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 Introduction to BIBD 1 Properties of BIBD with Repeated Blocks 2 Minimum Support Size 3 2015 WCA @SJTU Zongchen Chen Zhiyuan College 2/17

Introduction to BIBD Properties of BIBD with Repeated Blocks Minimum Support Size Outline Introduction to BIBD 1 Properties of BIBD with Repeated Blocks 2 Minimum Support Size 3 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. 2015 WCA @SJTU Zongchen Chen Zhiyuan College 4/17

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. 2015 WCA @SJTU Zongchen Chen Zhiyuan College 5/17

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 ) blocks. k − 1 Theorem k = λ ( v 2 − v ) A ( v , k , λ ) -BIBD has exactly b = vr blocks. k 2 − k 2015 WCA @SJTU Zongchen Chen Zhiyuan College 6/17

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 Introduction to BIBD 1 Properties of BIBD with Repeated Blocks 2 Minimum Support Size 3 2015 WCA @SJTU Zongchen Chen Zhiyuan College 8/17

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. 2015 WCA @SJTU Zongchen Chen Zhiyuan College 9/17

Introduction to BIBD Properties of BIBD with Repeated Blocks Minimum Support Size Some Properties Assume block B i is repeated exactly e i times, 1 ≤ i ≤ b ∗ . Theorem (Mann’s Inequality) e i ≤ r k = b v Let λ ij denote the number of points that blocks i and j have in common. Theorem (J.H. van Lint, H.J. Ryser) � r �� r � λ k − r λ ij � 2 � − k − k ≥ , i � = j e i e j r − λ 2015 WCA @SJTU Zongchen Chen Zhiyuan College 10/17

Introduction to BIBD Properties of BIBD with Repeated Blocks Minimum Support Size Outline Introduction to BIBD 1 Properties of BIBD with Repeated Blocks 2 Minimum Support Size 3 2015 WCA @SJTU Zongchen Chen Zhiyuan College 11/17

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 λ 0 = k ( k − 1 ) v − 1 , 0 < λ 0 < k − 1. 2015 WCA @SJTU Zongchen Chen Zhiyuan College 12/17

Introduction to BIBD Properties of BIBD with Repeated Blocks Minimum Support Size Minimum Support Size (1) Theorem If λ 0 < 1 (i.e., v > k 2 − k + 1 ), then b ∗ ≥ v ( v − 1 ) k ( k − 1 ) > v If b ∗ − v = a , then k ≤ a and v ≤ a 2 . An affine plane of order n , i.e. a ( n 2 , 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 λ 0 ≥ 1 and λ 0 / ∈ N , then � v � � ⌈ λ 0 ⌉ ( v − 1 ) � b ∗ ≥ > v k − 1 k 2015 WCA @SJTU Zongchen Chen Zhiyuan College 14/17

Introduction to BIBD Properties of BIBD with Repeated Blocks Minimum Support Size Minimum Support Size (3) Theorem If λ 0 = 1 , then b ∗ = v b ∗ � v + 2 ( k − 1 ) or 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. 2015 WCA @SJTU Zongchen Chen Zhiyuan College 15/17

Introduction to BIBD Properties of BIBD with Repeated Blocks Minimum Support Size λ 0 ≥ 1 and λ ∈ N We conjecture that if λ 0 ≥ 1 and λ ∈ N , then b ∗ = v b ∗ � v + 2 ( k − λ 0 ) or 2015 WCA @SJTU Zongchen Chen Zhiyuan College 16/17

Introduction to BIBD Properties of BIBD with Repeated Blocks Minimum Support Size Thank you! 2015 WCA @SJTU Zongchen Chen Zhiyuan College 17/17