scholarly journals Constructing the Tits ovoid from an elliptic quadric

2008 ◽  
Vol 8 (1) ◽  
pp. 73-79
Author(s):  
William Cherowitzo
Keyword(s):  
Elliptic Quadric ◽  
Tits Ovoid

scholarly journals THE VELDKAMP SPACE OF GQ(2, 4)

2010 ◽  
Vol 07 (07) ◽  
pp. 1133-1145 ◽  
Author(s):  
M. SANIGA ◽  
R. M. GREEN ◽  
P. LÉVAY ◽  
P. VRANA ◽  
P. PRACNA
Keyword(s):  
Information Theory ◽  
Black Hole ◽  
Quantum Information ◽  
Quantum Information Theory ◽  
Generalized Quadrangle ◽  
Combinatorial Properties ◽  
Elliptic Quadric

It is shown that the Veldkamp space of the unique generalized quadrangle GQ(2, 4) is isomorphic to PG(5, 2). Since the GQ(2, 4) features only two kinds of geometric hyperplanes, namely point's perp-sets and GQ(2, 2)s, the 63 points of PG(5, 2) split into two families; 27 being represented by perp-sets and 36 by GQ(2, 2)s. The 651 lines of PG(5, 2) are found to fall into four distinct classes: in particular, 45 of them feature only perp-sets, 216 comprise two perp-sets and one GQ(2, 2), 270 consist of one perp-set and two GQ(2, 2)s and the remaining 120 are composed solely of GQ(2, 2)s, according to the intersection of two distinct hyperplanes determining the (Veldkamp) line is, respectively, a line, an ovoid, a perp-set and a grid (i.e. GQ(2, 1)) of a copy of GQ(2, 2). A direct "by-hand" derivation of the above-listed properties is followed by their heuristic justification based on the properties of an elliptic quadric of PG(5, 2) and complemented by a proof employing combinatorial properties of a 2-(28, 12, 11)-design and associated Steiner complexes. Surmised relevance of these findings for quantum (information) theory and the so-called black hole analogy is also outlined.

scholarly journals Magic Three-Qubit Veldkamp Line and Veldkamp Space of the Doily

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 963
Author(s):  
Metod Saniga ◽  
Zsolt Szabó
Keyword(s):  
Quantum Information ◽  
Quadratic Cone ◽  
Hyperbolic Quadric ◽  
Elliptic Quadric ◽  
Parabolic Quadric ◽  
Parapolar Spaces

A magic three-qubit Veldkamp line of W ( 5 , 2 ) , i.e., the line comprising a hyperbolic quadric Q + ( 5 , 2 ) , an elliptic quadric Q − ( 5 , 2 ) and a quadratic cone Q ^ ( 4 , 2 ) that share a parabolic quadric Q ( 4 , 2 ) , the doily, is shown to provide an interesting model for the Veldkamp space of the doily. The model is based on the facts that: (a) the 20 off-doily points of Q + ( 5 , 2 ) form ten complementary pairs, each corresponding to a unique grid of the doily; (b) the 12 off-doily points of Q − ( 5 , 2 ) form six complementary pairs, each corresponding to a unique ovoid of the doily; and (c) the 15 off-doily points of Q ^ ( 4 , 2 ) , disregarding the nucleus of Q ( 4 , 2 ) , are in bijection with the 15 perp-sets of the doily. These findings lead to a conjecture that also parapolar spaces can be relevant for quantum information.

scholarly journals On Almost Complete Caps in PG(N, q)

2018 ◽  
Vol 18 (5) ◽  
pp. 54-62
Author(s):  
Alexander A. Davydov ◽  
Stefano Marcugini ◽  
Fernanda Pambianco
Keyword(s):  
Projective Space ◽  
Upper Bounds ◽  
Complete Caps ◽  
Elliptic Quadric ◽  
Complete Arc

Abstract We propose the concepts of almost complete subset of an elliptic quadric in the projective space PG(3, q) and of almost complete cap in the space PG(N, q), N ≥ 3, as generalizations of the concepts of almost complete subset of a conic and of almost complete arc in PG(2, q). Upper bounds of the smallest size of the introduced geometrical objects are obtained by probabilistic and algorithmic methods.

On the intersection of a Hermitian surface with an elliptic quadric

2015 ◽  
Vol 15 (2) ◽  
Author(s):  
Antonio Cossidente ◽  
Francesco Pavese
Keyword(s):  
Generalized Quadrangle ◽  
Elliptic Quadric ◽  
Hermitian Surface

AbstractWe investigate the intersection between the generalized quadrangle arising from a Hermitian surface H(3, q

scholarly journals Blocking sets of tangent and external lines to an elliptic quadric in PG(3, q)

2021 ◽  
Vol 131 (2) ◽  
Author(s):  
Bart De Bruyn ◽  
Puspendu Pradhan ◽  
Binod Kumar Sahoo
Keyword(s):  
Blocking Sets ◽  
Elliptic Quadric
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