行政院國家科學委員會補助專題研究計畫
成果報告
期中進度報告
(組合設計及其應用 (3/3)）
計畫類別：
個別型計畫 整合型計畫。
計畫編號：NSC 99-2115-M-009-005-MY3
執行期間：99 年 8 月 1 日至 102 年 7 月 31 日
執行機構及系所：國立交通大學應用數學系
計畫主持人：翁志文
共同主持人：無
計畫參與人員：博士生：黃喻培、李光祥、劉家安、鄭硯仁。碩士生：林
志嘉、劉侖欣、洪湧昇、施政成、蘇慧文、曾如汶、王稟鈞、徐志杰、許
博喻、林凡軒
成果報告類型 (依經費核定清單規定繳交)：
精簡報告
完整報告
本計畫除繳交成果報告外，另須繳交以下出國心得報告：
赴國外出差或研習心得報告
赴大陸地區出差或研習心得報告
出席國際學術會議心得報告
國際合作研究計畫國外研究報告
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涉及專利或其他智慧財產權， 一年 二年後可公開查詢
中
華
民
國
102
年
8
月
5
日
行政院國家科學委員會補助專題研究計畫結案報告
計畫編號：NSC 99-2115-M-009-005-MY3
計畫類別：個別型計畫
執行期間： 99 年 8 月 1 日至 102 年 7 月 31 日
計畫名稱: 具組合設計及其應用 (3/3)
計畫主持人：翁志文 (國立交通大學應用數學系)
[email protected] 計畫參與人員：博士生：黃喻培、李光祥、劉家安、鄭硯仁。碩士生：林志嘉、劉侖欣、洪湧昇、施政成、 蘇慧文、曾如汶、王稟鈞、徐志杰、許博喻、林凡軒。 102 年 8 月 5 日一、中文摘要
計畫提案人先前與黃大原教授於文獻 [T. Huang, C. Weng, Pooling spaces and non-adaptive pooling designs, Discrete Math. 282 (2004) 163–169] 中提出 一種稱為池空間的有秩偏序結構, 並舉許多例子. 在 此成果報告中, 我們提出池半格的結構, 也舉出相當 多對應的例子, 每一個例子都能用來建構一類與群試 設計有關的池設計. 關鍵詞: 池設計, 半格, 池半格, 正規池半格, 池空間二、英文摘要
In [T. Huang, C. Weng, Pooling spaces and non-adaptive pooling designs, Discrete Math. 282 (2004) 163–169], Huang and Weng introduced pool-ing spaces, and constructed poolpool-ing designs from a pooling space. In this report, we introduce the con-cept of pooling semilattices and prove that a pool-ing semilattice is a poolpool-ing space, then show how to construct pooling designs from a pooling semilattice. Moreover, we give many examples of pooling semi-lattices and thus obtain the corresponding pooling designs.
Keywords. Pooling design, semilattice, pooling semilattice, regular pooling semilattice, pooling space
2010 MSC: 05B30, 06A07
三、緣由與目的
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 out-comes 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 col-umn as a set of row indices intersecting the colcol-umn with a 1-entry, we can talk about the union of sev-eral 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 [15]. An s0-disjunct
ma-trix is called fully se-disjunct if it is neither (s + 1)e
-disjunct nor se+1-disjunct. An se-disjunct matrix is
⌊e/2⌋-error-correcting [5, 11].
Macula [14] proposed a novel way of constructing disjunct matrices by means of the containment rela-tion of subsets in a finite set. D’yachkov et al. [5] discussed the error-correcting capability of Macula’s designs. Ngo and Du [17] constructed a family of dis-junct 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], Jun Guo and Kaishun Wang proposed a new model for pooling designs–the inter-section 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 de-signs 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 that covers 0. The poset X is ranked and has rank
function, if there is a function ℓ from X to the
in-teger 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
lev-els) 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 ele-ment 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 Xk and columns
indexed with XN such 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] ∩ Xd such that y ̸≼ z for all
z∈ T .
Let (X,≼) be a finite poset with with the rank
function ℓ and fibers X0, . . . , XN. We call X a
semi-lattice, 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;
in-deed 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
num-ber |[u, z] ∩ Xr+1| is a constant µ(r, r + 1, t),
where 0 ≤ r < t ≤ N. Moreover, the
func-tion 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 pool-ing semilattice. In addition if X is a lattice, we use lattice to replace the above semilattice. We call X a geometric lattice if X is a finite atomic lattice and satisfies (A2).
