Contents lists available atScienceDirect
Discrete Applied Mathematics
journal homepage:www.elsevier.com/locate/damError-correcting pooling designs associated with some distance-regular
graphs
Yujuan Bai
a,b, Tayuan Huang
c, Kaishun Wang
a,∗ aSch. Math. Sci. & Lab. Math. Com. Sys., Beijing Normal University, Beijing, 100875, China bBeijing DongFangDeCai School, Beijing, 100026, ChinacDepartment 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 graphK2nconsidered 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. stronglyt-cliques) with various sizes of the Johnson graphJ(n,t)(resp. the Grassmann graphJq(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
nitems to be tested and that there are at mostddefective items among them. Eachtest(orpool) 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 outcomeif the subset contains no positive and apositive outcomeotherwise. 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
isnon-adaptiveif 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 iserror tolerantif it can detect some errors in test outcomes. A mathematical model of
error-tolerance designs is ade-disjunctmatrix. A binary matrixMis said to bede-disjunct if, given anyd
+
1 columns ofMwith onedesignated, there aree
+
1 rows with a 1 in the designated column and 0 in each of the otherdcolumns. Ade-disjunct matrixwithe
=
0 is said to bed-disjunct. Macula [12] proposed a novel way of constructingd-disjunct matrices by the containmentrelation of subsets in a finite set, while in [13] he constructedde-disjunct matrices for certain values ofe. 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 ad2e-disjunct
matrix ise-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 diameterD, and let∂(
x,
y)
denote the distance of the verticesxandy.Γis said tobedistance-regularwhenever for all non-negative integersh
,
i,
j, and for any two verticesxandyat distanceh, the numberphi,j
= |{
z∈
X|
∂(
x,
z)
=
i, ∂(
z,
y)
=
j}|
is independent of the choice ofxandy. For more information, the reader may consult [1].
For any positive integernwe shall use
[
n]
to denote the set{
1,
2, . . . ,
n}
. Also, given any setXand any vector spaceVover a finite fieldFq,
X k
denotes the collection of allk-subsets ofX, and
[
Vk
]
denotes the collection of allk-subspaces ofV.TheJohnson graph J
(
n,
t)
is defined on[n] t
such that two verticesAandBare adjacent if and only if
|
A∩
B| =
t−
1.Similarly, the Grassmann graph Jq
(
n,
t)
is defined onh
Fnq
t
i
such that two vertices Aand Bare 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. Anl-subset∆ofXis said to be at-cliqueofΓ with sizelif any two distinctvertices in∆are at distancet. 1-clique is the clique in traditional use. Astrongly t-cliqueofJq
(
n,
t)
with sizelis a subfamily{
A1,
A2, . . . ,
Al} ⊆
h
Fnq
t
i
such that dim
(
A1+
A2+ · · · +
Al)
=
tl. Note that anl-matching onK2nis a 2-clique ofJ(
n,
2)
withsizel. Hence at-clique ofJ
(
n,
t)
with sizelis a generalization of anl-matching.A class of pooling designs over the incidence matrices of matchings with various sizes of the complete graphK2n 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 graphJ
(
n,
2)
, the pooling designs by Ngo and Du are generalized tothe incidence matrices oft-cliques with various sizes of the Johnson graphJ
(
n,
t)
and stronglyt-cliques with various sizesof the Grassmann graphJq
(
n,
t)
, respectively. We show that our pooling designs have the same capability of error-detectingand 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 matrixMof orderN
×
T, letB(
D)
denote the Boolean sum of those columns indexed by elements ofD⊆ [
T]
,and letdH
(
B(
D),
B(
D0))
denote the Hamming distance betweenB(
D)
andB(
D0)
wheneverDandD0are two distinct subsetsof
[
T]
. Letes
=
min|D|=|D0|=sdH
(
B(
D),
B(
D 0)).
The larger the parameteres, 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 parameteresfor thosese-disjunct matrices.
2.1. Some known results
D’yachkov et al. [5] proposed the concept of fullyse-disjunct matrices. Anse-disjunct matrix isfully se-disjunctif it is not
de0-disjunct wheneverd
>
sore0>
e. D’yachkov et al. [4] gave the lower bounds ofesfor a fullyse-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] constructedd-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 integersd
<
k<
n, letJ(
n,
d,
k)
be the binary matrix with row-indexed (resp. column-indexed) by [n] d resp.
