# Eggs in finite projective spaces
and unitals in translation planes

Giusy Monzillo111The research was supported by the Italian National Group for
Algebraic and Geometric Structures and their Applications (GNSAGA-INdAM).
<EMAIL_ADDRESS>
Dipartimento di Matematica, Informatica ed Economia
Università degli Studi della Basilicata
Viale dell’Ateneo Lucano 10
85100 Potenza, Italy
Tim Penttila
<EMAIL_ADDRESS>
School of Mathematical Sciences
The University of Adelaide
Adelaide, South Australia
5005 Australia  Alessandro Siciliano∗
<EMAIL_ADDRESS>
Dipartimento di Matematica, Informatica ed Economia
Università degli Studi della Basilicata
Viale dell’Ateneo Lucano 10
85100 Potenza, Italy

###### Abstract

Inspired by the connection between ovoids and unitals arising from the
Buekenhout construction in the André/Bruck-Bose representation of translation
planes of dimension at most two over their kernel, and since eggs of ${\rm
PG}(4m-1,q)$, $m\geq 1$, are a generalization of ovoids, we explore the
relation between eggs and unitals in translation planes of higher dimension
over their kernel. By investigating such a relationship, we construct a unital
in the Dickson semifield plane of order $3^{10}$, which is represented in
${\rm PG}(20,3)$ by a cone whose base is a set of points constructed from the
dual of the Penttila-Williams egg in ${\rm PG}(19,3)$. This unital is not
polar; so, up to the knowledge of the authors, it seems to be a new unital in
such a plane.

Keywords: Unital, Blocking set, Egg, Projective plane

##  1 Introduction

Field reduction has become a theme of finite geometry which turned out very
fruitful in the last few decades. Given a construction of an interesting
object from a configuration in a vector space of dimension $r$ over a field of
order $q^{n}$, the question is raised as to which objects give rise similar
configurations in a vector space of dimension $rn$ over a field of order $q$.

The Buekenhout-Metz construction of unitals in finite translation planes [15,
33] (which gives all known unitals in Desarguesian planes) can be
recontextualized in this fashion, with cones projecting an ovoid as a base in
the André/Bruck-Bose representation of such planes.

It has long been known [14] that unitals are extremal in size among minimal
blocking sets (at the other end than that most studied - large rather than
small). The observation of Lunardon [29] at the turn of the millennium that
changing the field gave access to many more subspaces, some of which were
blocking sets, transformed the theory of blocking sets in the process giving
rise to the idea of linear sets. Thus, the idea of Buekenhout and Metz was
taken by Szőnyi et al. [40] and, later, by Mazzocca and Polverino [31] to
provide further minimal blocking sets, using cones rather than subspaces.

For the construction by Tits of generalized quadrangles from (ovals and)
ovoids, the configurations that arise by applying a field reduction are eggs,
and the similar objects are translation generalized quadrangles. Thus,
changing the field for ovoids and studying eggs gave the possibility of new
translation generalized quadrangles, first realized in work of Kantor [25]
from three decades past; the result of field reduction applied to the concept
of an ovoid is an egg.

Motivated by the relationship between ovoids and unitals via the Buekenhout-
Metz construction, and since eggs are generalization of ovoids, we explore
possible relationships between eggs and unitals. Putting all the above ideas
together, in this paper we construct a unital in the Dickson semifield plane
of order $3^{10}$, which is represented in ${\rm PG}(20,3)$ by a cone whose
base is a set of points constructed from the dual of the Penttila-Williams egg
in ${\rm PG}(19,3)$. This unital does not arise from a polarity; so it is a
new unital, up to the knowledge of the authors.

While field reduction is usually thought of in a projective setting, algebraic
dimensions are more amenable to an introductory discussion of it, so we will
take a vector space approach along all the paper.

##  2 Definitions and preliminary results

A unital in a finite projective plane $\pi$ of order $n^{2}$ is a set
$\mathcal{U}$ of $n^{3}+1$ points such that every line of $\pi$ meets
$\mathcal{U}$ in 1 or $n+1$ points. Therefore, $\mathcal{U}$ is equipped with
a family of subsets, each of size $n+1$, such that every pair of distinct
points of $\mathcal{U}$ is contained in exactly one subset of the family; such
subsets are usually called blocks, and $\mathcal{U}$ turns out to be a
2-$(n^{3}+1,n+1,1)$ design.

In a computer search, Brouwer [10, 11] found a large number of mutually non-
isomorphic 2-$(28,4,1)$ designs. Only a few of these are embeddable in a
projective plane of order 9 as unitals. One of the examples has been
generalized by Grüning [23], who constructed a unital of order $q$ for any odd
prime power in both the Hall plane and dual Hall plane of order $q$. An
infinite family of non-Buekenhout unitals in the Hall planes of order $q^{2}$
have been constructed in [19]. Other infinite families of unitals in various
square order planes are known to exist; see e.g. [1], [5], [6], [18], [36],
[37]. The only known 2-$(n^{3}+1,n+1,1)$ design with $n$ not a prime power is
the one found in [4] and [30] where $n=6$. For more on 2-$(n^{3}+1,n+1,1)$
designs embeddable as unitals in projective plane, see [8].

In the Desarguesian projective plane ${\rm PG}(2,q^{2})$, a unital can arise
from a unitary polarity: the points of the unital are the absolute points, and
the blocks are the intersections of the non-absolute lines of the polarity
with $\mathcal{U}$. These unitals are called classical  or Hermitian unitals.
By a result of Seib [39], the absolute points of a unitary polarity in any
square order projective plane form the point-set of a unital. Such unitals are
called polar unitals. So, classical unitals of ${\rm PG}(2,q^{2})$ are
examples of polar unitals, and Ganley [21] showed that polar unitals exist in
any Dickson commutative semifield plane of odd order.

