2010/7/12., 慶州, Pooling designs with d-disjunct property and block weight d + 1,

(1)

ALGEBRAIC AND GEOMETRIC COMBINATORICS CONFERENCE 2010

Pooling designs with d -disjunct property and block weight d + 1

Chih-wen Weng (

翁志文

₎

(with Yu-pei Huang and Wu, Hsin-Jung)

Department of Applied Mathematics, National Chiao Tung University, Taiwan

July 12, 2010

(2)

Definition

An incidence structure (P, B) is calledd -disjunctif any block in B is not covered by the union of d other blocks.

Assume P = {1, 2, . . . , v }, B = {B1, B2, . . . , Bb} and M is be the incidence matrix of (P, B), i.e.

M_ij =

1, i ∈ B_j; 0, i 6∈ Bj

for 1 ≤ i ≤ v and 0 ≤ j ≤ b.

The incidence matrix M of a d -disjunct incidence structure can be used in non-adaptive group testing programming, in which v << b is preferred.

(3)

Definition

M_ij =

1, i ∈ B_j; 0, i 6∈ Bj

for 1 ≤ i ≤ v and 0 ≤ j ≤ b.

(4)

Definition

M_ij =

1, i ∈ B_j; 0, i 6∈ Bj

for 1 ≤ i ≤ v and 0 ≤ j ≤ b.

(5)

1 Let M be a v × b incidence matrix of an incidence structure and set F2 = {0, 1}. Define theoutput function o_M : F₂^b → F₂^v by

o_M(P) := M ? P = [

P_i=1

M_i,

where ? is the matrix product by using Boolean sum to replace addition.

2 If the incidence structure is d -disjunct, then o_M F₂^b(≤ d ) is known to be injective, where F₂^b(≤ d ) is the set of binary vectors of length b and Hamming weight at most d .

3 This means that for each element u in the image of oM on F₂^b(≤ d ), we know which P ∈ F₂^b to have o_M(P) = u.

4 In application, P is interpreted as the unknown infected subset {j | P_j = 1} of a given set of b items, and u is interpreted as the sequence of test results. Then the injective property of o_M implies that the infected subset can be determined from the sequence of test results if the number of infected items is known in advance to be at most d .

(6)

o_M(P) := M ? P = [

P_i=1

M_i,

(7)

o_M(P) := M ? P = [

P_i=1

M_i,

(8)

o_M(P) := M ? P = [

P_i=1

M_i,

(9)

Example

The following 4 × 6 binary matrix is used to detect the infected item in {1, 2, 3, 4, 5, 6}, if the infected item is known to be at most one in advance (but do not know which one):







Tests/Items | 1 2 3 4 5 6 o_M((0, 0, 1, 0, 0, 0)^T)

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







If there are two infected items, the above 4 × 6 matrix does not work for detecting them. For example, both the infected sets {3, 4} and {1, 6} have the same output (1, 1, 1, 1)^T. So it is impossible to recover the infected set from the output (1, 1, 1, 1)^T.

(10)

Example

The following 4 × 6 binary matrix is used to detect the infected item in {1, 2, 3, 4, 5, 6}, if the infected item is known to be at most one in advance (but do not know which one):







Tests/Items | 1 2 3 4 5 6 o_M((0, 0, 1, 0, 0, 0)^T)

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







If there are two infected items, the above 4 × 6 matrix does not work for detecting them. For example, both the infected sets {3, 4} and {1, 6} have the same output (1, 1, 1, 1)^T. So it is impossible to recover the infected set from the output (1, 1, 1, 1)^T.

(11)

Relation to t-design

Definition

An incidence structure (P, B) is called a t-(v , k, λ) designif

1 |P| = v ,

2 |B| = k for and B ∈ B, and

3 any t-subset of P is contained in exactly λ blocks in B. Remark

1 A 2-(v , k, 1) design is (k − 1)-disjunct since a block has k points and it intersects another block in at most one point, so k − 1 other blocks can cover at most k − 1 points of a block, leaving at least one point uncovered.

2 If any point is incidence in at least two blocks, then any block in a d -disjunct matrix has size at least d + 1.

3 A d -disjunct incidence structure is called apooling design.

(12)

Relation to t-design

Definition

An incidence structure (P, B) is called a t-(v , k, λ) designif

1 |P| = v ,

2 |B| = k for and B ∈ B, and

3 any t-subset of P is contained in exactly λ blocks in B.

