# Clubs and their applications

Vito Napolitano  ,  Olga Polverino  ,  Paolo Santonastaso  and  Ferdinando
Zullo  Vito Napolitano, Olga Polverino, Paolo Santonastaso and Ferdinando
Zullo, Dipartimento di Matematica e Fisica, Università degli Studi della
Campania “Luigi Vanvitelli”, Viale Lincoln, 5, I– 81100 Caserta, Italy
<EMAIL_ADDRESS>
###### Abstract.

Clubs of rank $k$ are well-celebrated objects in finite geometries introduced
by Fancsali and Sziklai in 2006. After the connection with a special type of
arcs known as KM-arcs, they renewed their interest. This paper aims to study
clubs of rank $n$ in $\mathrm{PG}(1,q^{n})$. We provide a classification
result for $(n-2)$-clubs of rank $n$, we analyze the
$\operatorname{\Gamma\mathrm{L}}(2,q^{n})$-equivalence of the known subspaces
defining clubs, for some of them the problem is then translated in determining
whether or not certain scattered spaces are equivalent. Then we find a
polynomial description of the known families of clubs via some linearized
polynomials. Then we apply our results to the theory of blocking sets, KM-
arcs, polynomials and rank metric codes, obtaining new constructions and
classification results.

AMS subject classification: 51E20; 51E21; 94B05.

Keywords: Club; linear set; linearized polynomial; KM-arc; Blocking set; rank
metric code.

##  1\. Introduction

Linear sets have been shown to be a powerful tool in several classification
results and constructions of objects in finite geometry and coding theory,
such as semifields, blocking sets, translation ovoids, KM-arcs, rank metric
codes, etc. For more details on their applications we refer to [24, 34, 36].
The main topic of this paper regards a special class of linear sets on the
projective line called _clubs_. Let
$\Lambda=\mathrm{PG}(V,\mathbb{F}_{q^{n}})=\mathrm{PG}(1,q^{n})$, where $V$ is
an $\mathbb{F}_{q^{n}}$-vector space of dimension $2$. Let $U$ be an
${\mathbb{F}}_{q}$-subspace of $V$, then the set

$L_{U}=\\{\langle{u}\rangle_{\mathbb{F}_{q^{n}}}\colon{u}\in
U\setminus\\{{0}\\}\\}$

is said to be an ${\mathbb{F}}_{q}$-_linear set_ of rank
$\dim_{{\mathbb{F}}_{q}}(U)$ of $\Lambda$. Another important notion associated
with linear sets is those of _weight of a point_ $P$ which, roughly speaking,
represents a measure of how much of the linear set is contained in the point.
We say that a linear set is _scattered_ if all of its points have weight one,
that is its points share with the linear set the minimum possible. The term
scattered was introduced for the first time in [7] by Blokhuis and Lavrauw, in
which the authors studied this notion in a more general framework. Scattered
linear sets recently gained more attention due to their applications in the
theory of Maximum Rank Distance codes and more generally to rank metric codes;
see [36]. A quite close to the scattered linear sets family is the one of
_clubs_. An $i$-_club_ of rank $k$ is an ${\mathbb{F}}_{q}$-linear set $L_{U}$
of rank $k$ for which all but one of the points have weight one, the remaining
one has weight $i$. In the case in which the rank is $n$ we will simply say
that $L_{U}$ is an $i$-_club_. They were originally introduced by in Fancsali
and Sziklai in [19] (see also [20]) when studying maximal partial $2$-spreads
in $\mathrm{PG}(8,q)$. These linear sets renewed their interest when De Boeck
and Van de Voorde in [15] characterized the translation KM-arcs exactly as
those that can be described by $i$-clubs in even characteristic and such a
connection enabled them to give new constructions and classifications of KM-
arcs. Moreover, $i$-clubs also define linear blocking sets of Rédei type once
we look to the $i$-club as the set of determined directions of an affine
pointset. Apart for their own interesting combinatorial properties, these
pointsets turn out to be also very interesting since they also define Hamming
metric codes with _few weights_ ; see [32]. Furthermore, $i$-clubs have a very
natural algebraic description via linearized polynomials, which have been
recently investigated in [5], under the name of $1$-_fat polynomials_. In
particular, in [5] it has been proved that _exceptional_ $1$-fat polynomials
do not exist, which means that the nature of these polynomial heavily rely on
the extension field in which we are looking for this kind of polynomials.

In this paper we start by providing a classification for $(n-2)$-clubs in
$\mathrm{PG}(1,q^{n})$, by first studying a class of linear sets which, up to
equivalence, contains all the $(n-2)$-clubs and then using an algebraic result
which extends the linear analogue of Vosper’s theorem under certain
assumptions. Then we study the
$\operatorname{\Gamma\mathrm{L}}(2,q^{n})$-equivalence of the known subspaces
defining clubs (not necessarily $(n-2)$-clubs), in which the scattered spaces
play an important role. Then we find linearized polynomials defining the known
families of clubs, in particular for the $(n-2)$-clubs we find all of them (up
to equivalence). Finally we apply our results to the theory of blocking sets,
KM-arcs and rank metric codes, in such a way that we obtain constructions and
classification results. In particular, our results imply, as a special case,
the classification results of translation KM-arcs of type $q/4$ provided in
[15], but with an explicit description of the associated clubs.

The paper is organized as follows: in Section 2 we will briefly describe some
of the objects and related results we will use. More precisely, we recall
linearized polynomials, linear sets and clubs, dual basis and some other
algebraic results. In Section 3, we first detect a family of linear sets of
rank $h+2$ containing a point of weight $h$, then we show that, up to
equivalence, all the $(n-2)$-clubs belong to such a family. Therefore, we
characterize the linear sets belonging to such a family which turn out to be
clubs via an algebraic description which allowed us to use a generalization
(under certain assumption) of the linear analogue of the Vosper’s theorem.
Section 4 is devoted to the study of the
$\operatorname{\Gamma\mathrm{L}}(2,q^{n})$-equivalence of the known subspaces
whose associated linear set defines an $i$-club. Our analysis shows that there
is a wide variety of examples. Indeed, for the constructions shown by Gács and
Weiner in [21] we showed that the number of inequivalent subspaces is related
to the number of scattered subspaces in a smaller extension. We also detect
some inequivalence subspaces also when considering $(n-2)$-clubs. In Section 5
we describe linearized polynomials that define the known families of clubs and
in particular our description is complete when considering $(n-2)$-clubs,
answering to an open problem posed in [10]. Section 6 is devoted to the study
of blocking sets of Rédei type naturally associated with $i$-clubs and to
translation KM-arcs, for which we are able to provide classification results
and non-equivalent constructions, as a consequence of the results described
for $i$-clubs. In Section 8 we deal with rank metric codes. Using the
connection between $q$-systems and linear rank metric codes proved by
Randrianarisoa in [37], we are able to classify and construct linear rank
metric codes with certain parameters. The interest in these codes is again due
to the fact that the number of the non zero weights is only three, and for
these kind of codes it seems very difficult to find examples.

##  2\. Preliminaries

###  2.1. Linearized polynomials

Let $q=p^{h}$, where $h$ is a positive integer and $p$ is a prime. A
_linearized polynomial_ (or a $q$-_polynomial_) over ${\mathbb{F}}_{q^{n}}$ is
a polynomial of the form

$f(x)=\sum_{i=0}^{t}a_{i}x^{q^{i}},$

where $a_{i}\in{\mathbb{F}}_{q^{n}}$ and $t$ is a non-negative integer.
Furthermore, if $a_{t}\neq 0$ we say that $t$ is the $q$-_degree_ of $f(x)$.
The set of all $q$-polynomials over ${\mathbb{F}}_{q^{n}}$ will be denoted by
$\mathcal{L}_{n,q}$. A well-known example of linearized polynomial is given by
the trace, that is $\mathrm{Tr}_{q^{n}/q}(x)=x+x^{q}+\ldots+x^{q^{n-1}}$.

Sheekey in [38], in connection with linear sets and rank metric codes, gave
the following definition. Let $f(x)\in\mathcal{L}_{n,q}$. Then $f$ is said to
be _scattered_ if for every $x,y\in\mathbb{F}_{q^{n}}^{*}$

$\frac{f(x)}{x}=\frac{f(y)}{y}\,\,\Leftrightarrow\,\,\frac{y}{x}\in{\mathbb{F}}_{q},$

or equivalently

$\dim_{{\mathbb{F}}_{q}}(\ker(f(x)-mx))\leq 1,\,\,\,\text{for
every}\,\,\,m\in\mathbb{F}_{q^{n}}.$

The term _scattered_ arises from the theory of linear sets; see [7]. Scattered
polynomials have attracted a lot of attention because of their connections
with several objects, such as maximum distance rank metric codes as we will
see later.

For more details on linearized polynomials see [42].

###  2.2. Linear sets

In this paper, we deal with linear sets on the projective line and in some
cases also on the projective plane. More precisely, let $V$ be a
$r$-dimensional $\mathbb{F}_{q^{n}}$-vector space and let
$\Lambda=\mathrm{PG}(V,\mathbb{F}_{q^{n}})=\mathrm{PG}(r-1,q^{n})$. Let $U$ be
an ${\mathbb{F}}_{q}$-subspace of $V$, then the set

$L_{U}=\\{\langle{u}\rangle_{\mathbb{F}_{q^{n}}}\colon{u}\in
U\setminus\\{{0}\\}\\}$

is said to be an ${\mathbb{F}}_{q}$-_linear set_ of rank
$\dim_{{\mathbb{F}}_{q}}(U)$. The rank of $L_{U}$ will also be denoted by
$\mathrm{Rank}(L_{U})$. The _weight_ of a projective subspace
$S=\mathrm{PG}(W,\mathbb{F}_{q^{n}})\subseteq\Lambda$ in $L_{U}$ is defined
naturally as $w_{L_{U}}(S)=\dim_{{\mathbb{F}}_{q}}(U\cap W)$. In some cases,
linearized polynomials can be used to describe linear sets. Indeed, suppose
that $L_{U}$ is a linear set in $\mathrm{PG}(1,q^{n})$ and it has rank $n$,
then there exists a $q$-polynomial $f(x)\in\mathcal{L}_{n,q}$ such that
$L_{U}$ is mapped by a collineation of $\mathrm{PGL}(2,q^{n})$ into
$L_{f}=L_{U_{f}}$ where

$U_{f}=\\{(x,f(x))\colon x\in\mathbb{F}_{q^{n}}\\}.$

Nevertheless, it could be hard to find polynomials describing a fixed linear
set.

Let now recall some basics relations between the size of a linear set, the
number of point of a certain weight and its rank. If $N_{i}$ denotes the
number of points of $\Lambda$ having weight $i\in\\{0,\ldots,k\\}$ in $L_{U}$,
the following relations hold:

(1) $|L_{U}|\leq\frac{q^{k}-1}{q-1},$ (2) $|L_{U}|=N_{1}+\ldots+N_{k},$ (3)
$N_{1}+N_{2}(q+1)+\ldots+N_{k}(q^{k-1}+\ldots+q+1)=q^{k-1}+\ldots+q+1.$

Moreover, the following holds

(4) $w_{L_{U}}(P)+w_{L_{U}}(Q)\leq\mathrm{Rank}(L_{U}),$

for any $P,Q\in\mathrm{PG}(r-1,q^{n})$ with $P\neq Q$. An
${\mathbb{F}}_{q}$-linear set $L_{U}$ is $\mathrm{PG}(1,q^{n})$ for which all
of its points have weight one is called a _scattered_ linear set. If all of
its points have weight one except for one which has weight $i$, we call it
$i$-_club_.

By the above relations, it is easy to see that a scattered linear set of rank
$k$ has $\frac{q^{k}-1}{q-1}$ points and an $i$-club of rank $k$ has size
$q^{k-1}+\ldots+q^{i}+1$.

We refer to [24] and [34] for comprehensive references on linear sets.

###  2.3. Known examples of clubs

Up to now, very few examples of $i$-clubs are known, which have been found and
summarized in [15], where the authors investigated such a family of linear
sets in connection with KM-arcs; see Section 6.

The linear set $L_{\mathrm{Tr}_{q^{n}/q}(x)}$ is an example of $(n-1)$-club
and in [11, Theorem 3.7], it has been proved that every $(n-1)$-club is
$\mathrm{PGL}(2,q^{n})$-equivalent to $L_{\mathrm{Tr}_{q^{n}/q}(x)}$.

A further important example is the following, which extends the above example.
Let $n=\ell m$, $i=m(\ell-1)$, $\gcd(s,m)=1$ and $\sigma\colon
x\in\mathbb{F}_{q^{n}}\mapsto x^{q^{s}}\in\mathbb{F}_{q^{n}}$. Then the linear
set $L_{T}$ where

(5) $T(x)=\mathrm{Tr}_{q^{\ell m}/q^{m}}\circ\sigma(x)\in\mathcal{L}_{n,q}$

is an $i$-club in $\mathrm{PG}(1,q^{n})$ with $i=m(\ell-1)$. Apart from these
examples, polynomials defining $i$-clubs are not known except for [5,
Corollary 5.5] for $n=4$ (see also [13]) and [5, Corollary 6.3]. However, in
[15] two other examples of clubs were described.

######  Construction 2.1.

In [15, Lemma 2.12] it was proved that for any
$\lambda\in\mathbb{F}_{q^{n}}^{*}$ such that
$\\{1,\lambda,\ldots,\lambda^{n-1}\\}$ is an ${\mathbb{F}}_{q}$-basis of
$\mathbb{F}_{q^{n}}$, the ${\mathbb{F}}_{q}$-linear set

(6)
$L_{\lambda}=\\{\langle(t_{1}\lambda+\ldots+t_{n-1}\lambda^{n-1},t_{n-1}+t_{n}\lambda)\rangle_{\mathbb{F}_{q^{n}}}\colon(t_{1},\ldots,t_{n})\in{\mathbb{F}}_{q}^{n}\setminus\\{\mathbf{0}\\}\\},$

is an $(n-2)$-club of rank $n$ in $\mathrm{PG}(1,q^{n})$.

######  Construction 2.2.

In [15, Lemma 3.6] was detected the following $i$-clubs. Let $n=rt$, $t,r>1$
and $f$ a scattered polynomial in $\mathcal{L}_{t,q}$. Let

(7)
$U_{a,b}=\left\\{\left(f(x_{0})-ax_{0},bx_{0}+\sum_{i=1}^{r-1}x_{i}\omega^{i}\right)\colon
x_{i}\in{\mathbb{F}}_{q^{t}}\right\\},$

for some fixed $a,b\in{\mathbb{F}}_{q^{t}}$ with $b\neq 0$ and
$\\{1,\omega,\ldots,\omega^{r-1}\\}$ an ${\mathbb{F}}_{q^{t}}$-basis of
${\mathbb{F}}_{q^{n}}$. Then $L_{U_{a,b}}$ is an $i$-club, with

$i=\left\\{\begin{array}[]{ll}t(r-1),&\text{if}\,\,f(x)-ax\,\,\text{is
invertible over}\,\,{\mathbb{F}}_{q^{t}},\\\
t(r-1)+1,&\text{otherwise}.\end{array}\right.$

###  2.4. Algebraic preliminaries

In this section we will first recall the notion of dual basis and properties,
then we will briefly describe the linear analogue of Cauchy-Davenport
inequality and Vosper’s theorem and some related results.

We start by recalling that the trace function from $\mathbb{F}_{q^{n}}$ over
${\mathbb{F}}_{q}$ is a linear map that defines a nondegenerate symmetric
bilinear form as follows:

$(a,b)\in{\mathbb{F}}_{q^{n}}\times{\mathbb{F}}_{q^{n}}\mapsto\mathrm{Tr}_{q^{n}/q}(ab)\in{\mathbb{F}}_{q}.$

This allows us to give the following definition. Two ordered
${\mathbb{F}}_{q}$-bases $\mathcal{B}=(\xi_{0},\ldots,\xi_{n-1})$ and
$\mathcal{B}^{*}=(\xi_{0}^{*},\ldots,\xi_{n-1}^{*})$ of ${\mathbb{F}}_{q^{n}}$
are said to be _dual bases_ if

$\mathrm{Tr}_{q^{n}/q}(\xi_{i}\xi_{j}^{*})=\left\\{\begin{array}[]{ll}1&\text{if
}i=j,\\\ 0&\text{if }i\neq j.\end{array}\right.$

For any ${\mathbb{F}}_{q}$-basis $\mathcal{B}=(\xi_{0},\ldots,\xi_{n-1})$
there exists a unique dual basis
$\mathcal{B}^{*}=(\xi_{0}^{*},\ldots,\xi_{n-1}^{*})$ of $\mathcal{B}$, see
e.g. [25, Definition 2.30].

