# The minimum weight of the code of intersecting lines in
$\textnormal{PG}(3,q)$

Sam Adriaensen
Vrije Universiteit Brussel Robin Simoens
Ghent University
Universitat Politècnica de Catalunya
Leo Storme
Ghent University

###### Abstract

We characterise the minimum weight codewords of the $p$-ary linear code of
intersecting lines in $\textnormal{PG}(3,q)$, $q=p^{h}$, $q\geq 19$, $p$
prime, $h\geq 1$. If $q$ is even, the minimum weight equals $q^{3}+q^{2}+q+1$.
If $q$ is odd, the minimum weight equals $q^{3}+2q^{2}+q+1$. For $q$ even, we
also characterise the codewords of second smallest weight.

### Keywords.

Linear codes; Finite projective spaces; Klein quadric.

### MSC.

05B25, 51A50, 51E21, 94B05.

##  1 Introduction

Coding theory is the study of reliable communication through a noisy channel.
The challenge of coding theory is to construct codes with little redundancy
that have strong error-correcting capabilities. Finite geometry has
contributed in this endeavour in several ways, one of which is explored
further in this article. More specifically, we investigate linear codes
constructed from certain matrices arising from finite geometries. Almost 60
years ago, Weldon [Wel66] initiated the study of such codes by analysing codes
constructed from the incidence matrix of classical projective
planes111Actually, Weldon [Wel66] studied codes from projective planes arising
from perfect difference sets. Classical projective planes are known to belong
to this category. It is still an open conjecture that no other projective
planes do. This conjecture has been verified for planes up to order 2 billion
[BG04]., because they allow an efficient decoding algorithm. More general
classes of codes arising from finite geometries have been investigated since,
see e.g. [LSVdV10] for a detailed survey. Most attention has been paid to
codes arising from the incidence matrix of a finite projective space. This
works as follows. Choose integers $0\leq j<k<n$ and a prime power $q$.
Construct a $01$-matrix $M$ whose rows and columns are indexed by the
subspaces of the projective space $\textnormal{PG}(n,q)$ of dimension $k$ and
$j$ respectively. Let $M$ have entry 1 in a certain position if and only if
the corresponding $j$\- and $k$-space are incident. Then the code to be
studied is the row space of $M$ over a finite field, typically the prime
subfield of $\mathbb{F}_{q}$. The minimum weight of this code was determined
by Delsarte, Goethals, and MacWilliams [DGM70] in case $j=0$ and by Bagchi and
Inamdar [BI02] for general values of $j$. Stronger characterisations of the
codewords of small weight are known, see e.g. [AD23]. The dimension of these
codes was determined in case $j=0$ by Hamada [Ham68], but is still unknown if
$j>0$ and $k<n-1$.

In this paper, we study a similar code. Choose integers $0\leq j,k<n$ and a
prime power $q$. Construct a $01$-matrix $M$ whose rows and columns are
indexed by the subspaces of $\textnormal{PG}(n,q)$ of dimensions $k$ and $j$
respectively. Let $M$ have entry 1 in a certain position if and only if the
corresponding $j$\- and $k$-space have a non-empty intersection. Let
$\mathbb{F}_{p}$ denote the prime subfield of $\mathbb{F}_{q}$ and
$\mathcal{C}_{j,k}(n,q)$ the row space of $M$ over $\mathbb{F}_{p}$. The
length of this code equals the number of $j$-spaces of $\textnormal{PG}(n,q)$.
The dimension of this code was determined by Sin [Sin04]. The general formula
is quite involved, so we will not state it here. However, the case of interest
to us are the codes $\mathcal{C}_{1,1}(3,q)$, and for this fixed set of
dimensions $j,k,n$, there is a simple formula for the dimension of the code.

######  Result 1.1 ([Sin04]).

Let $p$ be a prime and $q=p^{h}$ with $h\geq 1$ an integer. Then

$\dim\mathcal{C}_{1,1}(3,q)=q\left(\frac{2p^{2}+1}{3}\right)^{h}+1.$

Yet, determining the minimum weight of $\mathcal{C}_{j,k}(n,q)$ is still an
open problem for $j,k>0$ and $j+k<n-1$ [LSVdV10, Open Problem 2.4]. In this
paper, we determine the minimum weight of $\mathcal{C}_{1,1}(3,q)$ for
sufficiently large $q$. For a set $S$ of lines in $\textnormal{PG}(3,q)$, we
call the $01$-vector whose positions are indexed by the lines of
$\textnormal{PG}(3,q)$ and that has a 1 exactly in the positions corresponding
to the lines of $S$, the characteristic vector of $S$. For any line $\ell$ of
$\textnormal{PG}(3,q)$, we refer to the characteristic vector of the set of
lines having non-empty intersection with $\ell$ simply as the characteristic
vector of $\ell$. Our main result is the following.

######  Theorem 1.2.

Suppose that $q\geq 19$ and let $c$ be a non-zero codeword of
$\mathcal{C}_{1,1}(3,q)$ of weight at most $q^{3}+2q^{2}+q+1$. Then either

  1. (1)

$c$ has weight $q^{3}+q^{2}+q+1$ and $c$ is the characteristic vector of the
set of absolute lines of a symplectic polar space $W(3,q)$,

  2. (2)

$c$ has weight $q^{3}+2q^{2}+q+1$ and $c$ is a scalar multiple of the
characteristic vector of a line.

Moreover, (1) occurs if and only if $q$ is even.

In order to prove this theorem, we use a classification of small blocking sets
on the Klein quadric. Metsch [Met00] proved the following result, where
$\delta$ denotes the Kronecker delta.

