On defining sets of full designs with block size three

Donovan, D., Lefevre, J., Waterhouse, M. and Yazici, E. S. (2009) On defining sets of full designs with block size three. Graphs and Combinatorics, 25 6: 825-839. doi:10.1007/s00373-010-0882-4

Author Donovan, D.
Lefevre, J.
Waterhouse, M.
Yazici, E. S.
Title On defining sets of full designs with block size three
Journal name Graphs and Combinatorics   Check publisher's open access policy
ISSN 0911-0119
Publication date 2009-12
Sub-type Article (original research)
DOI 10.1007/s00373-010-0882-4
Volume 25
Issue 6
Start page 825
End page 839
Total pages 15
Editor Mikio Kano
Place of publication Japan
Publisher Springer Japan KK
Collection year 2010
Language eng
Subject 010104 Combinatorics and Discrete Mathematics (excl. Physical Combinatorics)
970101 Expanding Knowledge in the Mathematical Sciences
Abstract A defining set of a t-(v, k, λ) design is a subcollection of its blocks which is contained in no other t-design with the given parameters, on the same point set. A minimal defining set is a defining set, none of whose proper subcollections is a defining set. The spectrum of minimal defining sets of a design D is the set {|M| | M is a minimal defining set of D}. We show that if a t-(v, k, λ) design D is contained in a design F, then for every minimal defining set d D of D there exists a minimal defining set d F of F such that $${d_D = d_F\cap D}$$. The unique simple design with parameters $${{\left(v,k, {v-2\choose k-2}\right)}}$$ is said to be the full design on v elements; it comprises all possible k-tuples on a v set. Every simple t-(v, k, λ) design is contained in a full design, so studying minimal defining sets of full designs gives valuable information about the minimal defining sets of all t-(v, k, λ) designs. This paper studies the minimal defining sets of full designs when t = 2 and k = 3. Several families of non-isomorphic minimal defining sets of these designs are found. For given v, a lower bound on the size of the smallest and an upper bound on the size of the largest minimal defining set are given. The existence of a continuous section of the spectrum comprising approximately v values is shown, where just two values were known previously.
Keyword Defining sets
Full designs
Q-Index Code C1
Q-Index Status Confirmed Code

Document type: Journal Article
Sub-type: Article (original research)
Collections: School of Mathematics and Physics
2010 Higher Education Research Data Collection
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Citation counts: TR Web of Science Citation Count  Cited 5 times in Thomson Reuters Web of Science Article | Citations
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Created: Sun, 28 Mar 2010, 00:05:11 EST