Pooling designs are standard experimental tools in many biotechnical
applications. It is well-known that all famous pooling designs are constructed
from mathematical structures by the "containment matrix" method. In particular,
Macula's designs (resp. Ngo and Du's designs) are constructed by the
containment relation of subsets (resp. subspaces) in a finite set (resp. vector
space). Recently, we generalized Macula's designs and obtained a family of
pooling designs with more high degree of error correction by subsets in a
finite set. In this paper, as a generalization of Ngo and Du's designs, we
study the corresponding problems in a finite vector space and obtain a family
of pooling designs with surprisingly high degree of error correction. Our
designs and Ngo and Du's designs have the same number of items and pools,
respectively, but the error-tolerant property is much better than that of Ngo
and Du's designs, which was given by D'yachkov et al. \cite{DF}, when the
dimension of the space is large enough

Guo, Jun

Wang, Kaishun

English

arXiv

AbstractPooling designs are standard experimental tools in many biotechnical applications. It is well-known that all famous pooling designs are constructed from mathematical structures by the “containment matrix” method. In particular, Macula’s designs (resp. Ngo and Du’s designs) are constructed by means of the containment relation of subsets (resp. subspaces) in a finite set (resp. vector space). In [J. Guo, K. Wang, A construction of pooling designs with high degree of error correction, J. Combin. Theory Ser. A 118 (2011) 2056–2058], we generalized Macula’s designs and obtained a family of pooling designs with higher degree of error correction. In this paper we consider, as a generalization of Ngo and Du’s designs, q-analogue of the above designs, and obtain a family of pooling designs with surprisingly high degree of error correction. Our designs and Ngo and Du’s designs have the same numbers of items and pools, but the error-tolerance property of our design is much better than that of Ngo and Du’s designs when the dimension of the space is large enough

Elsevier - Publisher Connector 

Discrete Applied Mathematics 160 (2012) 2172–2176
Contents lists available at SciVerse ScienceDirect
Discrete Applied Mathematics
journal homepage: www.elsevier.com/locate/dam
Pooling designs with surprisingly high degree of error correction in a
finite vector space
Jun Guo a, Kaishun Wang b,∗
aMath. and Inf. College, Langfang Teachers’ College, Langfang 065000, China
b Sch. Math. Sci. & Lab. Math. Com. Sys., Beijing Normal University, Beijing 100875, China
a r t i c l e i n f o
Article history:
Received 27 March 2011
Received in revised form 2 May 2012
Accepted 28 May 2012
Available online 27 June 2012
keywords:
Pooling design
Disjunct matrix
Error correction
a b s t r a c t
Pooling designs are standard experimental tools in many biotechnical applications. It is
well-known that all famous pooling designs are constructed frommathematical structures
by the ‘‘containment matrix’’ method. In particular, Macula’s designs (resp. Ngo and Du’s
designs) are constructed bymeans of the containment relation of subsets (resp. subspaces)
in a finite set (resp. vector space). In [J. Guo, K. Wang, A construction of pooling designs
with high degree of error correction, J. Combin. Theory Ser. A 118 (2011) 2056–2058], we
generalized Macula’s designs and obtained a family of pooling designs with higher degree
of error correction. In this paperwe consider, as a generalization of Ngo andDu’s designs, q-
analogue of the above designs, and obtain a family of pooling designswith surprisingly high
degree of error correction. Our designs and Ngo and Du’s designs have the same numbers
of items and pools, but the error-tolerance property of our design is much better than that
of Ngo and Du’s designs when the dimension of the space is large enough.
© 2012 Elsevier B.V. All rights reserved.
1. Introduction
A group test is applicable to an arbitrary subset of clones with two possible outcomes: a negative outcome indicates
that all clones 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 clones with a small
number of tests [1–3,7]. A 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 [9]. An
s0-disjunct matrix is also called s-disjunct. An se-disjunct matrix is called fully se-disjunct if it is not se11 -disjunct whenever
s1 > s or e1 > e. An se-disjunct matrix is ⌊e/2⌋-error-correcting [4].
For positive integers k ≤ n, let [n] = {1, 2, . . . , n} and

[n]
k

denote the collection of all k-subsets of [n].
Macula [8,9] proposed a novel way of constructing disjunct matrices by means of the containment relation of subsets
in [n].
Definition 1.1 ([8]). For positive integers 1 ≤ d < k < n, letM(d, k, n) be the binary matrix with rows indexed with

