Pooling semilattices and non-adaptive pooling designs

(1)

Pooling semilattices and non-adaptive pooling designs

Jun Guo

a,∗

, Kaishun Wang

b

, Chih-wen Weng

c a_{College of Math. and Info. Sci., Langfang Teachers’ College, Langfang 065000, China} b_{Sch. Math. Sci. & Lab. Math. Com. Sys., Beijing Normal University, Beijing 100875, 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 7 February 2013

Received in revised form 7 October 2013 Accepted 5 December 2013

Available online 22 December 2013

Keywords:

Pooling design Semilattice Pooling semilattice Regular pooling semilattice Pooling space

a b s t r a c t

In Huang and Weng (2004), Huang and Weng introduced pooling spaces, and constructed pooling designs from a pooling space. In this paper, we introduce the concept of pooling semilattices and prove that a pooling semilattice is a pooling space, then show how to construct pooling designs from a pooling semilattice. Moreover, we give many examples of pooling semilattices and thus obtain the corresponding pooling designs.

1. Introduction

The basic problem of group testing is to identify the set of defective items in a large population of items. A group testing algorithm is non-adaptive if all tests must be specified without knowing the outcomes of other tests. A group test is applicable to an arbitrary subset of items with two possible outcomes: a negative outcome indicates that all items in the subset are negative, and a positive outcome indicates otherwise. A pooling design is a specification of all tests such that they can be performed simultaneously with the goal being to identify all positive items with a small number of tests [3]. A non-adaptive pooling design is usually represented by a binary matrix with columns indexed with items and rows indexed with pools. A cell

(

i

,

j

)

contains a 1-entry if and only if the ith pool contains the jth item. By treating a column as a set of row indices intersecting the column with a 1-entry, we can talk about the union of several columns. A binary matrix is se-disjunct if every column has at least e

+

1 1-entries not contained in the union of any other s columns [13]. An s0_{-disjunct matrix is}

also called s-disjunct. An se_{-disjunct matrix is called fully s}e_{-disjunct if it is neither}

₍

_s

₊

₁

₎

e_{-disjunct nor s}e+1_{-disjunct. An}

se_{-disjunct matrix is}

_⌊

_e

_/

₂

_⌋

_{-error-correcting [}₅_,₁₁_].

Macula [12] proposed a novel way of constructing disjunct matrices by means of the containment relation of subsets in a finite set. D’yachkov et al. [5] discussed the error-correcting capability of Macula’s designs. Ngo and Du [14] constructed a family of disjunct matrices by means of the containment relation of subspaces in a finite vector space. D’yachkov et al. [4] discussed the error-tolerance capability of Ngo–Du’s designs. In [7,8], the first two authors of this paper proposed a new model for pooling designs—the intersection type incidence construction, and generalized Macula’s and Ngo–Du’s designs. Under this model, the pooling designs have surprisingly high degree of error correction. Huang and Weng [11] generalized the containment matrix construction of pooling designs to pooling spaces.

Let

(

X

, ≼)

be a finite partially ordered set (poset) with the least element 0. For x

,

y

∈

X , if x

≼

y, we say that y contains x. Moreover, if there does not exist element z such that x

≺

z

≺

y, we say that y covers x. An atom in X is an element in X

∗_{Corresponding author.}

(2)

Fig. 1. A pooling space that is not a pooling semilattice.

that covers 0. The poset X is ranked and has rank function, if there is a function

ℓ

from X to the integer set such that

ℓ(

0

) =

0 and

ℓ(

y

) = ℓ(

x

) +

1 if y covers x. The maximum value of

ℓ(

x

)

is called the rank of X , denoted by N. The fibers (or levels)

X0

,

X1

, . . . ,

XN of the poset are the subsets of X given by Xi

= {

x

∈

X

|

ℓ(

x

) =

i

}

. Pick any x

,

y

∈

X such that x

≼

y. By the interval

[

x

,

y

]

, we mean the subposet

[

x

,

y

] := {

z

∈

X

|

x

≼

z

≼

y

}

of X . A ranked poset X is called atomic whenever each element x

∈

X

\ {

0

}

is the least upper bound of the set

[

0

,

x

] ∩

X1. A pooling space is a finite poset

(

X

, ≼)

such that the

subposet induced on

w

+

_{= {}

_{w ≼}

_y

_|

_y

_∈

_X

_}

_{is atomic for each}

_{w ∈}

_{X . Huang and Weng [}₁₁_{] showed that how to construct} pooling designs from pooling spaces.

Theorem 1 ([11]). Let X be a pooling space with rank N

≥

1. For 1

≤

d

≤

k

≤

N, let M

(

k

,

N

)

be the binary matrix with rows indexed with Xkand columns indexed with XNsuch that M

(

x

,

y

) =

1 if and only if x

≼

y. Then M

(

k

,

N

)

is de-disjunct, where

e

=

min

| ∪

([

y

,

x

] ∩

Xk

)| −

1

,

the minimum is taken over all pairs

(

x

,

T

)

with T

⊆

XN

, |

T

| ≤

d and x

∈

XN

\

T ; the union is taken over all y

∈ [

0

,

x

] ∩

Xdsuch

that y

̸≼

z for all z

∈

T .

Let

(

X

, ≼)

be a finite poset with the rank function

ℓ

and fibers X0

, . . . ,

XN. We call X a semilattice, if any two elements x and y of X have the greatest lower bound, denoted by x

∧

y. As usual, we denote by x

∨

y the least upper bound of x and y

if it exists. Note that if X is a semilattice and x

,

y

∈

X have a common upper bound, then x

∨

y exists; indeed x

∨

y is the

greatest lower bound of the set of upper bounds of x and y. X is a lattice if x

∨

y exists for any x

,

y

∈

X .

Let X denote a semilattice with the rank function

ℓ

and fibers X0

, . . . ,

XN. We are concerned with the following axioms: (A1) For u

∈

Xr and z

∈

Xt with u

≺

z, the number

|[

u

,

z

] ∩

Xr+1

|

is a constant

µ(

r

,

r

+

1

,

t

)

, where 0

≤

r

<

t

≤

N.

