k_d



, where

(

N

,

T

) = (

u

(

n

,

t

,

d

),

u

(

n

,

t

,

k

))

and e

=



k−s

k−d



Proof. ByLemma 3.3, J

(

n

,

t

,

d

,

k

)

is an N

×

T matrix with row weight u

(

n

−

td

,

t

,

k

−

d

)

and column weight



k d



. Let C0

,

C1

, . . . ,

Csbe any s

+

1 distinct columns of J

(

n

,

t

,

d

,

k

)

. For each i

∈ [

s

]

, there exists Pi

∈

C0

\

Ci. Let E

= {

Pi

|

i

∈ [

s

]}

. Then

|

E

| ≤

s and E

⊆

C0but E

6⊆

Cifor each i

∈ [

s

]

. If

|

E

| =

j, the number of d-subsets of C0containing E is



k−j k−d



. Since



k−j k−d



≥



k−s k−d



whenever j

≤

s, the number of t-cliques with size d contained in C0but not contained in Cifor each i

∈ [

s

]

is at least



k−s k−d



. 

Corollary 3.5. Let 1

≤

s

≤

d

<

k and

(

k

+

1

)

t

≤

n. Then J

(

n

,

t

,

d

,

k

)

is fully se_{-disjunct with e}

₌



k−s k−d



−

1 and es

=

2



k−s k−d

 .

Proof. In order to prove that J

(

n

,

t

,

d

,

k

)

is fully se-disjunct, we only need to show that the maximum size of E is obtained

inTheorem 3.4. Since

(

k

+

1

)

t

≤

n, there exists a t-clique T

= {

A1

,

A2

, . . . ,

Ak+1

}

with size k

+

1. Let C0

=

T

− {

Ak+1

}

and

Ci

=

T

− {

Ai

}

for each i

∈ [

k

]

. Then

|

E

| = |{

Ai

|

i

∈ [

s

]}| =

s. Similar toTheorem 2.9, it is routine to compute es.  Similar results hold for Jq

(

n

,

t

,

d

,

k

)

too, and their proofs will be omitted.

Theorem 3.6. Let 1

≤

s

≤

d

<

k and kt

≤

n. Then the matrix Jq

(

n

,

t

,

d

,

k

)

is an se-disjunct matrix of order N

×

T with row weight uq

(

n

−

td

,

t

,

k

−

d

)

and column weight



k d



, where

(

N

,

T

) = (

uq

(

n

,

t

,

d

),

uq

(

n

,

t

,

k

))

and e

=



k

−

s k

−

d



−

1

.

Corollary 3.7. Let 1

≤

s

≤

d

<

k and

(

k

+

1

)

t

≤

n. Then Jq

(

n

,

t

,

d

,

k

)

is fully se-disjunct with e

=



k−s k−d



−

1 and es

=

2



k−s k−d

 .

Remarks. (i) By zigzag arguments similar toTheorem 3.1(ii), J

(

tn

,

t

,

d

,

n

)

is d-error-detecting and

b

d

/

2

c

error-correcting. (ii) The test-to-item ratio of J

(

tm

,

t

,

d

,

k

)

(resp. Jq

(

tm

,

t

,

d

,

k

)

) is much less than that of M

(

m

,

d

,

k

)

(resp. J

(

tm

,

t

,

d

,

k

)

). The following theorem tells us how to choose d and k such that the test-to-item ratio for J

(

tm

,

t

,

d

,

k

)

(resp. Jq

(

tm

,

t

,

d

,

k

)

) is minimized.

Theorem 3.8. For l goes from 1 to m

≥

3

,

(i) u

(

tm

,

t

,

l

)

is unimodal and get its peak when

b

l1

c ≤

l

≤ b

l2

c

where l1

,

l2satisfying l1

+

t

√

l1

+

1

=

m

,

l2

+

t

+√tl2+1−1

t

=

m.

(ii) uq

(

tm

,

t

,

l

)

is increasing and get its maximum at l

=

m.

Proof. (i) Suppose f

(

l

) =

u(_utm_(tm,t_,,l_t+_,l)1). Then f

(

l

)

is a decreasing number series while l goes from 1 to m

−

1. Since f

(

1

) >

1 and f

(

m

−

1

) =

_m1

<

1, u

(

tm

,

t

,

l

)

is unimodal. Note that

f

(

l

) =

u

(

tm

,

t

,

l

+

1

)

u

(

tm

,

t

,

l

)

=

tm

−

tl t

√

t l

+

1

·

tm

−

tl

−

1

(

t

−

1

)

√

t l

+

1

· · ·

tm

−

tl

−

t

+

1 t

√

l

+

1

.

Let ui

=

_(ttm_−i−₎√tlt−i l+1. Then

(

u0

)

t

_≤

_f

₍

_l

_{) ≤ (}

_u t−1

)

t. If u0

=

1, then l

+

t

√

l

+

1

=

m; if ut−1

=

1, then l

+

t+ t √ l+1−1 t

=

m. The desired results follow.

(ii) Suppose g

(

l

) =

uq_u(tm,t,l+1)

q(tm,t,l) . Then

g

(

l

) =

(

q

tm

₋

_qtl

_{) · · · (}

_qtm−t+1

₋

_qtl

₎

(

qt

₋

₁

) · · · (

_q

₋

₁

)(

_l

₊

₁

)

.

It follows that that g

(

l

)

is decreasing while l goes from 1 to m

−

1. Since g

(

1

) >

1 and g

(

m

−

1

) >

1, uq

(

tm

,

t

,

l

)

is increasing while l goes from 1 to m, and achieve the maximum value at l

=

m. 