######  Result 1.3 ([Met00, Theorem 1.1]).

Let $B$ be a minimal blocking set with respect to lines on the hyperbolic
quadric $Q^{+}(n,q)$, with $n\geq 5$. If

$|B|\leq\frac{q^{n-1}-1}{q-1}+q^{\frac{n-1}{2}}-\delta_{n,5},$

then $B$ is contained in a hyperplane section of $Q^{+}(n,q)$.

It is easy to check that a minimal blocking set $B$ with respect to the lines
of $Q^{+}(n,q)$, which is contained in a hyperplane section, can take two
forms:

  1. (1)

$B$ is a parabolic hyperplane section of $Q^{+}(n,q)$, in which case
$|B|=\frac{q^{n-1}-1}{q-1}$.

  2. (2)

There exists a point $P\in Q^{+}(n,q)$ with tangent hyperplane $P^{\perp}$
such that $B$ consists of the hyperplane section $P^{\perp}\cap Q^{+}(n,q)$,
where the point $P$ is removed. In this case
$|B|=\frac{q^{n-1}-1}{q-1}+q^{\frac{n-1}{2}}-1$.

We are specifically interested in the case $n=5$, but we need to strengthen
the above result slightly by removing the term $\delta_{n,5}$ from the upper
bound on $|B|$. We will do this on the condition that $q\geq 4$, and prove the
following theorem.

######  Theorem 1.4.

Suppose that $q\geq 4$. Let $B$ be a minimal blocking set with respect to the
lines of $Q^{+}(5,q)$. If $|B|\leq q^{3}+2q^{2}+q+1$, then $B$ is contained in
a hyperplane section of $Q^{+}(5,q)$.

### Overview.

In Section 2, we state the necessary background. In Section 3, we prove 1.4.
Finally, in Section 4, we prove 1.2.

##  2 Preliminaries

Throughout this article, $p$ denotes a prime number, $h\geq 1$ an integer, and
$q=p^{h}$ a prime power. The finite field of order $q$ will be denoted as
$\mathbb{F}_{q}$.

We recall the basics of the theory of linear codes, see e.g. [vL99, §3]. It
will be convenient to represent the ambient vector space of a linear code as
the vector space $\mathbb{F}_{q}^{S}$ of functions from some finite set $S$ to
$\mathbb{F}_{q}$. Given a vector $v\in\mathbb{F}_{q}^{S}$, its _support_ is
defined as

$\operatorname{supp}(v)=\left\\{s\in S\,\,||\,\,v(s)\neq 0\right\\},$

and its _(Hamming) weight_ as $\operatorname{wt}(v)=|\operatorname{supp}(v)|$.
The _(Hamming) distance_ between two vectors $v$ and $w$ is
$d_{H}(v,w)=\operatorname{wt}(v-w)$. A subspace $C$ of $\mathbb{F}_{q}^{S}$ is
said to be a _linear $[n,k,d]_{q}$ code_ if

  * its _length_ , which equals $|S|$, is $n$,

  * its dimension is $k$,

  * its _minimum distance_ $\min\left\\{d_{H}(v,w)\,\,||\,\,v,w\in C,\,v\neq w\right\\}$ equals $d$.

It is well-known that the minimum distance of $C$ equals its _minimum weight_
, which is defined as $\min\left\\{\operatorname{wt}(v)\,\,||\,\,v\in
C\setminus\\{\mathbf{0}\\}\right\\}$, where $\mathbf{0}$ denotes the zero
vector.

The standard scalar product on $\mathbb{F}_{q}^{S}$ is defined as follows, for
$v,w\in\mathbb{F}_{q}^{S}$:

$v\cdot w=\sum_{s\in S}v(s)w(s).$

If $C\subseteq\mathbb{F}_{q}^{S}$ is a linear code, then its orthogonal
complement with respect to the standard scalar product is denoted as
$C^{\perp}$ and is called the _dual code_ of $C$.

The projective space corresponding to the vector space $\mathbb{F}_{q}^{n+1}$
will be denoted as $\textnormal{PG}(n,q)$. From now on, all dimensions will be
projective dimensions. A space of dimension $k$ is called a $k$-space for
short. Let $\mathcal{G}_{k}(n,q)$ denote the set of $k$-spaces of
$\textnormal{PG}(n,q)$. For a subset $S$ of $\mathcal{G}_{k}(n,q)$, define its
_characteristic vector_ as

$\chi_{S}:\mathcal{G}_{k}(n,q)\to\\{0,1\\}:\kappa\mapsto\begin{cases}1&\text{if
}\kappa\in S,\\\ 0&\text{if }\kappa\notin S.\end{cases}$

For an integer $j$ and a $k$-space $\kappa$ of $\textnormal{PG}(n,q)$, let
$\chi_{\kappa}^{(j)}$ denote the characteristic vector of the set of
$j$-spaces of $\textnormal{PG}(n,q)$ having non-empty intersection with
$\kappa$. We refer to $\chi_{\kappa}^{(j)}$ as the characteristic vector of
$\kappa$.

######  Definition 2.1.

Let $q$ be a power of a prime $p$, and choose integers $0<j,k<n$. The code
$\mathcal{C}_{j,k}(n,q)$ is defined as the $\mathbb{F}_{p}$-span of
$\left\\{\chi_{\kappa}^{(j)}\,\,||\,\,\kappa\in\mathcal{G}_{k}(n,q)\right\\}$.

