Some Results on Minimum Support Size of ( v , k , ) -BIBD Zongchen - - PowerPoint PPT Presentation

Introduction to BIBD Properties of BIBD with Repeated Blocks Minimum Support Size

Some Results on Minimum Support Size of (v, k, λ)-BIBD

Zongchen Chen

Department of Mathematics, Zhiyuan College Shanghai Jiao Tong University, P.R.China

April 23, 2015

2015 WCA @SJTU Zongchen Chen Zhiyuan College 1/17

Introduction to BIBD Properties of BIBD with Repeated Blocks Minimum Support Size

Outline

1

Introduction to BIBD

2

Properties of BIBD with Repeated Blocks

3

Minimum Support Size

2015 WCA @SJTU Zongchen Chen Zhiyuan College 2/17

Introduction to BIBD Properties of BIBD with Repeated Blocks Minimum Support Size

Outline

1

Introduction to BIBD

2

Properties of BIBD with Repeated Blocks

3

Minimum Support Size

2015 WCA @SJTU Zongchen Chen Zhiyuan College 3/17

Introduction to BIBD Properties of BIBD with Repeated Blocks Minimum Support Size

Definition of BIBD

Definition A design is a pair (X, B) such that the following properties are satisfied: X is a set of elements called points, and B is a collection (i.e., multiset) of nonempty subsets of X called blocks.

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Introduction to BIBD Properties of BIBD with Repeated Blocks Minimum Support Size

Definition of BIBD

Definition Let v, k, and λ be positive integers such that v > k ≥ 2. A (v, k, λ)-balanced incomplete block design (which we abbreviate to (v, k, λ)-BIBD) is a design (X, B) such that the following properties are satisfied: |X| = v, each block contains exactly k points, and every pair of distinct points is contained in exactly λ blocks. A BIBD may possibly contain repeated blocks if λ > 1.

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Introduction to BIBD Properties of BIBD with Repeated Blocks Minimum Support Size

Basic Properties

Theorem In a (v, k, λ)-BIBD, every point occurs in exactly r = λ(v−1)

k−1

blocks. Theorem A (v, k, λ)-BIBD has exactly b = vr

k = λ(v2−v) k2−k

blocks.

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Introduction to BIBD Properties of BIBD with Repeated Blocks Minimum Support Size

Fisher’s Inequality

Theorem (Fisher’s Inequality) In any (v, b, r, k, λ)-BIBD, b ≥ v. If b = v, then it is called a symmetric BIBD (abbreviated to SBIBD).

2015 WCA @SJTU Zongchen Chen Zhiyuan College 7/17

Introduction to BIBD Properties of BIBD with Repeated Blocks Minimum Support Size

Outline

1

Introduction to BIBD

2

Properties of BIBD with Repeated Blocks

3

Minimum Support Size

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Introduction to BIBD Properties of BIBD with Repeated Blocks Minimum Support Size

Support Size of BIBD

The support size b∗ of a (v, k, λ)-BIBD is the number of distinct blocks in B. Theorem In a (v, k, λ)-BIBD, b∗ ≥ v, and b∗ = v if and only if it is some duplicates of an SBIBD.

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Introduction to BIBD Properties of BIBD with Repeated Blocks Minimum Support Size

Some Properties

Assume block Bi is repeated exactly ei times, 1 ≤ i ≤ b∗. Theorem (Mann’s Inequality) ei ≤ r k = b v Let λij denote the number of points that blocks i and j have in common. Theorem (J.H. van Lint, H.J. Ryser) r ei − k r ej − k

• ≥

λk − rλij r − λ 2 , i = j

2015 WCA @SJTU Zongchen Chen Zhiyuan College 10/17

Introduction to BIBD Properties of BIBD with Repeated Blocks Minimum Support Size

Outline

1

Introduction to BIBD

2

Properties of BIBD with Repeated Blocks

3

Minimum Support Size

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Introduction to BIBD Properties of BIBD with Repeated Blocks Minimum Support Size

Assumptions

Assume D is a (v, k, λ)-BIBD with repeated blocks and v > k + 1. b∗ is the support size of D. Let λ0 = k(k−1)

v−1 , 0 < λ0 < k − 1.

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Introduction to BIBD Properties of BIBD with Repeated Blocks Minimum Support Size

Minimum Support Size (1)

Theorem If λ0 < 1 (i.e., v > k2 − k + 1), then b∗ ≥ v(v − 1) k(k − 1) > v If b∗ − v = a, then k ≤ a and v ≤ a2. An affine plane of order n, i.e. a (n2, n, 1)-BIBD, if exists, has the property that b∗ − v = n.

2015 WCA @SJTU Zongchen Chen Zhiyuan College 13/17

Introduction to BIBD Properties of BIBD with Repeated Blocks Minimum Support Size

Minimum Support Size (2)

Theorem If λ0 ≥ 1 and λ0 / ∈ N, then b∗ ≥ ⌈λ0⌉ (v − 1) k − 1 v k

• > v

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Introduction to BIBD Properties of BIBD with Repeated Blocks Minimum Support Size

Minimum Support Size (3)

Theorem If λ0 = 1, then b∗ = v

• r

b∗ v + 2(k − 1) b∗ = v if and only if D is some duplicates of a (v, k, 1)-SBIBD. b∗ = v + 2(k − 1) if and only if D is the union of two adjacent (v, k, 1)-SBIBDs.

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Introduction to BIBD Properties of BIBD with Repeated Blocks Minimum Support Size

λ0 ≥ 1 and λ ∈ N

We conjecture that if λ0 ≥ 1 and λ ∈ N, then b∗ = v

• r

b∗ v + 2(k − λ0)

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Introduction to BIBD Properties of BIBD with Repeated Blocks Minimum Support Size

Thank you!

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