One-Factorisations of Complete Graphs Constructed in Desarguesian Planes of Certain Odd Square Orders

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 $K_{q(q-1)}$.

Blocking and Double Blocking Sets in Finite Planes

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.

On (q + t)-arcs of type (0, 2, t) in a desarguesian plane of order q

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].

The characterization of elliptic curves over finite fields

In 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.

Some translation planes constructed by multiple derivation

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.