[n] ksuch thatM
(
A,
B)
=
1 if and only ifA⊆
Band 0 otherwise.Theorem 2.2([5, Proposition 2]).Suppose1
≤
s≤
d<
k<
n and e=
e(
s)
=
k−sk−d
−
1. Then J(
n,
d,
k)
is fully se-disjunct.Ngo and Du [14] gave aq-analogue of Macula’s construction. The error-correcting property of Ngo and Du’s construction
Definition 2.2 ([14]). For positive integersd
<
k<
n, letJq(
n,
d,
k)
be the binary matrix with row-indexed (resp. column-indexed) byh
Fnq di
resp.
h
Fnq ki
such thatM
(
A,
B)
=
1 ifA⊆
Band 0 otherwise.LetFqbe the finite field withqelements, whereqis a prime power. LetFnqbe then-dimensional vector space overFq. For
a positive integern, theGaussian binomial coefficients with basis qis defined by
h
n ii
q=
i−1Y
j=0 n−
j i−
j,
ifq=
1,
i−1Y
j=0 qn−
qj qi−
qj,
ifq6=
1.
Naturally,n 0 q=
1 and n iq
=
0 ifi>
n. In the caseq=
1, we shall writen i
instead ofn i 1for convenience.In the rest of this paper, for positive integersd
<
kandk−
d≥
2r, we always assume thatpq
(
r)
=
h
k di
q−
h
k−r di
qh
k−r di
q−
h
k−2r di
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≥
2and 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 onJ
(
n,
d,
k)
, Macula [13] proposed another family ofse-disjunct matrices. D’yachkov et al. [5] also discussed theirerror-correcting property.
Definition 2.3. (i) A familyK
⊆
[n] k
is called an
{
r,
r+
1, . . . ,
k}
-cliqueofJ(
n,
k)
if|
K∩
K0| ≤
k−
rfor any two distinctK
,
K0∈
K. (ii) A familyF⊆
h
Fnq ki
is called an
{
r,
r+
1, . . . ,
k}
-cliqueofJq(
n,
k)
if dim(
K∩
K0)
≤
k−
rfor any two distinctK,
K0∈
F. Definition 2.4 ([13]). For positive integersd<
k<
n, letK⊆
[n] k
. SupposeJ
(
n,
d,
K)
denotes the binary matrix withrow-indexed (resp. column-indexed) by
[n] d
(resp.K) such thatM
(
A,
B)
=
1 ifA⊆
Band 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
≥
1with1+
rk−d
≤
r andα
d=
min(
rd
,
k−
d)
. Then J(
n,
d,
K)
is dαd−1-disjunct.(ii) J
(
n,
d,
K)
is se-disjunct where s∈ [
p1
(
r)
]
and e=
e1(
s,
r)
.As theq-analogue ofJ
(
n,
d,
K)
, we propose the following definition.Definition 2.5. For positive integersd
<
k<
n, letF⊆
h
Fnq
k
i
. SupposeJq
(
n,
d,
F)
denotes the binary matrix withrow-indexed (resp. column-row-indexed) by
h
Fnq
d
i
(resp.F) such thatM
(
A,
B)
=
1 ifA⊆
Band 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)
.LetX
= {
x=
(
x1,
x2, . . . ,
xn)
|
xi∈
F}
whereF= {
0,
1, . . . ,
q}
, and letPidenote the set of elements with weightiofX.For positive integers 1
≤
d≤
k≤
n, defineH(
n,
q,
d,
k)
to be the binary matrix with row-indexed (resp. column-indexed)byPd(resp.Pk) such thatM
(
x,
y)
=
1 ifxi=
0 orxi=
yiandxi6=
0.D’yachkov et al. [5] proposed the above matrix and discussed its disjunctness.