Moreover, the function

µ(

0

,

1

,

t

)

is strictly increasing about t, i.e. 1

=

µ(

0

,

1

,

1

) < µ(

0

,

1

,

2

) < · · · < µ(

0

,

1

,

N

)

. (A2) For x

,

y

∈

X , if x

∨

y exists, then

ℓ(

x

∨

y

) ≤ ℓ(

x

) + ℓ(

y

) − ℓ(

x

∧

y

)

.

(A3) For x

,

y

∈

X , if x

∨

y exists, then

ℓ(

x

∨

y

) = ℓ(

x

) + ℓ(

y

) − ℓ(

x

∧

y

)

.

We call X a pooling semilattice, if it satisfies (A1) and (A2). We call X a regular pooling semilattice, if it satisfies (A1) and (A3). Note that (A3) implies (A2) and thus a regular pooling semilattice is a pooling semilattice. In addition if X is a lattice, we use a lattice to replace the above semilattice. We call X a geometric lattice if X is a finite atomic lattice and satisfies (A2). In this paper, we focus on the construction of pooling designs from a pooling semilattice. In Section2, we first discuss some properties of pooling semilattices, then show how to construct pooling designs from a pooling semilattice. In Section3, we give many families of examples of pooling semilattices. They fall into three categories: regular pooling semilattices from sets, vector spaces and maps in Section 3.1, non-regular pooling semilattices from affine spaces in Section 3.2, pooling semilattices from distance-regular graphs in Section3.3. In Section4, we generalize the intersection type incidence construction to pooling lattices and give four families of examples of pooling lattices.

2. Pooling semilattices

In this section, we always assume that X denotes a pooling semilattice with the rank function

ℓ

and fibers X0

, . . . ,

XN. A poset can be described by a diagram in the plane in which y covers x if and only if there is a line moving upwards from

x to y.Fig. 1is a diagram of a pooling space with seven elements. It is not a pooling semilattice since z

∧

w

does not exist.

Lemma 2. A pooling semilattice X is atomic.

Proof. Pick any element

w ∈

X

\ {

0

}

. Suppose that u is the least upper bound of the set

[

0

, w] ∩

X1. Then u

≼

w

and

µ(

0

,

1

, ℓ(w)) ≤ µ(

0

,

1

, ℓ(

u

))

. By (A1), one gets

ℓ(w) = ℓ(

u

)

and u

=

w

, as desired. The usage of the term ‘‘pooling semilattice’’ is justified by the following proposition.

Proposition 3. Let X be a pooling semilattice with rank N. Then X is a pooling space. In particular, for each 1

≤

r

<

N, the function

µ(

r

,

r

+

1

,

t

)

is strictly increasing about t where r

+

1

≤

t

≤

N.

Proof. Let

w ∈

Xr be given. We shall prove that the subposet

w

+is atomic. Pick any x

∈

w

+. Then

[

0

,

x

]

is a semilattice. ByLemma 2,

[

0

,

x

]

is a atomic and hence is a geometric lattice. It is well-known that an interval in a geometric lattice is a

(3)

µ(

r

,

s

,

t

) := |[

u

,

z

] ∩

Xs

|

is a constant. Moreover for given r

,

s with 0

≤

r

≤

s

≤

N, the function

µ(

r

,

s

,

t

)

is strictly increasing

about t, where s

≤

t

≤

N.

Proof. Note that

µ(

r

,

r

,

t

) = µ(

r

,

t

,

t

) =

1

, µ(

r

,

r

+

1

,

t

)

is a constant by (A1) and

µ(

r

,

r

+

1

,

t

−

1

) < µ(

r

,

r

+

1

,

t

)

by

Proposition 3, where r

+

1

≤

t

≤

N. We prove the lemma by induction on t

−

r, and assume in the nontrivial situation r

+

2

≤

s

≤

t

−

1. Fix u

∈

Xrand z

∈

Xtwith u

≺

z. Counting pairs

(v, w) ∈

Xr+1

×

Xswith u

≺

v ≺ w ≺

z in two ways yields a constant

µ(

r

,

s

,

t

) = µ(

r

,

r

+

1

,

t

)µ(

r

+

1

,

s

,

t

)/µ(

r

,

r

+

1

,

s

)

(1)

by induction. Also

µ(

r

,

s

,

t

−

1

) = µ(

r

,

r

+

1

,

t

−

1

)µ(

r

+

1

,

s

,

t

−

1

)/µ(

r

,

r

+

1

,

s

) < µ(

r

,

r

+

1

,

t

)µ(

r

+

1

,

s

,

t

)/µ(

r

,

r

+

1

,

s

) =

µ(

r

,

s

,

t

)

since

µ(

r

,

r

+

1

,

t

−

1

) < µ(

r

,

r

+

1

,

t

)

byProposition 3, and

µ(

r

+

1

,

s

,

t

−

1

) < µ(

r

+

1

,

s

,

t

)

by induction. Lemma 5. Let X be a pooling semilattice and 1

≤

s

<

t

≤

N. Then the function

µ(

r

,

s

,

t

)

is strictly decreasing about r, where

1

≤

r

<

s, i.e.

µ(

1

,

s

,

t

) > µ(

2

,

s

,

t

) > · · · > µ(

s

−

1

,

s

,

t

) > µ(

s

,

s

,

t

) =

1.

Proof. As the above two-way counting argument,

µ(

r

,

s

,

t

)

µ(

r

+

1

,

s

,

t

)

=

µ(

r

,

r

+

1

,

t

)

µ(

r

,

r

+

1

,

s

)

>

1

byLemma 4.

Definition 1. Suppose that X is a pooling semilattice. For positive integers 1

≤

d

<

k

<

N, let M

(

d

,

k

;

N

)

be the binary matrix with rows indexed with Xdand columns indexed with Xksuch that M

(

x

,

y

) =

1 if and only if x

≼

y.

Theorem 6. Let X be a pooling semilattice. Then the following results hold.

(i) If 1

≤

s

≤

d, then M

(

d

,

k

;

N

)

is an se_{-disjunct matrix, where e}

₌

_µ(

_s

_,

_d

_,

_k

_{) −}

_1.

