The Nonexistence of Certain Finite Projective Planes

Bruck, R. H.; Ryser, H. J.

doi:10.4153/cjm-1949-009-2

Cited by 304 publications

(213 citation statements)

References 7 publications

Supporting

Mentioning

212

Contrasting

Order By: Relevance

“…A special case of this occurs for the values v = 15, b = 21, k = 5. (If a 2-design with these parameters existed, then by the Hall-Connor theorem [50] it would be the block residual of a 2-(22, 7, 2) design; but such a design does not exist, by the Bruck-Ryser-Chowla theorem [24,36].) This parameter set has been examined by Reck and Morgan [85,86], and the A-, D-and E-optimal designs determined.…”

Section: General Resultsmentioning

confidence: 99%

Combinatorics of optimal designs

Bailey¹,

Cameron²

2009

Surveys in Combinatorics 2009

View full text Add to dashboard Cite

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 ...

show abstract

Section: General Resultsmentioning

confidence: 99%

Combinatorics of optimal designs

Bailey¹,

Cameron²

2009

Surveys in Combinatorics 2009

View full text Add to dashboard Cite

show abstract

“…If N − 1 or N − 2 is divisible by four, and if N is not the sum of the squares of two integers, then M(N) is strictly less than N + 1. [19] 5. M(10) is strictly less than 11.…”

Section: Mutually Orthogonal Latin Squaresmentioning

confidence: 99%

Quantum Measurements and Finite Geometry

2006

View full text Add to dashboard Cite

A complete set of mutually unbiased bases for a Hilbert space of dimension N is analogous in some respects to a certain finite geometric structure, namely, an affine plane. Another kind of quantum measurement, known as a symmetric informationally complete positive-operator-valued measure, is, remarkably, also analogous to an affine plane, but with the roles of points and lines interchanged. In this paper I present these analogies and ask whether they shed any light on the existence or non-existence of such symmetric quantum measurements for a general quantum system with a finite-dimensional state space.

show abstract

“…Any two heptads have a unique point in common and each of the 28 points off the quadric is contained in two heptads. The existence of the Conwell heptads allows an 8-set description of much of the structure of PG (3,2). The lines of PG(3, 2) can be identified with the partitions of {1, 2, 3, .…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…Bruck conjectured that, in some instances, it might be possible to construct a projective plane of order q(q + 1) by extending the point-line geometry of PG(3, q) [1,2]. Of course, some cases can be immediately ruled out by the Bruck-Ryser theorem [3], in particular the case q = 2. In these cases, it is interesting to ask whether a near projective plane extension exists.…”

Section: Introductionmentioning

confidence: 99%