When $\mathcal{B}$ is a polynomial basis, then using the minimal polynomial of
the element defining the polynomial basis over ${\mathbb{F}}_{q}$, we can
easily construct the relative dual basis.

######  Corollary 2.3.

[31, Corollary 2.7] Let $\lambda\in\mathbb{F}_{q^{n}}$ such that
$\mathcal{B}=(1,\lambda,\ldots,\lambda^{n-1})$ is an ordered
${\mathbb{F}}_{q}$-basis of $\mathbb{F}_{q^{n}}$. Let
$f(x)=a_{0}+a_{1}x+\ldots+a_{n-1}x^{n-1}+x^{n}$ be the minimal polynomial of
$\lambda$ over ${\mathbb{F}}_{q}$. Then the dual basis $\mathcal{B}^{*}$ of
$\mathcal{B}$ is

$\mathcal{B}^{*}=(\delta^{-1}\gamma_{0},\ldots,\delta^{-1}\gamma_{n-1}),$

where $\delta=f^{\prime}(\lambda)$ and
$\gamma_{i}=\sum_{j=1}^{n-i}\lambda^{j-1}a_{i+j}$, for every
$i\in\\{0,\ldots,n-1\\}$.

In the case in which the minimal polynomial of $\lambda$ has degree two or
three, the dual basis has even a simpler description.

######  Corollary 2.4.

[31, Corollary 2.9] Let $\lambda\in\mathbb{F}_{q^{n}}$ such that
$\mathcal{B}=(1,\lambda,\ldots,\lambda^{n-1})$ is an ordered
${\mathbb{F}}_{q}$-basis of $\mathbb{F}_{q^{n}}$.

  * •

If $f(x)=x^{n}-d$ is the minimal polynomial of $\lambda$ over
${\mathbb{F}}_{q}$, then the dual basis of $\mathcal{B}$ is

$\mathcal{B}^{*}=\left(\frac{\lambda^{n}}{nd},\frac{\lambda^{n-1}}{nd},\ldots,\frac{\lambda}{nd},\frac{1}{nd}\right).$

  * •

If $f(x)=x^{n}-cx^{k}-1$ is the minimal polynomial of $\lambda$ over
${\mathbb{F}}_{q}$, then the dual basis of $\mathcal{B}$ is

$\mathcal{B}^{*}=(\delta^{-1}\lambda^{k-1},\delta^{-1}\lambda^{k-2},\ldots,\delta^{-1},\delta^{-1}\lambda^{n-1},\ldots,\delta^{-1}\lambda^{k}),$

where $\delta=\frac{n\lambda^{n-1}-ck\lambda^{k-1}}{-c+\lambda^{n-k}}$.

We also need the following result, which states that every hyperplane in
${\mathbb{F}}_{q^{n}}$ admits a polynomial basis.

######  Proposition 2.5.

Let $S$ be an $(n-1)$-dimensional ${\mathbb{F}}_{q}$-subspace of
${\mathbb{F}}_{q^{n}}$. Let $\gamma\in{\mathbb{F}}_{q^{n}}^{*}$ such that
${\mathbb{F}}_{q}(\gamma)={\mathbb{F}}_{q^{n}}$. Then there exists
$c\in{\mathbb{F}}_{q^{n}}^{*}$ such that $S=c\langle
1,\gamma,\ldots,\gamma^{n-2}\rangle_{{\mathbb{F}}_{q}}$.

###### Proof.

Since $\\{1,\gamma,\ldots,\gamma^{n-1}\\}$ is an ${\mathbb{F}}_{q}$-basis of
${\mathbb{F}}_{q^{n}}$, then the ${\mathbb{F}}_{q}$-subspace $T=\langle
1,\gamma,\ldots,\gamma^{n-2}\rangle_{{\mathbb{F}}_{q}}$ is an hyperplane of
$\mathbb{F}_{q^{n}}$. By [22, Propositon 3.5.16], for any hyperlane $S$ of
$\mathbb{F}_{q^{n}}$ there exists $c\in{\mathbb{F}}_{q^{n}}^{*}$ such that
$S=cT$ and the assertion is proved. ∎

We now recall some algebraic results which will be used in the rest of the
paper. The first two correspond to the linear analogue of Cauchy-Davenport
inequality and to the linear analogue of the Vosper’s Theorem. Let denote by
$S\cdot T$ the set of the products of any two elements in the subset $S$ and
$T$ of $\mathbb{F}_{q^{n}}$.

######  Theorem 2.6.

[2, Theorem 3] Let $S$ be an ${\mathbb{F}}_{q}$-subspace of
${\mathbb{F}}_{q^{n}}$. Then

  * i)

either for every ${\mathbb{F}}_{q}$-subspace $T\subseteq{\mathbb{F}}_{q^{n}}$
we have

$\dim_{{\mathbb{F}}_{q}}(\langle S\cdot
T\rangle_{{\mathbb{F}}_{q}})\geq\min\\{\dim_{{\mathbb{F}}_{q}}(S)+\dim_{{\mathbb{F}}_{q}}(T)-1,n\\},$

  * ii)

or there exists a positive integer $t>1$ that divides $n$, such that for every
${\mathbb{F}}_{q}$-subspace $T\subseteq{\mathbb{F}}_{q^{n}}$ satisfying

$\dim_{{\mathbb{F}}_{q}}(\langle S\cdot
T\rangle_{{\mathbb{F}}_{q}})<\dim_{{\mathbb{F}}_{q}}(S)+\dim_{{\mathbb{F}}_{q}}(T)-1,$

we have that $\langle S\cdot T\rangle_{{\mathbb{F}}_{q}}$ is also an
${\mathbb{F}}_{q^{t}}$-subspace.

The pairs $(S,T)$ satisfying the equality in i) of Theorem 2.6 with

$\dim_{{\mathbb{F}}_{q}}(\langle S\cdot
T\rangle_{{\mathbb{F}}_{q}})=\dim_{{\mathbb{F}}_{q}}(S)+\dim_{{\mathbb{F}}_{q}}(T)-1,$

are called _critical pairs_. The classification of critical pairs is a hard
problem in general and few results are known. The next result classify them
when $n$ is a prime number.

######  Theorem 2.7.

[1, Theorem 3] Suppose $n$ is a prime number. Let $S,T$ be
${\mathbb{F}}_{q}$-subspaces of ${\mathbb{F}}_{q^{n}}$ such that
$2\leq\dim_{{\mathbb{F}}_{q}}(S),\dim_{{\mathbb{F}}_{q}}(T)$ and
$\dim_{{\mathbb{F}}_{q}}(\langle S\cdot T\rangle_{{\mathbb{F}}_{q}})\leq n-2$.
If

$\dim_{{\mathbb{F}}_{q}}(\langle S\cdot
T\rangle_{{\mathbb{F}}_{q}})=\dim_{{\mathbb{F}}_{q}}(S)+\dim_{{\mathbb{F}}_{q}}(T)-1,$

then $S=g\langle
1,a,\ldots,a^{\dim_{{\mathbb{F}}_{q}}(S)-1}\rangle_{{\mathbb{F}}_{q}}$ and
$T=g^{\prime}\langle
1,a,\ldots,a^{\dim_{{\mathbb{F}}_{q}}(T)-1}\rangle_{{\mathbb{F}}_{q}}$, for
some $g,g^{\prime},a\in{\mathbb{F}}_{q^{n}}$.

Here we recall a result from [30], whose consequence is the classification of
critical pairs in the case in which one of the subspaces has dimension $2$;
see [30, Proposition 6.3].

######  Lemma 2.8.

[30, Lemma 3.1] Let $S$ be an ${\mathbb{F}}_{q}$-subspace of
$\mathbb{F}_{q^{n}}$ of dimension $k\geq 2$ and let
$\mu\in\mathbb{F}_{q^{n}}\setminus{\mathbb{F}}_{q}$. Let
$t=\dim_{{\mathbb{F}}_{q}}({\mathbb{F}}_{q}(\mu))$.

  * (a)

If $\dim_{{\mathbb{F}}_{q}}(S\cap\mu S)=k$, then $S$ is an
${\mathbb{F}}_{q}(\mu)$-subspace.

  * (b)

Suppose that $\dim_{{\mathbb{F}}_{q}}(S\cap\mu S)=k-1$.

    * (b1)

If $t\geq k$ then $S=b\langle
1,\mu,\ldots,\mu^{k-1}\rangle_{{\mathbb{F}}_{q}}$, for some
$b\in\mathbb{F}_{q^{n}}^{*}$ and $t\neq k$.

    * (b2)

If $t\leq k-1$, write $k=t\ell+m$ with $m<t$, then $m>0$ and
$S=\overline{S}\oplus b\langle
1,\mu,\ldots,\mu^{m-1}\rangle_{{\mathbb{F}}_{q}}$, where $\overline{S}$ is an
${\mathbb{F}}_{q^{t}}$-subspace of dimension $\ell$,
$b\in\mathbb{F}_{q^{n}}^{*}$ and
$b{\mathbb{F}}_{q^{t}}\cap\overline{S}=\\{0\\}$.

In particular, $t$ divides $n$.

##  3\. Classification of $(n-2)$-clubs

In this section we are going to classify $(n-2)$–clubs in
$\mathrm{PG}(1,q^{n})$. We will first detect a larger family of linear sets
which contains all the $(n-2)$–clubs (up to equivalence), then we will
describe how to find the number of points of a certain weight in relation to
the intersection of some subspaces. Using the above mentioned results together
with the results on critical pairs over finite fields, we are able to
completely classify $(n-2)$-clubs.

First observe that for $n=3$ all the $2$-clubs are $\mathrm{P\Gamma
L}(2,q^{3})$-equivalent to $L_{\mathrm{Tr}_{q^{3}/q}}$, for $n=4$ we only have
two not $\mathrm{P\Gamma L}(2,q^{4})$-equivalent $2$-clubs (that are
$L_{x^{q}-x^{q^{3}}}$ and the one from Construction 2.1) and the only $3$-club
is $L_{\mathrm{Tr}_{q^{4}/q}}$. So, in this section we will assume $n\geq 5$.

Let us start by studying the weight distribution of a class of linear sets
which contains the clubs we are interested.

######  Proposition 3.1.

Let $S$ be an $h$-dimensional ${\mathbb{F}}_{q}$-subspace of
${\mathbb{F}}_{q^{n}}$ such that $1\in S$ and $h\leq n-2$. Let
$(a,b)\in{\mathbb{F}}_{q^{n}}^{2}$, with $a\notin S$ and
$b\notin{\mathbb{F}}_{q}$. Let

$U=(S\times\\{0\\})+\langle(1,1)\rangle_{{\mathbb{F}}_{q}}+\langle(a,b)\rangle_{{\mathbb{F}}_{q}}\subseteq{\mathbb{F}}_{q^{n}}^{2}.$

Then

(8) $L_{U}=\\{\langle(s+\alpha+\beta a,\alpha+\beta b)\rangle\colon s\in
S,\alpha,\beta\in{\mathbb{F}}_{q},(s,\alpha,\beta)\neq(0,0,0)\\}$

is an ${\mathbb{F}}_{q}$-linear set of rank $h+2$ in $\mathrm{PG}(1,q^{n})$,
with $w_{L_{U}}(\langle(1,0)\rangle_{{\mathbb{F}}_{q^{n}}})=h$ and
$w_{L_{U}}(\langle(0,1)\rangle_{{\mathbb{F}}_{q^{n}}})=1$.

###### Proof.

Since $b\notin{\mathbb{F}}_{q}$, we have that $(a,b)\notin
S\times\\{0\\}+\langle(1,1)\rangle_{{\mathbb{F}}_{q}}$ and so $U$ is a direct
sum and $\dim_{{\mathbb{F}}_{q}}(U)=h+2$, that is $Rank(L_{U})=h+2$. From
$b\notin{\mathbb{F}}_{q}$, it also follows that $\alpha+\beta b=0$, if and
only if $\alpha=\beta=0$. Thus,
$w_{L_{U}}(\langle(1,0)\rangle_{{\mathbb{F}}_{q^{n}}})=h$. Finally,
$s+\alpha+\beta a=0$ implies $\beta=0$. Indeed, if $\beta\neq 0$, then
$a=-\beta^{-1}(s+\alpha)\in S$, a contradiction. So,
$w_{L_{U}}(\langle(0,1)\rangle_{{\mathbb{F}}_{q^{n}}})=1$. ∎

Note that, since the point $\langle(1,0)\rangle_{{\mathbb{F}}_{q^{n}}}$ has
weight $h$ in $L_{U}$ and the rank of $L_{U}$ is $h+2$, by (4) all the other
points can have weight at most two in $L_{U}$. In the next result we show how
to find the points of weight two.

######  Theorem 3.2.

Let $S$ be an $h$-dimensional ${\mathbb{F}}_{q}$-subspace of
${\mathbb{F}}_{q^{n}}$ such that $1\in S$ and $3\leq h\leq n-2$. Consider

$U=(S\times\\{0\\})\oplus\langle(1,1)\rangle_{{\mathbb{F}}_{q}}\oplus\langle(a,b)\rangle_{{\mathbb{F}}_{q}}\subseteq{\mathbb{F}}_{q^{n}}^{2},$

with $a\notin S$, and $b\notin{\mathbb{F}}_{q}$. Then the set of points of
weight $2$ in $L_{U}$ different from
$\langle(1,0)\rangle_{{\mathbb{F}}_{q^{n}}}$ is

$\\{P_{s}=\langle(-s+a,b)\rangle_{{\mathbb{F}}_{q^{n}}}\colon s\in
S\cap(a+bS)\\}$

and the size of such a set is $\lvert S\cap(a+bS)\rvert$. In particular,
$L_{U}$ is a $h$-club of rank $h+2$ if and only if $a\notin S+bS$.

###### Proof.

From Proposition 3.1, $L_{U}$ is a linear set of rank $h+2$, with
$w_{L_{U}}(\langle(1,0)\rangle_{{\mathbb{F}}_{q^{n}}})=h$ and
$w_{L_{U}}(\langle(0,1)\rangle_{{\mathbb{F}}_{q^{n}}})=1$. First note that, by
(4), $w_{L_{U}}(P)\leq 2$ for every point $P$ different from
$\langle(1,0)\rangle_{{\mathbb{F}}_{q^{n}}}$. Let consider the map

$\Phi:$ | $S\cap(a+bS)$ | $\longrightarrow$ | $\\{P\in L_{U}\colon w_{L_{U}}(P)=2,P\neq\langle(1,0)\rangle_{{\mathbb{F}}_{q^{n}}}\\}$
---|---|---|---
| $s$ | $\longmapsto$ |  $P_{s}=\langle(-s+a,b)\rangle_{{\mathbb{F}}_{q^{n}}}$.