四、結果與討論
In this project, we mainly focus on the construc-tion of pooling designs from a pooling semilattice. We first discuss some properties of pooling semilat-tices, then show how to construct pooling designs
from a pooling semilattice. We give many families of examples of pooling semilattices. They fall into three categories: (i) regular pooling semilattices from sets, vector spaces and maps; (ii) non-regular pool-ing semilattices from affine spaces; and (iii) poolpool-ing semilattices from distance-regular graphs. We gener-alize the intersection type incidence construction to pooling lattices and give four families of examples of pooling lattices. All these results will be appeared in an article coauthored with Jun Guo and Kaishun Wang. See 附錄 below for details.
五、附錄
1 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.
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. By Lemma 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 geometric lattice [12, Page 307], [9, Lemma 5.2]. Hence [w, x] is geometric. Theorem 5.4 in [9] tells us that a geometric lattice is a pooling space, which implies that [w, x] is a pooling space. In particular x is the least upper bound of [w, x]∩ Xr+1. This proves the first statement.
Fix u1∈ Xt−1 and u2∈ Xtwith w≺ u1≺ u2, Since u1 (resp. u2) is the least bound of the ℓ(r, r + 1, t− 1)
(resp. ℓ(r, r + 1, t)) elements in [w, u1]∩ Xr+1(resp. [w, u2]∩ Xr+1), we have ℓ(r, r + 1, t− 1) < ℓ(r, r + 1, t)
The following lemma says that the local assumption of (A1) can imply a global property on X.
Lemma 4. Let X be a pooling semilattice and 0≤ r ≤ s ≤ t ≤ N. Then for u ∈ Xrand z∈ Xtwith u≼ z,
the number 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× Xs
with 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) (1.1) 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) by Proposition 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
by Lemma 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 Xd and columns indexed with Xk such 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) is an se-disjunct matrix, where e = µ(0, d, k)−
sµ(0, d, k− 1) − 1.
Proof. (i) Let y0, y1, . . . , ys be any s + 1 distinct columns of M (d, k; N ). Note that ℓ(y0∧ yj)≤ 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 ∈ X1 such that aj ≼ y0 but aj ̸≼ yj for each j ∈ {1, 2, . . . , s}. Since y0 is a common upper bound of
a1, a2, . . . , as, the least upper bound of these elements exists. Suppose that x0 = a1∨ a2∨ · · · ∨ as. Then
x0 ≼ y0 and x0 ̸≼ yj for each j ∈ {1, 2, . . . , s}. By (A2) we have 1 ≤ ℓ(x0)≤ s. By Lemma 4, the size of
Xd∩ [x0, y0] is µ(ℓ(x0), d, k). From Lemma 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, . . . , ys be any s + 1 distinct columns of M (d, k; N ). Note that y0 contains µ(0, d, k) many
µ(0, d, k− 1) elements in Xd. Thus, the number of elements in Xd contained in y0 but not in yj for each
j∈ {1, 2, . . . , s} is at least µ(0, d, k) − sµ(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)−sµ(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 + 1 distinct columns of M (d, k; N ). Note that y0 contains 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)}, by Lemma 4 we may assume that y0∧ y1, . . . , y0∧ ys are s
distinct elements in Xk−1. Then the element y0∧ y1 contains 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. By Lemma 4, each of y0∧y2, . . . , y0∧yscan contain at most µ(0, d, k−1)−µ(0, d, k−2)
elements in Xd not contained in y0∧ y1. Thus, the number of elements in Xdcontained in y0 but not in yj
for 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. By
Lemma 4 the number of elements x ∈ Xk−1 such that u ≼ x ≼ y0 is 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 yj in Xk \ {y0} such that xj = y0 ∧ yj. Then the number of elements in Xd contained in y0
but not in yj for 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)− sµ(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.
2 Examples
In this section we give many examples of pooling semilattices, and give their parameters. By Theorems 6 and 7, 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 q̸= 1.
In the case q = 1, for convenience, we write(n i ) instead of[n i ] 1.
