Extending Large Sets oft-Designs

Ajoodani-Namini, S.

doi:10.1006/jcta.1996.0093

Cited by 27 publications

(83 citation statements)

References 14 publications

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“…However, the existence of t-designs without repeated blocks is a much more difficult question: none with t > 5 were known until the 1980s, and their existence for all t was shown by Teirlinck [96] in 1987. A different proof was found by Ajoodani-Namini [1] in 1996. Ajoodani-Namini's proof is very much simpler, but Teirlinck's gives designs where the value of λ (though large) is a fixed function of t, and does not grow with v. We refer to the paper by Khosrovshahi and Tayfeh-Reziae in this volume [64] for more information on t-designs.…”

Section: Combinatorial Designmentioning

confidence: 75%

“…But even the partition does not uniquely determine the block design. In our running example, two edges joining the vertices 1 and 2 form a weighted clique with weights (1, 2) or (2, 1), and so can arise from either the block [1,1,2] However, in the special case k = 2, then any graph satisfies the conditions: the blocks are the edges of the graph. (Our assumption that there is no block containing only a single treatment guarantees that every block is an edge of the graph rather than a "loop".…”

Section: Proposition 27mentioning

confidence: 99%

“…We have seen that, if v ≡ 2 mod 3, the best we can do is to take a block [1,1,2], and to choose the remaining (v + 1)(v − 2)/3 blocks to have all points distinct and to cover the pairs other than {1, 2} twice. The following DESIGN code produces a list of all such designs (up to isomorphism).…”

Section: The Design Packagementioning

confidence: 99%

“…To the output of the program, we can add either [1,1,2] or [1, 2, 2] to form a VB design. The resulting designs will be isomorphic if and only if the binary part has an automorphism interchanging 1 and 2.…”

Section: The Design Packagementioning

confidence: 99%

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Combinatorics of optimal designs

Bailey¹,

Cameron²

2009

Surveys in Combinatorics 2009

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To a combinatorialist, a design is usually a 2-design or balanced incompleteblock design. However, 2-designs do not necessarily exist in all cases where a statistician might wish to use one to design an experiment. As a result, statisticians need to consider structures much more general than the combinatorialist's designs, and to decide which one is "best" in a given situation. This leads to the theory of optimal design. There are several concepts of optimality, and no general consensus about which one to use in any particular situation.For block designs with fixed block size k, all these optimality criteria are determined by a graph, the concurrence graph of the design, and more specifically, by the eigenvalues of the Laplacian matrix of the graph. It turns out that the optimality criteria most used by statisticians correspond to properties of this graph which are interesting in other contexts: D-optimality involves maximizing the number of spanning trees; A-optimality involves minimizing the sum of resistances between all pairs of terminals (when the graph is regarded as an electrical circuit, with each edge being a one-ohm resistor); and E-optimality involves maximizing the smallest eigenvalue of the Laplacian (the corresponding graphs are likely to have good expansion and random walk properties). If you are familiar with these properties, you may expect that related "nice" properties such as regularity and large girth (or even symmetry) may tend to hold; some of our examples may come as a surprise! The aim of this paper is to point out that the optimal design point of view unifies various topics in graph theory and design theory, and suggests some interesting open problems to which combinatorialists of all kinds might turn their expertise. We describe in some detail both the statistical background and the mathematics of various topics such as Laplace eigenvalues of graphs. PreliminariesThis first section of the paper sets the scene. We look briefly at the different ways in which a combinatorialist and a statistician view a simple block design such as the Fano plane, and also introduce a running example which would not be recognised as a design under the standard combinatorial definition but is in fact the best design for a statistician in certain circumstances.Section 2 looks at the way that information about treatment differences is recovered from the results of an experiment conducted using a block design. We look briefly at what makes a good design, and show that all criteria for this can be expressed in terms of the concurrence graph of the design. Having thus focussed our attention on graphs, we give a brief survey of the Laplacian matrix of a graph and its eigenvalues, and mention the interesting question of which graphs can be concurrence graphs of block designs with given block size.Section 3 covers the three most important kinds of optimality, described by the letters A (average), D (determinant) and E (extreme), and how they look when expressed in terms of the concurrence matrix.In section ...

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Section: Combinatorial Designmentioning

confidence: 75%

Section: Proposition 27mentioning

confidence: 99%

Section: The Design Packagementioning

confidence: 99%

Section: The Design Packagementioning

confidence: 99%

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Combinatorics of optimal designs

Bailey¹,

Cameron²

2009

Surveys in Combinatorics 2009

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“…More precisely, an infinite series of halvings LS 3 [2](2, k, v) and LS 5 [2](2, k, v) with integers v ≥ 6, v ≡ 2 (mod 4) and 3 ≤ k ≤ v − 3, k ≡ 3 (mod 4) will be given. 1 The first step is the construction of the smallest members of both series (LS 3 [2](2, 3, 6) and LS 5 [2] (2,3,6)). In the first case, this large set is already known [8].…”

Section: 2mentioning

confidence: 99%

Large sets of subspace designs

Braun

Kiermaier

Kohnert³

et al. 2017

Journal of Combinatorial Theory, Series A

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Abstract. In this article, three types of joins are introduced for subspaces of a vector space. Decompositions of the Graßmannian into joins are discussed. This framework admits a generalization of large set recursion methods for block designs to subspace designs.We construct a 2-(6, 3, 78) 5 design by computer, which corresponds to a halving LS 5 [2](2, 3, 6). The application of the new recursion method to this halving and an already known LS 3 [2](2, 3, 6) yields two infinite two-parameter series of halvings LS 3 [2](2, k, v) and LS 5 [2](2, k, v) with integers v ≥ 6, v ≡ 2 (mod 4) and 3Thus in particular, two new infinite series of nontrivial subspace designs with t = 2 are constructed. Furthermore as a corollary, we get the existence of infinitely many nontrivial large sets of subspace designs with t = 2.

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