First, we prove that $\Phi$ is well-defined. If $S\cap(a+bS)=\emptyset$, then
$\Phi$ is not defined. Therefore, suppose that $S\cap(a+bS)\neq\emptyset$ and
let $\overline{s}\in S\cap(a+bS)$, so $\overline{s}=a+b\overline{s}^{\prime}$,
for some $\overline{s}^{\prime}\in S$. Then $(-\overline{s}+a,b)\in U$ and
$(-\overline{s}+a,b)=b(-\overline{s}^{\prime},1)=b(-1-\overline{s}^{\prime}+1,1)\in
U$. Since $b\notin{\mathbb{F}}_{q}$, the point $P_{\overline{s}}$ has weight 2
in $L_{U}$.
Now, we show that $\Phi$ is a one-to-one correspondence. Indeed, we only need
to prove that $\Phi$ is surjective. If $L_{U}$ has no point of weight 2, it is
clear. Otherwise, let $P$ be a point with weight $2$ in
$L_{U}\setminus\\{\langle(1,0)\rangle_{{\mathbb{F}}_{q^{n}}}\\}$. So

$P=\langle(s_{1}+\alpha_{1}+\beta_{1}a,\alpha_{1}+\beta_{1}b)\rangle_{{\mathbb{F}}_{q^{n}}}=\langle(s_{2}+\alpha_{2}+\beta_{2}a,\alpha_{2}+\beta_{2}b)\rangle_{{\mathbb{F}}_{q^{n}}},$

for some $s_{1},s_{2}\in S$ and
$\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}\in{\mathbb{F}}_{q}$, with
$(s_{1}+\alpha_{1}+\beta_{1}a,\alpha_{1}+\beta_{1}b)$ and
$(s_{2}+\alpha_{2}+\beta_{2}a,\alpha_{2}+\beta_{2}b)$
${\mathbb{F}}_{q^{n}}$-linearly dependent by a scalar
$t\in{\mathbb{F}}_{q^{n}}\setminus{\mathbb{F}}_{q}$, that is
$(s_{1}+\alpha_{1}+\beta_{1}a,\alpha_{1}+\beta_{1}b)=t(s_{2}+\alpha_{2}+\beta_{2}a,\alpha_{2}+\beta_{2}b)$.
This implies

$(s_{1}+\alpha_{1}+\beta_{1}a)(\alpha_{2}+\beta_{2}b)=(s_{2}+\alpha_{2}+\beta_{2}a)(\alpha_{1}+\beta_{1}b),$

and so

(9)
$s_{1}\alpha_{2}-s_{2}\alpha_{1}=a(\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1})-b(\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1})+b(s_{2}\beta_{1}-s_{1}\beta_{2})$

Now, we observe that $\gamma=\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\neq 0$,
that is $(\alpha_{1},\beta_{1})$ and $(\alpha_{2},\beta_{2})$ are
${\mathbb{F}}_{q}$-linearly independent. Indeed, suppose that
$(\alpha_{1},\beta_{1})$ and $(\alpha_{2},\beta_{2})$ are
${\mathbb{F}}_{q}$-proportional then there exists $\lambda\in{\mathbb{F}}_{q}$
such that $\alpha_{2}+\beta_{2}b=\lambda(\alpha_{1}+\beta_{1}b)=\lambda
t(\alpha_{2}+\beta_{2}b)$. This implies that $\lambda t=1$ and so
$t\in{\mathbb{F}}_{q}$, a contradiction.
Therefore (9) implies

(10)
$s_{1}\frac{\alpha_{2}}{\gamma}-s_{2}\frac{\alpha_{1}}{\gamma}=a+b\left(s_{2}\frac{\beta_{1}}{\gamma}-s_{1}\frac{\beta_{2}}{\gamma}-1\right).$

Let
$\overline{s}=s_{1}\frac{\alpha_{2}}{\gamma}-s_{2}\frac{\alpha_{1}}{\gamma}$,
because of $s_{2}\frac{\beta_{1}}{\gamma}-s_{1}\frac{\beta_{2}}{\gamma}-1\in
S$. By (10), $\overline{s}\in S\cap(a+bS)$. Since
$(s_{1}+\alpha_{1}+\beta_{1}a,\alpha_{1}+\beta_{1}b),(s_{2}+\alpha_{2}+\beta_{2}a,\alpha_{2}+\beta_{2}b)\in
U$ then

$\frac{\alpha_{2}}{\gamma}(s_{1}+\alpha_{1}+\beta_{1}a,\alpha_{1}+\beta_{1}b)-\frac{\alpha_{1}}{\gamma}(s_{2}+\alpha_{2}+\beta_{2}a,\alpha_{2}+\beta_{2}b)=(\overline{s}-a,-b)$

belongs to $U$ and defines again the point $P$ and hence $\Phi$ is surjective.
So,
$P=\langle(\overline{s}-a,-b)\rangle_{{\mathbb{F}}_{q^{n}}}=\Phi({-\overline{s}})$.
In particular, if there exists a point of weight 2 in $L_{U}$ then
$S\cap(a+bS)\neq\emptyset$.
Therefore, $L_{U}$ has only one point of weight greater than $1$ if and only
if $S\cap(a+bS)=\emptyset$, which happens if and only if $a\notin S+bS$. This
completes the proof. ∎

Also in the case in which the linear sets as in Theorem 3.2 are not
$(h-2)$-club, they have an interesting weight distribution if the points and
size.

######  Corollary 3.3.

Let $S$ be an $h$-dimensional ${\mathbb{F}}_{q}$-subspace of
${\mathbb{F}}_{q^{n}}$ such that $1\in S$ and $3\leq h\leq n-2$. Consider

$U=(S\times\\{0\\})\oplus\langle(1,1)\rangle_{{\mathbb{F}}_{q}}\oplus\langle(a,b)\rangle_{{\mathbb{F}}_{q}}\subseteq{\mathbb{F}}_{q^{n}}^{2},$

with $a\notin S$, and $b\notin{\mathbb{F}}_{q}$. If $a\in S+bS$, then $L_{U}$
has $q^{j}$ points of weight $2$, with

$j=\dim_{{\mathbb{F}}_{q}}(S\cap bS)=2h-\dim_{{\mathbb{F}}_{q}}(S+bS)$

and

$\lvert L_{U}\rvert=q^{h+1}+q^{h}-q^{j+1}+1.$

###### Proof.

Since $a\in S+bS$, we have that $S\cap(a+bS)\neq\emptyset$. So, by Theorem
3.2, we know that that number of points of weight 2 is $\lvert
S\cap(a+bS)\rvert$. Finally by (2) and (3), we get the assertion. ∎

Next result shows that $h$-clubs of rank $h+2$ (up to the action of
$\mathrm{PGL}(2,q^{n})$ on $\mathrm{PG}(1,q^{n})$) are as described in (8).

######  Lemma 3.4.

If $L_{W}$ is an $h$-club of rank $h+2\leq n$ in $\mathrm{PG}(1,q^{n})$. Then,
$L_{W}$ is $\mathrm{PGL}(2,q^{n})$-equivalent to a linear set $L_{U}$, where
$U=(S\times\\{0\\})\oplus\langle(1,1)\rangle_{{\mathbb{F}}_{q}}\oplus\langle(a,b)\rangle_{{\mathbb{F}}_{q}}$,
for some $h$-dimensional ${\mathbb{F}}_{q}$-subspace $S$ containing $1$, with
$a\notin S$, $b\notin{\mathbb{F}}_{q}$ and $a\notin S+bS$.

###### Proof.

Let $P=\langle\mathbf{v}_{1}\rangle_{{\mathbb{F}}_{q^{n}}}\in L_{W}$ be the
point of weight $h$, with $\mathbf{v}_{1}\in W$. Let
$Q=\langle\mathbf{v}_{2}\rangle_{{\mathbb{F}}_{q^{n}}}\in
L_{W}\setminus\\{P\\}$, with $\mathbf{v}_{2}\in W$ and let
$\psi\in\mathrm{PGL}(2,q^{n})$ be the projectivity of $\mathrm{PG}(1,q^{n})$
induced by the unique ${\mathbb{F}}_{q^{n}}$-isomorpishm $f$ of
${\mathbb{F}}_{q^{n}}^{2}$ that maps $\mathbf{v}_{1}$ in $(1,0)$ and
$\mathbf{v}_{2}$ in $(1,1)$. Then $\psi(L_{W})=L_{U}$ where $U=f(W)$ and
$(1,0),(1,1)\in U$. Clearly,
$\psi(P)=\langle(1,0)\rangle_{{\mathbb{F}}_{q^{n}}}$,
$\psi(Q)=\langle(1,1)\rangle_{{\mathbb{F}}_{q^{n}}}$ and
$w_{L_{U}}(\langle(1,0)\rangle_{{\mathbb{F}}_{q^{n}}})=h$. Let
$S^{\prime}=U\cap\langle(1,0)\rangle_{{\mathbb{F}}_{q^{n}}}$. We can write
$S^{\prime}=S\times\\{0\\}$, where $S$ is an ${\mathbb{F}}_{q}$-subspace of
$\mathbb{F}_{q^{n}}$,
$\dim_{{\mathbb{F}}_{q}}(S^{\prime})=\dim_{{\mathbb{F}}_{q}}(S)=h$ and $1\in
S$. Let $(a,b)\in U$ such that
$U=(S\times\\{0\\})\oplus\langle(1,1)\rangle_{{\mathbb{F}}_{q}}\oplus\langle(a,b)\rangle_{{\mathbb{F}}_{q}}$,
so

(11) $U=\\{(s+\alpha+\beta a,\alpha+\beta b)\colon s\in
S,\alpha,\beta\in{\mathbb{F}}_{q}\\}.$

Since
$(a,b)\notin(S\times\\{0\\})\oplus\langle(1,1)\rangle_{{\mathbb{F}}_{q}}$,
then either $a\notin S$ or $b\notin{\mathbb{F}}_{q}$. Suppose that
$b\in{\mathbb{F}}_{q}$. Then $(-b+a,0)\in U$ (choosing $s=0$, $\alpha=-b$ and
$\beta=1$ in (11)). So $-b+a\in S$ and then $a\in S$ (since
$b\in{\mathbb{F}}_{q}$), a contradiction. Therefore, we have that
$b\notin{\mathbb{F}}_{q}$. Suppose now that $a\in S$. Then, (choosing $s=-a$,
$\alpha=0$, $\beta=1$ in (11)) the point
$\langle(0,b)\rangle_{{\mathbb{F}}_{q^{n}}}=\langle(0,1)\rangle_{{\mathbb{F}}_{q^{n}}}$
is a point different from $\langle(1,0)\rangle_{{\mathbb{F}}_{q^{n}}}$ that
has weight $2$ in $L_{U}$, this is a contradiction since $L_{U}$ is an
$h$-club. Finally, by Theorem 3.2, we get $a\notin S+bS$. ∎

######  Remark 3.5.

Note that if $\dim_{{\mathbb{F}}_{q}}(S)=h<n/2$, then for every
$b\in{\mathbb{F}}_{q^{n}}\setminus{\mathbb{F}}_{q}$,
$\dim_{{\mathbb{F}}_{q}}(S+bS)<n$ and hence there exists
$a\in{\mathbb{F}}_{q^{n}}\setminus(S+bS)$. Therefore, the
${\mathbb{F}}_{q}$-subspace
$U=(S\times\\{0\\})\oplus\langle(1,1)\rangle_{{\mathbb{F}}_{q}}\oplus\langle(a,b)\rangle_{{\mathbb{F}}_{q}}$
defines an $h$-club of rank $h+2$.

Our aim is to give a complete classification of $(n-2)$-club of rank $n$. By
Lemma 3.4, we may assume that $(n-2)$-clubs are as described in (8) and we
start to prove under which conditions on $S$, $a$ and $b$ linear sets as in
Lemma 3.4 are clubs.

######  Theorem 3.6.

Let $S$ be an $(n-2)$-dimensional ${\mathbb{F}}_{q}$-subspace of
${\mathbb{F}}_{q^{n}}$, such that $1\in S$. Consider

$U=(S\times\\{0\\})\oplus\langle(1,1)\rangle_{{\mathbb{F}}_{q}}\oplus\langle(a,b)\rangle_{{\mathbb{F}}_{q}}\subseteq{\mathbb{F}}_{q^{n}}^{2},$

with $a\notin S$ and $b\notin{{\mathbb{F}}_{q}}$. Let $T=\langle
1,b\rangle_{{\mathbb{F}}_{q}}$ and ${\mathbb{F}}_{q^{t}}={\mathbb{F}}_{q}(b)$.
Then only one of the following three cases occurs:

  1. (1)

$\dim_{{\mathbb{F}}_{q}}(\langle S\cdot T\rangle_{{\mathbb{F}}_{q}})=n$ and
$L_{U}$ is not an $(n-2)$-club. In this case $L_{U}$ contains $q^{n-4}$ points
of weight 2 and $\lvert L_{U}\rvert=q^{n-1}+q^{n-2}-q^{n-3}+1$.

  2. (2)

$\dim_{{\mathbb{F}}_{q}}(\langle S\cdot T\rangle_{{\mathbb{F}}_{q}})=n-1$, and
in this case $L_{U}$ is an $(n-2)$-club of rank $n$ if and only if
$a\notin\langle S\cdot T\rangle_{{\mathbb{F}}_{q}}$. So, if $a\in\langle
S\cdot T\rangle_{{\mathbb{F}}_{q}}$, $L_{U}$ is an ${\mathbb{F}}_{q}$-linear
set having $q^{n-3}$ points of weight 2 and $\lvert L_{U}\rvert=q^{n-1}+1$.
While if $a\notin\langle S\cdot T\rangle_{{\mathbb{F}}_{q}}$:

    1. (2.1)

if $t\geq n-2$ there exists $c\in{\mathbb{F}}_{q^{n}}^{*}$, such that
$S=c\langle 1,b,\ldots,b^{n-3}\rangle_{{\mathbb{F}}_{q}}$,

$L_{U}=\left\\{\left\langle\left(\sum_{i=0}^{n-3}cx_{i}b^{i}+\alpha+\beta
a,\alpha+\beta b\ \right)\right\rangle_{{\mathbb{F}}_{q^{n}}}\colon
x_{i},\alpha,\beta\in{\mathbb{F}}_{q}\mbox{ not all zero}\right\\},$

and $t=n$.

    2. (2.2)

if $t\leq n-3$, there exist $\ell\in\mathbb{N},c\in{\mathbb{F}}_{q^{n}}^{*}$
such that $n=t(\ell+1)$, with $t\geq 3$ $S=\overline{S}\oplus c\langle
1,b,\ldots,b^{t-3}\rangle_{{\mathbb{F}}_{q}}$, with $\overline{S}$ an
${\mathbb{F}}_{q^{t}}$-subspace of dimension $\ell$ such that
$c{\mathbb{F}}_{q^{t}}\cap\overline{S}=\\{0\\}$, and

$L_{U}=\left\\{\left\langle\left(\overline{s}+\sum_{i=0}^{t-3}cx_{i}b^{i}+\alpha+\beta
a,\alpha+\beta b\right)\right\rangle_{{\mathbb{F}}_{q^{n}}}\colon
x_{i},\alpha,\beta\in{\mathbb{F}}_{q},\overline{s}\in\overline{S}\mbox{ not
all zero}\right\\}.$

  3. (3)

$\dim_{{\mathbb{F}}_{q}}(\langle S\cdot T\rangle_{{\mathbb{F}}_{q}})=n-2$,
then $n$ is even, ${\mathbb{F}}_{q}(b)={\mathbb{F}}_{q^{2}}$, $S$ is an
${\mathbb{F}}_{q^{2}}$-subspace. Moreover, there exists
$c\in{\mathbb{F}}_{q^{n}}^{*}$ such that $S=c\langle
1,\gamma,\ldots,\gamma^{n/2-2}\rangle_{{\mathbb{F}}_{q^{2}}}$ and

$L_{U}=\left\\{\left\langle\left(\sum_{i=0}^{n/2-2}cx_{i}\gamma^{i}+\alpha+\beta
a,\alpha+\beta b\ \right)\right\rangle_{{\mathbb{F}}_{q^{n}}}\colon
x_{i}\in{\mathbb{F}}_{q^{2}},\alpha,\beta\in{\mathbb{F}}_{q}\mbox{not all
zero}\right\\},$

for some $\gamma\in{\mathbb{F}}_{q^{n}}^{*}$ such that
${\mathbb{F}}_{q^{2}}(\gamma)={\mathbb{F}}_{q^{n}}$. Moreover, $L_{U}$ is an
$(n-2)$-club of rank $n$ if and only if $a\notin c\langle
1,\gamma,\ldots,\gamma^{n/2-2}\rangle_{{\mathbb{F}}_{q^{2}}}$. If $a\in
c\langle 1,\gamma,\ldots,\gamma^{n/2-2}\rangle_{{\mathbb{F}}_{q^{2}}}$, then
all the points of $L_{U}$ different from
$\langle(1,0)\rangle_{{\mathbb{F}}_{q^{n}}}$ have weight 2 and $\lvert
L_{U}\rvert=q^{n-2}+1$.

In particular, if $n$ is prime Cases (2.2) and (3) cannot occur.

###### Proof.

Case (1): This case is a consequence of Theorem 3.2. Indeed, note that
$\langle S\cdot T\rangle_{{\mathbb{F}}_{q}}=S+bS={\mathbb{F}}_{q^{n}}$ and
hence $a\in S+bS$, so that Theorem 3.2 implies that $L_{U}$ is not an
$(n-2)$-club. Also, by Corollary 3.3 the weight distribution and the size of
$L_{U}$ is determined.
Case (2): In the case that $\dim_{{\mathbb{F}}_{q}}(\langle S\cdot
T\rangle)=n-1$, we have $\dim_{{\mathbb{F}}_{q}}(S\cap bS)=n-3$, and so by
Lemma 2.8, we need to analyze the following two cases:

  * •

if $t\geq n-2$, there exists $c\in{\mathbb{F}}_{q^{n}}$ such that $S=c\langle
1,b,\ldots,b^{n-3}\rangle_{{\mathbb{F}}_{q}}$. Moreover, since $n>4$, we have
$t>n-2$ and so $t=n$.

  * •

If $t<n-2$, we write $n-2=t\ell+m$, with $0<m<t$. In this case, since $t\mid
n$ then $t\mid m+2$ and so there exists $s\in\mathbb{N}$ such that $m+2=st$.
Therefore, since $m<t$ then $2>t(s-1)$. Because of $t\geq 2$, $s=1$, hence
$n=t(\ell+1)$. Also since $t\mid m+2$ and $0<m<t$, we have $t\geq 3$.