Theorem 2.6 ([5]).For1
≤
s≤
d≤
k≤
n, H(
n,
q,
d,
k)
is fully se-disjunct, where e=
k−s k−d−
1.LetV be the vector space of dimensionn
+
r overFq, and letW be a fixedr-subspace ofV. Let Pi= {
A|
A∈
h
Vi
i
andA
∩
W=
0}
. For positive integers 1≤
d≤
k≤
n, defineBq(
n,
r,
d,
k)
to be the binary matrix with row-indexed(resp. column-indexed) byPd(resp.Pk) such thatM
(
A,
B)
=
1 ifA⊆
B.Huang and Weng [9] proved thatBq
(
n,
r,
d,
k)
is ade-disjunct matrix for somee. Similar toTheorem 2.3, we may obtainthe following results.
Theorem 2.7. Let q be a prime power. Suppose k
−
d≥
2and 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 complementMcof a binary matrixMis the matrix that results when one interchanges the 0’s and 1’s inM. LetK
be any subset of
[n] k
. Macula [13] considered the matrixJ∗
(
n,
d,
K)
as that which results by row augmenting the matrixJ
(
n,
d,
K)
withJc(
n,
1,
K)
. He claimed thate1
≥
4 forJ∗(
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 parameteresforJ
(
n,
d,
k)
andJq(
n,
d,
k)
, respectively. We begin withan example.
Example 2.1. Given a matrixJ
(
n,
d,
k)
with 1≤
d<
k<
n.
Fors∈ [
d]
, letD0
= {b
1,
b
2, . . . ,
s[−
1,
k\+
1}
and D00= {b
1,
b
2, . . . ,
s[−
1,
b
k}
where
b
i= [
k+
1] − {
i}
andi∈ [
k+
1]
. Then R|
R∈
[
k]
d,
R6⊆
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)
with1≤
s≤
d<
k<
n.
Then es=
2k−s k−d
.Proof. The upper bound foresis 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 matrixJq
(
n,
k,
d)
withk−
d≥
2.
For eachi∈ [
k+
1]
, leteibe the row vector ofV whosei-thcoordinate is 1 and all other coordinates are 0. SupposeFq
= {
a1,
a2, . . . ,
aq}
ands≤
q+
1.Fori
∈ [
s+
1]
, letD0
= {
C1, . . . ,
Cs−1,
Cs}
and D00= {
C1, . . . ,
Cs−1,
Cs+1}
,
whereCs
= h
e1,
e2, . . . ,
eki
,
Cs+1= h
e2,
e3, . . . ,
ek,
ek+1i
,
Ci= h
e1+
aie2,
e3, . . . ,
ek,
ek+1i
.
By the principle of inclusion and exclusion, R|
R∈
Cs d,
R6⊆
C1, . . . ,
Cs−1,
Cs+1=
k d q−
s 1 k−
1 d q+
k−
2 d q sX
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 foresis derived fromExample 2.2. ByTheorem 2.3andProposition 2.1, it is also a lower bound.
3. Disjunctness over matchings onK2mand its extensions
For positive integersd
<
k≤
m, letMbe a binary matrix with row-indexed (resp. column-indexed) byd-matchings(resp.k-matchings) ofK2msuch thatM
(
A,
B)
=
1 ifA⊆
Band 0 otherwise. This matrix is denoted byM(
2m,
d,
k)
. In [14],Ngo and Du proposed the matrix and discussed its disjunctness.
Theorem 3.1 ([14, Theorem 11,Corollary 12]).Let1
≤
d<
k≤
m. Then(i) M
(
2m,
d,
k)
is a d-disjunct matrix.(ii) M
(
2m,
d,
m)
is d-error-detecting andb
d/
2c
-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 andd-error-correcting.
With an interpretation of matchings as 2-cliques of Johnson graphJ
(
n,
2)
, we shall generalize Ngo and Du’s designs tothe incidence matrices oft-cliques with various sizes of the Johnson graphJ
(
n,
t)
and stronglyt-cliques with various sizesof the Grassmann graphJq
(
n,
t)
, respectively. We show that our pooling designs have the same capability of error-detectingand error-correcting as Ngo and Du’s. However, the test-to-item ratio of ours is much smaller.
Definition 3.1. Given positive integersd
<
kandkt≤
n.