4. Disjunctness over other distance-regular graphs

In this section, we shall give two constructions of pooling designs associated with antipodal distance-regular graphs and distance-regular graphs of order

(

r

,

t

)

, respectively. Since the results are similar to those in Section3, we shall omit all the proofs in this section.

A distance-regular graphΓ

=

(

X

,

R

)

of diameter D

≥

2 is said to be antipodal, if

∂(

x

,

y

) = ∂(

x

,

z

) =

D and y

6=

z implies

∂(

y

,

z

) =

D. Let kD

=

p0D,D. Then the number of maximal D-clique ofΓ is |X|

kD+1

.

Since any two distinct maximal D-cliques

have no common vertices, the number of D-clique with size l ofΓ is_k|X|

D+1

·



kD+1

Table 1

The parameters of se_{-disjunct matrices.}

Name Tests Items Ratio (test to item) s e Remarks

J(n,d,k) n_d
n k
(n−k)!k! (n−d)!d! s∈ [d]  k−s k−d  −1 Theorem 2.2 Jq(n,d,k) n d  q n k  q (qk−1)···(qd+1−1) (qn−k−1)···(qn−d+1−1) s∈ [pq(1)] eq(s,1) Theorem 2.3 J(n,d,K) n_d
|K| ( n d) |K| s∈ [p1(r)] e1(s,r) Theorem 2.4 Jq(n,d,F) n d  q |F| [n d]q |F| s∈ [pq(r)] eq(s,r) Corollary 2.5 H(n,q,d,k) n_d
qd n k
qk (n−k)!k!qd (n−d)!d!qk s∈ [d]  k−s k−d  −1 Theorem 2.6 Bq(n,r,d,k) n d  qq dr n k  qq kr (qk−1)···(qd+1−1)qdr (qn−k−1)···(qn−d+1−1)qdk s∈ [pq(1)] eq(s,1) Theorem 2.7 M(2m,d,k) (2d)! 2dd! · _2m 2d  ₍ 2k)! 2kk!· _2m 2k  ₍ 2m−2k)!k! (2m−2d)!d!2 k−d _{s∈ [d]} k−s k−d  −1 Theorem 3.1 J(n,t,d,k) _(t(_!td₎d)!d!· n td
(tk)! (t!)kk!· n tk
(n−tk)!k! (n−td)!d!(t!) k−d _{s∈ [d]} k−s k−d  −1 Theorem 3.4 Jq(n,t,d,k) uq(n,t,d) uq(n,t,k) h_n−qdtt(d−k)(k+d−1)/2k! t i q··· h_n−(k−1)t t i qd! s∈ [d]  k−s k−d  −1 Theorem 3.6 A(n,d,k) _kn D+1·  kD+1 d  n kD+1·  kD+1 k  (kD+1−k)!k! (kD+1−d)!d! s∈ [d]  k−s k−d  −1 Theorem 4.1 B(r,t;d,k) n(t+1) r+1 ·  r+1 d  n(t+1) r+1 ·  r+1 k  ₍ r+1−k)!k! (r+1−r)!d! s∈ [d]  k−s k−d  −1 Theorem 4.2

A distance-regular graph Γ

=

(

X

,

R

)

is said to be of order

(

r

,

t

)

if, for each vertex x

∈

X , the induced subgraph onΓ

(

x

)

is a disjoint union of t

+

1 cliques with size r. Then each maximal clique is of size r

+

1, and each vertex is contained in t

+

1 maximal cliques. Denote the set of all maximal cliques byC. By computing the number of the set

{

(

x

,

C

) |

x

∈

X

,

C

∈

C

,

x

∈

C

}

in two ways, the number of maximal cliques ofΓ is n(_rt₊+₁1); consequently the number of cliques with size l is



r+1 l



·

n(t+1) r+1 .

LetΓ be an antipodal distance-regular graph of diameter D with n vertices. For positive integers 1

<

d

<

k

<

kD

+

1, let M be the binary matrix whose row (resp. column) indexed by the D-cliques ofΓ with size d (resp. k) such that M

(

A

,

B

) =

1 if A

⊆

B and 0 otherwise. This matrix is denoted by A

(

n

,

d

,

k

)

.

Theorem 4.1. Let 1

≤

s

≤

d

<

k

<

kD

+

1. Then A

(

n

,

d

,

k

)

is a fully se-disjunct matrix of order N

×

T with row weight



kD+1−d

k−d



and column weight



k d



, where

(

N

,

T

) =



n kD

+

1

·



kD

+

1 d



,

n kD

+

1

·



kD

+

1 k



,

e

=



k

−

s k

−

d



−

1

.

Moreover, es

=

2



k−s k−d



.

LetΓ be a distance-regular graph of order

(

r

,

t

)

. For positive integers 1

<

d

<

k

<

r

+

1, let M be the binary matrix whose row (resp. column) indexed by the cliques ofΓ with size d (resp. k) such that M

(

A

,

B

) =

1 if A

⊆

B and 0 otherwise. This matrix is denoted by B

(

r

,

t

;

d

,

k

)

.

Theorem 4.2. Let 1

≤

s

≤

d

<

k

<

r

+

1. Then B

(

r

,

t

;

d

,

k

)

is a fully se_{-disjunct matrix of order N}

_×

_{T with row weight}



r+1−d k−d



and column weight



k d



, where

(

N

,

T

) =



n

(

t

+

1

)

r

+

1

·



r

+

1 d



,

n

(

t

+

1

)

r

+

1

·



r

+

1 k



,

e

=



k

−

s k

−

d



−

1

.

Moreover, es

=

2



k−s k−d



. Acknowledgment

The third author is partially supported by NCET-08-0052 and NSF of China (10871027).

Appendix

SeeTable 1.

References

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