The code of interest to us is $\mathcal{C}_{1,1}(3,q)$, which we will simply
denote as $\mathcal{C}(3,q)$. The code $\mathcal{C}_{0,1}(2,q)$ also plays an
important role in this article, as we will heavily rely on the
characterisation of its small weight codewords by Szőnyi and Weiner [SW18].

######  Result 2.2 ([SW18, §4]).

Suppose that $q\geq 19$ and that $c\in\mathcal{C}_{0,1}(2,q)$ with

$\operatorname{wt}(c)<\begin{cases}3q-3&\text{if $q$ is prime},\\\
3q-12&\text{otherwise.}\end{cases}$

Then $c$ is a linear combination of at most 2 characteristic vectors of lines.

We end the preliminaries by discussing quadrics and symplectic polar spaces.
We refer the reader to [HT16, §1] for an in-depth treaty of quadrics, and to
[Cam15] for a more general treatment of polar spaces. A form
$b:\mathbb{F}_{q}^{n+1}\times\mathbb{F}_{q}^{n+1}\to\mathbb{F}_{q}$ is called
_bilinear_ if it is linear in both arguments. We call a bilinear form $b$
_reflexive_ if either

  * $b$ is _symmetric_ , i.e. $b(x,y)=b(y,x)$ for all $x,y\in\mathbb{F}_{q}^{n+1}$, or

  * $b$ is _alternating_ , i.e. $b(x,y)=-b(y,x)$ and $b(x,x)=0$ for all $x,y\in\mathbb{F}_{q}^{n+1}$.

Now suppose that $b$ is a reflexive bilinear form. A non-zero vector
$x\in\mathbb{F}_{q}^{n+1}$ is called _singular_ with respect to $b$ if
$b(x,y)=0$ for all $y\in\mathbb{F}_{q}^{n+1}$. We call $b$ _degenerate_ if it
has a singular vector. If $b$ is non-degenerate, then it induces an involution
$\perp$ on the subspaces of $\mathbb{F}_{q}^{n+1}$ defined by

$\pi^{\perp}=\left\\{y\in\mathbb{F}_{q}^{n+1}\,\,||\,\,(\forall
x\in\pi)(b(x,y)=0)\right\\}.$

Since the subspaces of $\textnormal{PG}(n,q)$ are essentially the subspaces of
$\mathbb{F}_{q}^{n+1}$, we can interpret $\perp$ as being defined on
$\textnormal{PG}(n,q)$. Note that $\perp$ reverses inclusion. Any involution
on the subspaces of $\textnormal{PG}(n,q)$ that reverses inclusion is called a
_polarity_. A subspace $\pi$ of $\textnormal{PG}(n,q)$ is called _absolute_
with respect to the polarity $\perp$ if $\pi\subseteq\pi^{\perp}$.

If $b$ is a non-degenerate alternating form on $\mathbb{F}_{q}^{n+1}$, then
$n$ is necessarily odd, and up to a change of basis, $b$ is given by

$b(x,y)=x_{0}y_{1}-x_{1}y_{0}+\cdots+x_{n-1}y_{n}-x_{n}y_{n-1}.$

We call the corresponding polarity $\perp$ _symplectic_ , and the set of
absolute subspaces with respect to $\perp$ is called the _symplectic polar
space_ $W(n,q)$. It is worth noting that every point of $\textnormal{PG}(n,q)$
is absolute with respect to a given symplectic polarity $\perp$, and that a
line $\ell$ through a point $P$ is absolute if and only if $\ell\subset
P^{\perp}$.

A form $f:\mathbb{F}_{q}^{n+1}\to\mathbb{F}_{q}$ is called _quadratic_ if
$f(\alpha x)=\alpha^{2}f(x)$ for all $x\in\mathbb{F}_{q}^{n+1}$ and all
$\alpha\in\mathbb{F}_{q}$. A non-zero vector $x$ is called _isotropic_ with
respect to $f$ if $f(x)=0$ and _anisotropic_ otherwise. A point
$P\in\textnormal{PG}(n,q)$ is called (an)isotropic if its coordinate vectors
are (an)isotropic. The set of isotropic points of $\textnormal{PG}(n,q)$ with
respect to a quadratic form is called a _quadric_. Every quadratic form $f$
gives rise to a symmetric bilinear form $b(x,y)=f(x+y)-f(x)-f(y)$. We call $f$
and its corresponding quadric _degenerate_ if some isotropic vector $x$ is
singular with respect to $b$. The non-degenerate quadratic forms on
$\mathbb{F}_{q}^{n+1}$ have been classified.

  * If $n$ is odd, there are, up to a change of basis, two non-degenerate quadratic forms on $\mathbb{F}_{q}^{n+1}$. They are of the form

$f(x)=x_{0}x_{1}+\cdots+x_{n-3}x_{n-2}+g(x_{n-1},x_{n})$

for some non-degenerate quadratic form $g$. The corresponding quadric is the
_hyperbolic quadric_ $Q^{+}(n,q)$ or the _elliptic quadric_ $Q^{-}(n,q)$, if
$g$ is reducible or irreducible over $\mathbb{F}_{q}$ respectively. The
corresponding bilinear form $b$ is non-degenerate and defines a polarity
$\perp$. If $q$ is odd, then the set of absolute points with respect to
$\perp$ coincides with the quadric $Q^{\pm}(n,q)$. If $q$ is even, then
$\perp$ is a symplectic polarity.