[n]
d

and columns indexed with

[n]
k

such thatM(A, B) = 1 if and only if A ⊆ B.
∗ Corresponding author.
E-mail addresses: guojun_lf@163.com (J. Guo), wangks@bnu.edu.cn (K. Wang).
0166-218X/$ – see front matter© 2012 Elsevier B.V. All rights reserved.
doi:10.1016/j.dam.2012.05.018
J. Guo, K. Wang / Discrete Applied Mathematics 160 (2012) 2172–2176 2173
D’yachkov et al. [5] discussed the error-correcting property ofM(d, k, n).
Theorem 1.1 ([5]). For positive integers 1 ≤ d < k < n and 1 ≤ s ≤ d, M(d, k, n) is fully se1-disjunct, where e1 =

k−s
d−s

− 1.
In [6], we generalized Macula’s construction and obtained a family of pooling designs with higher degree of error
correction.
Definition 1.2 ([6]). For positive integers 1 ≤ d < k < n and 0 ≤ i ≤ d, let M(i; d, k, n) be the binary matrix with rows
indexed with

[n]
d

and columns indexed with

[n]
k

such thatM(A, B) = 1 if and only if |A ∩ B| = i.
Theorem 1.2 ([6]). Let 1 ≤ s ≤ i, ⌊(d+ 1)/2⌋ ≤ i ≤ d < k and n− k− s(k+ d− 2i) ≥ d− i. Then:
(i) M(i; d, k, n) is an se2-disjunct matrix, where e2 =

k−s
i−s
 
n−k−s(k+d−2i)
d−i

− 1;
(ii) for a given k, if i < d, then limn→∞ e2+1e1+1 = ∞.
LetFq be a finite fieldwith q elements, where q is a prime power. For a positive integer n, letFnq be an n-dimensional vector
space over Fq. For positive integers k ≤ n, let

[n]
k

q
be the set of all k-dimensional subspaces of Fnq . A matrix representation
of a subspace P is a matrix whose rows form a basis for P . When there is no danger of confusion, we use the same symbol to
denote a subspace and its matrix representation.
Letm1 andm2 be two integers. For brevity we use the Gaussian coefficient

m2
m1

q
=
m2
t=m2−m1+1
(qt − 1)
m1
t=1
(qt − 1)
.
For convenience, we set
m2
0

q = 1 and

m2
m1

q
= 0 wheneverm1 < 0 orm2 < m1. Then, by Wan [13],
 [n]
k

q
 = nk q.
As a q-analogue of Macula’s designs, Ngo and Du [12] constructed a family of disjunct matrices by using the containment
relation of subspaces in Fnq .
Definition 1.3 ([12]). For positive integers 1 ≤ d < k < n, let Mq(d, k, n) be the binary matrix with rows indexed with
[n]
d

q
and columns indexed with

[n]
k

q
such thatMq(A, B) = 1 if and only if A ⊆ B.
D’yachkov et al. [4] discussed the error-tolerance property ofMq(d, k, n).
Theorem 1.3 ([4]). For positive integers 1 ≤ d < k < n, k − d ≥ 2 and 1 ≤ s ≤ q(qk−1 − 1)/(qk−d − 1), Mq(d, k, n) is
se1-disjunct, where e1 = qk−d

k−1
d−1

q
− (s−1)qk−d−1

k−2
d−1

q
−1. In particular, if s ≤ q+1, then Mq(d, k, n) is fully se1-disjunct.
Nan and Guo [10] generalized Ngo and Du’s construction and obtained a family of pooling designs.
Definition 1.4 ([10]). For positive integers 1 ≤ d < k < n and max{0, d + k − n} ≤ i ≤ d, let Mq(i; d, k, n) be the binary
matrix with rows indexed with