A finite semifield is a finite set ${\mathbb{S}}$ with two binary operations
$+$ and $*$, such that $({\mathbb{S}},+)$ is an abelian group and
$({\mathbb{S}}\setminus\\{0\\},*)$ is a loop such that both distributive laws
hold.

Let $\pi({\mathbb{S}})$ be the point-line geometry whose points are the
elements in ${\mathbb{S}}\times{\mathbb{S}}$ and in
$\\{(m):m\in{\mathbb{S}}\cup\\{\infty\\}\\}$, and the lines are the sets

$[m,k]=\\{(y,x)\in{\mathbb{S}}\times{\mathbb{S}}:m*x+y=k\\}\cup\\{(m)\\},$
$[z]=\\{(y,z):y\in{\mathbb{S}}\\}\cup\\{(\infty)\\}$

and

$[\infty]=\\{(m):m\in{\mathbb{S}}\\}\cup\\{(\infty)\\}.$

with $m,k,z\in{\mathbb{S}}$, and $\infty$ a symbol not in ${\mathbb{S}}$.

It turns out that $\pi({\mathbb{S}})$ is a translation plane which is called
the semifield plane coordinatized by ${\mathbb{S}}$. We refer to [9] and [17]
for basic information on semifields and translation planes.

For any semifield ${\mathbb{S}}$, the subset
$\mathcal{N}_{l}=\\{a\in{\mathbb{S}}:a*(x*y)=(a*x)*y,\forall
x,y\in{\mathbb{S}}\\}$ is called the left nucleus of ${\mathbb{S}}$.
Similarly, the middle nucleus $\mathcal{N}_{m}$ and the right nucleus
$\mathcal{N}_{r}$ are defined. The set
$\mathcal{K}=\\{a\in\mathcal{N}_{l}\cap\mathcal{N}_{m}\cap\mathcal{N}_{r}:a*b=b*a,\forall
b\in{\mathbb{S}}\\}$ is called the center of ${\mathbb{S}}$. Each of these
four structures is a field, and a finite semifield is a left vector space over
its left nucleus and a two-sided vector space over its center [17]. Here,
$\mathcal{K}$ is isomorphic to the kernel of the translation plane
$\pi({\mathbb{S}})$.

For any element $b$ of the semifield ${\mathbb{S}}$ with center $\mathcal{K}$,
the map $\phi_{b}:x\in{\mathbb{S}}\mapsto xb\in{\mathbb{S}}$ is a linear map
when ${\mathbb{S}}$ is considered over its left nucleus $\mathcal{N}_{l}$. It
turns out that the set
$\mathcal{C}_{{\mathbb{S}}}=\\{\phi_{b}:b\in{\mathbb{S}}\\}$ is a
$\mathcal{K}$-vector subspace of the vector space of the
$\mathcal{N}_{l}$-linear maps of ${\mathbb{S}}$. Since ${\mathbb{S}}$ is
finite, we may assume $\mathcal{K}=\mathbb{F}_{q}$,
$\mathcal{N}_{l}=\mathbb{F}_{q^{n}}$ and ${\mathbb{S}}$ is an $t$-dimensional
left vector space over $\mathbb{F}_{q^{n}}$, for some positive integers $n$
and $t$.

Under the previous indentification, the set $\mathcal{C}_{{\mathbb{S}}}$
satisfies the following properties: (i) $\mathcal{C}_{{\mathbb{S}}}$ has
$q^{nt}$ elements; (ii) $\mathcal{C}_{{\mathbb{S}}}$ contains the zero and the
identity maps; (iii) $A-B$ is non-singular for all distinct
$A,B\in\mathcal{C}_{{\mathbb{S}}}$. A set of linear maps of $V(t,q^{n})$
satisfying the above properties is called a _spread set_ of $V(t,q^{n})$.

A $(t-1)$-spread of the $(r-1)$-dimensional projective space ${\rm PG}(r-1,q)$
over $\mathbb{F}_{q}$ is a set $\mathcal{S}$ of $(t-1)$-dimensional projective
subspaces such that every point is contained in exactly one subspace of
$\mathcal{S}$. It is known that a $(t-1)$-spread of ${\rm PG}(r-1,q)$ exists
if and only if $t$ divides $r$ [17].

Let $\mathcal{C}$ be a spread set of $V(t,q^{n})=\mathbb{F}_{q^{n}}^{\,t}$. In
${\rm PG}(2t-1,q^{n})$ consider the subspaces

$S_{\tau}=\\{((x_{1},\ldots,x_{t})^{\tau},x_{1},\ldots,x_{t}):x_{i}\in\mathbb{F}_{q^{n}}\\},$

for all $\tau\in\mathcal{C}$. Then, the set
$\mathcal{S}=\\{S_{\tau}:\tau\in\mathcal{C}\\}\cup\\{S_{\infty}\\}$, with
$S_{\infty}=\\{(x_{1},\ldots,x_{t},0,\ldots,0):x_{i}\in\mathbb{F}_{q^{n}}\\}$
forms a $(t-1)$-spread of ${\rm PG}(2t-1,q^{n})$.

