2009/7/28. 北京師大, Nontrivial Pooling Designs

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Nontrivial Pooling Designs

Chih-wen Weng

(with Huang, Tayuan & Wu, Hsin-Jung)

Department of Applied Mathematics, National Chiao Tung University

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Binary matrix for group testing

1 Let [n] := {1, 2, . . . , n} be a set ofitems containing a subset P ⊆ [n], the set of defecteditem.

2 We want to collect a group {T1, T2, . . . , Tt} of t tests, each test Ti is a subset of [n] for 1 ≤ i ≤ t.

3 We arrange such a group testing design by the following binary matrix M.

4 Let M be the t × n binary matrix defined by M_ij =

1, j ∈ T_i; 0, j 6∈ T_i for 1 ≤ i ≤ t and j ∈ [n].

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Binary matrix for group testing

2 We want to collect a group {T1, T2, . . . , Tt} of t tests, each test T_i is a subset of [n] for 1 ≤ i ≤ t.

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Binary matrix for group testing

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Binary matrix for group testing

1, j ∈ T_i; 0, j 6∈ Ti

for 1 ≤ i ≤ t and j ∈ [n].

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The output of a group testing

1 Let P ∈ F₂ⁿ denote the characteristic vector of P ⊆ [n].

2 The map P → P is a bijection from the power set of [n] to F₂ⁿ.

3 We use P ⊆ P⁰ if P ⊆ P⁰, and similar for using other set notations in vectors.

4 o_M(P) := S

i ∈P

M_i = M ? P, where ? is the matrix product by using Boolean sum to replace addition.

5 o_M : F₂ⁿ→ F₂^t is called the output function of M.

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The output of a group testing

4 o_M(P) := S

i ∈P

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The output of a group testing

4 o_M(P) := S

i ∈P

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The output of a group testing

4 o_M(P) := S

i ∈P

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The output of a group testing

4 o_M(P) := S

i ∈P

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The output of a group testing

4 o_M(P) := S

i ∈P

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Disjunct matrix

1 Note that o_M(P ∪ P⁰) = o_M(P) ∪ o_M(P⁰) for P, P⁰ ⊆ [n].

2 In particular if P ⊆ P⁰⁰ then o_M(P) ⊆ o_M(P⁰⁰).

3 For W ⊆ F₂ⁿ, M is W -disjunctif o_M(P) 6⊆ o_M(P⁰⁰) for any P, P⁰⁰∈ W with P 6⊆ P⁰⁰.

4 In the above definition, it suffices to assume |P| = 1.

5 M is d -disjunctif for any d + 1 distinct columns M_i₀, M_i₁, . . . , M_i_d, we have M_i₀ 6⊆ S^d

j =1

M_i_j

Exercise

Show that a d -disjunct matrix is

[n]

≤ d

-disjunct.

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Disjunct matrix

j =1

M_i_j

Exercise

[n]

≤ d

-disjunct.

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Disjunct matrix

j =1

M_i_j

Exercise

[n]

≤ d

-disjunct.

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Disjunct matrix

j =1

M_i_j

Exercise

[n]

≤ d

-disjunct.

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Disjunct matrix

j =1

M_i_j

Exercise

[n]

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Decidable matrix

1 Note that o_M(P)_i = 0 iff P ∩ T_i = ∅ iff P ⊆ T_i.

2 Hence P ⊆ T

oM(P)i=0

T_i.

3 For W ⊆ F₂ⁿ, M is W -decidableif P = T

oM(P)i=0

T_i for any P ∈ W .

4 A

[n] d

-decidable matrix is called d -decidable.

5 A

[n]

≤ d

Exercise

Show that a W -disjunct matrix is W -decidable. Problem

Find a W -decidable matrix which is not W -disjunct?

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Decidable matrix

2 Hence P ⊆ T

oM(P)i=0

T_i.

oM(P)i=0

4 A

[n] d

5 A

[n]

≤ d

Exercise

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Decidable matrix

2 Hence P ⊆ T

oM(P)i=0

T_i.

oM(P)i=0

4 A

[n] d

5 A

[n]

≤ d

Exercise

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Decidable matrix

2 Hence P ⊆ T

oM(P)i=0

T_i.

oM(P)i=0

4 A

[n]

d

5 A

[n]

≤ d

Exercise

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Decidable matrix

2 Hence P ⊆ T

oM(P)i=0

T_i.

oM(P)i=0

4 A

[n]

d

5 A

[n]

≤ d

Exercise

Show that a W -disjunct matrix is W -decidable.