Remark

1 A 2-(v , k, 1) design is (k − 1)-disjunct since a block has k points and it intersects another block in at most one point, so k − 1 other blocks can cover at most k − 1 points of a block, leaving at least one point uncovered.

2 If any point is incidence in at least two blocks, then any block in a d -disjunct matrix has size at least d + 1.

3 A d -disjunct incidence structure is called apooling design.

(13)

First result

Theorem

Let (P, B) be a d -disjunct pooling design with constant block size d + 1, and define v = |P| and b = |B|. Then b ≤ max{v (v − 1)/d (d + 1), v − d }. Moreover if v − d ≤ v (v − 1)/d (d + 1), then the above upper bound of b is reached if and only if (P, B) is a 2-(v , d + 1, 1) design.

The v × b incidence matrix

M =

I_b Jd

satisfies the equality b = v − d , where I_b is the b × b identity matrix and J_d is the d × d all 1’s matrix.

(14)

First result

Theorem

Let (P, B) be a d -disjunct pooling design with constant block size d + 1, and define v = |P| and b = |B|. Then b ≤ max{v (v − 1)/d (d + 1), v − d }.

Moreover if v − d ≤ v (v − 1)/d (d + 1), then the above upper bound of b is reached if and only if (P, B) is a 2-(v , d + 1, 1) design.

M =

I_b Jd

(15)

First result

Theorem

Let (P, B) be a d -disjunct pooling design with constant block size d + 1, and define v = |P| and b = |B|. Then b ≤ max{v (v − 1)/d (d + 1), v − d }.

Moreover if v − d ≤ v (v − 1)/d (d + 1), then the above upper bound of b is reached if and only if (P, B) is a 2-(v , d + 1, 1) design.

M =

I_b Jd

(16)

The following example gives the equality in previous theorem for d = q − 1.

Example

(2 − (q², q, 1) design)Let q be a prime power. The affine plane F_q² over Fq has q² points and q²+ q lines. Of course any line has q points and any two lines intersect at at most 1 point. Hence the points-lines incidence matrix is v × b d -disjunct with with constant weight w , where v = q², b = q²+ q and w = q = d + 1 satisfy

b = q²+ q = v (v − 1)/d (d + 1).

The first q which is not a prime power is when q = 6 = d + 1. In this case the equality does not hold by the Bruck-Ryser-Chowla Theorem. Then there is no 5-disjunct pooling design with 36 points, 42 blocks and constant bock size 6. We will construct a 5-disjunct pooling design with 36 points, 37 blocks and constant block size 6.

(17)

The following example gives the equality in previous theorem for d = q − 1.

Example

(2 − (q², q, 1) design)Let q be a prime power. The affine plane F_q² over Fq has q² points and q²+ q lines. Of course any line has q points and any two lines intersect at at most 1 point. Hence the points-lines incidence matrix is v × b d -disjunct with with constant weight w , where v = q², b = q²+ q and w = q = d + 1 satisfy

b = q²+ q = v (v − 1)/d (d + 1).

The first q which is not a prime power is when q = 6 = d + 1. In this case the equality does not hold by the Bruck-Ryser-Chowla Theorem. Then there is no 5-disjunct pooling design with 36 points, 42 blocks and constant bock size 6. We will construct a 5-disjunct pooling design with

(18)

Forward difference property

1 Let q be a prime power andm ≥ q be an integer.

2 Let Fq:= {0, a⁰, a¹, . . . , a^q−2} denote the finite field of q elements, where a is a generator of the cyclic multiplication group

F_q^∗:= F_q− {0}.

3 Let m ≥ q be an integer. Let Zm := {0, 1, . . . , m − 1} be the addition group of integers modulo m. We use the order of integers to order the elements in Zm, e.g. 0 < 1.

4 A subset T ⊆ Zm× F_q is said to have the forward difference distinct propertyin Zm× F_q if the forward difference set

FDT := {(j , y ) − (i , x ) | (i , x ), (j , y ) ∈ T with i < j } consists of |T |(|T |−1)

2 elements.

(19)

Forward difference property

F_q^∗:= F_q− {0}.

2 elements.

(20)

Forward difference property

F_q^∗:= F_q− {0}.

2 elements.