(ii) If 1

≤

s

< µ(

0

,

d

,

k

)/µ(

0

,

d

,

k

−

1

)

, then M

(

d

,

k

;

N

)

₌

_µ(

₀

_,

_d

_,

_k

_{) −}

_s

_µ(

₀

_,

_d

_,

_k

₋

₁

_{) −}

_1. Proof. (i) Let y₀

,

y₁

, . . . ,

y_sbe any s

+

1 distinct columns of M

(

d

,

k

;

N

)

. Note that

ℓ(

y₀

∧

y_j

) ≤

k

−

1 for each j

∈ {

1

,

2

, . . . ,

s

}

. By (A1) we have

µ(

0

,

1

,

k

) > µ(

0

,

1

, ℓ(

y0

∧

yj

))

, which implies that there exists some aj

∈

X1such that aj

≼

y0but aj

̸≼

yj for each j

∈ {

1

,

2

, . . . ,

s

}

. Since y0is a common upper bound of a1

,

a2

, . . . ,

as, the least upper bound of these elements exists. Suppose that x0

=

a1

∨

a2

∨ · · · ∨

as. Then x0

≼

y0and x0

̸≼

yjfor each j

∈ {

1

,

2

, . . . ,

s

}

. By (A2) we have 1

≤

ℓ(

x0

) ≤

s. ByLemma 4, the size of Xd

∩ [

x0

,

y0

]

is

µ(ℓ(

x0

),

d

,

k

)

. FromLemma 5, we deduce that

µ(ℓ(

x0

),

d

,

k

)

is

decreasing for 1

≤

ℓ(

x0

) ≤

s and gets its minimum at

ℓ(

x0

) =

s, which implies that the size of Xd

∩ [

x0

,

y0

]

is at least

µ(

s

,

d

,

k

)

, as desired.

(ii) Let y0

,

y1

, . . . ,

ysbe any s

+

(

d

,

k

;

N

)

. Note that y0contains

µ(

0

,

d

,

k

)

many elements in Xdand

ℓ(

y0

∧

yj

) ≤

k

−

1 for each j

∈ {

1

,

2

, . . . ,

s

}

. ByLemma 4, each y0

∧

yjcontains at most

µ(

0

,

d

,

k

−

1

)

elements in Xd. Thus, the number of elements in Xdcontained in y0but not in yjfor each j

∈ {

1

,

2

, . . . ,

s

}

is at least

µ(

0

,

d

,

k

) −

s

µ(

0

,

d

,

k

−

1

)

, as desired.

Theorem 7. Let X be a regular pooling semilattice. If k

−

d

≥

2 and 1

≤

s

≤

(µ(

0

,

d

,

k

) − µ(

0

,

d

,

k

−

1

))/(µ(

0

,

d

,

k

−

1

) −

µ(

0

,

d

,

k

−

2

))

, then M

(

d

,

k

;

N

)

is an se-disjunct matrix, where e

=

µ(

0

,

d

,

k

) −

s

µ(

0

,

d

,

k

−

1

) + (

s

−

1

)µ(

0

,

d

,

k

−

2

) −

1. In

particular, if s

≤

min

{

µ(

k

−

2

,

k

−

1

,

k

), (µ(

0

,

d

,

k

) − µ(

0

,

d

,

k

−

1

))/(µ(

0

,

d

,

k

−

1

) − µ(

0

,

d

,

k

−

2

))}

and

|

x+

_∩

_X k

|

>

1

for any x

∈

Xk−1, then M

(

d

,

k

;

N

)

is fully se-disjunct.

Proof. Let y0

,

y1

, . . . ,

ysbe any s

+

(

d

,

k

;

N

)

. Note that y0contains

µ(

0

,

d

,

k

)

many elements in Xd and

ℓ(

y0

∧

yj

) ≤

k

−

1 for each j

∈ {

1

,

2

, . . . ,

s

}

. To obtain the maximum elements with rank d in



s

j=1

{

x

∈

Xd

|

x

≼

(

y0

∧

yj

)}

, byLemma 4we may assume that y₀

∧

y₁

, . . . ,

y₀

∧

y_sare s distinct elements in X_k−1. Then the element y0

∧

y1contains at

most

µ(

0

,

d

,

k

−

1

)

elements in Xd. Since

(

y0

∧

y1

) ∨ (

y0

∧

yj

) ≼

y0and

ℓ(

y0

∧

y1

∧

yj

) ≤

k

−

2 for each j

∈ {

2

, . . . ,

s

}

, by (A3)

ℓ((

y0

∧

y1

) ∨ (

y0

∧

yj

)) =

k and

ℓ(

y0

∧

y1

∧

yj

) =

k

−

2. ByLemma 4, each of y0

∧

y2

, . . . ,

y0

∧

yscan contain at most

µ(

0

,

d

,

k

−

1

) − µ(

0

,

d

,

k

−

2

)

elements in Xdnot contained in y0

∧

y1. Thus, the number of elements in Xdcontained in y0

but not in yjfor each j

∈ {

1

,

2

, . . . ,

s

}

is at least

µ(

0

,

d

,

k

) − µ(

0

,

d

,

k

−

1

) − (

s

−

1

)(µ(

0

,

d

,

k

−

1

) − µ(

0

,

d

,

k

−

2

))

. Hence

M

(

d

,

k

;

N

)

is se_-disjunct.

Let s

≤

min

{

µ(

k

−

2

,

k

−

1

,

k

), (µ(

0

,

d

,

k

) − µ(

0

,

d

,

k

−

1

))/(µ(

0

,

d

,

k

−

1

) − µ(

0

,

d

,

k

−

2

))}

and

|

x+

_∩

_X k

|

>

1 for any x

∈

Xk−1. We show that M

(

d

,

k

;

N

)

is fully se-disjunct. Let u

∈

Xk−2 with u

≼

y0. ByLemma 4the number of

(4)

elements x

∈

Xk−1such that u

≼

x

≼

y0is

µ(

k

−

2

,

k

−

1

,

k

)

, and so we can choose s distinct ones among them, say

xj

(

1

≤

j

≤

s

)

. Since

|

x+j

∩

Xk

|

>

1 we can choose pairwise distinct yjin Xk

\ {

y0

}

such that xj

=

y0

∧

yj. Then the number of elements in Xdcontained in y0but not in yjfor each j

∈ {

1

,

2

, . . . ,

s

}

is e

+

1. Therefore M

(

d

,

k

;

N

)

is not se+1-disjunct. View the function e

=

e

(

s

) = µ(

0

,

d

,

k

) −

s

µ(

0

,

d

,

k

−

1

) + (

s

−

1

)µ(

0

,

d

,

k

−

2

) −

1 as a function of s, and notice that

e

(

s

+

1

) −

e

(

s

) = µ(

0

,

d

,

k

−

2

) − µ(

0

,

d

,

k

−

1

) <

0. The above argument with s

+

1 to replace s implies that M

(

d

,

k

;

N

)

is not

(

s

+

1

)

e-disjunct. 3. Examples

In this section we give many examples of pooling semilattices, and give their parameters. ByTheorems 6and7, we can construct pooling designs from these pooling semilattices.