2.1 Regular pooling semilattices from sets, vector spaces and maps
Example 1. ([5, 14] 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, 17] The Projective Geometry). LetFN
q be the N -dimensional vector space over the finite
field Fq and 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 ofFn+N
q . Let X be the collection of all subspaces x ofFn+Nq with 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 ofFn
q with 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 (qN−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) [2DN +1(q)] 2N + 2 quadratic (with rank N )
[N r ] q r∏−1 i=0 (qN−i+1+ 1) [2A2N(r)] 2N + 1 Hermitian (q = r2) [N r ] q r∏−1 i=0 (qN−i+1/2+ 1) [2A 2N−1(r)] 2N Hermitian (q = r2) [N r ] q r∏−1 i=0 (qN−i−1/2+ 1)
Let X be the collection of all totally isotropic subspaces ofFn
q. Ordered by inclusion, X is a regular pooling
semilattice with the rank function ℓ(x) = dim x and the parameters
|Xr| = [ N r ] q r∏−1 i=0 (qN +e−i−1+ 1), µ(r, s, t) = [ t− r s− r ] q .
where e = 1, 1, 0, 2, 3/2, 1/2 according to [CN(q)], [BN(q)], [DN(q)], [2DN +1(q)], [2A2N(r)], [2A2N−1(r)],
re-spectively.
For fixed positive integers n and m, let w be an l-dimensional subspace of Fn+m
q , 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
Example 5. (The Attenuated Classical Polar Space). For fixed positive integers n and m, let Fn q be the
classical polar space with rank N as in Example 4 and w = (0(m,n)I(m)). Then the quotient spaceFn+m q /w
is isomorphic toFn
q. Let X be the collection of all subspaces x = (x1x2) ofFn+mq with x∩ w = {0}, where
x1 is a totally isotropic subspace of Fnq and x2 is 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 in Example 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 ofFN q
and f : w → FN
q is a linear map. Ordered by inclusion, X is a strongly 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
ofFN
q 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
ofFN
q 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. 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 .
Example 12. (The Hermitian 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 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 .
2.2 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). LetFN
q and X be as in Example 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| = q N−r [ N r ] q , µ(r + 1, s + 1, t + 1) = [ t− r s− r ] q .
Example 15. (The Affine Attenuated Space). LetFn+N
q and X be as in Example 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| = q n+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). LetFn
q and X be as in Example 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 + 1 and the parameters
|X′ r+1| = q 2N +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)],
respec-tively, and e is as in Example 4.
Example 17. (The Affine Attenuated Classical Polar Space). Let Fn+m
q and X be as in Example 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 + 1 and the parameters
|X′ r+1| = q 2N +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 .
2.3 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.
Let Γ be a connected regular graph. We identify Γ with the set of vertices. For two vertices u and v, let ∂(u, v) denote the usual distance between u and v. The maximum value of the distance function in Γ is called the diameter of Γ, denoted by D(Γ). For vertices u and v at distance i, define
C(u, v) = Ci(u, v) ={w | ∂(u, w) = i − 1, ∂(w, v) = 1},
A(u, v) = Ai(u, v) ={w | ∂(u, w) = i, ∂(w, v) = 1}.
For the cardinalities of these sets we use lower case letters ci(u, v) and ai(u, v). A connected regular graph
Γ with diameter D is called distance-regular if ci(u, v) and ai(u, v) depend only on i for all 1≤ i ≤ D. The
reader is referred to [2] for general theory of distance-regular graphs.
Let Γ be a distance-regular graph. A r-subset{x1, x2, . . . , xr} ⊆ Γ is said to be a t-clique of Γ with size
r if any two distinct vertices in{x1, x2, . . . , xr} are at distance t.
Example 18. ([1, 20] 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∅. Ordered by inclusion, X is a regular pooling semilattice
with the rank function ℓ(x) =|x| and the parameters |Xr| = ( n rt ) (rt)!/(t!)rr!, µ(r, s, t) = ( t− r s− r ) .
A distance-regular graph Γ with diameter D ≥ 2 is said to be antipodal, if ∂(x, y) = ∂(x, z) = D and y̸= 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 that Γ is an antipodal distance-regular
graph with diameter D. Let N = kD+ 1 and X be the collection of all D-cliques of Γ 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 graph Γ is said to be of order (l, k) if, for each vertex x ∈ Γ, the induced subgraph
on Γ(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 that Γ is a distance-regular graph
of order (l, k). Let N = l + 1 and X be the collection of all cliques of Γ 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 ∆ of Γ is called strongly closed if C(u, v)∪ A(u, v) ⊆ ∆ for every pair of vertices u, v ∈ ∆. A distance-regular graph Γ 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 of Γ is called a subspace of Γ. For any two subspaces ∆1 and ∆2 of Γ, ∆1+ ∆2 denotes the minimum subspace containing ∆1 and ∆2.
Proposition 8. ([6, Lemma 2.1]). Let Γ 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 ∆⊆ ∆′, 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. ([19] The D-Bounded Distance-Regular Graph). Let Γ be a D-bounded distance-regular
graph with D = N . For x∈ Γ, let X be the collection of all subspaces ∆ containing x in Γ. 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 pooling semilattice.