Problem

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Decidable matrix

2 Hence P ⊆ T

oM(P)i=0

T_i.

oM(P)i=0

4 A

[n]

d

5 A

[n]

≤ d

Exercise

Show that a W -disjunct matrix is W -decidable.

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Separable matrix

1 For W ⊆ F₂ⁿ, M is W -separableif the restricted function oM W of oM to W is injective.

2 If M is W -separable then for each vector u in the output set oM(W ) there exists a unique vector P ∈ W such that o_M(P) = u, i.e. the set P of positive items can be decoded from the output vector u.

3 A

[n]

≤ d

-separable matrix is also calledd -separable.

4 A

[n] d

Exercise

A W -decidable matrix is W -separable for any W ⊆ F₂ⁿ.

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Separable matrix

2 If M is W -separable then for each vector u in the output set o_M(W ) there exists a unique vector P ∈ W such that o_M(P) = u, i.e. the set P of positive items can be decoded from the output vector u.

3 A

[n]

≤ d

4 A

[n] d

Exercise

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Separable matrix

3 A

[n]

≤ d

4 A

[n] d

Exercise

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Separable matrix

3 A

[n]

≤ d

4 A

[n]

d

Exercise

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Separable matrix

3 A

[n]

≤ d

4 A

[n]

d

Exercise

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Relations

'

&

$

% '

&

$

% '

&

$

% W -disjunct

W -decidable W -separable

Find the relation between the above three classes of binary matrices with slightly changing W and possibly adding or deleting a few rows.

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Relations

'

&

$

% '

&

$

% '

&

$

% W -disjunct

W -decidable W -separable

Find the relation between the above three classes of binary matrices with slightly changing W and possibly adding or deleting a few rows.

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A property distinguishes decidable matrix from others

Note that for each t × n binary matrix M there exists aunique maximal W_M ⊆ F₂ⁿ such that M is W_M-decidable, in fact, W_M = {P ∈ F₂ⁿ | P = T

oM(P)i=0

T_i}.

Problem

Study the map M → WM.

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A property distinguishes decidable matrix from others

Note that for each t × n binary matrix M there exists aunique maximal W_M ⊆ F₂ⁿ such that M is W_M-decidable, in fact, W_M = {P ∈ F₂ⁿ | P = T

oM(P)i=0

T_i}.

Problem

Study the map M → W_M.

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1-disjunct matrix

Example

A 1-disjunct matrix to detect the infected item3 from {1, 2, 3, 4, 5, 6} :







Tests/Items | 1 2 3 4 5 6 oM({3})

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







In fact the above 4 × 6 matrix M has W_M =

[6]

≤ 1

∪ {{3, 5, 6}, {2, 4, 6}, {1, 4, 5}, {4, 5, 6}, {1, 2, 3, 4, 5, 6}}.

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1-disjunct matrix

Example

A 1-disjunct matrix to detect the infected item3 from {1, 2, 3, 4, 5, 6} :







Tests/Items | 1 2 3 4 5 6 oM({3})

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







In fact the above 4 × 6 matrix M has W_M =

[6]

≤ 1

∪ {{3, 5, 6}, {2, 4, 6}, {1, 4, 5}, {4, 5, 6}, {1, 2, 3, 4, 5, 6}}.

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n-disjunct matrix

Example

For t = n the t × n identity matrix I is F₂ⁿ-disjunct.

In applying to a group testing, we need the number t of tests is smaller than the number n of items, otherwise we would rather test the items one by one.

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n-disjunct matrix

Example

For t = n the t × n identity matrix I is F₂ⁿ-disjunct.

In applying to a group testing, we need the number t of tests is smaller than the number n of items, otherwise we would rather test the items one by one.

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Nontrivial M

An t × n binary matrix isnontrivial if t < n.

Example

Let q be a prime power. The affine plane F_q² over Fq has q² points and q²+ q lines. Since any line has q points and any two lines intersect at most 1 point, the points of a line can not be covered by the union of other q − 1 lines. Hence the points-lines incidence matrix M is (q − 1)-disjunct matrix, and it is nontrivial since n = q²+ q > q² = t.