Case (3): Note that $\dim_{{\mathbb{F}}_{q}}(\langle S\cdot
T\rangle_{{\mathbb{F}}_{q}})=n-2$ implies $S\cap bS=S$. From (a) of Lemma 2.8,
we get that $S$ is an ${\mathbb{F}}_{q}(b)$-subspace and hence $t\mid n-2$.
Since $t\mid n$ then $t\mid 2$ and $t=2$. Hence
$\dim_{{\mathbb{F}}_{q^{2}}}(S)=n/2-1$ and fixed
$\gamma\in{\mathbb{F}}_{q^{n}}^{*}$ such that
${\mathbb{F}}_{q^{n}}={\mathbb{F}}_{q^{2}}(\gamma)$, we can now apply
Proposition 2.5 to obtain the desired form. In the case $a\in c\langle
1,\gamma,\ldots,\gamma^{n/2-2}\rangle_{{\mathbb{F}}_{q^{2}}}$ we have that
$a\in S+bS=S$ and so by Corollary 3.3 the weight distribution and the size of
$L_{U}$ are determined.

Finally, suppose that $n$ is a prime number. Then only the first case of
Theorem 2.6 can occur, which means that $\dim_{{\mathbb{F}}_{q}}(\langle
S\cdot T\rangle_{{\mathbb{F}}_{q}})\in\\{n-1,n\\}$ and correspond to Cases (1)
and (2.1). ∎

As a consequence of Lemma 3.4 and Theorem 3.6, we get the following theorem of
classification on $(n-2)$–clubs.

######  Corollary 3.7.

Let $L$ be a linear set of rank $n$ of $\mathrm{PG}(1,q^{n})$. Then $L$ is an
$(n-2)$–club if and only if it is $\mathrm{PGL}(2,q^{n})$–equivalent to a
linear set $L_{U}$ such that $U$ has the following form: $n=t(\ell+1)$, with
$\ell\in\mathbb{N}$, and

$U=\left\\{\left(\overline{s}+\sum_{i=0}^{t-3}cx_{i}b^{i}+\alpha+\beta
a,\alpha+\beta b\ \right)\colon
x_{i},\alpha,\beta\in{\mathbb{F}}_{q},\overline{s}\in\overline{S}\right\\},$

with $a,b,c\in{\mathbb{F}}_{q^{n}}^{*}$,
$\overline{S}\subseteq{\mathbb{F}}_{q^{n}}$ such that

  * •

${\mathbb{F}}_{q}(b)={\mathbb{F}}_{q^{t}}$,

  * •

$\overline{S}$ is an ${\mathbb{F}}_{q^{t}}$-subspace of dimension $\ell$ such
that $c{\mathbb{F}}_{q^{t}}\cap\overline{S}=\\{0\\}$,

  * •

$1\in\overline{S}\oplus c\langle
1,b,\ldots,b^{t-2}\rangle_{{\mathbb{F}}_{q}}$,

  * •

$a\notin\overline{S}\oplus c\langle
1,b,\ldots,b^{t-2}\rangle_{{\mathbb{F}}_{q}}$.

###### Proof.

Theorem 3.6 gives a complete characterization of subspaces defining
$(n-2)$-clubs. So, to prove the assertion, it is enough to observe that the
subspace $U$ in Case (2.1) of Theorem 3.6 is obtained choosing $\ell=0$, Case
(2.2) choosing $3\leq t\leq n-3$ and finally Case (3) choosing $t=2$. ∎

The known examples of $(n-2)$-clubs are described in the following remarks.

######  Remark 3.8.

Let $U$ be defined as follows

$U=\left\\{\left(\sum_{i=0}^{n-3}cx_{i}b^{i}+\alpha+\beta a,\alpha+\beta
b\right)\colon x_{i},\alpha,\beta\in{\mathbb{F}}_{q}\right\\},$

with $a,b,c\in{\mathbb{F}}_{q^{n}}^{*}$ such that

  * •

${\mathbb{F}}_{q}(b)={\mathbb{F}}_{q^{n}}$,

  * •

$a\notin c\langle 1,b,\ldots,b^{n-3},b^{n-2}\rangle_{{\mathbb{F}}_{q}}$.

Choosing $c=1$, $a=b-b^{n-1}$ and by applying the projectivity of
$\mathrm{PG}(1,q^{n})$ induced by the matrix $\left(\begin{matrix}b&-a\\\
0&1\end{matrix}\right)$, we obtain the family of linear sets described in
Construction 2.1.

######  Remark 3.9.

The families of $i$-clubs described in Construction 2.2 give $(n-2)$-clubs in
the following cases

  * •

$n=2r$, $r>1$, $f(x)=a_{0}x+a_{1}x^{q}\in{\mathbb{F}}_{q^{2}}[x]$ with
$a_{1}\neq 0$ and

$U_{a,b}=\left\\{\left(a_{0}x_{0}+a_{1}x_{0}^{q}-ax_{0},bx_{0}+\sum_{i=1}^{t-1}x_{i}\omega^{i}\right)\colon
x_{i}\in{\mathbb{F}}_{q^{2}}\right\\},$

for some fixed $a,b\in{\mathbb{F}}_{q^{2}}$ with $b\neq 0$ and
$\\{1,\omega,\ldots,\omega^{t-1}\\}$ an ${\mathbb{F}}_{q^{2}}$-basis of
${\mathbb{F}}_{q^{n}}$. Then $L_{U_{a,b}}$ is an $(n-2)$-club if
$a_{0}x+a_{1}x^{q}-ax$ is invertible over ${\mathbb{F}}_{q^{2}}$

  * •

$n=3r$, $r>1$,
$f(x)=a_{0}x+a_{1}x^{q}+a_{2}x^{q^{2}}\in{\mathbb{F}}_{q^{3}}[x]$ such that
either $a_{1}=0$ and $a_{2}\neq 0$ or $a_{1}\neq 0$ and
$\mathrm{N}_{q^{3}/q}(a_{2}/a_{1})\neq 1$, and

$U_{a,b}=\left\\{\left(a_{0}x_{0}+a_{1}x_{0}^{q}+a_{2}x_{0}^{q^{2}}-ax_{0},bx_{0}+\sum_{i=1}^{t-1}x_{i}\omega^{i}\right)\colon
x_{i}\in{\mathbb{F}}_{q^{3}}\right\\},$

for some fixed $a,b\in{\mathbb{F}}_{q^{3}}$ with $b\neq 0$ and
$\\{1,\omega,\ldots,\omega^{t-1}\\}$ an ${\mathbb{F}}_{q^{3}}$-basis of
${\mathbb{F}}_{q^{n}}$. Then $L_{U_{a,b}}$ is an $(n-2)$-club if
$a_{0}x+a_{1}x^{q}+a_{2}x^{q^{2}}-ax$ is not invertible over
${\mathbb{F}}_{q^{3}}$.

######  Remark 3.10.

The condition on being scattered on the polynomial $f$ for the subspaces in
(7) has been written down explicitly. Indeed, linearized polynomials which
results scattered in ${\mathbb{F}}_{q^{2}}$ and in ${\mathbb{F}}_{q^{3}}$ are
exactly those that define linear sets of pseudoregulus type. In particular,
for the case of ${\mathbb{F}}_{q^{3}}$, $f$ is scattered if and only if there
not exists an element $\alpha\in{\mathbb{F}}_{q^{3}}$ such that

$\dim_{{\mathbb{F}}_{q}}(\ker(f(x)-\alpha x))=2.$

By [25, Theorem 2.24], this is possible if and only if

$f(x)-\alpha x=\gamma\mathrm{Tr}_{q^{3}/q}(\beta x),$

for some nonzero $\beta,\gamma\in{\mathbb{F}}_{q^{3}}$. Clearly, this can
happen if and only if $a_{1}\neq 0$ and $\mathrm{N}_{q^{3}/q}(a_{2}/a_{1})=1$.

##  4\. Equivalence of clubs

In this section we will deal with the $\mathrm{\Gamma L}$-equivalence issue
for the known families of clubs. We start by studying when two examples of
Construction 2.2 are $\operatorname{\Gamma\mathrm{L}}(2,q^{n})$-equivalent.

######  Theorem 4.1.

Let $n=rt$, $t,r>1$, $f_{1},f_{2}\in\mathcal{L}_{t,q}$ such that $f_{1}$ and
$f_{2}$ are scattered $q$-polynomials. Let

$U_{1}=\left\\{\left(f_{1}(x_{0})-a_{1}x_{0},b_{1}x_{0}+\sum_{i=1}^{r-1}x_{i}\omega_{1}^{i}\right)\colon
x_{i}\in{\mathbb{F}}_{q^{t}}\right\\},$

and

$U_{2}=\left\\{\left(f_{2}(x_{0})-a_{2}x_{0},b_{2}x_{0}+\sum_{i=1}^{r-1}x_{i}\omega_{2}^{i}\right)\colon
x_{i}\in{\mathbb{F}}_{q^{t}}\right\\},$

for some fixed $a_{1},a_{2},b_{1},b_{2}\in{\mathbb{F}}_{q^{t}}$ with
$b_{1},b_{2}\neq 0$ and $\\{1,\omega_{1},\ldots,\omega_{1}^{r-1}\\}$ and
$\\{1,\omega_{2},\ldots,\omega_{2}^{r-1}\\}$ two ${\mathbb{F}}_{q^{t}}$-basis
of ${\mathbb{F}}_{q^{n}}$. Then $U_{1}$ and $U_{2}$ are
$\operatorname{\Gamma\mathrm{L}}(2,q^{n})$-equivalent if and only if the
spaces
$\overline{U}_{f_{1}}=\\{(f_{1}(x_{0})-a_{1}x_{0},b_{1}x_{0}):x_{0}\in{\mathbb{F}}_{q^{t}}\\}$
and
$\overline{U}_{f_{2}}=\\{(f_{2}(x_{0})-a_{2}x_{0},b_{2}x_{0}):x_{0}\in{\mathbb{F}}_{q^{t}}\\}$
are $\operatorname{\Gamma\mathrm{L}}(2,q^{t})$-equivalent via an element
$\phi\in\operatorname{\Gamma\mathrm{L}}(2,q^{t})$ such that $\phi$ fixes the
subspace $\langle(0,1)\rangle_{{\mathbb{F}}_{q^{t}}}$.

###### Proof.

Denote
$\overline{S}_{1}=\langle\omega_{1},\ldots,\omega_{1}^{r-1}\rangle_{{\mathbb{F}}_{q^{t}}}$
and
$\overline{S}_{2}=\langle\omega_{2},\ldots,\omega_{2}^{r-1}\rangle_{{\mathbb{F}}_{q^{t}}}$.
Suppose that $U_{1}$ and $U_{2}$ are
$\operatorname{\Gamma\mathrm{L}}(2,q^{n})$-equivalent. Then there exists a
matrix $\begin{pmatrix}A&B\\\ C&D\end{pmatrix}\in\mathrm{GL}(2,q^{n})$ and
$\rho\in\operatorname{Aut}({\mathbb{F}}_{q^{n}})$ such that

(12) $\begin{pmatrix}A&B\\\ C&D\end{pmatrix}U_{1}^{\rho}=U_{2}.$

Since $\langle(0,1)\rangle_{{\mathbb{F}}_{q^{n}}}$ is the only point in
$L_{U_{1}}$ and $L_{U_{2}}$ having weight greater than 1, then $B=0,D\neq 0$
and

$D\overline{S}_{1}^{\rho}=\overline{S}_{2}.$

Moreover, (12) is satisfied if and only if for every
$x_{0}\in{\mathbb{F}}_{q^{t}}$ and $s_{1}\in\overline{S}_{1}$ there exist
$y_{0}\in{\mathbb{F}}_{q^{t}}$ and $s_{2}\in\overline{S}_{2}$ such that

(13)
$C(f_{1}^{\rho}(x^{\rho}_{0})-a_{1}^{\rho}x^{\rho}_{0})+D(b^{\rho}_{1}x^{\rho}_{0}+s^{\rho}_{1})=b_{2}y_{0}+s_{2}$

and

(14)
$A(f_{1}^{\rho}(x^{\rho}_{0})-a_{1}^{\rho}x^{\rho}_{0})=f_{2}(y_{0})-a_{2}y_{0}.$

Note that $\\{1,\omega_{2},\ldots,\omega_{2}^{r-1}\\}$ is an
${\mathbb{F}}_{q^{t}}$-basis of ${\mathbb{F}}_{q^{n}}$, this allows us to
write $C=\sum_{i=0}^{r-1}C_{i}\omega_{2}^{i}$ and
$D=\sum_{i=0}^{r-1}D_{i}\omega_{2}^{i}$, with $C_{i}$’s and $D_{i}$’s in
${\mathbb{F}}_{q^{t}}$. Now, (13) reads as follows

$\sum_{i=0}^{r-1}C_{i}\omega_{2}^{i}(f_{1}^{\rho}(x^{\rho}_{0})-a_{1}^{\rho}x^{\rho}_{0})+\sum_{i=0}^{r-1}D_{i}\omega_{2}^{i}(b^{\rho}_{1}x^{\rho}_{0}+s^{\rho}_{1})=b_{2}y_{0}+s_{2}$

and since $Ds^{\rho}_{1}\in S_{2}$, the above equality implies

(15)
$C_{0}(f_{1}^{\rho}(x^{\rho}_{0})-a_{1}^{\rho}x^{\rho}_{0})+D_{0}(b^{\rho}_{1}x^{\rho}_{0})=b_{2}y_{0}.$

Note also that $A\in{\mathbb{F}}_{q^{t}}$, because of Equation (14). Moreover,
note that $(C_{0},D_{0})\neq(0,0)$. Suppose that $D_{0}=0$, then (15) together
with (14) implies that $f_{2}(y_{0})=\frac{A}{C_{0}}b_{2}y_{0}+a_{2}y_{0}$,
for every $y_{0}\in{\mathbb{F}}_{q^{t}}$. So $f_{2}$ is not a scattered
$q$-polynomial, a contradiction. So $D_{0}\neq 0$. This means that the spaces
$\overline{U}_{f_{1}}$ and $\overline{U}_{f_{2}}$ are
$\operatorname{\Gamma\mathrm{L}}(2,q^{t})$-equivalent via the map of
$\operatorname{\Gamma\mathrm{L}}(2,q^{t})$ defined by the matrix
$\begin{pmatrix}A_{0}&0\\\ C_{0}&D_{0}\end{pmatrix}$ and the restriction of
the automorphism $\rho$ to ${\mathbb{F}}_{q^{t}}$. Conversely, suppose that
$\overline{U}_{f_{1}}$ and $\overline{U}_{f_{2}}$ are
$\operatorname{\Gamma\mathrm{L}}(2,q^{t})$-equivalent via an element
$\phi\in\operatorname{\Gamma\mathrm{L}}(2,q^{t})$ such that
$\phi(\langle(0,1)\rangle_{{\mathbb{F}}_{q^{t}}})=\langle(0,1)\rangle_{{\mathbb{F}}_{q^{t}}}$.
Then $\phi$ can be represented by a matrix $\begin{pmatrix}A_{0}&0\\\
C_{0}&D_{0}\end{pmatrix}\in\mathrm{GL}(2,q^{t})$ and
$\rho\in\operatorname{Aut}({\mathbb{F}}_{q^{t}})$ such that

(16) $\begin{pmatrix}A_{0}&0\\\
C_{0}&D_{0}\end{pmatrix}\overline{U}_{f_{1}}^{\rho}=\overline{U}_{f_{2}}.$

Moreover, $\overline{S}^{\rho}_{1}$ is an $(r-1)$-dimensional
${\mathbb{F}}_{q^{t}}$-subspace of ${\mathbb{F}}_{q^{n}}$. By Lemma 2.5 there
exists $D^{\prime}\in{\mathbb{F}}_{q^{n}}^{*}$ such that
$D^{\prime}\overline{S}_{1}^{\rho}=\overline{S}_{2}$. Let write $D^{\prime}$
as an ${\mathbb{F}}_{q^{t}}$-linear combination of
$\\{1,\omega_{2},\ldots,\omega_{2}^{r-1}\\}$, namely
$D^{\prime}=\sum_{i=0}^{r-1}D_{i}^{\prime}\omega_{2}^{i}$. Then we also have
$\frac{D_{0}}{D_{0}^{\prime}}D^{\prime}\overline{S}_{1}^{\rho}=\overline{S}_{2}$
holds. Denote by $D=\frac{D_{0}}{D_{0}^{\prime}}D^{\prime}$. Therefore, since
$D=D_{0}+\sum_{i=1}^{r-1}\frac{D_{i}^{\prime}}{D_{0}^{\prime}}\omega_{2}^{i}$
and
$b_{1}^{\rho}x_{0}^{\rho}\sum_{i=1}^{r-1}\frac{D_{i}^{\prime}}{D_{0}^{\prime}}\omega_{2}^{i}\in\overline{S}_{2}$,
then the image of $U_{1}$ via the invertible map $\Phi$ represented by the
matrix $\begin{pmatrix}A_{0}&0\\\ C_{0}&D\end{pmatrix}$ and the automorphism
$\rho$ (whose action is extended to $\mathbb{F}_{q^{n}}$) is contained in
$U_{2}$. Since $U_{1}$ and $U_{2}$ have the same dimension we obtain that
$\Phi(U_{1})=U_{2}$ and hence $U_{1}$ and $U_{2}$ are
$\operatorname{\Gamma\mathrm{L}}(2,q^{n})$-equivalent. ∎

Due to the above result, we know that there are several
$\operatorname{\Gamma\mathrm{L}}(2,q^{n})$-inequivalent examples of $i$-club
in $\mathrm{PG}(1,q^{n})$, we refer to Table 1 provided in [31] for a list of
all the known families of scattered polynomials (for the more recent examples
see [27, 26, 33]).