(21)

Forward difference property

F_q^∗:= F_q− {0}.

2 elements.

(22)

The Set

m

T

q

Let _mT_q⊆ Zm× F_q be defined by

mTq = {(i , aⁱ) | i ∈ Zm, 0 ≤ i ≤ q − 1}.

` 0

` 1

`

2 · · ·

` `

q − 1

` m − 1

· · · ...

a⁰ ` a¹ ` a² ` a^q−2 `

s s

s

mT_q

(23)

The Set

5

T

5

For q = 5, a = 2,

5T5 = {(0, 1), (1, 2), (2, 4), (3, 3), (4, 1)}

and

FD₅_T₅ = { (1, 1), (1, 2), (1, 4), (1, 3) (2, 3), (2, 1), (2, 2) (3, 2), (3, 4) (4, 0) }.

Since |FD₅_T₅| = 10, the set ₅T₅ has the forward difference distinct property in Z5× F₅.

(24)

The Set

5

T

5

For q = 5, a = 2,

5T5 = {(0, 1), (1, 2), (2, 4), (3, 3), (4, 1)}

and

FD₅_T₅ = { (1, 1), (1, 2), (1, 4), (1, 3) (2, 3), (2, 1), (2, 2) (3, 2), (3, 4) (4, 0) }.

Since |FD₅_T₅| = 10, the set ₅T₅ has the forward difference distinct property in Z5× F₅.

(25)

m

T

q

has the forward difference distinct property

Lemma

The set _mT_q has the forward difference distinct property in Zm× T_q. Proof.

Given any pair (c, d ) ∈ Zm× F_q, solve the equations (c, d ) = (j , a^j) − (i , aⁱ)

for 0 ≤ i < j ≤ q − 1. Note that 1 ≤ c ≤ q − 1 to have a solution. If c = q − 1 then j = q − 1 and i = 0. If c 6= q − 1 then

aⁱ = d /(a^{j −i} − 1) = d /(a^c− 1) and j = c + i . In each case the pair (i , aⁱ), (j , a^j) is unique determined by the element (c, d ) ∈ Zm× F_q.

(26)

Difference Property

A subset T ⊆ Zm× F_q is said to have the difference distinct propertyin Zm× F_q if the difference setD_T := −FD_T∪ FD_T consists of |T |(|T | − 1) elements.

Since mTq intersects a vertical line in at most one point, we find (0, x ) 6∈ D_m_T_q for any x ∈ F_q.

(27)

Difference Property

A subset T ⊆ Zm× F_q is said to have the difference distinct propertyin Zm× F_q if the difference setD_T := −FD_T∪ FD_T consists of |T |(|T | − 1) elements.

Since mTq intersects a vertical line in at most one point, we find (0, x ) 6∈ D_m_T_q for any x ∈ F_q.

(28)

Non-example (m = q = 5)

We have seen

FD₅_T₅ = { (1, 1), (1, 2), (1, 4), (1, 3) (2, 3), (2, 1), (2, 2) (3, 2), (3, 4) (4, 0) }.

Hence

−FD₅_T₅ = { (4, 4), (4, 3), (4, 1), (4, 2) (3, 2), (3, 4), (3, 3) (2, 3), (2, 1) (1, 0) }.

Since |D₅_T₅| = 16 6= 20, the set ₅T₅ does not have the difference distinct property in Z5× F₅.

(29)

Example (m − 1 = q = 5)

FD₆_T₅ = { (1, 1), (1, 2), (1, 4), (1, 3) (2, 3), (2, 1), (2, 2) (3, 2), (3, 4) (4, 0) }.

Hence considering as the negative in Z6× F₅, we have

−FD₆_T₅ = { (5, 4), (5, 3), (5, 1), (5, 2) (4, 2), (4, 4), (4, 3) (3, 3), (3, 1) (2, 0) }.

Since |D | = 20 now, the set T has the difference distinct property in

(30)

2q−1

T

q

has the difference distinct property

Lemma

For m ≥ 2q − 1, the set _mT_q has the difference distinct property in Zm× T_q.

Proof.

We have |FD_mTq| = | − FD_m_T_q| = q(q − 1)/2. The first coordinate of an element in FD_2q−1_T_q runs from 1 to q − 1, and the first coordinate of an element in −FD_2q−1_T_q from m + 1 − q to m − 1. The assumption m ≥ 2q − 1 implies −FD_2q−1Tq ∩ FD_2q−1_T_q = ∅.