Problem

For each positive integer q find a nontrivial (q − 1)-disjunct matrix with t = q².

The first q which is not a prime power is when q = 6.

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Nontrivial M

Example

Let q be a prime power. The affine plane F_q² over Fq has q² points and q²+ q lines.

Since any line has q points and any two lines intersect at most 1 point, the points of a line can not be covered by the union of other q − 1 lines. Hence the points-lines incidence matrix M is (q − 1)-disjunct matrix, and it is nontrivial since n = q²+ q > q² = t.

Problem

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Nontrivial M

Example

Let q be a prime power. The affine plane F_q² over Fq has q² points and q²+ q lines. Since any line has q points and any two lines intersect at most 1 point, the points of a line can not be covered by the union of other q − 1 lines.

Hence the points-lines incidence matrix M is (q − 1)-disjunct matrix, and it is nontrivial since n = q²+ q > q² = t.

Problem

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Nontrivial M

Example

Let q be a prime power. The affine plane F_q² over Fq has q² points and q²+ q lines. Since any line has q points and any two lines intersect at most 1 point, the points of a line can not be covered by the union of other q − 1 lines. Hence the points-lines incidence matrix M is (q − 1)-disjunct matrix,

and it is nontrivial since n = q²+ q > q² = t.

Problem

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Nontrivial M

Example

Problem

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Nontrivial M

Example

Problem

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Nontrivial M

Example

Problem

For each positive integer q find a nontrivial (q − 1)-disjunct matrix

2

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Affine plane and projective plane

1 In general for any positive integer r , prime power or not, we can define affine plane using the language of designs.

2 Aprojective plane of orderr is a 2-(r²+ r + 1, r + 1, 1) design.

3 An affine plane of order r is a 2-(r², r , 1) design.

4 It is known that there is a projective plane of order r if and only if there is an affine plane of order r .

5 The points and lines structure in F_q² gives an affine plane of order q when q is a prime power.

6 The existence of finite projective planes of other orders is an open question.

7 The case r = 6 has been ruled out by Bruck-Ryser-Chowla theorem.

8 The next case r = 10 has been ruled out by massive computer calculations.

9 There is nothing more known, in particular r = 12 is still open.

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Affine plane and projective plane

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Affine plane and projective plane

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Affine plane and projective plane

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Affine plane and projective plane

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Affine plane and projective plane

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Affine plane and projective plane

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Affine plane and projective plane

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Affine plane and projective plane

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Nontrivial 5-disjunct matrix with 36 rows

Since there is no affine plane of order 6, we must find some other way to construct a nontrivial 5-disjunct matrix with 36 rows.

In the following we will give a system to construct nontrivial d -disjunct matrices including the above case.

Note that if there exists a nontrivial d -disjunct matrix with (d + 1)²− 1 rows then EFF’s conjecture is false. See page 29 of the book ”Pooling Designs and nonadaptive group testing” by Ding-Zhu Du and Frank K. Hwang for a description of EFF’s conjecture.

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Nontrivial 5-disjunct matrix with 36 rows

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Nontrivial 5-disjunct matrix with 36 rows

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A 36 × 37 5-disjunct matrix

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Forward difference property

1 Let q be a prime power and m ≥ q be an integer.

2 Let Fq:= {0, a⁰, a¹, . . . , a^p−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 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 property in Zm× F_q if the set

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

2 elements.

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Forward difference property

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

2 elements.

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Forward difference property

2 elements.

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Forward difference property

2 elements.

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The Set

m

T

q

LetmTq ⊆ 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

mTq

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The Set

m

T

q

LetmTq ⊆ 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

mTq

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A preview of 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

∞

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The Set

5

T

5

For q = 5, a = 2,

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

and

D₅_T₅ = { (1, 1), (1, 2), (1, 4), (1, 3) (2, 3), (2, 1), (2, 2) (3, 2), (3, 4) (4, 0) }.

Since |D₅T5| = 10, the set ₅T5 has the forward difference distinct property in Z5× F₅.

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The Set

5

T

5

For q = 5, a = 2,

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

and

D₅_T₅ = { (1, 1), (1, 2), (1, 4), (1, 3) (2, 3), (2, 1), (2, 2) (3, 2), (3, 4) (4, 0) }.