######  Remark 4.2.

The number of inequivalent subspaces of the recent family associated with the
scattered polynomials presented in [33], which extends those in [27, 26], is
computed in [33, Theorem 4.12] and a lower bound is given in [33, Theorem
4.14]. For the number of inequivalent subspaces associated with the more
classical family of Lunardon-Polverino polynomials see [41].

Now we proceed by showing that the two different types (according to the
behaviour of $\dim_{{\mathbb{F}}_{q}}(\langle S\cdot
T\rangle_{{\mathbb{F}}_{q}})$, cfr. Theorem 3.6) of
${\mathbb{F}}_{q}$-subspaces obtained in Theorem 3.7 are
$\Gamma\mathrm{L}(2,q^{n})$-inequivalent. To this aim the following lemma is
needed.

######  Lemma 4.3.

Consider the following ${\mathbb{F}}_{q}$-subspaces of ${\mathbb{F}}_{q^{n}}$
and let

  1. (1)

$S_{1}=\left\\{\sum_{i=0}^{n-3}c_{1}x_{i}b_{1}^{i}\colon
x_{i}\in{\mathbb{F}}_{q}\right\\},$

with $b_{1},c_{1}\in{\mathbb{F}}_{q^{n}}^{*}$ such that
${\mathbb{F}}_{q}(b_{1})={\mathbb{F}}_{q^{n}}$,

  2. (2)

if $n=t(\ell+1)$, with $\ell\in\mathbb{N}$,

$S_{2}=\left\\{\overline{s}+\sum_{i=0}^{t-3}c_{2}x_{i}b_{2}^{i}\colon
x_{i}\in{\mathbb{F}}_{q},\overline{s}\in\overline{S}\right\\},$

with $b_{2},c_{2}\in{\mathbb{F}}_{q^{n}}^{*}$ such that
${\mathbb{F}}_{q}(b_{2})={\mathbb{F}}_{q^{t}}$, with $3\leq t\leq n-3$, and
$\overline{S}\subseteq{\mathbb{F}}_{q^{n}}$ is an
${\mathbb{F}}_{q^{t}}$-subspace of dimension $\ell$ and
$c_{2}{\mathbb{F}}_{q^{t}}\cap\overline{S}=\\{0\\}$.

  3. (3)

if $n$ is even,

$S_{3}=\left\\{\sum_{i=0}^{n/2-2}c_{3}x_{i}b_{3}^{i}\colon
x_{i}\in{\mathbb{F}}_{q^{2}}\right\\},$

with $c_{3},b_{3}\in{\mathbb{F}}_{q^{n}}^{*}$, such that
${\mathbb{F}}_{q^{2}}(b_{3})={\mathbb{F}}_{q^{n}}$.

Then there cannot exist $\lambda\in{\mathbb{F}}_{q^{n}}^{*}$ such that
$S_{i}=\lambda S_{j}^{\rho}$, for some
$\rho\in\operatorname{Aut}({\mathbb{F}}_{q^{n}})$ and $i,j\in\\{1,2,3\\}$ such
that $i\neq j$.

###### Proof.

Let $n$ any positive integer for which we can consider $S_{i}$ and $S_{j}$ for
$i,j\in\\{1,2,3\\}$ with $i\neq j$. Suppose that there exists
$\lambda\in{\mathbb{F}}_{q^{n}}^{*}$ such that $S_{i}=\lambda S_{j}^{\rho}$
with $\rho\in\operatorname{Aut}({\mathbb{F}}_{q^{n}})$. Without loss of
generality, we may assume that $i<j$ and $\rho$ is the identity (up to replace
the $c_{i}$’s, $b_{i}$’s, $\gamma$ and $\overline{S}$) and hence
$S_{i}=\lambda S_{j}$. Now, we give a case-by-case analysis:

  * (i=1,j=2)

Here, we have that $\lambda\overline{S}\subseteq S_{1}$. It follows that
$S_{1}^{\perp}\subseteq(\lambda\overline{S})^{\perp}$ and
$\dim_{{\mathbb{F}}_{q}}((\lambda\overline{S})^{\perp})=n-\ell t=t$. So,
$(\lambda\overline{S})^{\perp}$ is an ${\mathbb{F}}_{q^{t}}$-subspace of
$\mathbb{F}_{q^{n}}$ of dimension one. Consider the ordered basis
$\mathcal{B}=(1,b_{1},\ldots,b_{1}^{n-1})$ and its dual basis
$\mathcal{B}^{*}=(\mu_{0}^{*},\mu_{1}^{*},\ldots,\mu_{n-1}^{*})$. It follows
that
$S_{1}^{\perp}=c_{1}^{-1}\langle\mu_{n-2}^{*},\mu_{n-1}^{*}\rangle_{{\mathbb{F}}_{q}}$
and by Lemma 2.3 we have that

$\mu_{n-2}^{*}=\delta^{-1}(a_{n-1}+b_{1}),$

and

$\mu_{n-1}^{*}=\delta^{-1},$

where $f(x)=a_{0}+a_{1}x+\ldots+a_{n-1}x^{n-1}+x^{n}$ is the minimal
polynomial of $b_{1}$ over ${\mathbb{F}}_{q}$ and $\delta=f^{\prime}(b_{1})$.
Now, since $\mu_{n-2}^{*},\mu_{n-1}^{*}\in(\lambda\overline{S})^{\perp}$ and
since $(\lambda\overline{S})^{\perp}$ is a $1$-dimensional
${\mathbb{F}}_{q^{t}}$-subspace, it follows

$\frac{\mu_{n-2}^{*}}{\mu_{n-1}^{*}}=a_{n-1}+b_{1}\in{\mathbb{F}}_{q^{t}},$

that is $b_{1}\in{\mathbb{F}}_{q^{t}}$, a contradiction.

  * (i=1,j=3)

Since $S_{1}=c_{1}\langle
1,b_{1}\ldots,b_{1}^{n-3}\rangle_{{\mathbb{F}}_{q}}$, then $S_{1}\cap
b_{1}S_{1}=c_{1}\langle b_{1},\ldots,b_{1}^{n-3}\rangle_{{\mathbb{F}}_{q}}$.
Since $S_{3}$ is an ${\mathbb{F}}_{q^{2}}$-subspace then also $S_{1}$ has to
be an ${\mathbb{F}}_{q^{2}}$-subspace and consequently $S_{1}\cap b_{1}S_{1}$
is an ${\mathbb{F}}_{q^{2}}$-subspace as well. Then $2$ divides
$\dim_{{\mathbb{F}}_{q}}(S_{1}\cap b_{1}S_{1})=n-3$, this is a contradiction
since $n$ is even.

  * (i=2,j=3)

Since $S_{2}=\overline{S}\oplus c_{2}\langle
1,b_{2},\ldots,b_{2}^{t-3}\rangle_{{\mathbb{F}}_{q}}$, then $S_{2}\cap
b_{2}S_{2}=\overline{S}\oplus c_{2}\langle
b_{2},\ldots,b_{2}^{t-3}\rangle_{{\mathbb{F}}_{q}}$. Since $S_{3}$ is an
${\mathbb{F}}_{q^{2}}$-subspace, then we can argue as in the previous case
getting a contradiction to the fact that $n$ is even.

∎

Consequently to Lemma 4.3, we show that any two different
${\mathbb{F}}_{q}$-subspaces obtained in Theorem 3.7 for different cases are
$\Gamma\mathrm{L}(2,q^{n})$-inequivalent.

######  Theorem 4.4.

Consider the following ${\mathbb{F}}_{q}$-subspaces:

  * (1)

Let

$U_{1}=\left\\{\left(\sum_{i=0}^{n-3}c_{1}x_{i}b_{1}^{i}+\alpha+\beta
a_{1},\alpha+\beta b_{1}\right)\colon
x_{i},\alpha,\beta\in{\mathbb{F}}_{q}\right\\},$

with $a_{1},b_{1},c_{1}\in{\mathbb{F}}_{q^{n}}^{*}$ such that:

    * –

${\mathbb{F}}_{q}(b_{1})={\mathbb{F}}_{q^{n}}$,

    * –

$1\in c_{1}\langle
1,b_{1},\ldots,b_{1}^{n-3},b_{1}^{n-2}\rangle_{{\mathbb{F}}_{q}}$

    * –

$a_{1}\notin c_{1}\langle
1,b_{1},\ldots,b_{1}^{n-3},b_{1}^{n-2}\rangle_{{\mathbb{F}}_{q}}$,

  * (2)

if $n=t(\ell+1)$, with $\ell\in\mathbb{N}$, let

$U_{2}=\left\\{\left(\overline{s}+\sum_{i=0}^{t-3}c_{2}x_{i}b_{2}^{i}+\alpha+\beta
a_{2},\alpha+\beta b_{2}\ \right)\colon
x_{i},\alpha,\beta\in{\mathbb{F}}_{q},\overline{s}\in\overline{S}\right\\},$

with $a_{2},b_{2},c_{2}\in{\mathbb{F}}_{q^{n}}^{*}$,
$\overline{S}\subseteq{\mathbb{F}}_{q^{n}}$ such that:

    * –

${\mathbb{F}}_{q}(b_{2})={\mathbb{F}}_{q^{t}}$, with $3\leq t\leq n-3$,

    * –

$\overline{S}$ ${\mathbb{F}}_{q^{t}}$-subspace of dimension $\ell$ such that
$c_{2}{\mathbb{F}}_{q^{t}}\cap\overline{S}=\\{0\\}$,

    * –

$1\in\overline{S}\oplus c_{2}\langle
1,b_{2},\ldots,b_{2}^{t-2}\rangle_{{\mathbb{F}}_{q}}$,

    * –

$a\notin\overline{S}\oplus c_{2}\langle
1,b_{2},\ldots,b_{2}^{t-2}\rangle_{{\mathbb{F}}_{q}}$,

  * (3)

if $n$ is even, let

$U_{3}=\left\\{\left(\sum_{i=0}^{n/2-2}c_{3}x_{i}\gamma^{i}+\alpha+\beta
a_{3},\alpha+\beta b_{3}\right)\colon
x_{i}\in{\mathbb{F}}_{q^{2}},\alpha,\beta\in{\mathbb{F}}_{q}\right\\},$

with $a_{3},b_{3},c_{3},\gamma\in{\mathbb{F}}_{q^{n}}^{*}$, such that:

    * –

${\mathbb{F}}_{q}(b_{3})={\mathbb{F}}_{q^{2}}$,

    * –

${\mathbb{F}}_{q^{2}}(\gamma)={\mathbb{F}}_{q^{n}}$,

    * –

$1\in c_{3}\langle
1,\gamma,\ldots,\gamma^{n/2-2}\rangle_{{\mathbb{F}}_{q^{2}}}$,

    * –

$a_{3}\notin c_{3}\langle
1,\gamma,\ldots,\gamma^{n/2-2}\rangle_{{\mathbb{F}}_{q^{2}}}$.

Then two ${\mathbb{F}}_{q}$-subspaces $U_{i}$ and $U_{j}$ (with $i\neq j$) are
$\Gamma\mathrm{L}(2,q^{n})$-inequivalent.

###### Proof.

Suppose that $U_{i}$ and $U_{j}$ are $\Gamma\mathrm{L}(2,q^{n})$-equivalent,
then there exist a matrix $\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right)\in\mathrm{GL}(2,q^{n})$ and
$\rho\in\mathrm{Aut}(\mathbb{F}_{q^{n}})$ such that

(17) $\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right)U_{i}^{\rho}=U_{j}.$

Since $\langle(1,0)\rangle_{\mathbb{F}_{q^{n}}}$ is the only point in
$L_{U_{i}}$ and $L_{U_{j}}$ having weight greater than 1 and the elements of
$\mathrm{\Gamma L}(2,q^{n})$ preserves the weight of points, then

$\left(\begin{matrix}A&B\\\ C&D\end{matrix}\right)\left(\begin{matrix}1\\\
0\end{matrix}\right)=\left(\begin{matrix}\rho\\\ 0\end{matrix}\right),$

for some nonzero $\rho\in{\mathbb{F}}_{q^{n}}$. This implies that $C=0$, and
therefore (17) implies $AS_{i}^{\rho}=S_{j}$, where $S_{i}$ and $S_{j}$ are as
in Lemma 4.3. By applying Lemma 4.3 we obtain that such $A$ cannot exist and
hence $U_{i}$ and $U_{j}$ cannot be
$\operatorname{\Gamma\mathrm{L}}(2,q^{n})$-equivalent. ∎

The above result allows us to prove the existence of new examples of
$(n-2)$-clubs.

######  Corollary 4.5.

If $6\nmid n$, the constructions of Case (2) in Corollary 3.7 are new.

###### Proof.

The previously known examples of $(n-2)$-clubs are those described in
Constructions 2.1 and 2.2. By Remark 3.8, the Construction 2.1 falls in Case
(1) of Corollary 3.7 and the Constructions 2.2, by Remark 3.9, give
$(n-2)$-club only when either $2$ or $3$ divides $n$. ∎

######  Remark 4.6.

Using Theorem 4.4, the $(n-2)$-clubs of Construction 2.2 (cf. Remark 3.9) are
not $\operatorname{\Gamma\mathrm{L}}(2,q^{n})$-equivalent to the family of
$(n-2)$-associated with $U_{1}$ in Theorem 4.4 (that is Construction 2.1).

##  5\. Polynomials defining clubs

In this section, we will provide a polynomial description for the known
families of clubs and for those we have found in the previous section. As in
the above section, in this section we will assume that $n\geq 5$ since if
$n\leq 4$ we already have a polynomial description of clubs.

We start with the family of $(n-2)$-clubs of Construction 2.1.

######  Theorem 5.1.

Let $\lambda\in\mathbb{F}_{q^{n}}^{*}$ such that
$\\{1,\lambda,\ldots,\lambda^{n-1}\\}$ is an ${\mathbb{F}}_{q}$-basis of
$\mathbb{F}_{q^{n}}$, the ${\mathbb{F}}_{q}$-linear set $L_{\lambda}$ defined
as in Construction 2.1 is $\mathrm{PG}\mathrm{L}(2,q^{n})$-equivalent to
$L_{U_{p}}$ with $U_{p}=\\{(x,p(x)\colon x\in{\mathbb{F}}_{q^{n}}\\}$ and

$p(x)=\mathrm{Tr}_{q^{n}/q}(c_{n-2}x)+\lambda\mathrm{Tr}_{q^{n}/q}(c_{n-1}x)$

where $\omega\in\mathbb{F}_{q^{n}}$ is such that
$(1,\lambda,\ldots,\lambda^{n-3},\lambda^{n-2}+\omega,\omega\lambda)$ is an
order ${\mathbb{F}}_{q}$-basis of $\mathbb{F}_{q^{n}}$ and
$(c_{0},\ldots,c_{n-1})$ is its dual basis. In particular, if $q$ is odd then
we may choose $\omega=\lambda^{n-2}$ and in this case

$c_{i}=\frac{1}{\delta}\sum_{j=0}^{n-i-1}\lambda^{j}a_{i+j+1},$

for $i\in\\{n-2,n-1\\}$, where $f(x)=a_{0}+a_{1}x+\ldots+a_{n-1}x^{n-1}+x^{n}$
is the minimal polynomial of $\lambda$ over ${\mathbb{F}}_{q}$ and
$\delta=f^{\prime}(\lambda)$.

