14 research outputs found
On Buekenhout-Metz unitals of even order
AbstractThe even order Buekenhout-Metz unitals are enumerated (up to projective equivalence) and their inherited collineation groups are computed. They are shown to be self-dual as designs, and certain related designs are also constructed
An alternative construction of B-M and B-T unitals in Desarguesian planes
We present a new construction of non-classical unitals from a classical
unital in . The resulting non-classical unitals are B-M unitals.
The idea is to find a non-standard model of with the
following three properties: 1. points of are those of ; 2.
lines of are certain lines and conics of ; 3. the points in
form a non-classical B-M unital in .
Our construction also works for the B-T unital, provided that conics are
replaced by certain algebraic curves of higher degree.Comment: Keywords: unital, desarguesian plane 11 pages; ISSN: 0012-365
On the Equivalence, Stabilisers, and Feet of Buekenhout-Tits Unitals
This paper addresses a number of problems concerning Buekenhout-Tits unitals
in , where and . We show that all
Buekenhout-Tits unitals are -equivalent (addressing an open problem in [S.
Barwick and G. L. Ebert. Unitals in projective planes. Springer Monographs in
Mathematics. Springer, New York, 2008.]), explicitly describe their -stabiliser (expanding Ebert's work in [G.L. Ebert. Buekenhout-Tits unitals.
J. Algebraic. Combin. 6.2 (1997), 133-140], and show that lines meet the feet
of points no on in at most four points. Finally, we show that
feet of points not on are not always a -set, in
contrast to what happens for Buekenhout-Metz unitals [N. Abarz\'ua, R.
Pomareda, and O. Vega. Feet in orthogonal-Buekenhout-Metz unitals. Adv. Geom.
18.2 (2018), 229-236]
Embedding of orthogonal Buekenhout-Metz unitals in the Desarguesian plane of order q^2
A unital, that is a 2-(q^3 + 1, q + 1, 1) block-design, is embedded in a projective plane Ï of order q^2 if its points are points of Ï and its blocks are subsets of lines of Ï, the point-block incidences being the same as in Ï. Regarding unitals U which are isomorphic, as a block-design, to the classical unital, T. Szonyi and the authors recently proved that the natural embedding is the unique embedding of U into the Desarguesian plane of order q^2. In this paper we extend this uniqueness result to all unitals which are isomorphic, as block-designs, to orthogonal Buekenhout-Metz unitals
On regular sets of affine type in finite Desarguesian planes and related codes
In this paper, we consider point sets of finite Desarguesian planes whose
multisets of intersection numbers with lines is the same for all but one
exceptional parallel class of lines. We call such sets regular of affine type.
When the lines of the exceptional parallel class have the same intersection
numbers, then we call these sets regular of pointed type. Classical examples
are e.g. unitals; a detailed study and constructions of such sets with few
intersection numbers is due to Hirschfeld and Sz\H{o}nyi from 1991. We here
provide some general construction methods for regular sets and describe a few
infinite families. The members of one of these families have the size of a
unital and meet affine lines of in one of possible
intersection numbers, each of them congruent to modulo . As a
byproduct, we determine the intersection sizes of the Hermitian curve defined
over with suitable rational curves of degree and
we obtain -divisible codes with non-zero weights. We also
determine the weight enumerator of the codes arising from the general
constructions modulus some -powers.Comment: 16 pages/revised and improved versio