Let q be a positive integer. Fix a positive integer n. The Gaussian binomial coefficients with basis q is defined by



n i



q

=











i−1



j=0 n

−

j i

−

j if q

=

1

,

i−1



j=0 qn

₋

_qj qi

₋

_qj if q

̸=

1

.

In the case q

=

1, for convenience, we write



n_i



instead of



n_i



1.

3.1. Regular pooling semilattices from sets, vector spaces and maps

In this subsection we give thirteen families of regular pooling semilattices with rank N.

Example 1 ([5,12] The Boolean Algebra). Let X be the collection of all subsets of

[

N

] := {

1

,

2

, . . . ,

N

}

. Ordered by inclusion,

X is a regular pooling semilattice with the rank function

ℓ(

x

) = |

x

|

and the parameters

|

Xr

| =



N r



,

µ(

r

,

s

,

t

) =



t

−

r s

−

r



.

Example 2 ([4,14] The Projective Geometry). Let FNq be the N-dimensional vector space over the finite field Fqand X be the collection of all subspaces of FNq. Ordered by inclusion, X is a regular pooling semilattice with the rank function

ℓ(

x

) =

dim x and the parameters

|

Xr

| =



N r



q

,

µ(

r

,

s

,

t

) =



t

−

r s

−

r



q

.

Example 3 ([11] The Attenuated Space). For fixed positive integers n and N, let

w

be a fixed n-dimensional subspace of Fnq+N. Let X be the collection of all subspaces x of Fnq+Nwith x

∩

w = {

0

}

. Ordered by inclusion, X is a regular pooling semilattice with the rank function

ℓ(

x

) =

dim x and the parameters

|

Xr

| =

qrn



N r



q

,

µ(

r

,

s

,

t

) =



t

−

r s

−

r



q

.

Example 4 ([10] The Classical Polar Space). Classical finite polar spaces are incidence structures, consisting of all the totally

isotropic subspaces of Fnqwith respect to a certain non-degenerate sesquilinear or quadratic form f . The rank of the polar space is the algebraic dimension of the maximal totally isotropic subspaces, denoted by N. The summary is given in the following table: Name n Form

|

Xr

|

[

CN

(

q

)]

2N Symplectic



N r



q



r−1 i=0

(

q N−i

₊

₁

₎

[

BN

(

q

)]

2N

+

1 Quadratic



N r



q



r−1 i=0

(

qN −i

₊

₁

₎

[

DN

(

q

)]

2N Quadratic (with rank N)



N r



q



r−1 i=0

(

qN −i−1

₊

₁

₎

[

2_D

N+1

(

q

)]

2N

+

2 Quadratic (with rank N)



N r



q



r−1 i=0

(

qN −i+1

₊

₁

₎

[

2_A 2N

(

r

)]

2N

+

1 Hermitian (q

=

r2)



N r



q



r−1 i=0

(

qN −i+1/2

₊

₁

₎

[

2_A 2N−1

(

r

)]

2N Hermitian (q

=

r2)



N r



q



r−1 i=0

(

qN −i−1/2

₊

₁

₎

(5)

For fixed positive integers n and m, let

w

be an l-dimensional subspace of Fq , denote also by

w

an l

×

(

n

+

m

)

matrix of rank l whose rows span the subspace

w

and call the matrix

w

a matrix representation of the subspace

w

.

Example 5 (The Attenuated Classical Polar Space). For fixed positive integers n and m, let Fn

qbe the classical polar space with rank N as inExample 4and

w = (

0(m,n)_I(m)

₎

_{. Then the quotient space F}n+m

q

/w

is isomorphic to Fnq. Let X be the collection of all subspaces x

=

(

x1x2

)

of Fnq+mwith x

∩

w = {

0

}

, where x1is a totally isotropic subspace of Fnqand x2is a matrix. Ordered

by inclusion, X is a regular pooling semilattice with the rank function

ℓ(

x

) =

dim x and the parameters

|

Xr

| =

qrm



N r



q r−1



i=0

(

qN+e−i−1

+

1

),

µ(

r

,

s

,

t

) =



t

−

r s

−

r



q where e as inExample 4.

Example 6 (The Map). Let X be the collection of all pairs

(w,

f

)

, where

w

is a subset of

[

N

] := {

1

,

2

, . . . ,

N

}

and f

:

w → [

N

]

is a map. Ordered by inclusion, that is

(w,

f

) ≼ (

u

,

g

)

if

w ⊆

u and g

|

_w

=

f , X is a regular pooling semilattice with the rank

function

ℓ(w,

f

) = |w|

and the parameters

|

Xr

| =

Nr



N r



,

µ(

r

,

s

,

t

) =



t

−

r s

−

r



.

Example 7 (The Injective Map). Let X be the collection of all pairs

(w,

f

)

, where

w

is a subset of

[

N

]

and f

:

w → [

N

]

is an injective map. Ordered by inclusion, X is a regular pooling semilattice with the rank function

ℓ(w,

f

) = |w|

and the parameters

|

Xr

| =



N r



N

(

N

−

1

) · · · (

N

−

r

+

1

),

µ(

r

,

s

,

t

) =



t

−

r s

−

r



.

Example 8 (The Bilinear Form). Let X be the collection of all pair

(w,

f

)

, where

w

is a subspace of FN

q and f

:

w →

FNq is a linear map. Ordered by inclusion, X is a regular pooling semilattice with the rank function

ℓ(w,

f

) =

dim

w

and the parameters

|

Xr

| =

qrN



N r



q

,

µ(

r

,

s

,

t

) =



t

−

r s

−

r



q

.

