Construct pooling designs from Hermitian forms graphs
Yu-Pei Huang, Chih-wen Weng(Speaker) Department of Applied Mathematics
National Chiao Tung University Hsinchu, Taiwan
June 7, 2007
b-disjunct matrix
Definition 0.1. An n × t matrix M over {0, 1} is b-disjunct if b < t and for any one column j and any other b columns j1, j2, . . . , jd, there exists a row i such that Mij = 1 and Mijs = 0 for s = 1, 2, . . . , b.
Example 0.2. A 2-disjunct matrix M =
1 0 0 0 1 0 0 0 1
.
Relation to Pooling Design
A 4 × 6 1-disjunct matrix to detect the infected item C from {A, B, C, D, E, F } :
Tests/Items A B C D E F Output
One 1 1 1 0 0 0 → 1
Two 1 0 0 1 1 0 → 0
Three 0 1 0 1 0 1 → 0
Four 0 0 1 0 1 1 → 1
Relation to Pooling Design (conti.)
If the size of defected items at most b, then a b-disjunct matrix works for finding the defected items.
Why?
Reason 1. All the subsets of the set of items with size at most b have different outputs.
Reason 2. The tests with 0 outputs determine all the non-infected items.
Reason 3. The infected columns of are exactly those
columns contained in the output vector (view vectors as
Construct b-disjunct matrices
Theorem 0.3. (Macula 1996) Let [m] := {1, 2, . . . , m}.
The incident matrix Wbk of b-subsets and k-subsets of [m]
is an
m b
×
m k
b-disjunct matrix.
The subsets of [m] when m = 4
c
c c c c
c c c c c c
c c c c
(1234)c
1 2 3 4
(12) (13) (14) (23) (24) (34)
(123) (124) (134) (234)
HH HH
HH
JJ J
©©©©©©
©©
©©
©©
JJJ
HHHH HH
QQ QQ
´´´´
´´
´´
PPPPPPPP PP
PP PP
PP QQ QQ
´´´´
³³³³³³³³
³³
³³
³³
³³
PPPPPPPP PP
PP PP
PP
³³³³³³³³
³³
³³
³³
³³
´´
´´
PPPPPPPP PP
PP PP
PP
´´´´
QQ QQ
´´
´´
W
1,2when m = 4
2−subsets
1−subsets (12) (13) (14) (23) (24) (34)
(1) 1 1 1 0 0 0
(2) 1 0 0 1 1 0
(3) 0 1 0 1 0 1
(4) 0 0 1 0 1 1
(b; d)-disjunct matrix
Definition 0.4. An n × t matrix M over {0, 1} is
(b; d)-disjunct if b < t and for any one column j and any other b columns j1, j2, . . . , jb, there exist d rows
i1, i2, . . . , id such that Miuj = 1 and Miujv = 0 for u = 1, 2, . . . , d and v = 1, 2, . . . , b.
A (b; d)-disjunct matrix can be used to construct a pooling design that can find the set of defected item of size at most b with bd−1c errors allowed in the output.
Example of (b; d)-disjunct matrix
Theorem 0.5. (Huang and Weng 2004) Macula’s b-disjunct matrix Wbk is (b − 1; k − b + 1)-disjunct.
Posets
Definition 0.6. A poset P is ranked if there exists a function rank : P → N ∪ {0} such that for all elements x, y ∈ P,
y covers x ⇒ rank(x) − rank(y) = 1.
Let Pi denote the elements of rank i in P . P is atomic if each elements w is the least upper bound of the set
P1 ∩ {y ≤ w|y ∈ P }.
Pooling Spaces
Definition 0.7. (Huang and Weng 2004) A pooling space is a ranked poset P that the for each element
w ∈ P the subposet induced on w+ := {y ≥ w|y ∈ P } is atomic.