  * If $n$ is even, there is up to a change of basis a unique non-degenerate quadratic form on $\mathbb{F}_{q}^{n+1}$, given by

$f(x)=x_{0}x_{1}+\cdots+x_{n-2}x_{n-1}+x_{n}^{2}.$

The corresponding quadric is called _parabolic_ and denoted as $Q(n,q)$. Let
$b$ denote the corresponding bilinear form. Then $b$ is non-degenerate if and
only if $q$ is odd, in which case $Q(n,q)$ is the set of absolute points of
the corresponding polarity.

We also use the notation $Q^{\varepsilon}(n,q)$, where $\varepsilon$ is either
$-1$, $0$, or $+1$, in which case the quadric is respectively elliptic,
parabolic, or hyperbolic. In other words, $Q^{-1}(n,q)=Q^{-}(n,q)$,
$Q^{0}(n,q)=Q(n,q)$ and $Q^{1}(n,q)=Q^{+}(n,q)$.

Let $\pi$ and $\rho$ be two disjoint subspaces of $\textnormal{PG}(n,q)$ and
let $S$ be a set of points in $\pi$. The set $\rho
S=\bigcup_{P\in\pi}\left\langle\rho,P\right\rangle$ is called the _cone_ with
_base_ $S$ and _vertex_ $\rho$. By convention, $\rho S=S$ if $\rho=\emptyset$
and $\rho S=\rho$ if $S=\emptyset$. Every quadric is a cone with as base a
non-degenerate quadric. We call a subspace $\pi$ _singular_ with respect to a
quadric $Q$ if $\pi\cap Q$ is a degenerate quadric. If $P\in
Q^{\varepsilon}(n,q)$ and $Q^{\varepsilon}(n,q)$ has a polarity $\perp$, then
$P^{\perp}$ intersects $Q^{\varepsilon}(n,q)$ in a quadric
$PQ^{\varepsilon}(n-2,q)$. We call $P^{\perp}$ the _tangent hyperplane_ of
$P$. A subspace contained in $Q^{\varepsilon}(n,q)$ of maximal dimension is
called a _generator_.

######  Result 2.3.

  1. (1)

The quadric $Q^{\varepsilon}(n,q)$ has $\frac{q^{n}-1}{q-1}+\varepsilon
q^{\frac{n-1}{2}}$ points.

  2. (2)

The quadric $Q^{\varepsilon}(n,q)$ has

$\prod_{i=1-\varepsilon}^{\frac{n-\varepsilon}{2}}(q^{i}+1)$

generators, and their dimension equals $\frac{n+\varepsilon}{2}-1$.

  3. (3)

Let $\mathcal{G}$ denote the set of generators of $Q^{+}(n,q)$. Then the
relation

$\left\\{(\pi,\rho)\in\mathcal{G}^{2}\,\,||\,\,\dim(\pi\cap\rho)\equiv\frac{n-1}{2}\pmod{2}\right\\}$

is an equivalence relation on $\mathcal{G}$, having two classes of equal size.

An equivalence class of generators of $Q^{+}(3,q)$ is called a _regulus_ , and
two distinct reguli of the same quadric $Q^{+}(3,q)$ are called _opposite_.

Lastly, we discuss the _Klein correspondence_. This is a bijection between the
points of $Q^{+}(5,q)$ and the lines of $\textnormal{PG}(3,q)$. For this
reason, the quadric $Q^{+}(5,q)$ is also called the _Klein quadric_. The lines
contained in $Q^{+}(5,q)$ correspond to the sets
$\left\\{\ell\in\mathcal{G}_{1}(3,q)\,\,||\,\,P\in\ell\subset\pi\right\\}$ of
lines in $\textnormal{PG}(3,q)$, with $P$ and $\pi$ an incident point and
plane of $\textnormal{PG}(3,q)$. The equivalence classes of generators of
$Q^{+}(5,q)$ are called the classes of _Greek_ and _Latin_ planes. The Greek
planes correspond to the sets
$\left\\{\ell\in\mathcal{G}_{1}(3,q)\,\,||\,\,\ell\subset\pi\right\\}$ with
$\pi$ a plane, and the Latin planes correspond to the sets
$\left\\{\ell\in\mathcal{G}_{1}(3,q)\,\,||\,\,P\in\ell\right\\}$ with $P$ a
point. A parabolic hyperplane section of $Q^{+}(5,q)$ corresponds to the set
of lines of a symplectic polar space $W(3,q)$. A tangent hyperplane section of
$Q^{+}(5,q)$ corresponds to the set of all lines having non-empty intersection
with a fixed line of $\textnormal{PG}(3,q)$.

##  3 Blocking sets of the Klein quadric

In this section, we extend 1.3 in the case $n=5$ to 1.4. We will closely
follow the original proof by Metsch [Met00].

######  Definition 3.1.

A _blocking set_ of a quadric $Q$ with respect to the $k$-spaces is a set $B$
of points of $Q$ that intersects every $k$-space of $Q$. We call $B$ _minimal_
if none of its proper subsets is a blocking set.

From now on, blocking sets are always considered with respect to lines. We
focus on $Q^{+}(5,q)$, which will simply be denoted as $Q$. The corresponding
polarity will be denoted as $\perp$.

Metsch proved the following result.

######  Result 3.2 ([Met00, Lemma 2.2]).

Let $B$ be a blocking set of $Q$ with $|B|\leq q^{3}+2q^{2}+q$. Let $H$ be a
hyperplane of $\textnormal{PG}(5,q)$ such that $H\cap B$ is not a blocking set
of $Q$.

  1. (1)

If $H$ is a tangent hyperplane, then $|H\cap B|\leq|B|-q(q^{2}-q+1)\leq
3q^{2}$.

