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This article studies conics and subconics of $PG(2,q^2)$ and their representation in the Andr\'e/Bruck-Bose setting in $PG(4,q)$. In particular, we investigate their relationship with the transversal lines of the regular spread. The main result is to show that a conic in a tangent Baer subplane of $PG(2,q^2)$ corresponds in $PG(4,q)$ to a normal rational curve that meets the transversal lines of the regular spread. Conversely, every 3 and 4-dimensional normal rational curve in $PG(4,q)$ that meets the transversal lines of the regular spread corresponds to a conic in a tangent Baer subplane of $PG(2,q^2)$.

Let $${{\mathcal {K}}}$$ be a set of $$q^2+2q+1$$ points in $$\text {PG}(4,q)$$ . We show that if every 3-space meets $${{\mathcal {K}}}$$ in either one, two or three lines, a line and a non-degenerate conic, or a twisted cubic, then $${{\mathcal {K}}}$$ is a ruled cubic surface. Moreover, $${{\mathcal {K}}}$$ corresponds via the Bruck–Bose representation to a tangent Baer subplane of $$\text {PG}(2,q^2)$$ . We use this to prove a characterisation in $$\text {PG}(2,q^2)$$ of a set of points $${{\mathcal {B}}}$$ as a tangent Baer subplane by looking at the intersections of $${{\mathcal {B}}}$$ with Baer-pencils.

Let π be an order- q -subplane of P G ( 2 , q 3 ) that is exterior to l ∞ . Then the exterior splash of π is the set of q 2 + q + 1 points on l ∞ that lie on an extended line of π . Exterior splashes are projectively equivalent to scattered linear sets of rank 3, covers of the circle geometry C G ( 3 , q ) , and hyper-reguli in P G ( 5 , q ) . We use the Bruck–Bose representation in P G ( 6 , q ) to investigate the structure of π , and the interaction between π and its exterior splash. We show that the point set of P G ( 6 , q ) corresponding to π is the intersection of nine quadrics, and that there is a unique tangent plane at each point, namely the intersection of the tangent spaces of the nine quadrics. In P G ( 6 , q ) , an exterior splash S has two sets of cover planes (which are hyper-reguli) and we show that each set has three unique transversal lines in the cubic extension P G ( 6 , q 3 ) . These transversal lines are used to characterise the carriers and the sublines of S .

This article looks at subconics of order q of $$\mathrm{PG}(2,q^2)$$ in the Bruck–Bose representation in $$\mathrm{PG}(4,q)$$. A unified characterisation is given, namely a normal rational curve $${{{\mathcal {N}}}}$$ in $$\mathrm{PG}(4,q)$$ corresponds to an $${\mathbb {F}}_{\!q\!\;}$$-conic of $$\mathrm{PG}(2,q^2)$$ if and only if $${{{\mathcal {N}}}}$$ is 2-special - a notion related to how $${{{\mathcal {N}}}}$$ interacts with the regular 1-spread of the Bruck–Bose representation.

A simple counting argument is used to show that for all q, an Andre hyper-regulus $${\mathbb {X}}$$ in $$\mathrm{PG}(5,q)$$ has exactly two switching sets. Moreover, there are exactly $$2(q^2+q+1)$$ planes in $$\mathrm{PG}(5,q)$$ that meet every plane of $${\mathbb {X}}$$ in a point, namely the planes in the switching sets.

In this article, we begin with arcs in PG(2, qn) and show that they correspond to caps in PG(2n, q) via the Andre/Bruck–Bose representation of PG(2, qn) in PG(2n, q). In particular, we show that a conic of PG(2, qn) that meets l∞ in x points corresponds to a (qn + 1 − x)-cap in PG(2n, q). If x = 0, this cap is the intersection of n quadrics. If x = 1 or 2, this cap is contained in the intersection of n quadrics and we discuss ways of extending these caps. We also investigate the structure of the n quadrics.

In this article, the lines not meeting an oval cone in PG(3, q) (q even) are characterised by their intersection properties with points and planes.

In this article, the lines not meeting a quadric cone in [math] ( [math] odd) are characterised by their intersection properties with points and planes.

A linear (qd, q, t)-perfect hash family of size s consists of a vector space V of order qd over a field F of order q and a sequence ϕ1,…,ϕs of linear functions from V to F with the following property: for all t subsets X ⊆ V, there exists i ∈ {1,·,s} such that ϕi is injective when restricted to F. A linear (qd, q, t)-perfect hash family of minimal size d( – 1) is said to be optimal. In this paper, we prove that optimal linear (q2, q, 4)-perfect hash families exist only for q = 11 and for all prime powers q > 13 and we give constructions for these values of q. © 2004 Wiley Periodicals, Inc. J Comb Designs 12: 311–324, 2004

This article proves a characterisation of the classical unital that is a generalisation of a characterisation proved in 1982 by Lefevre-Percsy. It is shown that if $ {\cal U} $ is a Buekenhout-Metz unital with respect to a line $ \ell_\infty $ in $ PG(2,q^2) $ such that a line of $ PG(2,q^2) $ not through $ \cal U cap \ell_\infty $ meets $ {\cal U} $ in a Baer subline, then $ {\cal U} $ is classical. An immediate corollary is that if $ {\cal U} $ is a unital in PG $ (2,q^2) $ such that $ {\cal U} $ is Buekenhout-Metz with respect to two distinct lines, then $ {\cal U} $ is classical.

We prove that a parabolic unitalU in a translation plane π of orderq2 with kernel containing GF(q) is a Buekenhout-Metz unital if and only if certain Baer subplanes containing the translation line of π meetU in 1 moduloq points. As a corollary we show that a unital 16-03 in PG(2,q2) is classical if and only if it meets each Baer subplane of PG(2,q2) in 1 moduloq points.

The unitals in the Hall plane are studied by deriving PG(2,q2)and observing the effect on the unitals of PG(2,q2).The number of Buekenhout and Buekenhout-Metz unitals in the Hall plane is determined. As a corollary we show that the classical unital is not embeddble in the Hall plane as a Buekenhout unital and that the Buekenhout unitals of H(q2)are not embeddable as Buekenhout unitals in the Desarguesian plane. Finally, we generalize this technique to other translation planes.

We define Buekenhout unitals in derivable translation planes of dimension 2 over their kernel and provide a characterization of these unitals. We use this result to improve the characterization of classical unitals given by Lefevre-Percsy [13] and Faina and Korchmaros [7].

Let U be the classical unital in PG(2;q 2 ) secant to ‘1. By deriving PG(2;q 2 ) with respect to a derivation set disjoint from U we obtain a new unital U 0 in the Hall plane of order q 2 . We show that this unital contains O’Nan congurations and is not isomorphic to the known unitals of the Hall plane, hence it forms a new class of unitals in the Hall plane.