Non-classical polar unitals in finite Figueroa planes

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Some partitions in Figueroa planes

Using Grundhöfer's construction of the Figueroa planes from Pappian planes  which  have an order $3$ planar collineation ${\widehat \alpha }$, we show that any  Figueroa plane (finite or infinite) has a partition of the complement of any proper (${\widehat \alpha }$)-invariant triangle mostly into subplanes together with a few  collinear  point sets (from the point set view) and a few concurrent line sets (from the  line set  view).  The partition shows that each Figueroa line (regarded as a set of  points) is  either the same as a Pappian line or consists mostly of a disjoint union of  subplanes of the Pappian plane (most lines are of this latter type) anddually. This last sentence is true with "Figueroa" and "Pappian" interchanged. There are many collinear subsets of Figueroa points which are a subset of the set of points of a Pappian conic and dually

An upper bound for the minimum weight of the dual codes of desarguesian planes

AbstractWe show that a construction described in [K.L. Clark, J.D. Key, M.J. de Resmini, Dual codes of translation planes, European J. Combin. 23 (2002) 529–538] of small-weight words in the dual codes of finite translation planes can be extended so that it applies to projective and affine desarguesian planes of any order pm where p is a prime, and m≥1. This gives words of weight 2pm+1−pm−1p−1 in the dual of the p-ary code of the desarguesian plane of order pm, and provides an improved upper bound for the minimum weight of the dual code. The same will apply to a class of translation planes that this construction leads to; these belong to the class of André planes.We also found by computer search a word of weight 36 in the dual binary code of the desarguesian plane of order 32, thus extending a result of Korchmáros and Mazzocca [Gábor Korchmáros, Francesco Mazzocca, On (q+t)-arcs of type (0,2,t) in a desarguesian plane of order q, Math. Proc. Cambridge Philos. Soc. 108 (1990) 445–459]

Unitals in projective planes revisited

This thesis revisits the topic of unitals in finite projective planes. A unital U in a projective plane of order q2 is a set of q3 + 1 points, such that every line meets U in one or q + 1 points. Unitals are an important class of point-set in finite projective planes, whose combinatorial and algebraic properties have been the subject of considerable study. In this work, we summarise, revise, and extend contemporary research on unitals. Chapter 1 covers the necessary prerequisites to study unitals and related objects in finite geometry. In Chapter 2, we focus on Buekenhout-Tits unitals and answer some open problems regarding their equivalence, stabilisers and feet. The results presented in Chapter 2 are also available in a preprint paper [22]. Following this, Chapter 3 summarises recent results on Buekenhout- Metz unitals, and presents a small result on the intersection of ovoidal-Buekenhout-Metz unitals and Buekenhout-Metz unitals. Chapter 4 highlights Kestenband arcs and their relationship to Hermitian unitals, and makes explicit a proof of their equivalence. Finally in Chapter 5, we review our understanding of Figueroa planes. Beyond describing ovals and unitals in Figueroa planes, we also suggest generalisations of their constructions to semi-ovals

On the structure of the Figueroa unital and the existence of O’Nan configurations

AbstractThe finite Figueroa planes are non-Desarguesian projective planes of order q3 for all prime powers q>2, constructed algebraically in 1982 by Figueroa, and Hering and Schaeffer, and synthetically in 1986 by Grundhöfer. All Figueroa planes of finite square order are shown to possess a unitary polarity by de Resmini and Hamilton in 1998, and hence admit unitals. Hui and Wong (2012) have shown that these polar unitals do not satisfy a necessary condition, introduced by Wilbrink in 1983, for a unital to be classical, and hence they are not classical. In this article we introduce and make use of a new alternative synthetic description of the Figueroa plane and unital to demonstrate the existence of O’Nan configurations, thus providing support to Piper’s conjecture (1981)

Hyperovals and unitals in Figueroa planes

In [3], W. M. Cherowitzo constructed ovals in all finite Figueroa planes of odd order. Here a class of hyperovals is constructed in the finite Figueroa planes of even order. These hyperovals are inherited from regular hyperovals in the associated desarouesian planes. It is also shown that all Figueroa planes of finite square order possess a unitary polarity, and hence admit unitals. (C) 1998 Academic Press Limited