[n]
d

q
and columns indexed with

[n]
k

q
such thatMq(A, B) = 1 if and only if dim(A∩B) = i.
Note that Mq(i; d, k, n) and Mq(d, k, n) have the same size. In [10], the error-tolerance property of Mq(i; d, k, n) is not
well expressed. In this paper, we discuss again the error-tolerance property ofMq(i; d, k, n).
2. The main results
In this section, we discuss the error-tolerance property ofMq(i; d, k, n). We begin with a useful lemma.
Lemma 2.1. Suppose max{0, r + m − n} ≤ j ≤ r and j ≤ m ≤ n. Let P be an m-dimensional subspace of Fnq and let W
be a j-dimensional subspace of P. Then the number of r-dimensional subspaces of Fnq intersecting P at W is f (j, r, n;m) =
q(r−j)(m−j)

n−m
r−j

q
.Moreover the function f (j, r, n;m+ α) about α is decreasing for 0 ≤ α ≤ n+ j−m− r.
2174 J. Guo, K. Wang / Discrete Applied Mathematics 160 (2012) 2172–2176
Proof. Since the general linear group GLn(Fq) acts transitively on the set of such pairs (P,W ), we may assume that
P = (I(m) 0(m,n−m)),W = (I(j) 0(j,n−j)). Let Q be any r-dimensional subspace of Fnq satisfying P ∩ Q = W . Then Q has
a matrix representation of the form
I(j) 0(j,m−j) 0(j,n−m)
0(r−j,j) A2 A3

,
where A2 is an (r − j) × (m − j) matrix and A3 is an (r − j)-dimensional subspace of Fn−mq . Therefore, f (j, r, n;m) =
q(r−j)(m−j)

n−m
r−j

q
.
Since
f (j, r, n;m)− f (j, r, n;m+ 1) = q(r−j)(m−j)

n−m
r − j

q
− q(r−j)(m+1−j)

n−m− 1
r − j

q
= (qr−j − 1)
q(r−j)(m−j)
n−m−1
l=n−m−(r−j)+1
(ql − 1)
r−j
l=1
(ql − 1)
≥ 0,
the desired result follows. 
Theorem 2.2. Let i, d, k, n be positive integers with ⌊(d+ 1)/2⌋ ≤ i ≤ d < k and n− k− s(k+ d− 2i) ≥ d− i. If k− i ≥ 2
and 1 ≤ s ≤ q(qk−1 − 1)/(qk−i − 1), then the following hold:
(i) Mq(i; d, k, n) is an se2-disjunct matrix, where
e2 = q(d−i)(k+s(k+d−2i)−i)

n− k− s(k+ d− 2i)
d− i

q

qk−i

k− 1
i− 1

q
− (s− 1)qk−i−1

k− 2
i− 1

q

− 1;
(ii) for a given k, if i < d, then limn→∞ e2+1e1+1 = ∞.
Proof. (i) Let B0, B1, . . . , Bs ∈

[n]
k

q
be any s + 1 distinct columns of Mq(i; d, k, n). Note that B0 contains

k
i

q
many
i-dimensional subspaces. To obtain the minimum number of i-dimensional subspaces of B0 not contained in Bj for each
1 ≤ j ≤ s¯, we may assume that dim(B0 ∩ Bj) = k− 1 and dim(B0 ∩ Bj ∩ Bl) = k− 2 for any two distinct j, l ∈ {1, 2, . . . , s¯}.
Then B1 contains

k−1
i

q
many i-dimensional subspaces of B0, while each of B2, B3, . . . , Bs contains at most

k−1
i

q
−

k−2
i

q
many i-dimensional subspaces of B0 not contained in B1. Consequently, the number of i-dimensional subspaces of B0 not
contained in B1, B2, . . . , Bs is at least
α =

k
i

q
−

k− 1
i

q
− (s− 1)

k− 1
i

q
−

k− 2
i

q

= qk−i

k− 1
i− 1

q
− (s− 1)qk−i−1

k− 2
i− 1

q
.
LetD ∈

[n]
d

q
with dim(D∩ B0) = i. If there exists j ∈ {1, 2, . . . , s} such that dim(D∩ Bj) = i, by (D∩ B0)+(D∩ Bj) ⊆ D,
we have
dim(B0 ∩ Bj) ≥ dim

D ∩ B0 ∩ Bj

= dim(D ∩ B0)+ dim(D ∩ Bj)− dim

(D ∩ B0)+ (D ∩ Bj)