(i) LetJ
(
n,
t,
d,
k)
be the binary matrix with row-indexed (resp. column-indexed) byt-cliques with sized(resp.k) ofJ(
n,
t)
such thatM
(
A,
B)
=
1 ifA⊆
Band 0 otherwise.(ii) LetJq
(
n,
t,
d,
k)
be the binary matrix with row-indexed (resp. column-indexed) by stronglyt-cliques with sized(resp.k) ofJq
(
n,
t)
such thatM(
A,
B)
=
1 ifA⊆
Band 0 otherwise.SinceJ
(
2m,
2,
d,
k)
=
M(
m,
d,
k)
,J(
n,
t,
d,
k)
is a generalization ofM(
m,
d,
k)
.Lemma 3.2. Let W be a given k-subspace ofFn
q. Then the number of d-subspaces ofFnqintersecting trivially with W is
h
n−k di
q qdk.
Proof. LetD= {
A|
A∈
h
Fnq di
,
A∩
W=
0}
. Counting the set{
(v
1, v
2, . . . , v
d)
|
v
i6∈ h
W, v
1, v
2, . . . , v
i−1i
,
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 di
q qdkas 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!
)
ll!
.
(1)(ii) The number of strongly t-cliques of Jq
(
n,
t)
with size l isuq
(
n,
t,
l)
=
h
n ti
q n−
t t q· · ·
n−
(
l−
1)
t t q·
q t2l(l−1)/2 l!
.
(2)Proof. (i) Since everytl-subset of
[
n]
forms((tl)!t!)ll!manyt-cliques ofJ
(
n,
t)
with sizel,(
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.
LetN
(
n,
t,
l)
be the number of ordered tuples(
A1,
A2, . . . ,
Al)
oft-subspaces ofFnqsuch that dim(
A1+
A2+ · · ·
Al)
=
tl. Thenuq
(
n,
t,
l)
=
N(nl,!t,l).
Hence, if we want to getuq(
n,
t,
l)
, it suffices to computeN(
n,
t,
l)
. There are n tqways to chooseA1,
then
n−ttqqt2ways to chooseA2byLemma 3.2and so on. It follows thatN
(
n,
t,
l)
=
h
n ti
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 weightu
(
n−
td,
t,
k−
d)
and column weightkd, where(
N,
T)
=
(
u(
n,
t,
d),
u(
n,
t,
k))
and e=
k−sk−d
−
1.Proof. ByLemma 3.3,J
(
n,
t,
d,
k)
is anN×
Tmatrix with row weightu(
n−
td,
t,
k−
d)
and column weight k d . LetC0,
C1, . . . ,
Csbe anys+
1 distinct columns ofJ(
n,
t,
d,
k)
. For eachi∈ [
s]
, there existsPi∈
C0\
Ci. LetE= {
Pi|
i∈ [
s]}
.Then
|
E| ≤
sandE⊆
C0butE6⊆
Cifor eachi∈ [
s]
. If|
E| =
j, the number ofd-subsets ofC0containingEis k−j k−d . Since k−j k−d≥
k−s k−dwheneverj
≤
s, the number oft-cliques with sizedcontained inC0but not contained inCifor eachi∈ [
s]
is at least
k−s k−d .Corollary 3.5. Let1
≤
s≤
d<
k and(
k+
1)
t≤
n. Then J(
n,
t,
d,
k)
is fully se-disjunct with e=
k−s k−d−
1and es=
2 k−s k−d.
Proof. In order to prove thatJ(
n,
t,
d,
k)
is fullyse-disjunct, we only need to show that the maximum size ofEis obtained inTheorem 3.4. Since(
k+
1)
t≤
n, there exists at-cliqueT= {
A1,
A2, . . . ,
Ak+1}
with sizek+
1. LetC0=
T− {
Ak+1}
andCi
=
T− {
Ai}
for eachi∈ [
k]
. Then|
E| = |{
Ai|
i∈ [
s]}| =
s. Similar toTheorem 2.9, it is routine to computees.Similar results hold forJq
(
n,
t,
d,
k)
too, and their proofs will be omitted.Theorem 3.6. Let1
≤
s≤
d<
k and kt≤
n. Then the matrix Jq(
n,
t,
d,
k)
is an se-disjunct matrix of order N×
T with rowweight 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. Let1
≤
s≤
d<
k and(
k+
1)
t≤
n. Then Jq(
n,
t,
d,
k)
is fully se-disjunct with e=
k−s k−d−
1and es=
2 k−s k−d.