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Difference Property

A subset T ⊆ Zm× F_q is said to have thedifference distinct propertyin Zm× F_q if the set −D_T∪ D_T consists of |T |(|T | − 1) elements.

From the structure of D_mTq we find (0, x ) 6∈ −D_mTq∪ D_m_T_q for any x ∈ Fq. This property will be used later.

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Difference Property

A subset T ⊆ Zm× F_q is said to have thedifference distinct propertyin Zm× F_q if the set −D_T∪ D_T consists of |T |(|T | − 1) elements.

From the structure of D_mTq we find (0, x ) 6∈ −D_mTq∪ D_m_T_q for any x ∈ Fq. This property will be used later.

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Non-example

We have seen

D₅_T₅ = { (1, 1), (1, 2), (1, 4), (1, 3) (2, 3), (2, 1), (2, 2) (3, 2), (3, 4) (4, 0) }.

Hence

−D₅_T₅ = { (4, 4), (4, 3), (4, 1), (4, 2) (3, 2), (3, 4), (3, 3) (2, 3), (2, 1) (1, 0) }.

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

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Non-example

We have seen

D₅_T₅ = { (1, 1), (1, 2), (1, 4), (1, 3) (2, 3), (2, 1), (2, 2) (3, 2), (3, 4) (4, 0) }.

Hence

−D₅_T₅ = { (4, 4), (4, 3), (4, 1), (4, 2) (3, 2), (3, 4), (3, 3) (2, 3), (2, 1)

}.

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Non-example

We have seen

D₅_T₅ = { (1, 1), (1, 2), (1, 4), (1, 3) (2, 3), (2, 1), (2, 2) (3, 2), (3, 4) (4, 0) }.

Hence

−D₅_T₅ = { (4, 4), (4, 3), (4, 1), (4, 2) (3, 2), (3, 4), (3, 3) (2, 3), (2, 1) (1, 0) }.

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Embedding

For positive integers n < m, the set Zn can be viewed as a subset of Zm in the usual way. Hence we have Zn× F_q⊆ Zm× F_q.

In this setting, again

D₆_T₅= D₅_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

−D₆_T₅ = { (5, 4), (5, 3), (5, 1), (5, 2) (4, 2), (4, 4), (4, 3) (3, 3), (3, 1) (2, 0) }.

Since | − D₆_T₅∪ D₆_T₅| = 20 now, the set ₆T₅ has the difference distinct property in Z6× F₅.

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Embedding

For positive integers n < m, the set Zn can be viewed as a subset of Zm in the usual way. Hence we have Zn× F_q⊆ Zm× F_q. In this setting, again

D₆_T₅= D₅_T₅ = { (1, 1), (1, 2), (1, 4), (1, 3) (2, 3), (2, 1), (2, 2) (3, 2), (3, 4) (4, 0) }.

−D₆_T₅ = { (5, 4), (5, 3), (5, 1), (5, 2) (4, 2), (4, 4), (4, 3) (3, 3), (3, 1) (2, 0) }.

Since | − D₆_T₅∪ D₆_T₅| = 20 now, the set ₆T₅ has the difference distinct property in Z6× F₅.

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Embedding

For positive integers n < m, the set Zn can be viewed as a subset of Zm in the usual way. Hence we have Zn× F_q⊆ Zm× F_q. In this setting, again

D₆_T₅= D₅_T₅ = { (1, 1), (1, 2), (1, 4), (1, 3) (2, 3), (2, 1), (2, 2) (3, 2), (3, 4) (4, 0) }.

−D₆_T₅ = { (5, 4), (5, 3), (5, 1), (5, 2) (4, 2), (4, 4), (4, 3) (3, 3), (3, 1)

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Problem

Determine the prime power integer q such that with a suitable choice of a generator a ∈ Fq, the setq+1Tq has the difference distinct property in Zq+1× F_q.

By direct computing by hands, we find the above statement is true for q = 2, 4, 5 and is false for q = 3, 7 (First two primes in 4k + 3 form).

Example Note that

4T₃ = {(0, 1), (1, 2), (2, 1)}, D₄_T₃ = {(1, 1), (1, 2), (2, 0)},