Example 9 (The Injective Linear Map). Let X be the collection of all pair

(w,

f

)

, where

w

is a subspace of FNq and f

:

w →

FNq is an injective linear map. Ordered by inclusion, X is a regular pooling semilattice with the rank function

ℓ(w,

f

) =

dim

w

and the parameters

|

Xr

| =

qr(r−1)/2 N



i=N−r+1

(

qi

−

1

)



N r



q

,

µ(

r

,

s

,

t

) =



t

−

r s

−

r



q

.

Example 10 (The Square Bilinear Form). Let X be the collection of all pair

(w,

f

)

, where

w

is a subspace of FNq and f

:

w → w

is a bilinear form on

w

. Ordered by inclusion, X is a regular pooling semilattice with the rank function

ℓ(w,

f

) =

dim

w

and the parameters

|

Xr

| =

qr 2



N r



q

,

µ(

r

,

s

,

t

) =



t

−

r s

−

r



q

.

Example 11 (The Alternating Form). Let X be the collection of all pair

(w,

f

)

, where

w

is a subspace of FN

q and f

:

w → w

is an alternating bilinear form on

w

(6)

ℓ(w,

f

) =

dim

w

and the parameters

|

Xr

| =

qr(r−1)/2



N r



q

,

µ(

r

,

s

,

t

) =



t

−

r s

−

r



q

.

Example 12 (The Hermitian Form). Let X be the collection of all pair

(w,

f

)

, where

w

is a subspace of FN

qand f

:

w → w

is a Hermitian form on

w

, where q

=

r2_{is square. Ordered by inclusion, X is a regular pooling semilattice with the rank function}

ℓ(w,

f

) =

dim

w

and the parameters

|

Xr

| =

qr 2_/₂



N r



q

,

µ(

r

,

s

,

t

) =



t

−

r s

−

r



q

.

Example 13 (The Symmetric Bilinear Form). Let X be the collection of all pair

(w,

f

)

, where

w

is a subspace of FN q and

f

:

w → w

is a symmetric bilinear form on

w

. Ordered by inclusion, X is a regular pooling semilattice with the rank function

ℓ(w,

f

) =

dim

w

and the parameters

|

Xr

| =

qr(r+1)/2



N r



q

,

µ(

r

,

s

,

t

) =



t

−

r s

−

r



q

.

3.2. Pooling semilattices from affine spaces

In this subsection we give four families of examples of non-regular pooling semilattices with rank N

+

1. These examples are from an affine space.

Example 14 ([9,10] The Affine Geometry). Let FN

qand X be as inExample 2. Let X

′_{be the collection of all cosets of subspaces} in X together with the empty set

∅

. We define

ℓ(∅) =

0. Ordered by inclusion, X′is a pooling semilattice with the rank function

ℓ(

x

) =

dim x

+

1 and the parameters

|

X_r′₊₁

| =

qN−r



N r



q

,

µ(

r

+

1

,

s

+

1

,

t

+

1

) =



t

−

r s

−

r



q

.

Example 15 (The Affine Attenuated Space). Let Fn+N

q and X be as inExample 3. Let X

′_{be the collection of all cosets of subspaces} in X together with the empty set

∅

. Ordered by inclusion, X′_{is a pooling semilattice with the rank function}

_ℓ(

_x

_{) =}

_{dim x}

₊

₁ and the parameters

|

X_r′₊₁

| =

qn+N+rn−r



N r



q

,

µ(

r

+

1

,

s

+

1

,

t

+

1

) =



t

−

r s

−

r



q

.

Example 16 ([10] The Affine Classical Polar Space). Let Fn

qand X be as inExample 4. Let X

′_{be the collection of all cosets} of subspaces in X together with the empty set

∅

. Ordered by inclusion, X′_{is a pooling semilattice with the rank function}

ℓ(

x

) =

dim x

+

|

X_r′+1

| =

q2N +δ−r



N r



q r−1



i=0

(

qN+e−i−1

+

1

),

µ(

r

+

1

,

s

+

1

,

t

+

1

) =



t

−

r s

−

r



q

,

where

δ =

0

,

1

,

0

,

2

,

1

,

0 according to

[

CN

(

q

)], [

BN

(

q

)], [

DN

(

q

)], [

2DN+1

(

q

)], [

2A2N

(

r

)], [

2A2N−1

(

r

)]

, respectively, and e is as

inExample 4.

Example 17 (The Affine Attenuated Classical Polar Space). Let Fnq+mand X be as inExample 5. Let X ′

be the collection of all cosets of subspaces in X together with the empty set

∅

. Ordered by inclusion, X′is a pooling semilattice with the rank function

ℓ(

x

) =

dim x

+

|

X_r′₊₁

| =

q2N+δ+m+rm−r



N r



q r−1



i=0

(

qN+e−i−1

+

1

),

µ(

r

+

1

,

s

+

1

,

t

+

1

) =



t

−

r s

−

r



q

.

3.3. Pooling semilattices from distance-regular graphs

In this subsection, we give four families of examples of pooling semilattices with rank N. These examples are from distance-regular graphs.

(7)

i

(

, v)

i

(

, v)

theory of distance-regular graphs.

LetΓ be a distance-regular graph. A r-subset

{

x1

,

x2

, . . . ,

xr

} ⊆

Γ is said to be a t-clique ofΓ with size r if any two distinct vertices in

{

x1

,

x2

, . . . ,

xr

}

are at distance t.

Example 18 ([1,17] The Johnson Graph). Let N

= ⌊

n

/

t

⌋

and X be the collection of all t-cliques of the Johnson graph J

(

n

,

t

)

together with the empty set

∅

ℓ(

x

) = |

x

|

and the parameters

|

Xr

| =



n rt

 (

rt

)!/(

t

!

)

r_r

_!

_,

_µ(

_r

_,

_s

_,

_t

_{) =}



t

−

r s

−

r



.

A distance-regular graphΓ with diameter D

≥

2 is said to be antipodal, if

∂(

x

,

y

) = ∂(

x

,

z

) =

D and y

̸=

z implies

∂(

y

,

z

) =

D. For u

∈

Γ, the size of the set

{

v ∈

Γ

|

∂(

u

, v) =

D

}

depends only on D, denoted by kD.