Conversely, let $\mathcal{S}$ be a $(t-1)$-spread of ${\rm PG}(2t-1,q^{n})$.
Then, it is possible to choose homogeneus coordinates in ${\rm
PG}(2t-1,q^{n})$ such that there is a spread set $\mathcal{C}$ of $V(t,q^{n})$
from which $\mathcal{S}$ is constructed as above. Thanks to the André/Bruck-
Bose construction, the spread $\mathcal{S}$ defines a translation plane
$\Pi(\mathcal{S})$ [3, 12, 13]. If the set $\mathcal{C}$ is closed under the
sum, then there is a (finite) semifield ${\mathbb{S}}$ that coordinatizes
$\Pi(\mathcal{S})$ such that $\mathcal{C}=\mathcal{C}_{{\mathbb{S}}}$; the
left nucleus of ${\mathbb{S}}$ is $\mathbb{F}_{q^{n}}$ and ${\mathbb{S}}$ can
be viewed as a $t$-dimensional left vector space over $\mathbb{F}_{q^{n}}$
[17]. In addition, if $\mathbb{F}_{q}$ is the largest subfield $K$ of
$\mathbb{F}_{q^{n}}$ such that $\mathcal{C}$ is a $K$-vector subspace of the
vector space of the $\mathbb{F}_{q^{n}}$-linear maps of $V(t,q^{n})$, the
center of ${\mathbb{S}}$ is $\mathbb{F}_{q}$. Therefore, there exists a
canonical correspondence between translation planes coordinatized over a
semifield ${\mathbb{S}}$ with dimension $t$ over its left nucleus
$\mathbb{F}_{q^{n}}$ and center $\mathbb{F}_{q}$, and the $(t-1)$-spreads of
${\rm PG}(2t-1,q^{n})$ arising from a spread set of $V(t,q^{n})$, that is
closed under the sum. Moreover, it is well-known that the resulting plane is
Desarguesian if and only if $\mathcal{S}$ is a Desarguesian spread [13].

Buekenhout [15], and Metz [33] (by refining Buekenhout’s idea), constructed
unitals in any translation planes with dimension at most two over their kernel
by using the André/Bruck-Bose representation of such planes. These unitals are
cones of ${\rm PG}(4,q)$ projecting an ovoid in a 3-dimensional subspace of
${\rm PG}(4,q)$ from a point at infinity. These unitals are called Buekenhout-
Metz unitals. Since classical unitals can be obtained in this way, they fall
in the class of Buekenhout-Metz unitals which, so far, are the only known
unitals of ${\rm PG}(2,q^{2})$.

Many other authors have used the above representation of ${\rm PG}(2,q^{n})$
in ${\rm PG}(2n,q)$ to study objects in the Desarguesian plane in order to
determine whether this higher dimensional representation provides additional
information about those objects in the plane. In particular, the projective
plane ${\rm PG}(2,q^{4})$, modelled in ${\rm PG}(8,q)$, has been considered in
[7] to study the representation of classical unitals, and the representation
of ${\rm PG}(2,q^{2m})$ in ${\rm PG}(4m,q)$, for $m>1$, have been considered
to study other geometric objects of the plane; see [31, 32, 38, 40] just to
cite some.

A blocking set in a projective plane $\pi$ is a set of points such that every
line of $\pi$ has a non-empty intersection with the set. A blocking set is
said to be minimal if through any of its points there is a line of $\pi$
intersecting it precisely in that point.

In the paper [14], Bruen and Thas proved that, when the order of the
projective plane is a square, say $n^{2}$, then the size of a minimal blocking
set is bounded by $n^{3}+1$. This size is reached if and only if the minimal
blocking set is a unital.

In [31] the following geometric setting was introduced to construct large
minimal blocking sets of ${\rm PG}(2,q^{2m})$ from cones in its André/Bruck-
Bose representation in ${\rm PG}(4m,q)$. Let $\mathbf{z}$ be a fixed element
of a $(2m-1)$-spread $\mathcal{S}$ of $\Sigma_{\infty}$ and $\mathcal{V}$ an
$(m-1)$-dimensional subspace of $\mathbf{z}$. Let $\Gamma$ be a
$(3m-1)$-dimensional subspace of $\Sigma_{\infty}$ disjoint from
$\mathcal{V}$. For every $\mathbf{x}\in\mathcal{S}$,
$\mathbf{x}\neq\mathbf{z}$, let $I(\mathbf{x})$ be the $(2m-1)$-dimensional
subspace $\langle\mathbf{x},\mathcal{V}\rangle\cap\Gamma$. We denote by
$\mathcal{I}(\mathcal{V})$ the set of all the subspaces $I(\mathbf{x})$,
$\mathbf{x}\in\mathcal{S}$. Let $\Gamma^{\prime}$ be an affine
$3m$-dimensional subspace of ${\rm PG}(4m,q)$ through $\Gamma$, and denote by
$\mathcal{F}(\mathcal{V})$ the set of all affine $2m$-dimensional subspaces of
$\Gamma^{\prime}$ containing an element of $\mathcal{I}(\mathcal{V})$.

Let $\mathcal{F}$ be a family of $2m$-dimensional subspaces of
$\Gamma^{\prime}$. An $\mathcal{F}$-blocking set of $\Gamma^{\prime}$ is a set
$\mathcal{B}$ of affine points such that every element of $\mathcal{F}$ has a
non-empty intersection with $\mathcal{B}$. The blocking set $\mathcal{B}$ is
said to be minimal if through any point of $\mathcal{B}$ there is an element
in $\mathcal{F}$ intersecting $\mathcal{B}$ precisely in that point.

By keeping the above geometric setting in mind, the following result, which is
a sharpening of Corollary 3.3 in [31], is crucial for our succeeding
considerations.