###### Proof.

Consider

$U_{\lambda}=\\{(t_{1}\lambda+\ldots+t_{n-1}\lambda^{n-1},t_{n-1}+t_{n}\lambda)\colon(t_{1},\ldots,t_{n})\in{\mathbb{F}}_{q}^{n}\\}.$

Note that such $\omega$ as in the statement there exists since the set

$Y=\left\\{\frac{-\alpha_{0}-\ldots-\alpha_{n-2}\lambda^{n-2}}{\alpha_{n-2}+\lambda\alpha_{n-1}}\colon\alpha_{0},\ldots,\alpha_{n-1}\in{\mathbb{F}}_{q},(\alpha_{n-2},\alpha_{n-1})\neq(0,0)\right\\}$

does not cover all the elements of $\mathbb{F}_{q^{n}}$. Indeed, $\lvert
Y\rvert=\lvert L_{U_{\lambda}}\rvert-1=q^{n-1}+q^{n-2}$. Now, applying
$\left(\begin{array}[]{cc}\lambda^{-1}&\omega\\\ 0&1\end{array}\right)$ to
$U_{\lambda}$ we obtain the following subspace

$U=\\{(t_{1}+\ldots+t_{n-1}\lambda^{n-2}+\omega(t_{n-1}+t_{n}\lambda),t_{n-1}+t_{n}\lambda)\\}.$

Since $(c_{0},\ldots,c_{n-1})$ is the dual basis of
$(1,\lambda,\ldots,\lambda^{n-3},\lambda^{n-2}+\omega,\omega\lambda)$, then

$p(t_{1}+\ldots+t_{n-1}\lambda^{n-2}+\omega(t_{n-1}+t_{n}\lambda))=t_{n-1}+\lambda
t_{n},$

that is $U=U_{p}$ and the first part of the assertion is proved. The second
part follows from the fact that when $q$ is odd we may choose
$\omega=\lambda^{n-2}$ so that $U$ can be written as follows

$\\{(t_{1}+\ldots+2t_{n-1}\lambda^{n-2}+t_{n}\lambda^{n-1},t_{n-1}+t_{n}\lambda)\colon(t_{1},\ldots,t_{n})\in{\mathbb{F}}_{q}^{n}\\}.$

Since now the first component can be expressed through a polynomial basis the
assertion follows by Corollary 2.3. ∎

######  Remark 5.2.

In the case in which the minimal polynomial of $\lambda$ has degree two or
three we may use Corollary 2.4 to obtain nicer expressions for $p(x)$.

Regarding Construction 2.2 we can give the relative polynomial description
which will rely on the associated scattered polynomial (for $q=2$ see also
[16]).

######  Theorem 5.3.

Let $n=tr$, $t,r>1$, $f\in\mathcal{L}_{t,q}$ a scattered polynomial. Let

$U_{a,b}=\left\\{\left(f(x_{0})-ax_{0},bx_{0}+\sum_{i=1}^{r-1}x_{i}\omega^{i}\right)\colon
x_{i}\in{\mathbb{F}}_{q^{t}}\right\\},$

for some fixed $a,b\in{\mathbb{F}}_{q^{t}}$ with $b\neq 0$ and
$\\{1,\omega,\ldots,\omega^{r-1}\\}$ an ${\mathbb{F}}_{q^{t}}$-basis of
${\mathbb{F}}_{q^{n}}$. Then $L_{U_{a,b}}$ is
$\mathrm{PG}\mathrm{L}(2,q^{n})$-equivalent to $L_{U_{p}}$ with
$U_{p}=\\{(x,p(x)\colon x\in{\mathbb{F}}_{q^{n}}\\}$ and

$p(x)=\mathrm{Tr}_{q^{n}/q^{t}}(f(x)-x).$

###### Proof.

First observe that $U_{a,b}$ is $\mathrm{GL}(2,q^{n})$-equivalent to

$U=\left\\{\left(f(x_{0})-ax_{0},x_{0}+\sum_{i=1}^{r-1}x_{i}\omega^{i}\right)\colon
x_{i}\in{\mathbb{F}}_{q^{t}}\right\\}.$

Then consider $(c_{0},\ldots,c_{t-1})$ be the dual basis of
$(1,\omega,\ldots,\omega^{t-1})$. Noting that if
$x=\sum_{i=0}^{t-1}x_{i}\omega^{i}$ then
$x_{0}=\mathrm{Tr}_{q^{n}/q^{r}}(c_{0}x)$ and applying
$\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right)$ we obtain that
$U_{a,b}$ is $\mathrm{GL}(2,q^{n})$-equivalent to

$U^{\prime}=\\{(x,f(\mathrm{Tr}_{q^{n}/q^{t}}(c_{0}x))-a\mathrm{Tr}_{q^{n}/q^{t}}(c_{0}x))\colon
x\in{\mathbb{F}}_{q^{t}}\\},$

since $f\circ\mathrm{Tr}_{q^{n}/q^{t}}=\mathrm{Tr}_{q^{n}/q^{t}}\circ f$, and
by applying $\left(\begin{array}[]{cc}c_{0}&0\\\ 0&1\end{array}\right)$ to
$U^{\prime}$ we obtain that $U^{\prime}$ is $\mathrm{GL}(2,q^{n})$-equivalent
to $U_{p}$ and hence the assertion. ∎

Now, we give the polynomial description of any possible $(n-2)$-clubs.

######  Theorem 5.4.

Let $L_{W}$ be an $(n-2)$-club in $\mathrm{PG}(1,q^{n})$. There exist
$b\in\mathbb{F}_{q^{n}}\setminus{\mathbb{F}}_{q}$ and
$\xi,\eta\in\mathbb{F}_{q^{n}}^{*}$ such that $L_{W}$ is
$\mathrm{PGL}(2,q^{n})$-equivalent to $L_{p}$, where

$p(x)=\mathrm{Tr}_{q^{n}/q}(\xi x)+b\mathrm{Tr}_{q^{n}/q}(\eta x),$

with $\langle\xi,\eta\rangle_{{\mathbb{F}}_{q}}^{\perp}=\overline{S}\oplus
c\langle 1,b,\ldots,b^{t-2}\rangle_{{\mathbb{F}}_{q}}$, where
$t=[{\mathbb{F}}_{q}(b),{\mathbb{F}}_{q}]$, $c\in\mathbb{F}_{q^{n}}^{*}$ and
$\overline{S}$ an ${\mathbb{F}}_{q^{t}}$-subspace of $\mathbb{F}_{q^{n}}$ of
dimension $n/t-1$.

###### Proof.

Let $L_{W}$ be an $(n-2)$-club, then by Corollary 3.7 $L_{W}$ is
$\mathrm{PGL}(2,q^{n})$-equivalent to $L_{U}$, where $n=t(\ell+1)$, with
$\ell\in\mathbb{N}_{0}$, and

$U=\left\\{\left(\overline{s}+\sum_{i=0}^{t-3}cx_{i}b^{i}+\alpha+\beta
a,\alpha+\beta b\right)\colon
x_{i},\alpha,\beta\in{\mathbb{F}}_{q},\overline{s}\in\overline{S}\right\\},$

with $a,b,c\in{\mathbb{F}}_{q^{n}}^{*}$,
$\overline{S}\subseteq{\mathbb{F}}_{q^{n}}$ such that

  * •

${\mathbb{F}}_{q}(b)={\mathbb{F}}_{q^{t}}$,

  * •

$\overline{S}$ is an ${\mathbb{F}}_{q^{t}}$-subspace of dimension $\ell$ such
that $c{\mathbb{F}}_{q^{t}}\cap\overline{S}=\\{0\\}$,

  * •

$1\in\overline{S}\oplus c\langle
1,b,\ldots,b^{t-2}\rangle_{{\mathbb{F}}_{q}}$,

  * •

$a\notin\overline{S}\oplus c\langle
1,b,\ldots,b^{t-2}\rangle_{{\mathbb{F}}_{q}}$.

Now, we prove the existence of $\omega\in\mathbb{F}_{q^{n}}^{*}$ such that
$(s_{1},\ldots,s_{\ell t},c,cb,\ldots,cb^{t-3},1+\omega,a+\omega b)$ is an
ordered ${\mathbb{F}}_{q}$-basis, where $\\{s_{1},\ldots,s_{\ell t}\\}$ is an
${\mathbb{F}}_{q}$-basis of $\overline{S}$. Indeed, $\omega$ exists if and
only if the set

$\left\\{\frac{d_{1}s_{1}+d_{2}s_{2}+\ldots+d_{n-2}cb^{n-3}+d_{n-1}+ad_{n}}{d_{n-1}+d_{n}b}\colon
d_{1},\ldots,d_{n}\in{\mathbb{F}}_{q},(d_{n-1},d_{n})\neq(0,0)\right\\}$

does not cover all the elements of ${\mathbb{F}}_{q^{n}}$ and this is
equivalent to the fact that there exists a point of the form
$\langle(1,\omega)\rangle_{\mathbb{F}_{q^{n}}}$ which is not in $L_{U}$. This
is always true since $L_{U}$ has size $q^{n-1}+q^{n-2}+1$ which is always less
than $q^{n}-1$. Finally, applying $\left(\begin{array}[]{cc}1&\omega\\\
0&1\end{array}\right)$ to $U$ we obtain $U_{p}$, where
$p(x)=\mathrm{Tr}_{q^{n}/q}(\xi_{n-2}x)+b\mathrm{Tr}_{q^{n}/q}(\xi_{n-1}x)$
and $(\xi_{0},\ldots,\xi_{n-1})$ is the dual basis of $(s_{1},\ldots,s_{\ell
t},c,cb,\ldots,cb^{t-3},1+\omega,a+\omega b)$. This is due to the fact that
every element $\delta$ of $\mathbb{F}_{q^{n}}$ may be written as
$\delta=\overline{s}+\sum_{i=0}^{t-3}cx_{i}b^{i}+\alpha+\beta a$ for some
$\overline{s}\in\overline{S}$,
$x_{0},\ldots,x_{t-3},\alpha,\beta\in{\mathbb{F}}_{q}$ and
$p(\delta)=\alpha+b\beta$. Therefore $\xi$ and $\eta$ of the statement
correspond to $\xi_{n-2}$ and to $\xi_{n-1}$. ∎

##  6\. Blocking sets and KM-arcs

A well-studied topic in finite geometry is the theory of blocking set. A
_blocking set_ $\mathcal{B}$ in $\mathrm{PG}(2,q)$ is a set of points with the
property that every line meets $\mathcal{B}$ in at least one point. An easy
example is given by a line, which is also called a _trivial blocking set_. An
important class of blocking sets is given by the blocking sets of Rédei type.
If $\mathcal{B}$ is a blocking set in $\mathrm{PG}(2,q)$ of size $q+t$ with
$t<q$ for which there exists a line $\ell$ meeting $\mathcal{B}$ in exactly
$t$ points, then $\mathcal{B}$ is called a blocking set of _Rédei type_.
Related to the celebrated linearity conjecture are the linear blocking sets.
An _${\mathbb{F}}_{q}$ -linear blocking set_ $L_{W}$ in $\mathrm{PG}(2,q^{n})$
is any ${\mathbb{F}}_{q}$-linear set of rank $n+1$, and those of Rédei type
are the one that can be described as follows: let $\ell$ be the Rédei line of
$\mathcal{B}$ and consider $L_{U}=\ell\cap L_{W}$ and $v\in W\notin\langle
U\rangle_{{\mathbb{F}}_{q^{n}}}$. Then $L_{W}=L_{U\oplus\langle
v\rangle_{{\mathbb{F}}_{q}}}$ and $L_{U}$ is an ${\mathbb{F}}_{q}$-linear set
of rank $n$ contained in the line $\ell$. The Rédei line $\ell$ is unique if
$L_{U}$ is a strictly ${\mathbb{F}}_{q}$-linear set which is not an
$(n-1)$-club, equivalently $L_{U}$ is not $\mathrm{P\Gamma
L}(2,q^{n})$-equivalent to $L_{\mathrm{Tr}_{q^{n}/q}}$.

We are now interested in studying those associated with $i$-clubs.

######  Remark 6.1.

Let $\ell_{\infty}$ be the line in
$\mathrm{PG}(2,q^{n})=\mathrm{PG}(V,{\mathbb{F}}_{q^{n}})$ with equation
$X_{2}=0$ and consider $L_{U}\subseteq\ell_{\infty}$ to be an $i$-club with
$i\leq n-2$. Consider $v\in V\setminus\langle U\rangle_{{\mathbb{F}}_{q^{n}}}$
and $W=U\oplus\langle v\rangle_{{\mathbb{F}}_{q}}$. Then $L_{W}$ is an
${\mathbb{F}}_{q}$-linear blocking set of Rédei type in $\mathrm{PG}(2,q^{n})$
with size $q^{n}+q^{n-1}+\ldots+q^{i}+1$ in which all but one of the points
have weight one and the remaining one has weight $i$. Moreover, for any line
$\ell$ then

$w_{L_{U}}(\ell)\in\\{1,2,i+1,n\\}.$

In particular, since $i<n-1$, there exists only one line having weight $n$ and
exactly $q^{n-i}$ lines having weight $i+1$. Moreover, $L_{W}$ has a
$(q+1)$-secant line since $L_{U}$ contains at least one point of weight one.

Using the classification proved for $(n-2)$-clubs we obtain the classification
of linear blocking sets of small Rédei type with some constraints on the
weight of the points. Recall that a blocking set in $\mathrm{PG}(2,q^{n})$ is
said to be _small_ if $|\mathcal{B}|\leq 3(q^{n}+1)/2$.

######  Theorem 6.2.

Suppose that $\mathrm{char}(\mathbb{F}_{q^{n}})>2$ and $n\geq 5$. Let
$\mathcal{B}$ be a small minimal blocking set in $\mathrm{PG}(2,q^{n})$ of
Rédei type and suppose that for every line $s$ different from the Rédei line
$\ell_{\infty}$ we have

$|s\cap\mathcal{B}|\in\\{1,q+1,q^{n-2}+1\\},$

and for all of the values there exists at least one line meeting $\mathcal{B}$
in this number of points.

Then $\mathcal{B}$ is $\mathrm{PGL}(3,q^{n})$-equivalent to $L_{\overline{U}}$
where $\overline{U}$ has the following form: let $n=t(\ell+1)$, with
$\ell\in\mathbb{N}_{0}$ and

$\overline{U}=\left\\{\left(\overline{s}+\sum_{i=0}^{t-3}cx_{i}b^{i}+\alpha+\beta
a,\alpha+\beta b,\delta\right)\colon
x_{i},\alpha,\beta,\delta\in{\mathbb{F}}_{q},\overline{s}\in\overline{S}\right\\},$

with $a,b,c\in{\mathbb{F}}_{q^{n}}^{*}$,
$\overline{S}\subseteq{\mathbb{F}}_{q^{n}}$ such that

  * •

${\mathbb{F}}_{q}(b)={\mathbb{F}}_{q^{t}}$,

  * •

$\overline{S}$ is an ${\mathbb{F}}_{q^{t}}$-subspace of dimension $\ell$ such
that $c{\mathbb{F}}_{q^{t}}\cap\overline{S}=\\{0\\}$,

  * •

$1\in\overline{S}\oplus c\langle
1,b,\ldots,b^{t-2}\rangle_{{\mathbb{F}}_{q}}$,

  * •

$a\notin\overline{S}\oplus c\langle
1,b,\ldots,b^{t-2}\rangle_{{\mathbb{F}}_{q}}$.

###### Proof.

Since $\mathcal{B}$ is a small minimal blocking set in $\mathrm{PG}(2,q^{n})$
of Rédei type with a $(q+1)$-secant line, then $\mathcal{B}$ is an
${\mathbb{F}}_{q}$-linear set, that is $\mathcal{B}=L_{W}$, see [8, Theorem
4.3 (iv)] (which relies on the well-celebrated results in [3, 4]). Since
$L_{W}$ is an ${\mathbb{F}}_{q}$-linear blocking set of Rédei type, then we
have $W=U\oplus\langle v\rangle_{{\mathbb{F}}_{q}}$ and $L_{U}$ is an
${\mathbb{F}}_{q}$-linear set of rank $n$ contained in the Rédei line
$\ell_{\infty}$. Since every line different from the Rédei line has weight
$1$, $2$ or $n-1$ in $L_{W}$, it follows that the points of $L_{U}$ have
weight either $1$ or $n-2$ and $n\geq 5$, it results that $L_{U}$ is an
$(n-2)$-club and the assertion now follows by Corollary 3.7. ∎

######  Remark 6.3.

Since [8, Theorem 4.3 (iv)] needs the assumption
$\mathrm{char}(\mathbb{F}_{q^{n}})>2$, in the above statement we have to add
such a condition. However, if we add the assumption on $\mathcal{B}$ to be
linear, then the above result holds for every characteristic.