  2. (2)

If $H$ is a non-tangent hyperplane, then $|H\cap B|\leq 4q^{2}-2q$. If in
addition $|H\cap B|>3q^{2}-q$, then every line of $H\cap Q$ meets $B$ in at
least two points.

We extend this result using similar arguments to the following lemma.

######  Lemma 3.3.

Suppose that $q\geq 3$. Let $B$ be a blocking set of $Q$ with $|B|\leq
q^{3}+2q^{2}+q+1$. Let $H$ be a hyperplane of $\textnormal{PG}(5,q)$ such that
$H\cap B$ is not a blocking set of $Q$. If $|H\cap B|\geq 3q^{2}+2$, then $H$
is a non-tangent hyperplane, $|H\cap B|\leq 4q^{2}-2q+1$, and every line of
$H\cap Q$ meets $B$ in at least two points.

###### Proof.

Note that the case $|B|\leq q^{3}+2q^{2}+q$ follows from 3.3, so we may
suppose that $|B|=q^{3}+2q^{2}+q+1$.

In case that $H$ is a non-tangent hyperplane, put ${\mathcal{P}}\coloneqq
H\cap Q$. In case that $H$ is the tangent hyperplane of a point $P_{0}\in Q$,
put ${\mathcal{P}}\coloneqq(H\cap Q)\setminus\\{P_{0}\\}$. In both cases,
${\mathcal{P}}$ is a minimal blocking set of $Q$.

There exist integers $\delta,c_{0},c_{1},d,\Delta$ such that the following
properties hold:

  1. (a)

every point $P\in{\mathcal{P}}$ belongs to $c_{0}$ lines of $Q$ that do not
lie in $H$;

  2. (b)

every line contained in ${\mathcal{P}}$ lies in $c_{1}$ planes of $Q$ that do
not lie in $H$;

  3. (c)

$\delta$ points of ${\mathcal{P}}$ are not in $B$;

  4. (d)

$d$ is the maximal number of points of $Q\setminus B$ on a line contained in
${\mathcal{P}}$;

  5. (e)

$\Delta=|B|-|{\mathcal{P}}|$, which implies that $|B\setminus H|=|B|-|B\cap
H|\leq\delta+\Delta$.

Note that $|B\setminus H|=\delta+\Delta$ or $|B\setminus H|=\delta+\Delta-1$
and that the latter occurs only when $H$ is the tangent hyperplane of a point
$P_{0}\in B$.

Since $B$ meets every line of $Q$, $d\neq q+1$. The definition of
${\mathcal{P}}$ implies that ${\mathcal{P}}$ meets every line of $Q$. Since,
by hypothesis, the points of $H\cap B$ do not meet every line of $Q$, we see
that ${\mathcal{P}}$ is not contained in $B$. Hence, $d\neq 0$. Thus $1\leq
d\leq q$.

If $X$ is a point of $B$ that does not belong to $H$, then $X^{\perp}$ meets
$H$ in a quadric $Q_{X}$ of type $Q^{+}(3,q)$. Denote by $z$ the maximal
number of points of $Q\setminus B$ that lie in such a quadric $Q_{X}$. In case
that $H$ is the tangent hyperplane of a point $P_{0}\in Q$, note that $P_{0}$
does not occur in any of the quadrics $Q_{X}$.

Then every point $X$ of $B\setminus H$ is perpendicular to at most $z$ points
of ${\mathcal{P}}\setminus B$. On the other hand, a point $P$ of
${\mathcal{P}}\setminus B$ belongs to $c_{0}$ lines of $Q$ that are not
contained in $H$ and as each of these lines meets $B$, the point $P$ is
perpendicular to at least $c_{0}$ points of $B\setminus H$. We now count the
pairs $(X,P)$ of perpendicular points $X\in B\setminus H$ and
$P\in{\mathcal{P}}\setminus B$. We obtain that $|B\setminus H|\cdot
z\geq\delta c_{0}$.

Since $|B\setminus H|\leq\Delta+\delta$, this implies that

$(\Delta+\delta)\cdot z\geq\delta c_{0}.$ (1)

Consider a point $X\in B\setminus H$ for which the quadric $Q_{X}$ has exactly
$z$ points in $Q\setminus B$. Since every line of $Q_{X}$ meets $Q\setminus B$
in at most $d$ points, a simple double counting argument (cf. [Met00, Lemma
2.1]) gives

$z\leq\frac{|Q_{X}|d}{q+1}.$

So this upper and lower bound for $z$ imply that

$\frac{\delta c_{0}}{\delta+\Delta}\leq z\leq\frac{|Q_{X}|d}{q+1}.$ (2)

By definition (d) of $d$, ${\mathcal{P}}$ contains a line $\ell$ with exactly
$d$ points $P_{1},\dots,P_{d}$ in $Q\setminus B$. The point $P_{1}$ belongs to
$c_{0}$ lines of $Q$ that do not lie in $H$ and each of these lines meets
$B\setminus H$. In this way, the point $P_{1}$ gives rise to at least $c_{0}$
points of $B\setminus H$. For the point $P_{2}$, the same holds, but $c_{1}q$
of these lines on $P_{2}$ lie in a plane of $Q$ on $Q^{+}(5,q)$ through
$\ell$, so that the corresponding points of $B\setminus H$ might have already
been counted using the lines on $P_{1}$. Thus $P_{2}$ gives rise to at least
$c_{0}-c_{1}q$ new points in $B\setminus H$ and the same is true for
$P_{3},\dots,P_{d}$. This shows that

$|B\setminus H|\geq dc_{0}-(d-1)c_{1}q.$ (3)

But $|B\setminus H|\leq\Delta+\delta$, so $\delta\geq
dc_{0}-(d-1)c_{1}q-\Delta$.