######  Proposition 2.1.

Let $\mathcal{B}$ be a set of affine points of $\Gamma^{\prime}$ and

$\mathcal{B}^{*}=\bigcup_{P\in\mathcal{B}}{\langle\mathcal{V},P\rangle}\cup\\{\mathbf{z}\\}.$
(1)

If $\mathcal{B}$ is a minimal $\mathcal{F}(\mathcal{V})$-blocking set, then
$\mathcal{B}^{*}$ is a minimal blocking set of size
$|\mathcal{B}^{*}|=q^{m}|\mathcal{B}|+1$ in the translation plane
$\Pi(\mathcal{S})$.

###### Proof.

Construction 2 in [31] works perfectly well under the milder hypothesis that
$\mathcal{S}$ is any $(2m-1)$-spread of $\Sigma_{\infty}$. The details are
left to the reader. ∎

By combining the above result of Bruen and Thas with Proposition 2.1, we get
the following theorem.

######  Theorem 2.2.

Let $\mathcal{B}$ be a minimal $\mathcal{F}(\mathcal{V})$-blocking set of size
$q^{2m}$. Then, the cone $\mathcal{B}^{*}$ defined in Proposition 2.1 is a
unital in $\Pi(\mathcal{S})$.

If $\mathcal{S}$ is a Desarguesian $(2m-1)$-spread of $\Sigma_{\infty}$, then
there is a unique Desarguesian $(m-1)$-spread, say $\mathcal{T}$, that fills
every element of $\mathcal{S}$, i.e., $\mathcal{T}$ induces a $(m-1)$-spread
in each spread element of $\mathcal{S}$ [20]. The following result gives a
characterization of Buekenhout-Metz unitals as cones in ${\rm PG}(4m,q)$.

######  Proposition 2.3.

[31] Let $\mathcal{S}$ be a Desarguesian $(2m-1)$-spread of $\Sigma_{\infty}$
and $\mathcal{B}$ a minimal $\mathcal{F}(\mathcal{V})$-blocking set of size
$q^{2m}$. Then, the cone $\mathcal{B}^{*}$ is a Buekenhout-Metz unital in
${\rm PG}(2,q^{2m})$ if and only if $\mathcal{V}$ is an element of the spread
$\mathcal{T}$.

##  3 Unitals from eggs

An egg in ${\rm PG}(4m-1,q)$ is a set $\mathcal{E}$ of $q^{2m}+1$ pairwise
disjoint $(m-1)-$dimensional subspaces such that any three egg elements span a
$(3m-1)-$dimensional subspace. When $m=1$, this definition recovers indeed the
notion of ovoid in ${\rm PG}(3,q)$. Therefore, since the notion of an egg,
introduced by J.A. Thas in [41], generalizes that of an ovoid, it make sense
to investigate whether it is possible to mimic Buekenhout’s construction to
get unitals in translation planes with dimension over their kernel greater
than two, by using eggs. Apart from the so-called elementary eggs, which are
obtained by applying the field reduction to an ovoid in ${\rm PG}(3,q^{m})$,
there are few other known examples of eggs, namely, the Kantor-Knuth eggs, the
Cohen-Ganley eggs and the (sporadic) Penttila-Williams egg; see [27] for an
explicit description of these objects.

Let $\mathcal{E}$ be an egg in ${\rm PG}(4m-1,q)$. For every egg element $E$
there exists a unique $(3m-1)$-dimensional subspace, denoted by $E^{*}$,
containing $E$ and disjoint from any other egg element; it is called the
tangent space of $\mathcal{E}$ at $E$. Therefore, the egg $\mathcal{E}$
defines an egg in the dual space of ${\rm PG}(4m-1,q)$, called the dual egg of
$\mathcal{E}$ and denoted by $\mathcal{E}^{D}$.

The following result is a corollary of Theorem 2.2.

######  Theorem 3.1.

Let $\mathcal{E}$ be an egg in ${\rm PG}(4m-1,q)$, and $E_{\infty}$ a fixed
egg element. Let $\Gamma^{\prime}$ be a $3m$-dimensional subsubspace of ${\rm
PG}(4m-1,q)$ containing the tangent space $E_{\infty}^{*}$ at $E_{\infty}$. In
$\Gamma^{\prime}$ we consider the sets:

$\mathcal{B}_{\mathcal{E}}=\\{E\cap\Gamma^{\prime}:E\in\mathcal{E},E\neq
E_{\infty}\\}$

and

$\mathcal{I}_{\mathcal{E}}=\\{E^{*}\cap E_{\infty}^{*}:E\in\mathcal{E},E\neq
E_{\infty}\\}.$

Let $\mathcal{F}_{\mathcal{E}}$ be the family of all affine $2m$-dimensional
subspaces of $\Gamma^{\prime}$ containing an element of
$\mathcal{I}_{\mathcal{E}}$, and assume that $\mathcal{B}_{\mathcal{E}}$ is a
minimal $\mathcal{F}_{\mathcal{E}}$-blocking set.

Embed $\Gamma^{\prime}$ in ${\rm PG}(4m,q)$ in such a way that
$E_{\infty}^{*}$ is a subspace of the hyperplane at infinity $\Sigma_{\infty}$
of ${\rm PG}(4m,q)$, and $\Gamma^{\prime}$ is an affine subspace.

If there exist a $(2m-1)$-spread $\mathcal{S}$ of $\Sigma_{\infty}$ and a
$(m-1)$-dimensional subspace $\mathcal{V}$ disjoint from $E_{\infty}^{*}$ and
contained in a spread element $\mathbf{z}$ such that
$\mathcal{I}_{\mathcal{E}}=\mathcal{I}(\mathcal{V})$, then the cone

$\mathcal{B}^{*}=\bigcup_{P\in\mathcal{B}_{\mathcal{E}}}{\langle
P,\mathcal{V}\rangle}\cup\\{\mathbf{z}\\}$

is a unital in $\Pi(\mathcal{S})$.