Moreover, from the Construction 2.2, we have examples of linear blocking sets
of Rédei type of the following form.

######  Remark 6.4.

Let $n=rt$, $t,r>1$, $f\in\mathcal{L}_{t,q}$ a scattered polynomial. Let

$U_{a,b}=\left\\{\left(f(x_{0})-ax_{0},bx_{0}+\sum_{i=1}^{r-1}x_{i}\omega^{i},0\right)\colon
x_{i}\in{\mathbb{F}}_{q^{t}}\right\\},$

for some fixed $a,b\in{\mathbb{F}}_{q^{t}}$ with $b\neq 0$ and
$\\{1,\omega,\ldots,\omega^{r-1}\\}$ an ${\mathbb{F}}_{q^{t}}$-basis of
${\mathbb{F}}_{q^{n}}$. Let

$i=\left\\{\begin{array}[]{ll}t(r-1),&\text{if}\,\,f(x)-ax\,\,\text{is
invertible over}\,\,{\mathbb{F}}_{q^{t}},\\\
t(r-1)+1,&\text{otherwise}.\end{array}\right.$

Then

$W=U_{a,b}\oplus\langle(0,0,1)\rangle_{{\mathbb{F}}_{q}}$

defines an ${\mathbb{F}}_{q}$-linear blocking set of Rédei type with size
$q^{n}+q^{n-1}+\ldots+q^{i}+1$.

We now prove that in the family of blocking set described in the above
proposition, there are several inequivalent examples. To this aim we recall
the following result.

######  Lemma 6.5.

[9, Proposition 2.3] Let $L_{W}$ be an ${\mathbb{F}}_{q}$-linear blocking set
with a $(q+1)$-secant line. Then $L_{W}=L_{W^{\prime}}$ if and only if there
exists $\lambda\in\mathbb{F}_{q^{n}}^{*}$ such that $W^{\prime}=\lambda W$.

Let $L_{W}$ and $L_{W^{\prime}}$ be two ${\mathbb{F}}_{q}$-linear sets of
Rédei type with Rédei lines $\ell_{1}$ and $\ell_{2}$, respectively, and let
$\ell_{1}\cap L_{W}=L_{U}$ and $\ell_{2}\cap L_{W^{\prime}}=L_{U^{\prime}}$.
Then by the above lemma $L_{W}$ and $L_{W^{\prime}}$ are $\mathrm{P\Gamma
L}(3,q^{n})$-equivalent if and only if $W$ and $W^{\prime}$ are
$\mathrm{\Gamma L}(3,q^{n})$-equivalent. Moreover, if $L_{W}$ and
$L_{W^{\prime}}$ has only one Rédei line, then they are $\mathrm{P\Gamma
L}(3,q^{n})$-equivalent if and only if $U$ and $U^{\prime}$ are
$\mathrm{\Gamma L}(2,q^{n})$-equivalent. Therefore, from Theorem 4.1 we have
the following result.

######  Corollary 6.6.

The number of $\mathrm{P\Gamma L}(3,q^{n})$-inequivalent blocking sets of the
family described in Remark 6.4 coincide with the number of $\mathrm{\Gamma
L}(2,q^{t})$-inequivalent scattered ${\mathbb{F}}_{q}$-subspaces in
${\mathbb{F}}_{q^{t}}^{2}$ of dimension $t$.

KM-arcs have been originally introduced in [23] by Korchmarós and Mazzocca. A
_KM-arc_ $\mathcal{A}$ of type $t$ is a set of $q+t$ points in
$\mathrm{PG}(2,q)$ such that every line contains $0,2$ or $t$ points. When
$2<t<q$, in [23, Theorem 2.5] it has been proved that if KM-arc of type $t$
exists then $q$ is even and $t\mid q$. Moreover, $\mathcal{A}$ is called a
_translation_ KM-arc if there exists a line $\ell$ of $\mathrm{PG}(2,q)$ such
that the group of elations with axis $\ell$ and fixing $\mathcal{A}$ acts
transitively on the points of $\mathcal{A}\setminus\ell$. In [15], De Boeck
and Van de Voorde proved several and important results on translation KM-arcs
and pointed out that the main question on studying this object regards for
which values of $q$ and $t$ a KM-arc of type $t$ exists in $\mathrm{PG}(2,q)$
and which nonequivalent one there exist. Using the link between translation
KM-arcs and $i$-clubs established in [15, Theorem 2.2] we will give a
classification for translation KM-arcs of type $2^{n-2}$ in
$\mathrm{PG}(2,2^{n})$. Although this result has been already established in
[15, Theorem 4.12], our result give a more direct expression of these sets. To
this aim let’s recall [15, Theorem 2.2].

######  Construction 6.7.

Let $\ell\colon
X_{2}=0\subset\mathrm{PG}(2,2^{n})=\mathrm{PG}(V,{\mathbb{F}}_{2^{n}})$ and
consider $L_{U}\subset\ell$ to be an $i$-club. Let $v\in V\notin\langle
U\rangle_{{\mathbb{F}}_{2^{n}}}$ and let $U^{\prime}=U\oplus\langle
v\rangle_{{\mathbb{F}}_{2}}$. Consider the following set of points

$\mathcal{A}(U,v)=(L_{U^{\prime}}\setminus\ell)\cup(\ell\setminus L_{U}).$

In [15, Theorem 2.1] it has been proved that $\mathcal{A}(U,v)$ is a
translation KM-arc of type $2^{i}$ in $\mathrm{PG}(2,2^{n})$.

Actually, all the translation KM-arcs can be constructed as before.

######  Theorem 6.8.

[15, Theorem 2.2] Every translation KM-arc of type $2^{i}$ in
$\mathrm{PG}(2,2^{n})$ can be constructed as in Construction 6.7 by using an
$i$-club.

The first example of KM-arc of type $2^{m(\ell-1)}$ in $\mathrm{PG}(2,2^{\ell
m})$ was presented in [23] and can be described via the club $L_{T}$, where
$T(x)$ is as in Equation 5; see [15, Theorem 3.2]. The clubs arising from
Construction 2.2 define the KM-arcs found by Gács and Weiner in [21].

We are now able to prove the following classification result for translation
KM-arcs of type $2^{n-2}$ in $\mathrm{PG}(2,2^{n})$. These KM-arcs have been
already classified in [15, Theorem 4.12], but our result give a more explicit
description.

######  Theorem 6.9.

Let $\mathcal{A}$ be a translation KM-arcs of type $2^{n-2}$ in
$\mathrm{PG}(2,2^{n})$. Then $\mathcal{A}$ is
$\mathrm{PGL}(2,q^{n})$-equivalent to $\mathcal{A}(U,(0,0,1))$, where $U$ is
described as in Corollary 3.7.

###### Proof.

Since $\mathcal{A}$ be a translation KM-arcs of type $2^{n-2}$ in
$\mathrm{PG}(2,2^{n})$, by Theorem 6.8 $\mathcal{A}$ is
$\mathrm{PGL}(2,2^{n})$-equivalent to $\mathcal{A}(U,(0,0,1))$ where $L_{U}$
is an $(n-2)$-club. Using the classification of $(n-2)$-clubs provided in
Corollary 3.7 we obtain the assertion. ∎

##  7\. Linearized polynomials with conditions on its value set

The problem of estimating the size of the value set of a polynomial over
finite field or finding polynomials with large/small value set is a quite
classical problem in the theory of polynomials over finite field; see [29].
When considering linearized polynomials, we have more information when
considering its quotient with $x$. More precisely, let $f(x)$ be a linearized
polynomial, then the value set we are interested in is

$\mathcal{V}\left(\frac{f(x)}{x}\right)=\left\\{\frac{f(\alpha)}{\alpha}\colon\alpha\in{\mathbb{F}}_{q^{n}}^{*}\right\\}.$

If $f(x)$ is a $q$-polynomial having ${\mathbb{F}}_{q}$ as maximum field of
linearity, i.e. $f(x)$ does not define an ${\mathbb{F}}_{q^{i}}$-linear map
from ${\mathbb{F}}_{q^{n}}$ to ${\mathbb{F}}_{q^{n}}$ for $1<i<n$ and $i\mid
n$, by the results in [3, 4], we have the following bounds

(18)
$q^{n-1}+1\leq\left|\mathcal{V}\left(\frac{f(x)}{x}\right)\right|\leq\frac{q^{n}-1}{q-1}.$

This is indeed connected with the directions of $f$. Consider the projective
plane $\mathrm{PG}(2,q^{n})$ as the projective closure of
$\mathrm{AG}(2,q^{n})$ via the line $\ell_{\infty}(=\mathrm{PG}(1,q^{n}))$,
that without loss of generality we may assume to have equation $X_{2}=0$. Let
$f(x)$ be any linearized polynomial in $\mathcal{L}_{n,q}$, then

$\mathcal{V}\left(\frac{f(x)}{x}\right)=\mathcal{D}_{f}=\left\\{\frac{f(x)-f(y)}{x-y}\colon
x,y\in\mathbb{F}_{q^{n}},x\neq y\right\\}$

is the set $\mathcal{D}_{f}$ of directions determined by the graph of $f$,
that is the set
$\mathcal{G}_{f}=\\{\langle(x,f(x),1)\rangle_{\mathbb{F}_{q^{n}}}\colon
x\in\mathbb{F}_{q^{n}}\\}\subseteq\mathrm{AG}(2,q^{n})$. In the case in which
$f(x)$ is a $q$-polynomial having ${\mathbb{F}}_{q}$ as maximum field of
linearity, then by (18) we have

$q^{n-1}+1\leq\left|\mathcal{D}_{f}\right|\leq\frac{q^{n}-1}{q-1}.$

If $\mathcal{D}_{f}$ is a scattered ${\mathbb{F}}_{q}$-linear set in
$\mathrm{PG}(1,q^{n})$, then
$\left|\mathcal{V}\left(\frac{f(x)}{x}\right)\right|=\frac{q^{n}-1}{q-1}$,
whereas if $f(x)=\mathrm{Tr}_{q^{n}/q}(x)$ or $f(x)$ is as in [30] then
$\left|\mathcal{V}\left(\frac{f(x)}{x}\right)\right|=q^{n-1}+1$. So, both the
bounds in (18) are sharp.

Two polynomials $f$ and $g$ are said to be _equivalent_ if there exists
$\varphi\in\mathrm{\Gamma L}(3,q^{n})$ such that
$\varphi(\mathcal{G}_{f})=\mathcal{G}_{g}$, see [10]. Using this definition of
equivalence, in [10, Problem 4] the authors asked to classify and to find more
examples, up to equivalence, of $q$-polynomials $f(x)\in\mathcal{L}_{n,q}$
such that in the multiset
$\\{f(\alpha)/\alpha\colon\alpha\in\mathbb{F}_{q^{n}}^{*}\\}$ there is a
unique element which is represented more than $q-1$ times, namely $q^{i}-1$
times for some $i\in\\{1,\ldots,n-1\\}$. We call these polynomials _$i$ -club
polynomials_ because they correspond to those polynomials for which $L_{f}$ is
an $i$-club.

Up to our knowledge, only two examples of $i$-clubs polynomials were
previously known: the one presented in (5) and some examples when $n\leq 5$
(see [9, 13, 5]).

We will first show the new examples of club polynomials.

######  Proposition 7.1.

Let $\lambda\in\mathbb{F}_{q^{n}}^{*}$ such that
$\\{1,\lambda,\ldots,\lambda^{n-1}\\}$ is an ${\mathbb{F}}_{q}$-basis of
$\mathbb{F}_{q^{n}}$, $\omega\in\mathbb{F}_{q^{n}}$ is such that
$(1,\lambda,\ldots,\lambda^{n-3},\lambda^{n-2}+\omega,\omega\lambda)$ is an
order ${\mathbb{F}}_{q}$-basis of $\mathbb{F}_{q^{n}}$ and
$(b_{0},\ldots,b_{n-1})$ is its dual basis. Then

$p(x)=\mathrm{Tr}_{q^{n}/q}(b_{n-2}x)+\lambda\mathrm{Tr}_{q^{n}/q}(b_{n-1}x)$

is an $(n-2)$-club polynomial. Let $n=rt$, $t,r>1$, $f\in\mathcal{L}_{t,q}$ be
a scattered $q$-polynomial. Then

$p(x)=\mathrm{Tr}_{q^{n}/q^{t}}(f(x)-x),$

is an $i$-club polynomial with

$i=\left\\{\begin{array}[]{ll}t(r-1),&\text{if}\,\,f(x)-x\,\,\text{is
invertible over}\,\,{\mathbb{F}}_{q^{t}},\\\
t(r-1)+1,&\text{otherwise}.\end{array}\right.$

###### Proof.

It follows from the fact that if $L_{f}$ is an $i$-club, then $f(x)$ is an
$i$-club polynomial. ∎

We are now able to give a classification of $(n-2)$-club polynomials.

######  Theorem 7.2.

Let $f\in\mathcal{L}_{n,q}$ an $(n-2)$-club polynomial. Then $f$ is equivalent
to $p(x)=\mathrm{Tr}_{q^{n}/q}(\xi x)+b\mathrm{Tr}_{q^{n}/q}(\eta x)$, where
$b,\xi$ and $\eta$ are as in Theorem 5.4.

###### Proof.

The proof again follows from the fact that $f$ is an $(n-2)$-club if and only
if $U_{f}$ defines an $(n-2)$-club. This can happen if and only if $U_{f}$ is
$\mathrm{GL}(2,q^{n})$-equivalent to $U_{g}$, where $g$ is one of the
polynomials described in Theorem 5.4. ∎

##  8\. Linear rank metric codes

Rank metric codes were introduced by Delsarte [18] in 1978 as subsets of
matrices and they have been intensively investigated in recent years because
of their applications; we refer to [39, 36]. In this section we will be
interested in rank metric codes in ${\mathbb{F}}_{q^{m}}^{n}$. The _rank_
(weight) $w(v)$ of a vector
$v=(v_{1},\ldots,v_{n})\in{\mathbb{F}}_{q^{m}}^{n}$ is defined as the
dimension of the vector space generated over ${\mathbb{F}}_{q}$ by its
entries, i.e $w(v)=\dim_{{\mathbb{F}}_{q}}(\langle
v_{1},\ldots,v_{n}\rangle_{{\mathbb{F}}_{q}})$.

A _(linear vector) rank metric code_ $\operatorname{\mathcal{C}}$ is an
${\mathbb{F}}_{q^{m}}$-subspace of ${\mathbb{F}}_{q^{m}}^{n}$ endowed with the
rank distance, where such a distance is defined as

$d(x,y)=w(x-y),$

where $x,y\in{\mathbb{F}}_{q^{m}}^{n}$.

Let $\operatorname{\mathcal{C}}\subseteq{\mathbb{F}}_{q^{m}}^{n}$ be a linear
rank metric code. We will write that $\operatorname{\mathcal{C}}$ is an
$[n,k,d]_{q^{m}/q}$ code (or $[n,k]_{q^{m}/q}$ code) if
$k=\dim_{{\mathbb{F}}_{q^{m}}}(\operatorname{\mathcal{C}})$ and $d$ is its
minimum distance, that is

$d=\min\\{d(x,y)\colon x,y\in\operatorname{\mathcal{C}},x\neq y\\}.$

By the classification of ${\mathbb{F}}_{q^{m}}$-linear isometry of
${\mathbb{F}}_{q^{m}}^{n}$ (see [6]), we say that two rank metric codes
$\operatorname{\mathcal{C}},\operatorname{\mathcal{C}}^{\prime}\subseteq{\mathbb{F}}_{q^{m}}^{n}$
are _(linearly) equivalent_ if and only if there exists a matrix
$A\in\mathrm{GL}(n,q)$ such that
$\operatorname{\mathcal{C}}^{\prime}=\operatorname{\mathcal{C}}A=\\{vA:v\in\operatorname{\mathcal{C}}\\}$.
Moreover, we will say that $\operatorname{\mathcal{C}}$ and
$\operatorname{\mathcal{C}}^{\prime}$ are _semilinearly equivalent_ if and
only if there exist a matrix $A\in\mathrm{GL}(n,q)$ and an automorphism
$\rho\in\mathrm{Aut}({\mathbb{F}}_{q^{m}})$ such that
$\operatorname{\mathcal{C}}^{\prime}=\operatorname{\mathcal{C}}^{\rho}A=\\{v^{\rho}A:v\in\operatorname{\mathcal{C}}\\}$,
where the action of $\rho$ is extended entrywise. Most of the codes we will
consider are _non-degenerate_ , i.e. those for which the columns of any
generator matrix of $\operatorname{\mathcal{C}}$ are
${\mathbb{F}}_{q}$-linearly independent. Denote by
$\mathfrak{C}[n,k,d]_{q^{m}/q}$ the set of all linear $[n,k,d]_{q^{m}/q}$ rank
metric codes in ${\mathbb{F}}_{q^{m}}^{n}$.

