Introduction via the Fano Plane

1998 ◽  
pp. 3-17
Author(s):  
Burkard Polster
Keyword(s):  
Fano Plane

scholarly journals Turán’s Theorem for the Fano Plane

COMBINATORICA ◽  
2019 ◽  
Vol 39 (5) ◽  
pp. 961-982 ◽  
Author(s):  
Louis Bellmann ◽  
Christian Reiher
Keyword(s):  
Fano Plane ◽  
Turan's Theorem ◽  
Turán’S Theorem

The Turán Number Of The Fano Plane

COMBINATORICA ◽  
2005 ◽  
Vol 25 (5) ◽  
pp. 561-574 ◽  
Author(s):  
Peter Keevash ◽  
Benny Sudakov*
Keyword(s):  
Fano Plane ◽  
Turán Number

scholarly journals Residual $q$-Fano Planes and Related Structures

10.37236/7107 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Tuvi Etzion ◽  
Niv Hooker
Keyword(s):  
Block Design ◽  
Infinite Family ◽  
Steiner System ◽  
Interesting Case ◽  
Common Belief ◽  
Steiner Systems ◽  
Fano Plane ◽  
Open Problems ◽  
One Step ◽  
Existence Question

One of the most intriguing problems for $q$-analogs of designs, is the existence question of an infinite family of $q$-Steiner systems that are not spreads. In particular the most interesting case is the existence question for the $q$-analog of the Fano plane, known also as the $q$-Fano plane. These questions are in the front line of open problems in block design. There was a common belief and a conjecture that such structures do not exist. Only recently, $q$-Steiner systems were found for one set of parameters. In this paper, a definition for the $q$-analog of the residual design is presented. This new definition is different from previous known definition, but its properties reflect better the $q$-analog properties. The existence of a design with the parameters of the residual $q$-Steiner system in general and the residual $q$-Fano plane in particular are examined. We construct different residual $q$-Fano planes for all $q$, where $q$ is a prime power. The constructed structure is just one step from a construction of a $q$-Fano plane.

Rainbow Erdös--Rothschild Problem for the Fano Plane

2021 ◽  
Vol 35 (3) ◽  
pp. 1927-1951
Author(s):  
Lucas de Oliveira Contiero ◽  
Carlos Hoppen ◽  
Hanno Lefmann ◽  
Knut Odermann
Keyword(s):  
Fano Plane

scholarly journals FINITE GEOMETRIES WITH QUBIT OPERATORS

2009 ◽  
Vol 12 (02) ◽  
pp. 359-366 ◽  
Author(s):  
AMBAR N. SENGUPTA
Keyword(s):  
Clifford Algebras ◽  
Quantum Gate ◽  
Finite Geometries ◽  
Fano Plane ◽  
Finite Projective Geometries ◽  
Composite Systems ◽  
Projective Geometries ◽  
Higher Dimensional

Finite projective geometries, especially the Fano plane, have been observed to arise in the context of certain quantum gate operators. We use Clifford algebras to explain why these geometries, both planar and higher dimensional, appear in the context of multi-qubit composite systems.

scholarly journals Veldkamp spaces: From (Dynkin) diagrams to (Pauli) groups

2017 ◽  
Vol 14 (05) ◽  
pp. 1750080
Author(s):  
Metod Saniga ◽  
Frédéric Holweck ◽  
Petr Pracna
Keyword(s):  
Incidence Structure ◽  
Polar Space ◽  
Exceptional Point ◽  
Specific Point ◽  
Fano Plane ◽  
Magic Square ◽  
Dynkin Diagrams ◽  
Pauli Matrix ◽  
Matrix Formula ◽  
Point Line

Regarding a Dynkin diagram as a specific point-line incidence structure (where each line has just two points), one can associate with it a Veldkamp space. Focusing on extended Dynkin diagrams of type [Formula: see text], [Formula: see text], it is shown that the corresponding Veldkamp space always contains a distinguished copy of the projective space PG[Formula: see text]. Proper labeling of the vertices of the diagram (for [Formula: see text]) by particular elements of the two-qubit Pauli group establishes a bijection between the 15 elements of the group and the 15 points of the PG[Formula: see text]. The bijection is such that the product of three elements lying on the same line is the identity and one also readily singles out that particular copy of the symplectic polar space [Formula: see text] of the PG[Formula: see text] whose lines correspond to triples of mutually commuting elements of the group; in the latter case, in addition, we arrive at a unique copy of the Mermin–Peres magic square. In the case of [Formula: see text], a more natural labeling is that in terms of elements of the three-qubit Pauli group, furnishing a bijection between the 63 elements of the group and the 63 points of PG[Formula: see text], the latter being the maximum projective subspace of the corresponding Veldkamp space; here, the points of the distinguished PG[Formula: see text] are in a bijection with the elements of a two-qubit subgroup of the three-qubit Pauli group, yielding a three-qubit version of the Mermin–Peres square. Moreover, save for [Formula: see text], each Veldkamp space is also endowed with some exceptional point(s). Interestingly, two such points in the [Formula: see text] case define a unique Fano plane whose inherited three-qubit labels feature solely the Pauli matrix [Formula: see text].

Multiplication of integral octonions

2016 ◽  
Vol 15 (08) ◽  
pp. 1650144
Author(s):  
Myeung Su Kim ◽  
Yeonjun Kim ◽  
Jae-Hyouk Lee ◽  
Seungbum Nam ◽  
Kiho Park
Keyword(s):  
Real Numbers ◽  
Fano Plane ◽  
Elementary Methods

The integral subsets of octonions are an analog of integers in real numbers and related to many interesting topics in geometry and physics via [Formula: see text]-lattices. In this paper, we study the properties of the multiplication of the integral subsets of octonions by studying configuration of Fano plane via blocks and operations on them. And we show that the integral subsets are integral indeed by introducing new and elementary methods.

scholarly journals The order of the automorphism group of a binary $${\varvec{q}}$$ q -analog of the Fano plane is at most two

2017 ◽  
Vol 86 (2) ◽  
pp. 239-250 ◽  
Author(s):  
Michael Kiermaier ◽  
Sascha Kurz ◽  
Alfred Wassermann
Keyword(s):  
Automorphism Group ◽  
Fano Plane
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