###### Proof.

Here, $\mathcal{B}_{\mathcal{E}}$ is a set of $q^{2m}$ points of
$\Gamma^{\prime}\setminus E^{*}_{\infty}$, and hence it consists of affine
points of ${\rm PG}(4m,q)$. Furthermore, every element in
$\mathcal{I}_{\mathcal{E}}$ is a $(2m-1)$-dimensional subspace of
$E^{*}_{\infty}$. By Theorem 2.2, if $\mathcal{F}_{\mathcal{E}}$ coincides
with the family $\mathcal{F}(\mathcal{V})$ previously defined, then
$\mathcal{B}^{*}$ is a unital in the semifield plane $\Pi(\mathcal{S})$. Since
$\mathcal{F}_{\mathcal{E}}$ consists of all affine $2m$-dimensional subspaces
of $\Gamma^{\prime}$ through an element of $\mathcal{I}_{\mathcal{E}}$, we get
that $\mathcal{F}_{\mathcal{E}}=\mathcal{F}(\mathcal{V})$ if and only if
$\mathcal{I}_{\mathcal{E}}=\mathcal{I}(\mathcal{V})$. ∎

An egg is said to be good at an element  $E$ if every $(3m-1)$-dimensional
subspace containing $E$ and at least two other egg elements, contains exactly
$q^{m}+1$ egg elements [42].

Let $\mathcal{K}$ be the quadratic cone in ${\rm PG}(3,q^{m})$ with equation
$X_{0}X_{1}=X_{2}^{2}$. A flock of $\mathcal{K}$ is a set of $q^{m}$ planes
partitioning the cone minus its vertex $V=\langle(0,0,0,1)\rangle$ into
disjoint conics. In accordance with this choice of coordinates, the planes of
a flock of $\mathcal{K}$ can be written as
$tX_{0}+f(t)X_{1}+g(t)X_{2}+X_{3}=0$, for all $t\in\mathbb{F}_{q^{m}}$, for
some $f,g:\mathbb{F}_{q^{m}}\rightarrow\mathbb{F}_{q^{m}}$. We denote this
flock by $\mathcal{F}(f,g)$. If $f$ and $g$ are linear over a subfield of
$\mathbb{F}_{q^{m}}$, then the flock is called a semifield flock. The maximal
subfield with this property is called the kernel of the flock.

From now on, we assume that the kernel of a semifield flock $\mathcal{F}(f,g)$
is $\mathbb{F}_{q}$. This implies that the $f$ and $g$ are
$\mathbb{F}_{q}$-linearized polynomials, i.e.

$f(t)=\sum_{i=0}^{m-1}{c_{i}t^{q^{i}}},\,\,\,\,\,g(t)=\sum_{i=0}^{m-1}{b_{i}t^{q^{i}}},$

for some $b_{i},c_{i}\in\mathbb{F}_{q^{m}}$, $i=0,\ldots,m-1$.

If a basis of $\mathbb{F}_{q^{m}}$ over $\mathbb{F}_{q}$ is fixed, then every
$r$-ple $(x_{1},\ldots,x_{r})\in\mathbb{F}_{q^{m}}^{r}$ can be viewed as a
$rm$-ple over $\mathbb{F}_{q}$, which will be denoted by
$(x_{1},\ldots,x_{r})_{q}$. In the paper [27] it was shown that for every
semifield flock $\mathcal{F}(f,g)$ there corresponds an egg in ${\rm
PG}(4m-1,q)$ whose dual, say $\mathcal{E}$, is good at an element, which can
be assumed to be $E_{\infty}$. Then, the elements and the tangent spaces of
$\mathcal{E}$ have the following form, respectively:

$\begin{array}[]{ll}E(a,b)=\\{(t,-g_{(a,b)}(t),-at,-bt)_{q}:t\in\mathbb{F}_{q^{m}}\\},\mathrm{\
for\ all\ }a,b\in\mathbb{F}_{q^{m}},\\\\[7.22743pt]
E_{\infty}=\\{(0,t,0,0)_{q}:t\in\mathbb{F}_{q^{m}}\\},\\\\[7.22743pt]
E^{*}(a,b)=\\{(t,h_{(a,b)}(r,s)+g_{(a,b)}(t),r,s)_{q}:t,r,s\in\mathbb{F}_{q^{m}}\\},\mathrm{\
for\ all\ }a,b\in\mathbb{F}_{q^{m}},\\\\[7.22743pt]
E^{*}_{\infty}=\\{(0,t,r,s)_{q}:t,r,s\in\mathbb{F}_{q^{m}}\\},\end{array}$ (2)

with

$g_{(a,b)}(t)=a^{2}t+\sum_{i=0}^{m-1}{(b_{i}ab+c_{i}b^{2})^{1/{q^{i}}}t^{1/{q^{i}}}},$

and

$h_{(a,b)}(r,s)=2ar+\sum_{i=0}^{m-1}{(b_{i}(as+br)+2c_{i}bs)^{1/{q^{i}}}}.$

Because of the expression of the polynomials $g(a,b)$ and $h(a,b)$, such an
egg will be denoted by $\mathcal{E}(\mathbf{b},\mathbf{c})$.

######  Theorem 3.2.

Let $\mathcal{E}=\mathcal{E}(\mathbf{b},\mathbf{c})$ be a good egg of ${\rm
PG}(4m-1,q)$, which is good at $E_{\infty}$. Then, the set
$\mathcal{B}_{\mathcal{E}}$ is a minimal $\mathcal{F}_{\mathcal{E}}$-blocking
set in $\Gamma^{\prime}={\rm PG}(3m,q)$ if and only if
$X^{2}+\sum_{i=0}^{m-1}{(b_{i}XY+c_{i}Y^{2})^{1/{q^{i}}}}+c=0$ has a solution
for all $c\in\mathbb{F}_{q^{m}}$.