Inequality (2) remains true, if we replace $\delta$ by this lower bound for
$\delta$. This gives, after simplifications using that $|Q_{X}|=(q+1)^{2}$,
that

$(dc_{0}-(d-1)c_{1}q-\Delta)c_{0}\leq(q+1)d(dc_{0}-(d-1)c_{1}q).$ (4)

There are two possibilities for the intersection of the hyperplane $H$ with
the quadric $Q$. Either $H$ intersects $Q$ in a non-degenerate parabolic
quadric $Q(4,q)$ or $H$ is the tangent hyperplane $P^{\perp}$ of a point $P\in
Q$.

Suppose by contradiction that $H$ is a tangent hyperplane. Then $c_{0}=q^{2}$
and $c_{1}=1$. The bound on $|B|$ gives $\Delta\leq 1$. From (4), we obtain
$\left(dq^{2}-(d-1)q\right)\left(q^{2}(q+1)-d(q+1)^{2}\right)\leq q^{2}(q+1)$,
so $d=q$. Then $|B\setminus H|\geq dc_{0}-(d-1)c_{1}q=q^{3}-q^{2}+q$, and thus
$|B\cap H|\leq 3q^{2}+1$, a contradiction.

Therefore, $H$ is a non-tangent hyperplane. Then $c_{0}=q^{2}+q$ and
$c_{1}=2$. The bound on $|B|$ gives $\Delta\leq q^{2}$. From (4), we obtain
$\left(dq-d+2\right)\left(q-d\right)\leq q^{2}$, so either $d=1$, $d=q-1$ or
$d=q$.

If $d=1$, then (2) implies that $\delta\leq q+1$ and therefore $|B\setminus
H|\leq q^{2}+q+1$. Also, (3) implies that $|B\setminus H|\geq q^{2}+q$ and
therefore $\delta\geq q$. The latter means that there are at least $q$ points
in $H\cap Q$ not in $B$, and since $d=1$, these points are pairwise non-
collinear in $Q$. Take two of these points. The number of points of
$B\setminus H$ that are collinear with both points is at least $q^{2}+q-1$,
since each point belongs to $c_{0}=q^{2}+q$ lines of $Q^{+}(5,q)\setminus H$,
and $|B\setminus H|\leq q^{2}+q+1$. There are at most $2$ points not collinear
with both. The $q^{2}+q-1$ points that are, belong to a hyperbolic quadric
$Q^{+}(3,q)$. If we take a third point of $B\setminus H$, then it can be
collinear with at most $2q+1$ points of that hyperbolic quadric $Q^{+}(3,q)$,
i.e. at most $2q+1+4=2q+5$ lines of the $c_{0}=q^{2}+q$ lines of
$Q^{+}(5,q)\setminus H$ through that third point can be blocked by $B$, a
contradiction.

If $d=q-1$, then (3) implies that $|B\setminus H|\geq q^{3}-2q^{2}+3q$, so
$|B\cap H|\leq 4q^{2}-2q+1$. Moreover, $d=q-1$ means that every line of $H\cap
Q$ contains at least two points of $B$.

If $d=q$, then (3) implies that $|B\setminus H|\geq q^{3}-q^{2}+2q$, so
$|B\cap H|\leq 3q^{2}-q+1$, a contradiction. ∎

######  Result 3.4 ([Met00, Lemma 2.4]).

Consider in $\textnormal{PG}(4,q)$ a degenerate quadric
$Q^{\prime}=PQ^{+}(3,q)$ for some point $P$. Assume that $B$ is a set of at
most $(q+1)^{2}+(2q-3)$ points of $Q^{\prime}$ such that $P\notin B$ and such
that every line of $Q^{\prime}$ meets $B$. Then there exists a solid $S$ with
$P\notin S$ and $S\cap Q^{\prime}\subseteq B$.

We are now ready to prove 1.4, following a similar strategy as [Met00, Theorem
3.1].

###### Proof of 1.4.

If $|B|\leq q^{3}+2q^{2}+q$, then this is follows from 1.3, so suppose that
$|B|=q^{3}+2q^{2}+q+1$.

Let ${\mathcal{F}}$ be one of the two equivalence classes of planes of $Q$.
Then ${|\mathcal{F}}|=q^{3}+q^{2}+q+1$, and every point of $Q$ belongs to
$q+1$ planes of ${\mathcal{F}}$. It follows that
$\sum_{\pi\in{\mathcal{F}}}|\pi\cap B|=|B|(q+1)$. Since $|B|\leq
q(q+1)^{2}+1$, this implies that there exists a plane $\pi\in{\mathcal{F}}$
with $|\pi\cap B|<q+2$. It follows from a result of Bruen [Bru70] that
$|\pi\cap B|=q+1$ and that $\ell\coloneqq\pi\cap B$ is a line.

Consider a point $P$ of $Q\setminus B$. Then 3.4 shows that $|P^{\perp}\cap
B|\geq(q+1)^{2}$. It also shows that $|P^{\perp}\cap B|<(q+1)^{2}+2q-2$
implies that there exists a solid $S$ in the hyperplane $P^{\perp}$, with
$P\not\in S$ and $S\cap Q\subseteq B$. Since $P\not\in S$, the subspace $S$
meets $Q$ in a quadric of type $Q^{+}(3,q)$, which has $(q+1)^{2}$ points. For
every point $P\in Q\setminus B$, put $e_{P}\coloneqq|P^{\perp}\cap
B|-(q+1)^{2}$.