(31)

2q−3

T

q

has the difference distinct property

Lemma

The set _mT_q has the difference distinct property for m = 2q − 3.

Proof.

We have |FD_T_m,q| = | − FD_T_m,q| = q(q − 1)/2. Let (c, d ) ∈ FD_T_m,q. If m = 2q − 3, then 1 ≤ c ≤ q − 1 and q − 2 ≤ −c ≤ 2q − 4. Thus the repetition of differences occurs only when c = q − 2 or c = q − 1. Note that d = 0 iff c = q − 1, and −d = 0 iff −c = q − 2. For c = q − 2, suppose (c⁰, d⁰) ∈ −FD_mTq and (c⁰, d⁰) = (c, d ). Then we have c⁰= q − 2 and d⁰= 0. Hence d = 0, a contradiction. Similarly for c = q − 1, we have d = 0 but (q − 1, 0) /∈ −FD_T_m,q.

(32)

2q−4

T

q

has the difference distinct property

Lemma

The set _mT_q has the difference distinct property for m = 2q − 4.

Proof.

Let (c, d ) ∈ FD_T_m,q. Since m = 2q − 4, we have 1 ≤ c ≤ q − 1 and q − 3 ≤ −c ≤ 2q − 5. Thus the repetition of differences occurs only when c = q − 3, q − 2 or q − 1. Note that d = 0 iff c = q − 1, and −d = 0 iff

−c = q − 3. For c = q − 1 or c = q − 3, similar process as the above m = 2q − 3 case can be applied to get contradictions. For c = q − 2,

−c = q − 2. Thus a repetition implies that there are

(q − 2, d₁), (q − 2, d₂) ∈ FD_T_m,q such that d₁ = −d₂. Note that the only two elements of FDTm,q with the first coordinate q − 2 are (q − 2, a^q−2− 1) and (q − 2, a^q−1− a), where a is the generator chosen for F_q^∗. So we have a^q−2− 1 = −(a^q−1− a) and this implies a = −1, also a contradiction.

(33)

Lines with any two intersecting in at most a point

Proposition

Suppose that mTq ⊆ Zm× F_q has the difference distinct property in Zm× F_q. Set B = {u +_mT_q | u ∈ Zm× F_q}. Then |L ∩ L⁰| ≤ 1 for any distinct L, L⁰ ∈ B.

Proof.

Routine.

1 Note that there are mq lines and mq points in Zm× F_q, and a line has q = |T | points with q different first coordinates.

2 Apparently more lines can be added to B still having the conclusion of the above proposition, for example, adding vertical lines to B.

3 We will add m more points to P, add m + 1 lines to B, and add one more point to each original line in B.

(34)

Lines with any two intersecting in at most a point

Proposition

Proof.

Routine.

(35)

Lines with any two intersecting in at most a point

Proposition

Proof.

Routine.

(36)

Lines with any two intersecting in at most a point

Proposition

Proof.

Routine.

(37)

A picture for the finial result

` 0

` 1

`

2 · · ·

` `

q − 1

` q

` m − 1

· · · ...

a⁰ ` a¹ ` a² ` a^q−2 `

s

s s

s

s s

s

s s

s

s s s s

s

∞

Lines in Z_m× (F_q∪ {∞})

(38)

Second and final result

Theorem

There exists a q-disjunct pooling design (P, B) with |P| = m(q + 1),

|B| = m(q + 1) + 1 and constant block weight q + 1, where q is a prime power, and m is an integer at least three satisfying m = 2q − 4,

m = 2q − 3 or m ≥ 2q − 1.

By choosing q = 5 and m = 2q − 4 = 6, there exists a 5-disjunct pooling design with 36 points, 37 blocks and constant block size 6.

(39)

Second and final result

Theorem

There exists a q-disjunct pooling design (P, B) with |P| = m(q + 1),

|B| = m(q + 1) + 1 and constant block weight q + 1, where q is a prime power, and m is an integer at least three satisfying m = 2q − 4,

m = 2q − 3 or m ≥ 2q − 1.

By choosing q = 5 and m = 2q − 4 = 6, there exists a 5-disjunct pooling design with 36 points, 37 blocks and constant block size 6.

(40)

The end

Thank you for your attention.