###### Proof.

Let $\Gamma^{\prime}=\\{(u,t,r,s)_{q}:u\in\mathbb{F}_{q}\hbox{ and
}r,s,t\in\mathbb{F}_{q^{m}}\\}$. It is evident that $\Gamma^{\prime}$ is a
projective space of dimension $3m$ over $\mathbb{F}_{q}$ and it contains
$E^{*}_{\infty}$. By taking into account the general form of the elements of
$\mathcal{E}=\mathcal{E}(\mathbf{b},\mathbf{c})$, we get

$\mathcal{B}_{\mathcal{E}}=\\{\langle(1,-g_{(a,b)}(1),-a,-b)_{q}\rangle:a,b\in\mathbb{F}_{q^{m}}\\}$

and

$\mathcal{I}_{\mathcal{E}}=\\{I(a,b):a,b\in\mathbb{F}_{q^{m}}\\},$

where

$I(a,b)=E^{*}(a,b)\cap
E^{*}_{\infty}=\\{(0,h_{(a,b)}(r,s),r,s)_{q}:r,s\in\mathbb{F}_{q^{m}}\\}.$ (3)

All the affine $2m$-dimensional subspaces of $\Gamma^{\prime}$ through an
$I(a,b)$ are determined by joining it with an affine point of the affine
$m$-dimensional subspace spanned by $E_{\infty}$ and
$O=\langle(1,0,0,0,0)\rangle$. Therefore, the elements of
$\mathcal{F}_{\mathcal{E}}$ have the form

$F(a,b,c)=\\{(u,uc+h_{(a,b)}(r,s),r,s)_{q}:u\in\mathbb{F}_{q}\hbox{ and
}r,s\in\mathbb{F}_{q^{m}}\\},$

for all $a,b,c\in\mathbb{F}_{q^{m}}$.

A point
$P(x,y)=\langle(1,-g_{(x,y)}(1),-x,-y)_{q}\rangle\in\mathcal{B}_{\mathcal{E}}$
lies in $F(a,b,c)$ if and only if

$-g_{(x,y)}(1)=-h_{(a,b)}(x,y)+c$

or, equivalently, if and only if $(x,y)$ is a solution of

$X^{2}+\sum_{i=0}^{m-1}{(b_{i}XY+c_{i}Y^{2})^{1/{q^{i}}}}-2aX-\sum_{i=0}^{m-1}{(b_{i}(aY+bX)+2c_{i}bY)^{1/{q^{i}}}}+c=0.$
(4)

We refer to the polynomial on the left-hand side of the equation as
$H_{(a,b,c)}(x,y)$. Since $\mathcal{E}$ is an egg, for any given
$a,b\in\mathbb{F}_{q^{m}}$, the intersection of the tangent space $E^{*}(a,b)$
with $\Gamma^{\prime}$ is the $2m$-dimensional subspace
$F(a,b,c^{\prime})\in\mathcal{F}_{\mathcal{E}}$, with
$c^{\prime}=g_{(a,b)}(1)$. Therefore, through the point
$P(a,b)\in\mathcal{B}_{\mathcal{E}}$ there is the element
$F(a,b,c^{\prime})\in\mathcal{F}_{\mathcal{E}}$ intersecting
$\mathcal{B}_{\mathcal{E}}$ precisely at $P(a,b)$. This implies that
$\mathcal{B}_{\mathcal{E}}$ is a minimal $\mathcal{F}_{\mathcal{E}}$-blocking
set if and only if Eq. (4) has a solution
$(x,y)\in\mathbb{F}_{q^{m}}\times\mathbb{F}_{q^{m}}$ for any given elements
$a,b,c\in\mathbb{F}_{q^{m}}$.

From [26, Lemma 1.4], for any $a,b\in\mathbb{F}_{q^{m}}$, the linear
collineation

$\begin{array}[]{cccc}\psi_{a,b}:&{\rm PG}(4m-1,q)&\longrightarrow&{\rm
PG}(4m-1,q)\\\
&\langle(u,t,r,s)_{q}\rangle&\mapsto&\langle(u,t+h_{(a,b)}(r,s)-g_{(a,b)}(u),r-ua,s-ub)_{q}\rangle\end{array}$

fixes $E_{\infty}$ pointwise and maps $E(a^{\prime},b^{\prime})$ to
$E(a^{\prime}+a,b^{\prime}+b)$. In addition, $\psi_{a,b}$ fixes
$\Gamma^{\prime}$, and hence $\mathcal{B}_{\mathcal{E}}$. A straightforward,
though tedious, calculation shows that $\psi_{a,b}$ acts also on the set of
tangent spaces by fixing $E^{*}_{\infty}$ setwise and mapping
$E^{*}(a^{\prime},b^{\prime})$ to $E^{*}(a+a^{\prime},b+b^{\prime})$. This
implies that $\psi_{a,b}$ fixes both $\mathcal{I}_{\mathcal{E}}$ and
$\mathcal{F}_{\mathcal{E}}$ setwise; in particular,
$F(a^{\prime},b^{\prime},c)$ is mapped to
$F(a+a^{\prime},b+b^{\prime},c^{\prime})$, with
$c^{\prime}=c-g_{a,b}(1)+h_{(a^{\prime}+a,b^{\prime}+b)}(a,b)$. This means
that, because of the linearity of the second sum in Eq. (4),
$H_{(a^{\prime}+a,b^{\prime}+b,c)}(x,y)=0$ has a solution for all
$c\in\mathbb{F}_{q^{m}}$ if and only if $H_{(a,b,c)}(x,y)=0$ has a solution
for all $c\in\mathbb{F}_{q^{m}}$. Therefore, $\mathcal{B}_{\mathcal{E}}$ is a
minimal $\mathcal{F}_{\mathcal{E}}$-blocking set if and only if, for a fixed
pair $(a,b)\in\mathbb{F}_{q^{m}}\times\mathbb{F}_{q^{m}}$, Eq. (4) has at
least one solution $(x,y)\in\mathbb{F}_{q^{m}}\times\mathbb{F}_{q^{m}}$, for
all $c\in\mathbb{F}_{q^{m}}$. In particular, we can chose $(a,b)=(0,0)$ so
that Eq. (4) reduces to