Example 19 ([1] The Antipodal Distance-Regular Graph). Suppose thatΓis an antipodal distance-regular graph with diameter

D. Let N

=

kD

+

1 and X be the collection of all D-cliques ofΓ together with the empty set

∅

. Ordered by inclusion, X is a regular pooling semilattice with the rank function

ℓ(

x

) = |

x

|

and the parameters

|

Xr

| =



kD

+

1 r



|

Γ

|

/(

kD

+

1

),

µ(

r

,

s

,

t

) =



t

−

r s

−

r



.

A distance-regular graphΓis said to be of order

(

l

,

k

)

if, for each vertex x

∈

Γ, the induced subgraph onΓ

(

x

)

is a disjoint union of k

+

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

+

1, and each vertex is contained in k

+

1 maximal cliques.

Example 20 ([1] The Distance-Regular Graph of Order

(

l

,

k

)

). Suppose thatΓ is a distance-regular graph of order

(

l

,

k

)

. Let

N

=

l

+

1 and X be the collection of all cliques ofΓ together with the empty set

∅

. Ordered by inclusion, X is a regular pooling semilattice with the rank function

ℓ(

x

) = |

x

|

and the parameters

|

Xr

| =



l

+

1 r



n

(

k

+

1

)/(

l

+

1

),

µ(

r

,

s

,

t

) =



t

−

r s

−

r



.

Recall that a subgraph induced on a subset∆ofΓ is called strongly closed if C

(

u

, v) ∪

A

(

u

, v) ⊆

∆for every pair of vertices u

, v ∈

∆. A distance-regular graphΓ with diameter D is called D-bounded, if every strongly closed subgraph of

Γ is regular, and any two vertices x and y are contained in a common strongly closed subgraph with diameter

∂(

x

,

y

)

. A regular strongly closed subgraph ofΓ is called a subspace ofΓ. For any two subspaces∆1and∆2ofΓ

,

∆1

+

∆2denotes

the minimum subspace containing∆1and∆2.

Proposition 8 ([6, Lemma 2.1]). LetΓ be a D-bounded distance-regular graph with diameter D

≥

2. For 1

≤

i

+

1

≤

i

+

s

≤

i

+

s

+

t

≤

D, suppose that∆and∆′_{are two subspaces satisfying}₁

_⊆

_∆′_{, D}

_{(∆) =}

_{i and D}

_(∆

′

_{) =}

_i

₊

_s

₊

_{t. Then the number}

of the subspaces with diameter i

+

s containing∆and contained in∆′_{, denoted by N}

₍

_i

_,

_i

₊

_s

_,

_i

₊

_s

₊

_t

₎

_{, is}

(

bi

−

bi+s+t

)(

bi+1

−

bi+s+t

) · · · (

bi+s−1

−

bi+s+t

)

(

bi

−

bi+s

)(

bi+1

−

bi+s

) · · · (

bi+s−1

−

bi+s

)

.

Example 21 ([16] The D-Bounded Distance-Regular Graph). LetΓ be a D-bounded distance-regular graph with D

=

N. For x

∈

Γ, let X be the collection of all subspaces∆containing x inΓ. Ordered by inclusion, X is a pooling semilattice with the rank function

ℓ(∆) =

D

(∆)

and the parameters

|

Xr

| =

N

(

0

,

r

,

D

),

µ(

r

,

s

,

t

) =

N

(

r

,

s

,

t

).

In particular, if D

(∆

1

) +

D

(∆

2

) =

D

(∆

1

+

∆2

) +

D

(∆

1

∩

∆2

)

for any∆1

,

∆2

∈

X , then the pooling semilattice X is a regular

(8)

4. Pooling lattices

In this section, we show how to construct pooling designs from the pooling lattices by the intersection type incidence method.

Lemma 9. Let X be a pooling lattice with rank N and 0

≤

r

≤

s

,

t

≤

N. For u

∈

Xr

,

x

∈

Xswith u

≼

x, the number of elements

z

∈

Xtsuch that x

∧

z

=

u is a constant

π(

r

,

s

,

t

)

. Moreover, for given r and t, the function

π(

r

,

s

,

t

)

is decreasing about s and

is indeed strictly decreasing until its value is zero, i.e.

π(

r

,

r

,

t

) > π(

r

,

r

+

1

,

t

) > · · · > π(

r

,

p

,

t

) > π(

r

,

p

+

j

,

t

) =

0 for

some p

≥

r and any 1

≤

j

≤

N

−

p.

Proof. We prove the first statement by induction on s

−

r. The case s

−

r

=

0 follows fromLemma 4with

π(

r

,

r

,

t

) =

|[

u

,

1

] ∩

Xt

| =

µ(

r

,

t

,

N

)

, where the element 1 is the greatest element of X . Suppose s

−

r

≥

1. Choose any u

∈

Xr and

x

∈

Xswith u

≼

x. Note that the set u+

∩

Xtis partitioned into Ui

= {

z

|

z

∈

u+

∩

Xt

, ℓ(

x

∧

z

) =

i

}

for r

≤

i

≤

s. Since each element z

∈

Uihas the greatest lower bound x

∧

z

∈ [

u

,

x

] ∩

Xi

, |

Ui

| =

µ(

r

,

i

,

s

)π(

i

,

s

,

t

)

by induction for r

<

i. Hence

π(

r

,

s

,

t

) = µ(

r

,

t

,

N

) −

s



i=r+1

µ(

r

,

i

,

s

)π(

i

,

s

,

t

)

(2)

is a constant, where

π(

i

,

s

,

t

) =

0 if i

>

t. The first statement follows. Let x1

∈

Xsand x2

∈

Xs+1with u

≼

x1

≺

x2, where

r

≤

s

≤

N

−

1. Then

{

z

∈

Xt

|

x1

∧

z

=

u

} ⊇ {

z

∈

Xt

|

x2

∧

z

=

u

}

, which implies that

π(

r

,

s

,

t

) ≥ π(

r

,

s

+

1

,

t

)

. Choose the

largest p

≤

N such that

π(

r

,

p

,

t

) >

0, and restrict to s

≤

p

−

1 in the above proof. Pick y

∈

Xtwith x1

∧

y

=

u

,

a

∈ [

u

,

y

]∩

Xr+1

and let x2

=

x1

∨

a. Then x1

∧

a

=

u and a

≼

x2

∧

y, which implies that

ℓ(

x2

) =

s

+

1. Hence y

̸∈ {

z

∈

Xt

|

x2

∧

z

=

u

}

. The

second statement follows.