Now we prove the following results.

(1) The set $\pi\setminus\ell$ contains at least one point $P$ with $e_{P}\leq
q-1$, at least two points $P$ with $e_{P}\leq q$, and, if $q\geq 4$, at least
$q+2$ points $P$ with $e_{P}<2q-2$, unless $e_{P}=q$ for all the points
$P\in\pi\setminus\ell$.

Recall that $\pi\cap B=\ell$. Every point $P$ of $\pi\setminus\ell$ is
perpendicular to $|P^{\perp}\cap B|=e_{P}+(q+1)^{2}$ points of $B$. But then
$P$ is collinear to $e_{P}+(q+1)^{2}-(q+1)=q^{2}+q+e_{P}$ points of
$B\setminus\ell$. Then every point of $B\setminus\ell$ is perpendicular to $q$
points of $\pi\setminus\ell$, unless this point of $B$ lies in the second
plane of $Q$ through the line $\ell$.

Therefore

$\sum_{P\in\pi\setminus\ell}(q^{2}+q+e_{P})\leq(|B|-q-1)q.$

So

$\sum_{P\in\pi\setminus\ell}e_{P}\leq(|B|-q-1)q-(q^{2}+q)q^{2}\leq q^{3},$

since $|B|\leq q^{3}+2q^{2}+q+1$.

Since $|\pi\setminus\ell|=q^{2}$ and since the upper bound is $q^{3}$, there
exists at least one point $P\in\pi\setminus\ell$ with $e_{P}\leq q-1$, unless
$e_{P}=q$ for every point $P\in\pi\setminus\ell$. It is also valid that there
exist at least two points $P\in\pi\setminus\ell$ with $e_{P}\leq q$, and, if
$q\geq 4$, at least $q+2$ points $P\in\pi\setminus\ell$ with $e_{P}<2q-2$.

Case 1. Assume that it is not so that for all points $P\in\pi\setminus\ell$,
$e_{P}=q$.

By (1), we find $q+2$ points $P_{1},\dots,P_{q+2}$ in $\pi\setminus\ell$, with
$e_{P_{i}}<2q-2$. For each of these points $P_{i}$, the hyperplane
$P_{i}^{\perp}$ contains a solid $S_{i}$, with $P_{i}\not\in S_{i}$, and such
that all the points of the quadric $S_{i}\cap Q$ belong to $B$. These quadrics
$S_{i}\cap Q$ are hyperbolic quadrics $Q^{+}(3,q)_{i}$.

Here, $\pi\subseteq P_{i}^{\perp}$. Since all the intersection points of the
plane $\pi$ with $S_{i}$ belong to $B$, and $\pi\cap B=\ell$, necessarily the
line $\ell\subseteq S_{i}$.

For $i\neq j$, the subspace $S_{i}\cap S_{j}$ is either the line $\ell$ or it
is a plane that meets $Q$ in two lines, one of which is the line $\ell$. In
particular, at most $2q+1$ points of $Q$ lie in $S_{i}\cap S_{j}$. It follows
that the union $S_{i}\cup S_{j}$ contains at least
$2(q+1)^{2}-(2q+1)=2q^{2}+2q+1$ points of $B$. Also, as $\ell$ belongs to all
the subspaces $S_{i}$, the union of three of the solids $S_{i}$ contains at
least $3(q+1)^{2}-3(2q+1)+(q+1)=3q^{2}+q+1$ points of $B$.

By (1), we may assume that $e_{P_{1}}\leq q-1$. Consider a point $P_{i}$ with
$i\geq 2$. The line $P_{1}P_{i}$ is a line of $Q$; so it lies in two planes of
$Q$. One of these planes is $\pi$. Let $\pi^{\prime}$ be the other plane. Then
$\pi^{\prime}$ meets $S_{i}$ in a line and this line lies in $B$. As
$|P_{1}^{\perp}\cap B|\leq(q+1)^{2}+q-1$, at least two of the points of this
line must lie in $S_{1}$. Hence, the line $\pi^{\prime}\cap S_{i}$ lies in
$S_{1}$. This shows that $S_{1}\cap S_{i}$ is a plane that meets $Q$ in two
lines $\pi\cap S_{i}=\ell$ and $\pi^{\prime}\cap S_{i}$.

The solid $S_{1}$ lies in $q+1$ hyperplanes. One of these hyperplanes is
$P_{1}^{\perp}$. As $|P_{1}^{\perp}\cap
B|=(q+1)^{2}+e_{P_{1}}\leq(q+1)^{2}+q-1<2q^{2}+2q+1$ , the hyperplane
$P_{1}^{\perp}$ cannot contain one of the solids $S_{2},\dots,S_{q+2}$. Since
$S_{1}$ spans a hyperplane with each of the $q+1$ solids
$S_{2},\dots,S_{q+2}$, it follows that there exists a hyperplane $H$ on
$S_{1}$ that contains at least two of the solids $S_{2},\dots,S_{q+2}$. Then
$|H\cap B|\geq 3q^{2}+q+1$. We may assume that $S_{2},S_{3}\subseteq H$.

In order to prove the theorem, it suffices to show that $H\cap B$ meets all
the lines of $Q$. Assume that this is not the case. As $|H\cap B|\geq
3q^{2}+q+1$, 3.3 shows that $H$ is a non-tangent hyperplane, that $|H\cap
B|\leq 4q^{2}-2q+1$, and that every line of $H\cap Q$ meets $B$ in at least
two points.