A Nonexample of Pooling Spaces
c
c c
c c
@@
@
¡¡¡
©©©©©© HH HH
HH
0
12 13
1234 1235
Every interval in P is atomic, but P is not a pooling space.
b-Disjunct Matrices in Pooling Spaces
Theorem 0.8. (Huang and Weng 2004) Let P be a pooling space. Then the incident matrix Mbk of rank b elements Pb and rank k elements Pk is a b-disjunct
matrix. In fact, Mbk is (b0; db0)-disjunct matrix for some large integer db0 depending on b0 ≤ b and P. (We can reduce the disjunct value b to increase the
error-correctable value d)
Examples of Pooling Spaces
Hamming matroids, the attenuated spaces, quadratic polar spaces, alternating polar spaces, quadratic polar spaces (two types), Hermitian polar spaces (two types). These are called quantum matroids. More generally, projective and affine
geometries, contraction lattices of a graph are also pooling spaces. All these are called geometric lattices.
Hermitian Forms Graphs
Let q denote a prime power, and let H denote the set of D × D Hermitian matrices over the field GF (q2). The
Hermitian forms graph Herq(D) = (X, R) is the graph with vertex set X = H and vertices x, y ∈ R iff rk(x − y) = 1 for x, y ∈ X.
Properties
It is well known that Herq(D) is distance-regular with diameter D and intersection numbers
ci = qi−1(qi − (−1)i) q + 1 , bi = q2D − q2i
q + 1 for 0 ≤ i ≤ D. Note that
|X| = |H| = qD2.
Many Hermitian Forms Graphs
Let Γ = (X, R) be the Hermitian forms graph Herq(D).
Then for two vertices x, y ∈ X with distance t, there exists a subgraph ∆(x, y) such that ∆(x, y) is isomorphic to the Hermitian forms graph Herq(t).
The Poset P
Fix a Hermitian forms graph Γ = Herq(D), and let
P = P (Herq(D)) denote the poset consisting of ∆(x, y) for any x, y ∈ X, and ∆ ≤ ∆0 in P iff ∆ ⊇ ∆0 for ∆, ∆0 ∈ P.
Note that ∆ is isomorphic to Herq(t) iff ∆ had rank D − t for ∆ ∈ P.
The Binary Matrix M
q(D, k, r)
Let Pr and Pk denote the rank r elements and rank k elements of P = P (Herq(D)).
Let M = Mq(D, k, r) denote the incidence matrix of Pr and Pk, i.e. M is a binary matrix with rows and columns indexed by Pr and Pk respectively such that
MΩ∆ =
0, if Ω 6≤ ∆, (i.e. ∆ 6⊆ Ω);
Main Result
Suppose k − r ≥ 2 and set p := q2(q2k−2 − 1)
q2k−2r − 1 + 1. Then Mq(D, k, r) is (b; d)-disjunct of size
D r
q2
qr(2D−r) ×
D k
q2
qk(2D−k),
for any 1 ≤ b < p and d = q2k−2r
k − 1 r − 1
− (b − 1)q2k−2r−2
k − 2 r − 1
.
A Special Case
Suppose D ≥ k ≥ 2. Then Mq(D, k, 1) is (q2; q2k−4)-disjunct matrix of size
D 1
q2
q(2D−1) ×
D k
q2
qk(2D−k).
The Transpose of M
q(D, k, r)
Suppose k − r ≥ 2. Then the transpose of Mq(D, k, r) is (b; d)-disjunct of size
D k
q2
qk(2D−k) ×
D r
q2
qr(2D−r),
where b, d are defined in the next page.
The Transpose of M
q(D, k, r) (conti.)
b is any positive integer such that
d = q(k−r)(2D−k−r)
D − r k − r
q2
−bq
(k−r−1)(2D−k−r−1)
D − r − 1 k − r − 1
q2
Another Special Case
The transpose of Mq(D, D, D − 1) is (q − s; s)-disjunct of size
qD2 ×
D 1
q2
qD2−1, where 1 ≤ s ≤ q − 1.