$X^{2}+\sum_{i=0}^{m-1}{(b_{i}XY+c_{i}Y^{2})^{1/{q^{i}}}}+c=0.$ (5)

∎

##  4 A new unital in a Dickson commutative semifield plane

In [35], Penttila and Williams constructed an ovoid of the parabolic quadric
$Q(4,3^{5})$ in ${\rm PG}(4,3^{5})$, i.e., a set $\mathcal{O}$ of $3^{10}+1$
points having exactly one point on each generator of the quadric. Moreover,
$\mathcal{O}$ is a translation ovoid, meaning that the points of $\mathcal{O}$
can be coordinatized by using functions that are additive over
$\mathbb{F}_{3}$. According to a construction given in [28], such a
translation ovoid corresponds to a semifield flock of the quadratic cone in
${\rm PG}(3,3^{5})$, which, in turn, corresponds to a generalized quadrangle
with parameters $(3^{10},3^{5})$, whose point-line dual is a translation
generalized quadrangle. By a result of Payne and Thas [34, 8.7.1], the latter
generalized quadrangle is isomorphic to $T(\mathcal{E})$ for some egg
$\mathcal{E}$ in ${\rm PG}(19,3)$. By Theorem 3.4 in [27], the dual egg of
$\mathcal{E}$ forms a good egg $\mathcal{E}^{D}$ in ${\rm PG}(19,3)$. Whence,
via the above correspondences, the Penttila-Williams ovoid of $Q(4,3^{5})$
gives rise to a good (dual) egg in ${\rm PG}(19,3)$. In order to simplify the
notation, we will refer to it as
$\mathcal{E}=\mathcal{E}(\mathbf{b},\mathbf{c})$ with
$\mathbf{b}=(0,1,0,0,0)$, $\mathbf{c}=(0,0,0,-1,0)$; see [27].

According to the expressions of the polynomials $g_{(a,b)}(t)$ and
$h_{(a,b)}(r,s)$ in this case, the egg elements of $\mathcal{E}$ are defined
by the polynomials

$g_{(a,b)}(t)=a^{2}t-(b^{2})^{3^{2}}t^{3^{2}}+(ab)^{3^{4}}t^{3^{4}}$ (6)

and

$h_{(a,b)}(r,s)=-ar+b^{3^{2}}s^{3^{2}}+(br+as)^{3^{4}},$ (7)

for all $a,b\in\mathbb{F}_{3^{5}}$.

Let $p$ be an odd prime and $\xi$ a non-square in $\mathbb{F}_{p^{m}}$. By
[17, p.241], the multiplication defined by

$(x,y)*(a,b)=(ax+\xi b^{\alpha}y^{\alpha},bx+ay)$

with $\alpha\in{\rm Aut}(\mathbb{F}_{p^{m}})$ not the identity, turns
$\mathbb{F}_{p^{m}}^{2}$ into a Dickson commutative semifield of order
$p^{2m}$ which we denote by ${\mathbb{D}}={\mathbb{D}}(p^{m},\xi,\alpha)$. In
particular, its middle nucleus is
$\mathcal{N}_{m}=\\{(a,0):a\in\mathbb{F}_{p^{m}}\\}$, and its left nucleus is
$\mathcal{N}_{l}=\\{(a,0):a\in\mathrm{Fix}(\alpha)\\}$, coinciding with its
center $\mathcal{K}$.

Now, let $p=3$ and $m=5$. For any pair $(a,b)\in\mathbb{F}_{3^{5}}^{2}$, we
consider the following map

$\begin{array}[]{cccc}\tau_{(a,b)}:&(x,y)&\mapsto&(bx+ay,-ax+b^{3^{2}}y^{3^{2}}),\end{array}$

which defines the subspaces
$S(a,b)=\\{((x,y)^{\tau_{(a,b)}},x,y)_{3}:x,y\in\mathbb{F}_{3^{5}}\\}$ of
${\rm PG}(19,3)$. Set
$\mathcal{S}=\\{S(a,b):a,b\in\mathbb{F}_{3^{5}}\\}\cup\\{S_{\infty}\\}$, where
$S_{\infty}=\\{(x,y,0,0)_{3}:x,y\in\mathbb{F}_{3^{5}}\\}$.

Let $\varphi$ be the linear map $\varphi:(x,y)\mapsto(-y,x)$. Then, the set
$\\{\varphi\tau_{(a,b)}:a,b\in\mathbb{F}_{3^{5}}\\}$ is precisely the spread
set of $\mathbb{F}_{3}^{10}$ associated with the Dickson commutative semifield
${\mathbb{D}}={\mathbb{D}}(3^{5},-1,3^{2})$.