Lemma 10. Let X be a pooling lattice with rank N and 1

≤

r

≤

s

,

t

≤

N. For x

∈

Xs, the number of elements z

∈

Xtsuch that

ℓ(

x

∧

z

) =

r is

µ(

0

,

r

,

s

)π(

r

,

s

,

t

)

.

Proof. This is clear byLemma 9, since for x

∈

Xsthe number

µ(

0

,

r

,

s

)π(

r

,

s

,

t

) = |[

0

,

x

] ∩

Xr

|

π(

r

,

s

,

t

)

counts the desired

z.

Definition 2. Suppose that X is a pooling semilattice. For positive integers i

,

d

,

k

,

N with 1

≤

i

≤

d

<

k

<

N, let M

(

i

;

d

,

k

;

N

)

be the binary matrix with rows indexed with Xdand columns indexed with Xksuch that M

(

x

,

y

) =

1 if and only if

ℓ(

x

∧

y

) =

i. Theorem 11. Suppose that X is a pooling lattice and 1

≤

i

≤

d

<

k

<

N. Then the following results hold.

(i) Let s satisfy 1

≤

s

≤

i and N

−

(

s

+

1

)

k

≥

d

−

i. Then M

(

i

;

d

,

k

;

N

)

₌

_µ(

_s

_,

_i

_,

_k

_)π(

_i

_{, (}

_s

₊

1

)

k

,

d

) −

1.

(ii) Let s satisfies 1

≤

s

< µ(

0

,

i

,

k

)/µ(

0

,

i

,

k

−

1

)

and N

−

(

s

+

1

)

k

≥

d

−

i. Then M

(

i

;

d

,

k

;

N

)

is an se-disjunct matrix, where e

=

(µ(

0

,

i

,

k

) −

s

µ(

0

,

i

,

k

−

1

))π(

i

, (

s

+

1

)

k

,

d

) −

1.

Proof. (i) Let y0

,

y1

, . . . ,

ysbe any s

+

(

i

;

d

,

k

;

N

)

. Similar to the proof ofTheorem 6(i), there exists an

aj

∈

X1such that aj

≼

y0but aj

̸≼

yjfor each j

∈ {

1

,

2

, . . . ,

s

}

. Suppose a0

=

a1

∨

a2

∨ · · · ∨

as. By the proof ofTheorem 6(i), the size of Xi

∩ [

a0

,

y0

]

is at least

µ(

s

,

i

,

k

)

. Let x0

∈ [

a0

,

y0

] ∩

Xiand x

∈

Xdsatisfy x

∧

(

y0

∨

y1

∨ · · · ∨

ys

) =

x0. Then x

∧

y0

=

x0

and x

∧

yj

≼

x0for each j

∈ {

1

,

2

, . . . ,

s

}

, which implies that

ℓ(

x

∧

yj

) <

i by a0

̸≼

yj. Since x

∧

(

y0

∨

y1

∨ · · · ∨

ys

) =

x0,

by (A2)

ℓ(

y0

∨

y1

∨ · · · ∨

ys

) ≤ (

s

+

1

)

k and

ℓ(

x

∨

y0

∨

y1

∨ · · · ∨

ys

) ≤

d

+

(

s

+

1

)

k

−

i

≤

N. ByLemma 9, the number of elements x

∈

Xdsatisfying x

∧

(

y0

∨

y1

∨ · · · ∨

ys

) =

x0is at least

π(

i

, (

s

+

1

)

k

,

d

)

. Therefore, the number of elements

x

∈

Xdsatisfying

ℓ(

x

∧

y0

) =

i and

ℓ(

x

∧

yj

) ̸=

i for each j

∈ {

1

,

2

, . . . ,

s

}

is at least

µ(

s

,

i

,

k

)π(

i

, (

s

+

1

)

k

,

d

)

, as desired. (ii) Let y0

,

y1

, . . . ,

ysbe any s

+

(

i

;

d

,

k

;

N

)

. ByTheorem 6(ii), the number of elements in Xi contained in y0but not in yjfor each j

∈ {

1

,

2

, . . . ,

s

}

is at least

µ(

0

,

i

,

k

) −

s

µ(

0

,

i

,

k

−

1

)

. Given x0

∈

Xi with x0

≼

y0

but x0

̸≼

yjfor each j

∈ {

1

,

2

, . . . ,

s

}

. By the proof of (i), the number of elements x

∈

Xdsatisfying x

∧

y0

=

x0 and

ℓ(

x

∧

yj

) <

i for each j

∈ {

1

,

2

, . . . ,

s

}

is at least

π(

i

, (

s

+

1

)

k

,

d

)

. Therefore, the desired result follows.

Theorem 12. Suppose that X is a regular pooling lattice. Let s

,

i

,

d

,

k and N satisfy k

−

i

≥

2

,

1

≤

s

≤

(µ(

0

,

i

,

k

) − µ(

0

,

i

,

k

−

1

))/(µ(

0

,

i

,

k

−

1

) − µ(

0

,

i

,

k

−

2

))

and N

−

k

−

s

(

k

−

max

{

2i

−

d

,

0

}

) ≥

d

−

i. Then M

(

i

;

d

,

k

;

N

)

is an se_{-disjunct matrix,}

where e

=

(µ(

0

,

i

,

k

) − µ(

0

,

i

,

k

−

1

) − (

s

−

1

)(µ(

0

,

i

,

k

−

1

) − µ(

0

,

i

,

k

−

2

)))π(

i

,

k

+

s

(

k

−

max

{

2i

−

d

,

0

}

),

d

) −

1.

