desarguesian plane
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10.37236/8378 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Nicola Pace ◽  
Angelo Sonnino

A geometric construction of one-factorisations of complete graphs $K_{q(q-1)}$ is provided for the case when either $q=2^d +1$ is a Fermat prime, or $q=9$. This construction uses the affine group $\mathrm{AGL}(1,q)$, points and ovals in the Desarguesian plane $\mathrm{PG}(2,q^{2})$ to produce one-factorisations of the complete graph $K_{q(q-1)}$.


2019 ◽  
Vol 16 (2) ◽  
pp. 609-623
Author(s):  
Gábor Korchmáros ◽  
Alessandro Siciliano
Keyword(s):  

10.37236/5717 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Jan De Beule ◽  
Tamás Héger ◽  
Tamás Szőnyi ◽  
Geertrui Van de Voorde

In this paper, by using properties of Baer subplanes, we describe the construction of a minimal blocking set in the Hall plane of order $q^2$ of size $q^2+2q+2$ admitting $1-$, $2-$, $3-$, $4-$, $(q+1)-$ and $(q+2)-$secants. As a corollary, we obtain the existence of a minimal blocking set of a non-Desarguesian affine plane of order $q^2$ of size at most $4q^2/3+5q/3$, which is considerably smaller than $2q^2-1$, the Jamison bound for the size of a minimal blocking set in an affine Desarguesian plane of order $q^2$.We also consider particular André planes of order $q$, where $q$ is a power of the prime $p$, and give a construction of a small minimal blocking set which admits a secant line not meeting the blocking set in $1$ mod $p$ points. Furthermore, we elaborate on the connection of this problem with the study of value sets of certain polynomials and with the construction of small double blocking sets in Desarguesian projective planes; in both topics we provide some new results.


2011 ◽  
Vol 102 (1-2) ◽  
pp. 179-188 ◽  
Author(s):  
Markus Stroppel
Keyword(s):  

Author(s):  
M. Giulietti ◽  
J. W. P. Hirschfeld ◽  
G. Korchmáros
Keyword(s):  

1990 ◽  
Vol 108 (3) ◽  
pp. 445-459 ◽  
Author(s):  
Gábor Korchmáros ◽  
Francesco Mazzocca

This paper is concerned with certain point-sets T in a projective plane PG (2, q) over GF (q) which have only three characters with respect to the lines. We assume throughout this paper that for any line l of πwhere It is easily seen that if t = 1 then T is a (q + 1)-arc, i.e. an oval; otherwise T is a (q+t, t)-arc of type (0, 2, t). Therefore (q+t, t)-arcs of type (0, 2, t) appear to be a generalization of ovals and there are interesting connections between ovals and (q + t, t)-arcs of type (0, 2, t) from various points of view. Our purpose is to investigate such particular (k, t)-arcs using some ideas of B. Segre developed for ovals in three fundamental papers [16, 17, 18]. For these papers and more recent results in this direction the reader is referred to [6], chapter 10 and [9]. General results concerning (k, n)-arcs may be found in [6], chapter 12; see also [4, 7, 20, 23].


Author(s):  
J. W. P. Hirschfeld ◽  
J. F. Voloch

AbstractIn a finite Desarguesian plane of odd order, it was shown by Segre thirty years ago that a set of maximum size with at most two points on a line is a conic. Here, in a plane of odd or even order, sufficient conditions are given for a set with at most three points on a line to be a cubic curve. The case of an elliptic curve is of particular interest.


1981 ◽  
Vol 23 (2) ◽  
pp. 313-315
Author(s):  
N. L. Johnson

It is noted that the translation planes of Rao and Rao may be constructed from a Desarguesian plane by the replacement of a set of disjoint derivable nets. Their plane of order 25 which admits a collineation group splitting the infinite points into orbits of lengths 18 and 8 may be obtained by replacing exactly three disjoint derivable nets and may be viewed as being derived from the Andre nearfield plane of order 25.


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