Remarks. (i) By zigzag arguments similar toTheorem 3.1(ii),J(
tn,
t,
d,
n)
isd-error-detecting andb
d/
2c
error-correcting. (ii) The test-to-item ratio ofJ(
tm,
t,
d,
k)
(resp.Jq(
tm,
t,
d,
k)
) is much less than that ofM(
m,
d,
k)
(resp.J(
tm,
t,
d,
k)
).The following theorem tells us how to choosedandksuch that the test-to-item ratio forJ
(
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 whenb
l1c ≤
l≤ b
l2c
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) Supposef
(
l)
=
u(tm,t,l+1)u(tm,t,l) . Thenf
(
l)
is a decreasing number series whilelgoes from 1 tom−
1. Sincef(
1) >
1andf
(
m−
1)
=
1m
<
1,u(
tm,
t,
l)
is unimodal. Note thatf
(
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.
Letui=
(ttm−i−)√tlt−i l+1. Then(
u0)
t≤
f(
l)
≤
(
u t−1)
t. Ifu0=
1, thenl+
t√
l+
1=
m; ifut−1=
1, thenl+
t+ t √ l+1−1 t=
m. Thedesired results follow.
(ii) Supposeg
(
l)
=
uq(tm,t,l+1)uq(tm,t,l) . Then
g
(
l)
=
(
qtm
−
qtl)
· · ·
(
qtm−t+1−
qtl)
(
qt−
1)
· · ·
(
q−
1)(
l+
1)
.
It follows that thatg
(
l)
is decreasing whilelgoes from 1 tom−
1. Sinceg(
1) >
1 andg(
m−
1) >
1,uq(
tm,
t,
l)
is increasingwhilelgoes from 1 tom, and achieve the maximum value atl
=
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 theproofs in this section.
A distance-regular graphΓ
=
(
X,
R)
of diameterD≥
2 is said to beantipodal, if∂(
x,
y)
=
∂(
x,
z)
=
Dandy6=
zimplies∂(
y,
z)
=
D. LetkD=
p0D,D. Then the number of maximalD-clique ofΓ is|X|
kD+1
.
Since any two distinct maximalD-cliqueshave no common vertices, the number ofD-clique with sizelofΓ isk|X|
D+1
·
kD+1 l.
Table 1
The parameters ofse-disjunct matrices.
Name Tests Items Ratio (test to item) s e Remarks
J(n,d,k) nd 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) nd |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) nd 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 (2 k)! 2kk!· 2m 2k (2 m−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) q t(d−k)(k+d−1)/2k! hn−dt t i q··· hn−(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 oforder(
r,
t)
if, for each vertexx∈
X, the induced subgraphonΓ
(
x)
is a disjoint union oft+
1 cliques with sizer. Then each maximal clique is of sizer+
1, and each vertex iscontained int
+
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 ofcliques with sizelis
r+1 l·
n(t+1) r+1 .LetΓ be an antipodal distance-regular graph of diameterDwithnvertices. For positive integers 1
<
d<
k<
kD+
1, letMbe the binary matrix whose row (resp. column) indexed by theD-cliques ofΓ with sized(resp.k) such thatM
(
A,
B)
=
1ifA
⊆
Band 0 otherwise. This matrix is denoted byA(
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 weightkD+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, letMbe the binary matrixwhose row (resp. column) indexed by the cliques ofΓ with sized(resp.k) such thatM
(
A,
B)
=
1 ifA⊆
Band 0 otherwise.This matrix is denoted byB
(
r,
t;
d,
k)
.Theorem 4.2. Let1
≤
s≤
d<
k<
r+
1. Then B(
r,
t;
d,
k)
is a fully se-disjunct matrix of order N×
T with row weightr+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 . AcknowledgmentThe third author is partially supported by NCET-08-0052 and NSF of China (10871027).
Appendix
SeeTable 1.
References
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