It turns out that $\mathcal{S}$ is a 9-spread of $\Sigma_{\infty}={\rm
PG}(19,3)$ and, by [2], the translation plane $\Pi(\mathcal{S})$ is isomorphic
to the Dickson commutative semifield plane $\pi({\mathbb{D}})$.

Let $\mathcal{V}=\\{(t,-t^{3^{4}},0,0)_{3}:t\in\mathbb{F}_{3^{5}}\\}$. Then,
$\mathcal{V}$ is contained in the spread element $\mathbf{z}=S_{\infty}$ and
it intersects trivially the subspace
$\Gamma=E^{*}_{\infty}=\\{(0,t,r,s)_{3}:t,r,s\in\mathbb{F}_{3^{5}}\\}$. We
also have

$\langle
S(a,b),\mathcal{V}\rangle=\\{(br+as+t,-ar+b^{3^{2}}s^{3^{2}}-t^{3^{4}},r,s)_{3}:t,r,s\in\mathbb{F}_{3^{5}}\\},$

giving

$\langle
S(a,b),\mathcal{V}\rangle\cap\Gamma=\\{(0,-ar+b^{3^{2}}s^{3^{2}}+(br+as)^{3^{4}},r,s)_{3}:r,s\in\mathbb{F}_{3^{5}}\\}$

which is precisely the subspace $I(a,b)$ defined by expression (3), with
$h_{(a,b)}(r,s)$ as in (7).

######  Proposition 4.1.

The set $\mathcal{B}_{\mathcal{E}}$ defined by the Penttila-Williams egg
$\mathcal{E}=\mathcal{E}(\mathbf{b},\mathbf{c})$ is a minimal
$\mathcal{F}(\mathcal{E})$-blocking set.

###### Proof.

By taking into account Theorem 3.2, $\mathcal{B}_{\mathcal{E}}$ is a minimal
$\mathcal{F}_{\mathcal{E}}$-blocking set if and only if

$X^{2}+(XY)^{3^{4}}-(Y^{2})^{3^{2}}=-c$ (8)

has a solution for all $c\in\mathbb{F}_{q^{m}}$.

We distinguish two cases: $-c$ is a square in $\mathbb{F}_{q^{m}}$ or not. If
$-c$ is a square, then $(\pm\sqrt{-c},0)$ are solutions of Eq. (8); if $-c$ is
not a square, then $(0,\pm\sqrt{c^{3^{3}}})$ are solutions of Eq. (8). ∎

By Theorem 3.1, the cone

$\mathcal{B}^{*}_{\mathcal{E}}=\\{\langle(1,c,-g_{(a,b)}(1)-c^{3^{4}},-a,-b)_{3}\rangle:a,b,c\in\mathbb{F}_{3^{5}}\\}\cup\\{S_{\infty}\\},$

with $g_{(a,b)}(t)$ as in (6), is a unital in the translation plane
$\Pi(\mathcal{S})$.

Consider the collineation of ${\rm PG}(20,3)$ defined as
$\overline{\varphi}:\langle(u,v,t,r,s)_{3}\rangle\mapsto\langle(u,-t,v,r,s)_{3}\rangle$.
Then, $\Pi(\mathcal{S})^{\overline{\varphi}}$ represents the Dickson
commutative semifield plane $\pi({\mathbb{D}})$. It turns out that the set

$\mathcal{U}=\\{(g_{(a,b)}(1)+c^{3^{4}},c,-a,-b):a,b,c\in\mathbb{F}_{3^{5}}\\}\cup\\{(\infty)\\}$

is a unital in $\pi({\mathbb{D}})$. Note that $\mathcal{U}$ cannot be a
Buekenhout-Metz unital since $\pi({\mathbb{D}})$ is a 10-dimensional
translation plane over its kernel $\mathbb{F}_{3}$. On the other hand, as
$\pi({\mathbb{D}})$ admits unitary polarities [21], $\mathcal{U}$ might be a
polar unital. The following result shows that this is not the case.

######  Theorem 4.2.

The unital $\mathcal{U}$ is not a polar unital in $\pi({\mathbb{D}})$.

###### Proof.

Since the tangent space at the egg element $E(0,0)$ is $E^{*}(0,0)$, the
tangent line of $\Pi(\mathcal{S})$ at the point
$O=\langle(1,0,0,0,0)_{3}\rangle\in\mathcal{B}^{*}_{\mathcal{E}}$ is the
subspace spanned by $S(0,0)$ and $O$. Then, the tangent line of
$\pi({\mathbb{D}})$ at $(0,0)\in\mathcal{U}$ is $[0,0]$.

From [24, Theorem 2.1], any unitary polarity of $\pi({\mathbb{D}})$ mapping
$(0,0)$ to $[0,0]$ is given by

$\begin{array}[]{cccc}\rho_{a}:&(x_{1},x_{2},y_{1},y_{2})&\leftrightarrow&[ax_{1},-ax_{2},-y_{1},y_{2}],\\\
&(m_{1},m_{2})&\leftrightarrow&(a^{-1}m_{1},-a^{-1}m_{2})\\\
&(\infty)&\leftrightarrow&[\infty].\end{array}$

for some non-zero $a\in\mathbb{F}_{3^{5}}$.

The unital $\mathcal{U}$ is a polar unital with respect to $\rho_{a}$, for
some $a\in\mathbb{F}_{3^{5}}$, if and only if each of its points is an
absolute point. Straightforward calculations show that the point
$(1,1,0,0)\in\mathcal{U}$ is not incident with $\rho_{a}(1,1,0,0)=[a,-a,0,0]$
for all non-zero $a\in\mathbb{F}_{3^{5}}$, showing that $\mathcal{U}$ is not a
polar unital. ∎

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