Proof. Let y0

,

y1

, . . . ,

ysbe any s

+

(

i

;

d

,

k

;

N

)

. ByTheorem 7the number of elements of Xicontained in y0but not in yjfor each 1

≤

j

≤

s is at least

µ(

0

,

i

,

k

) − µ(

0

,

i

,

k

−

1

) − (

s

−

1

)(µ(

0

,

i

,

k

−

1

) − µ(

0

,

i

,

k

−

2

))

. Let x

∈

Xd satisfy

ℓ(

x

∧

y0

) =

i. If there exists j

∈ {

1

,

2

, . . . ,

s

}

such that

ℓ(

x

∧

yj

) =

i, by

(

x

∧

y0

) ∨ (

x

∧

yj

) ≼

x and (A3), we have

ℓ(

y0

∧

yj

) ≥ ℓ(

x

∧

y0

∧

yj

)

=

ℓ(

x

∧

y0

) + ℓ(

x

∧

yj

) − ℓ((

x

∧

y0

) ∨ (

x

∧

yj

))

(9)

Given x0

∈

Xiwith x0

≼

y0but x0

̸≼

yjfor each j

∈ {

1

,

2

, . . . ,

s

}

. By the proof ofTheorem 11, the number of elements x

∈

Xd satisfying x

∧

y0

=

x0and

ℓ(

x

∧

yj

) <

i for each j

∈ {

1

,

2

, . . . ,

s

}

is at least

π(

i

,

k

+

s

(

k

−

max

{

2i

−

d

,

0

}

),

d

)

. Therefore, the desired result follows.

Now we give four families of pooling lattices. ByTheorems 11and12, we can construct pooling designs from these lattices.

Example 22 ([7] The Boolean Algebra). Let X be as inExample 1. Then X is a regular pooling lattice with the parameters

|

Xr

| =



N r



,

µ(

r

,

s

,

t

) =



t

−

r s

−

r



,

π(

r

,

s

,

t

) =



N

−

s t

−

r



.

Example 23 ([8] The Projective Geometry). Let X be as inExample 2. Then X is a regular pooling lattice with the parameters

|

X_r

| =



N r



q

,

µ(

r

,

s

,

t

) =



t

−

r s

−

r



q

,

π(

r

,

s

,

t

) =

q(s−r)(t−r)



N

−

s t

−

r



q

.

Example 24 (The Affine Geometry). Let X′_{be as in}_{Example 14}_{. Then X}′_{is a pooling lattice with the parameters}

|

X_r′+1

| =

qN −r



N r



q

,

µ(

r

+

1

,

s

+

1

,

t

+

1

) =



t

−

r s

−

r



q

,

π(

r

+

1

,

s

+

1

,

t

+

1

) =

q(s−r)(t−r)+s−r



N

−

s t

−

r



q

.

Example 25 (The D-Bounded Distance-Regular Graph). Let X be as in Example 21. Then X is a pooling lattice with the parameters

|

Xr

| =

N

(

0

,

r

,

D

),

µ(

r

,

s

,

t

) =

N

(

r

,

s

,

t

),

π(

r

,

s

,

t

),

where

π(

r

,

s

,

t

)

can be computed using(2). In particular, if D

(∆

1

)+

D

(∆

2

) =

D

(∆

1

+

∆2

)+

D

(∆

1

∩

∆2

)

for any∆1

,

∆2

∈

X ,

then the pooling lattice X is a regular pooling lattice.

Acknowledgments

This research is supported by NSFC (11271047, 11371204), NSF of Hebei Province (A2012408003, A2013408009), NSF of Hebei Education Department (ZH2012082), the Fundamental Research Funds for the Central University of China, the Fund for Hundreds of Excellent Innovative Talents in Higher Education of Hebei Province (BR2-235), TPF-2011-11 of Hebei Province and NSC (99-2115-M-009-005-MY3) of Taiwan.

References

[1]Y. Bai, T. Huang, K. Wang, Error-correcting pooling designs associated with some distance regular graphs, Discrete Appl. Math. 157 (2009) 3038–3045. [2]A.E. Brouwer, A.M. Cohen, A. Neumaier, Distance-Regular Graphs, Springer-Verlag, Berlin, Heidelberg, 1989.

[3]D. Du, F.K. Hwang, Pooling Designs and Nonadaptive Group Testing: Important Tools for DNA Sequencing, World Scientific, 2006.

[4]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) 1127–1134.

[5]A.G. D’yachkov, A.J. Macula, P.A. Vilenkin, Nonadaptive and trivial two-stage group testing with error-correcting de_{-disjunct inclusion matrices,}

in: Entropy, Search, Complexity, in: Bolyai Society Mathematical Studied, vol. 16, Springer, Berlin, 2007, pp. 71–83.

[6]S. Gao, J. Guo, W. Liu, Lattices generated by strongly closed subgraphs in d-bounded distance-regular graphs, European J. Combin. 28 (2007) 1800–1813. [7]J. Guo, K. Wang, A construction of pooling designs with high degree of error correction, J. Combin. Theory Ser. A 118 (2011) 2056–2058.

[8]J. Guo, K. Wang, Pooling designs with surprisingly high degree of error correction in a finite vector space, Discrete Appl. Math. 160 (2012) 2172–2176. [9]H. Huang, Y. Huang, C. Weng, More on pooling spaces, Discrete Math. 308 (2008) 6330–6338.

[10]T. Huang, K. Wang, C. Weng, Pooling spaces associated with finite geometry, European J. Combin. 29 (2008) 1483–1491. [11]T. Huang, C. Weng, Pooling spaces and non-adaptive pooling designs, Discrete Math. 282 (2004) 163–169.

[12]A.J. Macula, A simple construction of d-disjunct matrices with certain constant weights, Discrete Math. 162 (1996) 311–312. [13]A.J. Macula, Error-correcting non-adaptive group testing with de_{-disjunct matrices, Discrete Appl. Math. 80 (1997) 217–222.}

[14]H. Ngo, D. Du, New constructions of non-adaptive and error-tolerance pooling designs, Discrete Math. 243 (2002) 161–170. [15] J.H. van Lint, R.M. Wilson, A Course in Combinatorics, Cambridge, Victoria, 2001.

[16]X. Zhang, J. Guo, S. Gao, Two new error-correcting pooling designs from d-bounded distance-regular graphs, J. Comb. Optim. 17 (2009) 339–345. [17]P. Zhao, K. Diao, K. Wang, A generalization of Macula’s disjunct matrices, J. Comb. Optim. 22 (2011) 495–498.