The geometric counterpart of rank metric are the systems. An
$[n,k,d]_{q^{m}/q}$ _system_ $U$ is an ${\mathbb{F}}_{q}$-subspace of
${\mathbb{F}}_{q^{m}}^{k}$ of dimension $n$, such that $\langle
U\rangle_{{\mathbb{F}}_{q^{m}}}={\mathbb{F}}_{q^{m}}^{k}$ and

$d=n-\max\left\\{\dim_{{\mathbb{F}}_{q}}(U\cap H)\mid H\textnormal{ is an
${\mathbb{F}}_{q^{m}}$-hyperplane of }{\mathbb{F}}_{q^{m}}^{k}\right\\}.$

Moreover, two $[n,k,d]_{q^{m}/q}$ systems $U$ and $U^{\prime}$ are
_equivalent_ if there exists an ${\mathbb{F}}_{q^{m}}$-isomorphism
$\varphi\in\mathrm{GL}(k,{\mathbb{F}}_{q^{m}})$ such that

$\varphi(U)=U^{\prime}.$

We denote the set of equivalence classes of $[n,k,d]_{q^{m}/q}$ systems by
$\mathfrak{U}[n,k,d]_{q^{m}/q}$.

######  Theorem 8.1.

(see [37]) Let $\operatorname{\mathcal{C}}$ be a non-degenerate
$[n,k,d]_{q^{m}/q}$ rank metric code and let $G$ be an its generator matrix.
Let $U\subseteq{\mathbb{F}}_{q^{m}}^{k}$ be the ${\mathbb{F}}_{q}$-span of the
columns of $G$. The rank weight of an element
$xG\in\operatorname{\mathcal{C}}$, with $x\in{\mathbb{F}}_{q^{m}}^{k}$ is

(19) $w(xG)=n-\dim_{{\mathbb{F}}_{q}}(U\cap x^{\perp}),$

where $x^{\perp}=\\{y\in{\mathbb{F}}_{q^{m}}^{k}\colon x\cdot y=0\\}.$ In
particular,

(20) $d=n-\max\left\\{\dim_{{\mathbb{F}}_{q}}(U\cap H)\colon H\mbox{ is an
}{\mathbb{F}}_{q^{m}}\mbox{-hyperplane of }{\mathbb{F}}_{q^{m}}^{k}\right\\}.$

Actually, the above result allows us to give a one-to-one correspondence
between equivalence classes of non-degenerate $[n,k,d]_{q^{m}/q}$ codes and
equivalence classes of $[n,k,d]_{q^{m}/q}$ systems, see [37]. The system $U$
and the code $\operatorname{\mathcal{C}}$ and in Theorem 8.1 are said to be
_associated_.

Moreover, the semilinear inequivalence on linear rank metric codes can be read
also on the associated systems via the action of $\mathrm{\Gamma L}(k,q^{m})$
on the ${\mathbb{F}}_{q}$-subspaces of ${\mathbb{F}}_{q^{m}}^{n}$.

######  Theorem 8.2.

(see [38] and [40]) Let $\operatorname{\mathcal{C}}$ and
$\operatorname{\mathcal{C}}^{\prime}$ two linear $[n,k,d]_{q^{m}/q}$ non-
degenerate rank metric codes and let $U$ and $U^{\prime}$ be two associated
systems with $\operatorname{\mathcal{C}}$ and
$\operatorname{\mathcal{C}}^{\prime}$, respectively. Then
$\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{C}}^{\prime}$ are
semilinearly equivalent if and only if $U$ and $U^{\prime}$ are
$\mathrm{\Gamma L}(k,q^{m})$-equivalent.

###  8.1. $2$-dimensional linear rank metric codes

Let $U$ be an ${\mathbb{F}}_{q}$-subspace of ${\mathbb{F}}_{q^{n}}^{2}$ of
dimension $n$ such that $\langle
U\rangle_{{\mathbb{F}}_{q^{n}}}={\mathbb{F}}_{q^{n}}^{2}$. Then $U$ is an
$[n,2,d]_{q^{n}/q}$ system where $d=n-\max\\{w_{L_{U}}(P)\colon
P\in\mathrm{PG}(1,q^{n})\\}$. Up to the action of $\mathrm{GL}(2,q^{n})$, as
already said in Section 2.2, we can assume that

$U=\\{(x,f(x))\colon x\in\mathbb{F}_{q^{n}}\\},$

for some $f\in\mathcal{L}_{n,q}$. Then a code associated with $U$ is the code
$\mathcal{C}_{f}$ whose generator matrix is

$G=\left(\begin{array}[]{cccc}\xi_{1}&\xi_{2}&\cdots&\xi_{n}\\\
f(\xi_{1})&f(\xi_{2})&\cdots&f(\xi_{n})\end{array}\right),$

where $\xi_{1},\ldots,\xi_{n}$ is an ${\mathbb{F}}_{q}$-basis of
$\mathbb{F}_{q^{n}}$. This means that a code associated with $U$ is given by
the evaluation of the polynomials in $\langle
x,f(x)\rangle_{\mathbb{F}_{q^{n}}}$.

Let determine the weight distribution of the codes associated with $i$-clubs.

######  Proposition 8.3.

Let $U$ be an ${\mathbb{F}}_{q}$-subspace of ${\mathbb{F}}_{q^{n}}^{2}$ of
dimension $n$ such that $\langle
U\rangle_{{\mathbb{F}}_{q^{n}}}={\mathbb{F}}_{q^{n}}^{2}$. If $L_{U}$ is an
$i$-club, then the weight distribution of an associated code is the following:

  * •

$q^{n}-1$ codewords of weight $n-i$;

  * •

$(q^{n}-1)(q^{n-1}+\ldots+q^{i})$ of weight $n-1$;

  * •

$(q^{n}-1)(q^{n}-q^{n-1}-\ldots-q^{i})$ of weight $n$.

Conversely, let $\operatorname{\mathcal{C}}$ be a linear $[n,2,n-i]_{q^{n}/q}$
non-degenerate rank metric code with $q^{n}-1$ codewords of weight $n-i$ and
all the remaining nonzero codewords having weight either $n-1$ or $n$, then
any associate system with $\mathcal{C}$ defines an $i$-club.

###### Proof.

The proof follows by Theorem 8.1 and from the fact that an $i$-club has size
$q^{n-1}+\ldots+q^{i}+1$ points, one point of weight $i$ and all the remaining
points have weight one. ∎

Maximum rank distance codes of dimension $2$ are exactly those that have only
codewords of weight $n-1$ and $n$. However, the codes associated with
$i$-clubs are very _close_ to them and for this reason they are certainly of
interest. Also, they are codes for which the nonzero weights are only three,
see [35].

Therefore, we call the examples of rank metric codes as in Proposition 8.3
_$i$ -club rank metric codes_. We list now the examples of $i$-club rank
metric codes using the polynomial description we found in Section 5.

Examples of $(n-1)$-club rank metric codes.
Up to equivalence, they are all of the form $\mathcal{C}_{f}$ with
$f(x)=\mathrm{Tr}_{q^{n}/q}(x)$. This is a consequence of [11, Theorem 3.7].

Examples of $(n-2)$-club rank metric codes.
Up to equivalence, they are all of the form $\mathcal{C}_{f}$ with
$f(x)=\mathrm{Tr}_{q^{n}/q}(\xi_{n-2}x)+b\mathrm{Tr}_{q^{n}/q}(\xi_{n-1}x)$,
where $n=t(\ell+1)$ and there exist $a,b,c,\omega\in{\mathbb{F}}_{q^{n}}^{*}$,
$\overline{S}\subseteq{\mathbb{F}}_{q^{n}}$ such that:

  * •

${\mathbb{F}}_{q}(b)={\mathbb{F}}_{q^{t}}$, with $1\leq t\leq n$,

  * •

$\overline{S}$ ${\mathbb{F}}_{q^{t}}$-subspace of dimension $\ell$ such that
$c{\mathbb{F}}_{q^{t}}\cap\overline{S}=\\{0\\}$,

  * •

$a\notin\overline{S}\oplus c\langle
1,b,\ldots,b^{t-2}\rangle_{{\mathbb{F}}_{q}}$,

  * •

$(s_{1},\ldots,s_{\ell t},c,cb,\ldots,cb^{t-3},1+\omega,a+\omega b)$ is an
ordered ${\mathbb{F}}_{q}$-basis and $(\xi_{0},\ldots,\xi_{n-1})$ is its dual
basis.

This follows by Theorem 5.4.

Examples of $i$-club rank metric codes.
Let $n=rt$ with $t,r>1$. Examples of $i$-club rank metric codes are the code
$\mathcal{C}_{f}$ with $f(x)=\mathrm{Tr}_{q^{n}/q^{t}}(g(x)-x)$, where
$g(x)\in\mathcal{L}_{t,q}$ is a scattered $q$-polynomial. This follows by
Theorem 5.3.

###  8.2. $3$-dimensional linear rank metric codes with minimum distance $1$

As a consequence on the classification of the linear Rédei blocking sets of
Theorem 6.2 we also obtain the following classification of rank metric codes
with the following properties: let $\operatorname{\mathcal{C}}$ be a linear
$[n+1,3,1]_{q^{n}/q}$ non-degenerate rank metric code with at least a codeword
of weight $n-1$. Let $U$ be any system associated with
$\operatorname{\mathcal{C}}$ and let $W$ be a $2$-dimensional
${\mathbb{F}}_{q^{n}}$-subspace such that $\dim_{\mathbb{F}_{q^{n}}}(\langle
U\cap W\rangle_{\mathbb{F}_{q^{n}}})=2$, then we say that $U$ is _$q$ -non-
degenerate_.

######  Theorem 8.4.

Let $\operatorname{\mathcal{C}}$ be a linear $[n+1,3,1]_{q^{n}/q}$ $q$-non-
degenerate rank metric code. Then any associated system with
$\operatorname{\mathcal{C}}$ is a linear Rédei type blocking set in
$\mathrm{PG}(2,q^{n})$. Then

  * •

If $\operatorname{\mathcal{C}}$ has more than $q^{n}-1$ codewords of weight
one, then $\operatorname{\mathcal{C}}$ is semilinearly equivalent to the code
generated by

$G=\left(\begin{array}[]{ccccc}\xi_{1}&\xi_{2}&\cdots&\xi_{n}&0\\\
\mathrm{Tr}_{q^{n}/q}(\xi_{1})&\mathrm{Tr}_{q^{n}/q}(\xi_{2})&\cdots&\mathrm{Tr}_{q^{n}/q}(\xi_{n})&0\\\
0&0&\cdots&0&1\end{array}\right),$

where $\xi_{1},\ldots,\xi_{n}$ is an ${\mathbb{F}}_{q}$-basis of
$\mathbb{F}_{q^{n}}$.

  * •

If $n=4$, then $\operatorname{\mathcal{C}}$ is semilinearly equivalent to the
code generated by

$G=\left(\begin{array}[]{ccccc}\xi_{1}&\xi_{2}&\cdots&\xi_{n}&0\\\
f(\xi_{1})&f(\xi_{2})&\cdots&f(\xi_{n})&0\\\
0&0&\cdots&0&1\end{array}\right),$

where $\xi_{1},\ldots,\xi_{n}$ is an ${\mathbb{F}}_{q}$-basis of
$\mathbb{F}_{q^{n}}$ and $f$ is one of the polynomial described in [31,
Theorem A.3].

  * •

If $\operatorname{\mathcal{C}}$ has exactly $q^{n}-1$ codewords of weight one,
$(q^{n}-1)q^{2}$ codewords of weight two and all the remaining nonzero
codewords have weight greater than or equal to $n$, then
$\operatorname{\mathcal{C}}$ is semilinearly equivalent to the code generated
by

$G=\left(\begin{array}[]{ccccc}\xi_{1}&\xi_{2}&\cdots&\xi_{n}&0\\\
f(\xi_{1})&f(\xi_{2})&\cdots&f(\xi_{n})&0\\\
0&0&\cdots&0&1\end{array}\right),$

where $\xi_{1},\ldots,\xi_{n}$ is an ${\mathbb{F}}_{q}$-basis of
$\mathbb{F}_{q^{n}}$ and $f$ is as described in Theorem 5.4.

###### Proof.

Let $U$ be any associated system with $\operatorname{\mathcal{C}}$. Then
$\dim_{{\mathbb{F}}_{q}}(U)=n+1$ and, since the minimum distance is one, by
Theorem 8.1 there exists at least one $\mathbb{F}_{q^{n}}$-subspace of
dimension $2$ such that $\dim_{{\mathbb{F}}_{q}}(U\cap W)=n$. Therefore,
$L_{U}$ is a linear blocking set of Rédei type with
$\ell=\mathrm{PG}(W,\mathbb{F}_{q^{n}})$ an its Rédei line. Moreover, since
$\operatorname{\mathcal{C}}$ is $q$-non-degenerate, then there exists
$\overline{W}$ $\mathbb{F}_{q^{n}}$-subspace of dimension $2$ such that
$\dim_{{\mathbb{F}}_{q}}(U\cap\overline{W})=2$ and
$\dim_{\mathbb{F}_{q^{n}}}(\langle
U\cap\overline{W}\rangle_{\mathbb{F}_{q^{n}}})=2$, that is $L_{U}$ has a
$(q+1)$-secant line.

  * •

If $\operatorname{\mathcal{C}}$ has more than $q^{n}-1$ codewords of weight
one, then there is more than one Rédei line to $L_{U}$. Moreover, since there
exists a codeword of weight $n-1$. By [14, Theorem 4.1] we have that $L_{U}$
has size at least $q^{n}+q^{n-1}+1$. In [28, Theorem 5], it has been proved
that an ${\mathbb{F}}_{q}$-linear blocking set in $\mathrm{PG}(2,q^{n})$ of
Rédei type having size at least $q^{n}+q^{n-1}+1$ and with at least two Rédei
lines is $\mathrm{PGL}(2,q^{n})$-equivalent to $L_{U}^{\prime}$, where

$U^{\prime}=U_{\mathrm{Tr}_{q^{n}/q}}\oplus\langle(0,0,1)\rangle_{{\mathbb{F}}_{q}}.$

By Lemma 6.5, then $U$ and $U^{\prime}$ are $\mathrm{GL}(3,q^{n})$-equivalent
and hence the assertion.

  * •

This point follows from the classification result [9, Section 4] and from
Lemma 6.5 and Theorem 8.2.

  * •

Because of the conditions on the weight distribution of
$\operatorname{\mathcal{C}}$, we have that $L_{U\cap W}$ is an $(n-2)$-club
contained in the line $\mathrm{PG}(W,\mathbb{F}_{q^{n}})$. Therefore, the
result follows from Theorem 5.4 and again from Lemma 6.5 and Theorem 8.2.

∎

##  9\. Conclusions and open problems

In this paper we have investigated clubs of rank $n$ in
$\mathrm{PG}(1,q^{n})$. We have been able to provide a classification result
for $(n-2)$-clubs. Then we have analyzed the
$\operatorname{\Gamma\mathrm{L}}(2,q^{n})$-equivalence of the known subspaces
defining clubs, for some of them the problem is then translated in determining
whether or not certain scattered spaces are equivalent. Then we have detected
all the linearized polynomials defining the known families of clubs. Then we
have applied our results to the theory of blocking sets, KM-arcs and rank
metric codes.

Here, we list some open problems and questions that naturally arise:

  * •

To classify $i$-clubs for $i<n-2$.

  * •

Can a coding-theoretical approach be used to the first problem?

  * •

Recently De Boeck and Van de Voorde in [17] proved that $2$-clubs do not exist
in $\mathrm{PG}(1,q^{5})$. Can this result be extended to
$\mathrm{PG}(1,q^{n})$ for any value of $n\geq 5$?

  * •

As for the blocking sets, are KM-arcs constructed from two
$\operatorname{\Gamma\mathrm{L}}(2,q^{n})$-inequivalent clubs $\mathrm{P\Gamma
L}(3,q^{n})$-inequivalent?

  * •

Can we find other conditions on the parameters of a linear rank metric code in
such a way that results on linear blocking sets can be still used to classify
them?

  * •

Similarly to what happens to the traditional KM-arcs, can linear sets be used
to construct example of generalized KM-arcs introduced in [12]?

## Acknowledgement

The authors were supported by the project “VALERE: VAnviteLli pEr la RicErca”
of the University of Campania “Luigi Vanvitelli” and by the Italian National
Group for Algebraic and Geometric Structures and their Applications (GNSAGA -
INdAM).

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