Finite Generalized Quadrangles

doi:10.4171/066

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Finite Generalized Quadrangles

Monograph•DOI•

Finite Generalized Quadrangles

08 Apr 2009-

About: The article was published on 2009-04-08. It has received 888 citations till now. The article focuses on the topics: Generalized polygon & Generalized quadrangle.

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Book•

Basic Algebra = 代数学入門

[...]

庸三公庄

01 Jan 2002

TL;DR: In this paper, the value of the variable in each equation is determined by a linear combination of the values of the variables in the equation and the variable's value in the solution.

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Abstract: Determine the value of the variable in each equation.

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635 citations

Journal Article•DOI•

Which graphs are determined by their spectrum

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Edwin van Dam¹, Willem H. Haemers¹•Institutions (1)

Tilburg University¹

01 Nov 2003-Linear Algebra and its Applications

TL;DR: For almost all graphs the answer to the question in the title is still unknown as mentioned in this paper, and the cases for which the answer is known are surveyed in the survey of cases where the Laplacian matrix is known.

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605 citations

Journal Article•DOI•

Interlacing eigenvalues and graphs

[...]

Willem H. Haemers¹•Institutions (1)

Tilburg University¹

01 Sep 1995-Linear Algebra and its Applications

TL;DR: In this paper, bounds are obtained for characteristic numbers of graphs, such as the size of a maximal clique, the chromatic number, the diameter, and the bandwidth, in terms of the eigenvalues of the standard adjacency matrix or the Laplacian matrix.

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395 citations

Book•

Designs and their codes

[...]

Jr. E. F. Assmus¹•Institutions (1)

Lehigh University¹

01 Jan 1992

TL;DR: The standard geometric codes are presented, followed by a list of recommended designs and some examples of how these designs might be implemented in the real world.

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Abstract: Algebraic coding theory has in recent years been increasingly applied to the study of combinatorial designs. This book gives an account of many of those applications together with a thorough general introduction to both design theory and coding theory - developing the relationship between the two areas. The first half of the book contains background material in design theory, including symmetric designs and designs from affine and projective geometry, and in coding theory, coverage of most of the important classes of linear codes. In particular, the authors provide a new treatment of the Reed-Muller and generalized Reed-Muller codes. The last three chapters treat the applications of coding theory to some important classes of designs, namely finite planes, Hadamard designs and Steiner systems, in particular the Witt systems. The book is aimed at mathematicians working in either coding theory or combinatorics - or related areas of algebra. The book is, however, designed to be used by non-specialists and can be used by those graduate students or computer scientists who may be working in these areas.

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374 citations

Journal Article•DOI•

Combinatorial design of key distribution mechanisms for wireless sensor networks

[...]

Seyit Camtepe¹, Bülent Yener¹•Institutions (1)

Rensselaer Polytechnic Institute¹

01 Apr 2007-IEEE ACM Transactions on Networking

TL;DR: Novel deterministic and hybrid approaches based on Combinatorial Design are presented for deciding how many and which keys to assign to each key-chain before the sensor network deployment to obtain efficient key distribution schemes.

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Abstract: Secure communications in wireless sensor networks operating under adversarial conditions require providing pairwise (symmetric) keys to sensor nodes. In large scale deployment scenarios, there is no priory knowledge of post deployment network configuration since nodes may be randomly scattered over a hostile territory. Thus, shared keys must be distributed before deployment to provide each node a key-chain. For large sensor networks it is infeasible to store a unique key for all other nodes in the key-chain of a sensor node. Consequently, for secure communication either two nodes have a key in common in their key-chains and they have a wireless link between them, or there is a path, called key-path, among these two nodes where each pair of neighboring nodes on this path have a key in common. Length of the key-path is the key factor for efficiency of the design. This paper presents novel deterministic and hybrid approaches based on Combinatorial Design for deciding how many and which keys to assign to each key-chain before the sensor network deployment. In particular, Balanced Incomplete Block Designs (BIBD) and Generalized Quadrangles (GQ) are mapped to obtain efficient key distribution schemes. Performance and security properties of the proposed schemes are studied both analytically and computationally. Comparison to related work shows that the combinatorial approach produces better connectivity with smaller key-chain sizes.

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371 citations

Cites background or methods from "Finite Generalized Quadrangles"

...A point is in the projective space with the canonical equation if . Points and are said to be collinear if where for . Let represents the set of points collinear with point , then is a line connecting two distinct and collinear points and [ 22 ]....
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...In this work, we are interested in three known GQs as defined in [ 22 ] and [23]: 1) from projective space ;2 ) from projective space ; and 3) from projective space ....
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References

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Open Access

More filters

Book•

Finite Geometries

[...]

Peter Dembowski

01 Jan 1968

1,788 citations

"Finite Generalized Quadrangles" refers background in this paper

...Now the internal (or residual) [50] structure of the inversive plane π(x∞, x′) at z is an affine plane of order s which is a substructure of Sz and contains the points u− 1, u2, u3....
[...]
...On the other hand, the corresponding spreads of W (q) are the regular spread [50] and the Lüneburg-spread [100] of PG(3, q)....
[...]
...of π(x∞, z) with carriers [50, 58] u and y....
[...]
...Hence no three elements of O∞ are collinear in PG(3, s), if s 6= 2 [50]....
[...]
...Moreover, he showed that the corresponding spread of W (q) gives rise to a Knuth semifield plane [50]....
[...]

Book•

Projective geometries over finite fields

[...]

J. W. P. Hirschfeld

24 Jan 1980

TL;DR: The first properties of the plane can be found in this article, where the authors define the following properties: 1. Finite fields 2. Projective spaces and algebraic varieties 3. Subspaces 4. Partitions 5. Canonical forms for varieties and polarities 6. The line 7. Ovals 9. Arithmetic of arcs of degree two 10. Cubic curves 12. Arcs of higher degree 13. Blocking sets 14. Small planes 15.

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Abstract: 1. Finite fields 2. Projective spaces and algebraic varieties 3. Subspaces 4. Partitions 5. Canonical forms for varieties and polarities 6. The line 7. First properties of the plane 8. Ovals 9. Arithmetic of arcs of degree two 10. Arcs in ovals 11. Cubic curves 12. Arcs of higher degree 13. Blocking sets 14. Small planes Appendix Notation References

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1,593 citations

"Finite Generalized Quadrangles" refers background or methods or result in this paper

...Hence {L0, L1} and {L0, L1} are the reguli of an hyperbolic quadric Q+ of PG(3, q) [80]....
[...]
...a (q + 2)-arc [80], of the projective plane PG(2, q), q = 2h, and let PG(2, q) = H be embedded as a plane in PG(3, q) = P ....
[...]
...Then O′ is an oval with nucleus x [80]....
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...Since each plane of order s, s 6 8, is desarguesian [80], the result follows....
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...,Mq are contained in a three dimensional space P , and moreover {L0, L1} and {L0, L1} are the reguli of an hyperbolic quadric Q+ of P [80]....
[...]

Book•

Basic algebra

[...]

Nathan Jacobson

01 Jan 1974

1,462 citations

Book•

Basic Algebra II

[...]

Nathan Jacobson

01 Sep 1989

1,175 citations

"Finite Generalized Quadrangles" refers background in this paper

...Then by (ii) G is a Frobenius group on xG (cf [87])....
[...]

Book•

Basic Algebra = 代数学入門

[...]

庸三公庄

01 Jan 2002

TL;DR: In this paper, the value of the variable in each equation is determined by a linear combination of the values of the variables in the equation and the variable's value in the solution.

...read moreread less

Abstract: Determine the value of the variable in each equation.

...read moreread less

635 citations