Unitals in the code of the Hughes plane

2003 ◽  
Vol 12 (1) ◽  
pp. 35-38
Author(s):  
R. D. Baker ◽  
K. L. Wantz
Keyword(s):  
Hughes Plane

Pseudo-Homogeneous Coordinates for Hughes Planes

1996 ◽  
Vol 39 (3) ◽  
pp. 330-345 ◽  
Author(s):  
Peter Maier ◽  
Markus Stroppel
Keyword(s):  
Projective Planes ◽  
Baer Subplane ◽  
Homogeneous Coordinates ◽  
Desargues Configuration ◽  
Coset Spaces ◽  
Insight Into ◽  
Hughes Plane

AbstractAmong the projective planes, the class of Hughes planes has received much interest, for several good reasons. However, the existing descriptions of these planes are somewhat unsatisfactory. We introduce pseudo-homogeneous coordinates which at the same time are easy to handle and give insight into the action of the group that is generated by all elations of the desarguesian Baer subplane of a Hughes plane. The information about the orbit decomposition is then used to give a description in terms of coset spaces of this group. Finally, we exhibit a non-closing Desargues configuration in terms of coordinates.

A Family of Unitals in the Hughes Plane

1990 ◽  
Vol 42 (6) ◽  
pp. 1067-1083 ◽  
Author(s):  
Barbu C. Kestenband
Keyword(s):  
Present Article ◽  
English Translation ◽  
Symmetric Matrices ◽  
Configurational Property ◽  
The Future ◽  
The Matrix ◽  
Classical Unitals ◽  
Hughes Plane

We construct a family of unitals in the Hughes plane. We prove that they are not isomorphic with the classical unitals, and in so doing we exhibit a configuration that exists in the latter, but not in the former. This new configurational property of the classical unitals might serve in the future again as an isomorphism test.A particular instance of our construction has appeared in [11]. But it only concerns itself with the case where the matrix involved is the identity, whereas the present article treats the general case of symmetric matrices over a suitable field. Furthermore, [11] does not answer the isomorphism question. It states that (the English translation is ours) “It remains to be seen whether the unitary designs constructed in this note are isomorphic or not with known designs”.

scholarly journals A unital in the hughes plane of order nine

1989 ◽  
Vol 77 (1-3) ◽  
pp. 55-56 ◽  
Author(s):  
A.E. Brouwer
Keyword(s):  
Hughes Plane

Veblen–Wedderburn hybrid planes

1969 ◽  
Vol 309 (1497) ◽  
pp. 157-170 ◽  
Keyword(s):  
Projective Plane ◽  
Galois Field ◽  
Real Field ◽  
Type I ◽  
Type Ii ◽  
Desarguesian Plane ◽  
The Real ◽  
Desarguesian Planes ◽  
Hughes Plane

The paper describes a method of redistributing the points of the collinear sets in a Desarguesian plane so as to produce a (hybrid) projective plane which is non-Desarguesian. The method is applied to the construction: (i) of a plane over a prescribed subfield of the real field, and (ii) of a plane (over a Galois field) which is proved to be identical with the Hughes plane. On the basis of this construction algebraic relations in the field can be interpreted as incidence relations in the hybrid plane. In order to verify that the planes of type (i) are not isomorphic with Desarguesian planes, some conditions are established which show that all planes of this type (as well as of type (ii)) contain Fano subplanes.

scholarly journals Subplanes of Order $3$ in Hughes Planes

10.37236/489 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Cafer Caliskan ◽  
G. Eric Moorhouse
Keyword(s):  
Prime Power ◽  
Linear Spaces ◽  
Partial Linear ◽  
Hughes Plane

In this study we show the existence of subplanes of order $3$ in Hughes planes of order $q^2$, where $q$ is a prime power and $q \equiv 5 \ (mod \ 6)$. We further show that there exist finite partial linear spaces which cannot embed in any Hughes plane.

Subplanes of the Hughes plane of order 9

1968 ◽  
Vol 64 (3) ◽  
pp. 589-598 ◽  
Author(s):  
R. H. F. Denniston
Keyword(s):  
Translation Plane ◽  
Projective Planes ◽  
Point Of View ◽  
Finite Plane ◽  
Field Of Study ◽  
Finite Projective Planes ◽  
Hughes Plane

The literature of finite projective planes consists largely of general investigations, taking in as many as possible of these systems at once. However, the geometry in a specific finite plane may well be an amusing, and not entirely trivial, field of study on its own. Some papers (5, 9, 13) have in fact appeared on the geometry of the translation plane of order 9: but the Hughes plane of the same order has comparatively been neglected. The object of the present paper is to make a beginning with the study of this plane from a synthetic point of view.

scholarly journals Polarities and ovals in the Hughes plane

1972 ◽  
Vol 13 (2) ◽  
pp. 196-204 ◽  
Author(s):  
T. G. Room
Keyword(s):  
Projective Plane ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Proper Subset ◽  
Small Subset ◽  
Order Form ◽  
The Absolute ◽  
Hughes Plane

SummaryIn 1946 Baer (Polarities infinite projective planes, Bull. Am. Math. Soc. 52, 77–93) showed that the absolute points of a polarity in a finite projective plane of odd non-square order always form an oval, that is, in a plane of order n there are exactly n+ 1 absolute points and no three are collinear. It is well known that the absolute points of polarities in planes of odd square order form ovals in some cases.If the oval is a subset of the set of absolute points, then the oval itself determines the polarity, and this makes it appear unlikely that the oval could be a proper subset. Among other results in the paper it is to be proved that in the regular Hughes plane there is a polarity which is determined by an oval which is a relatively small subset of the set of absolute points. Explicitly, if Ω is the Hughes plane of order q2 and A is the central subplane of order q, then every conic in Δ can be extended to an oval in Ω, and this oval determines a polarity in which there are ½(q3–q) additional absolute points.

Polarities in the Hughes Plane

1970 ◽  
Vol 2 (2) ◽  
pp. 209-213 ◽  
Author(s):  
Fred Piper
Keyword(s):  
Hughes Plane
Sign in / Sign up

Export Citation Format

Share Document