3 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 ∈ Xs with
u≼ x, the number of elements z ∈ Xt such 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 from Lemma 4
with 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 ∈ Xs with u ≼ x. Note that the set u+∩ Xt is partitioned into
Ui ={z | z ∈ u+∩ Xt, ℓ(x∧ z) = i} for r ≤ i ≤ s. Since each element z ∈ Ui has 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) (3.1)
is a constant, where π(i, s, t) = 0 if i > t. The first statement follows. Let x1 ∈ Xs and x2 ∈ Xs+1 with
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 y̸∈ {z ∈ Xt| x2∧ z = u}. The second statement
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 by Lemma 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 Xd and columns indexed with Xk such
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 satisfies 1≤ s ≤ i and N − (s + 1)k ≥ d − i. Then M(i; d, k; N) is an se-disjunct matrix, where
e = 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)− sµ(0, i, k − 1))π(i, (s + 1)k, d) − 1.
Proof. (i) Let y0, y1, . . . , ysbe any s + 1 distinct columns of M (i; d, k; N ). Similar to the proof of Theorem 6
(i), there exists an aj ∈ X1 such that aj ≼ y0 but aj ̸≼ yj for each j ∈ {1, 2, . . . , s}. Suppose a0 =
a1∨a2∨· · ·∨as. By the proof of Theorem 6 (i), the size of Xi∩[a0, y0] is at least µ(s, i, k). Let x0∈ [a0, y0]∩Xi
and x∈ Xdsatisfy x∧(y0∨y1∨· · ·∨ys) = x0. Then x∧y0= x0and x∧yj ≼ x0for each j∈ {1, 2, . . . , s}, which
implies that ℓ(x∧yj) < i by a0̸≼ 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. By Lemma 9, the number of elements x ∈ Xdsatisfying
x∧ (y0∨ y1∨ · · · ∨ ys) = x0 is at least π(i, (s + 1)k, d). Therefore, the number of elements x∈ Xd satisfying
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, . . . , ys be any s + 1 distinct columns of M (i; d, k; N ). By Theorem 6 (ii), the number of
elements in Xi contained in y0 but not in yj for each j∈ {1, 2, . . . , s} is at least µ(0, i, k) − sµ(0, i, k − 1).
Given x0 ∈ Xi with x0 ≼ y0 but x0 ̸≼ yj for 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+1 distinct columns of M (i; d, k; N ). By Theorem 7 the number of elements
of Xi contained in y0but not in yj for each 1≤ j ≤ s is at least µ(0, i, k) − µ(0, i, k − 1) − (s − 1)(µ(0, i, k −
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))
≥ max{2i − d, 0}.
Suppose ℓ(y0∧ yj)≥ max{2i − d, 0} for each j ∈ {1, 2, . . . , s}. By (A3) we have
ℓ(y0∨ y1∨ · · · ∨ ys)
= ℓ(y0∨ y1∨ · · · ∨ ys−1) + ℓ(ys)− ℓ((y0∨ y1∨ · · · ∨ ys−1)∧ ys)
≤ ℓ(y0∨ y1∨ · · · ∨ ys−1) + ℓ(ys)− ℓ(y0∧ ys)
≤ ℓ(y0∨ y1∨ · · · ∨ ys−1) + k− max{2i − d, 0}
≤ ℓ(y0) + s(k− max{2i − d, 0})
= k + s(k− max{2i − d, 0}).
Given x0 ∈ Xi with x0 ≼ y0 but x0 ̸≼ yj for each j ∈ {1, 2, . . . , s}. By the proof of Theorem 11, the
number of elements x ∈ Xd satisfying x∧ y0 = x0 and 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. By Theorems 11 and 12, we can construct pooling designs from these lattices.
Example 22. ([7] The Boolean Algebra). Let X be as in Example 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 in Example 2. Then X is a regular pooling
lattice with the parameters
|Xr| = [ 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| = q N−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 (3.1). In particular, if D(∆1) + D(∆2) = D(∆1+ ∆2) + D(∆1∩ ∆2)
Acknowledgment
This research is supported by NSFC (11271047), NSF of Hebei Province (A2012408003), TPF-2011-11 of Hebei Province, NSF of Hebei Education Department (ZH2012082), the Fundamental Research Funds for the Central University of China and NSC (99-2115-M-009-005-MY3) of Taiwan.
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