≥ 2i− d.
Suppose dim(B0 ∩ Bj) ≥ 2i− d for each j ∈ {1, 2, . . . , s}. Then
dim(B0 + B1 + · · · + Bs) = dim(B0 + B1 + · · · + Bs−1)+ dim Bs − dim((B0 + B1 + · · · + Bs−1) ∩ Bs)
≤ dim(B0 + B1 + · · · + Bs−1)+ dim Bs − dim(B0 ∩ Bs)
≤ dim(B0 + B1 + · · · + Bs−1)+ k+ d− 2i
≤ dim B0 + s(k+ d− 2i)
= k+ s(k+ d− 2i).
J. Guo, K. Wang / Discrete Applied Mathematics 160 (2012) 2172–2176 2175
Table 1
Disjunct property ofM2(d, 8, 60) andM2(i; d, 8, 60).
(i, d) s e1 e2 Remarks
(1, 2) 2 6111 36893488146882232319 Theorem2.2
(1, 3) 2 74927 3544607988605033156167647492927651839 Theorem2.3
(2, 3) 4 54095 599519146661432524799 Theorem2.2
(1, 4) 2 177815 284599986330728289752034695103377217756856319 Theorem2.3
(2, 4) 4 155495 28799857511436549196854689617936383999 Theorem2.2
(3, 4) 8 110855 800925501358079 Theorem2.2
Let P be a given i-dimensional subspace of B0 not contained in B1, B2, . . . , Bs. By Lemma 2.1, the number of d-dimensional
subspaces D in Fnq satisfying D ∩ (B0 + B1 + · · · + Bs) = P is at least
q(d−i)(k+s(k+d−2i)−i)

n− k− s(k+ d− 2i)
d− i

q
.
Observe that D ∩ B0 = P and dim(D ∩ Bj) ≠ i for each j ∈ {1, 2, . . . , s}. Therefore, the number of d-dimensional subspaces
D in Fnq satisfying dim(D ∩ B0) = i and dim(D ∩ Bj) ≠ i for each j ∈ {1, 2, . . . , s} is at least
αq(d−i)(k+s(k+d−2i)−i)

n− k− s(k+ d− 2i)
d− i

q
.
Since e2 ≥ 0, we have α > 0, which implies that
s ≤
qk−i

k−1
i−1

q
qk−i−1

k−2
i−1

q
= q(q
k−1 − 1)
qk−i − 1 .
Hence, (i) holds.
(ii) is straightforward by (i) and Theorem 1.3. 
Theorem 2.3. Let i, d, k, n be positive integers with 1 ≤ i < ⌊(d + 1)/2⌋ and d < k, n − (s + 1)k ≥ d − i. If
1 ≤ s ≤ q(qk−1 − 1)/(qk−i − 1), then the following hold:
(i) Mq(i; d, k, n) is an se2-disjunct matrix, where
e2 = q(d−i)((s+1)k−i)

n− (s+ 1)k
d− i

q

qk−i

k− 1
i− 1

q
− (s− 1)qk−i−1

k− 2
i− 1

q

− 1;
(ii) for a given k, limn→∞ e2+1e1+1 = ∞.
Proof. The proof is similar to that of Theorem 2.2, and will be omitted. 
For q = 2, k = 8, n = 60, Table 1 shows the disjunct property of our designs and Ngo and Du’s designs for small i, d, s.
3. Concluding remarks
(i) For given positive integers d < k, limn→∞
[ nd ]q
[ nk ]q
= 0. This shows that the test-to-item ratio of Mq(i; d, k, n) is small
enoughwhen n is large enough. By Theorems 2.2 and 2.3, our pooling designs aremuch better thanNgo andDu’s designs
when n is large enough.
(ii) Ngo [11] improved the error-tolerance property ofMq(d, k, n) for s ≥ q+ 2, s ≥ q+ 3 and s ≥ q+ 4, respectively. By
a similar method, we also can improve the error-tolerance property ofMq(i; d, k, n) for these cases.
(iii) For positive integers 1 ≤ d < k < n, it seems to be interesting to consider the error-tolerance property ofMq(0; d, k, n).
Acknowledgments
J. Guo was partially supported by NSF of Hebei Province (A2012408003), TPF-2011-11 of Hebei Province, NSF of China
(10971052), and Langfang Teachers’ College (LSZB201104). K. Wangwas supported by NSF of China, NCET-08-0052, and the
Fundamental Research Funds for the Central Universities of China.
2176 J. Guo, K. Wang / Discrete Applied Mathematics 160 (2012) 2172–2176
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[3] D. Du, F.K. Hwang, Pooling Designs and Nonadaptive Group Testing: Important Tools for DNA Sequencing, World Scientific, 2006.
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