Consider the three solids $S_{1},S_{2},S_{3}$ of $H$. The line $\ell$ belongs
to all three of them. As $H$ is a non-tangent hyperplane, it meets $Q$ in a
quadric of type $Q(4,q)$. Thus $\ell$ meets $(q+1)q$ other lines of $H\cap
Q=Q(4,q)$. Each of these lines meets $B$ in at least two points. One of these
two points belongs to the line $\ell$. Of the $(q+1)q$ lines of $H\cap Q$ that
meet $\ell$, at most $3(q+1)$ lie in one of the solids $S_{1},S_{2},S_{3}$.
Therefore, at least $(q+1)(q-3)$ lines of $H\cap Q$ meet $\ell$ and are not
contained in $S_{1},S_{2},S_{3}$. Since each line of $H$ meets $B$ in at least
two points, each of these $(q+1)(q-3)$ lines contains a point of $B$ that is
not in $S_{1},S_{2}$ or $S_{3}$. Different lines yield different points of
$B$, since $H$ does not contain a plane of $Q$ (or, equivalently, $H\cap
Q=Q(4,q)$ is a generalised quadrangle) [PT09]. Thus we have found $(q+1)(q-3)$
extra points in $B$ so that $|H\cap B|\geq 3q^{2}+q+1+(q+1)(q-3)=4q^{2}-q-2$.
But $|B\cap H|\leq 4q^{2}-2q+1$, a contradiction.

Case 2. Assume that for all the points $P\in\pi\setminus\ell$, $e_{P}=q$.

If $e_{P}=q$ for every point $P\in\pi\setminus\ell$, then $|P^{\perp}\cap
B|=(q+1)^{2}+q\leq(q+1)^{2}+2q-3$. So for every such point
$P\in\pi\setminus\ell$, $P^{\perp}$ contains a solid $S$, with $P\notin S$ and
$S\cap Q=Q^{+}(3,q)\subseteq B$ by Lemma 3.4. Moreover, $S\cap\pi=\ell$ for
each such solid $S$. We want to find points $P_{1},P_{2},\dots,P_{q+2}$ such
that $S_{1}\cap S_{i}$ is a plane for all $i=2,\dots,q+2$, since then we can
continue the proof of Case 1.

First choose $q+2$ arbitrary points $P_{i}\in\pi\setminus\ell$. The line
$P_{1}P_{i}$ lies in two planes of $Q$, one of which is $\pi$. Let $\pi_{i}$
be the other plane. Its intersection with $S_{i}$ is a line $\ell_{i}$ that is
contained in $B$. If this line $\ell_{i}$ also lies in $S_{1}$, then
$S_{1}\cap S_{i}$ is a plane. If this is the case for all
$i\in\\{2,\dots,q+2\\}$, then we are done. So suppose that this is not the
case. Then $\ell_{i}\not\subseteq S_{1}$ for a certain
$i\in\\{2,\dots,q+2\\}$. W.l.o.g. this is the case for $i=2$. Since
$\ell_{2}\subseteq P_{1}^{\perp}\cap B$ and $e_{P}=q$, $P_{1}^{\perp}\cap
B=Q^{+}(3,q)_{1}\cup\ell_{2}$. Now, for any other point
$P_{i}\in\pi\setminus(\ell\cup\pi_{2})$, $\ell_{i}$ must lie in $S_{1}$, since
otherwise $P_{1}^{\perp}\cap B$ contains at least $q-1$ more points (those
extra points on the line $\ell_{i}$) and therefore $e_{P_{1}}\geq 2q-1$, a
contradiction. So $S_{1}\cap S_{i}$ is a plane for all points
$P_{i}\in\pi\setminus(\ell\cup\pi_{2})$.

The remainder of the proof is very similar, but replacing 3.2 by 3.3 and
replacing $4q^{2}-2q$ by $4q^{2}-2q+1$. ∎

##  4 Small weight codewords of $\mathcal{C}(3,q)$

In this section, we determine the small weight codewords of
$\mathcal{C}(3,q)$. We do this by investigating the small weight codewords of
a code that contains $\mathcal{C}(3,q)$.

######  Definition 4.1.

Let $C(q)$ denote the code spanned by $\mathcal{C}(3,q)$ and the
characteristic vectors of the lines of the symplectic polar spaces $W(3,q)$
embedded in $\textnormal{PG}(3,q)$.

First, we prove that $C(q)=\mathcal{C}(3,q)$ if and only if $q$ is even. We
actually prove a similar result for a more general class of codes.

######  Definition 4.2.

Let $0<i<j<n$. For $c:\mathcal{G}_{j}(n,q)\to\mathbb{F}_{p}$, define

$\operatorname{proj}^{(i)}(c):\mathcal{G}_{i}(n,q)\to\mathbb{F}_{p}:\iota\mapsto\sum_{\begin{subarray}{c}\lambda\in\mathcal{G}_{j}(n,q)\\\
\iota\subseteq\lambda\end{subarray}}c(\lambda).$

######  Lemma 4.3.

Let $0<i,j,k<n$, $i<j$. A function $c:\mathcal{G}_{j}(n,q)\to\mathbb{F}_{p}$
belongs to $\mathcal{C}_{j,k}(n,q)^{\perp}$ if and only if
$\operatorname{proj}^{(i)}(c)$ belongs to $\mathcal{C}_{i,k}(n,q)^{\perp}$.

###### Proof.

It is valid that $\operatorname{proj}^{(i)}(c)$ belongs to
$\mathcal{C}_{i,k}(n,q)^{\perp}$ if and only if for each $k$-space $\kappa$,

$\sum_{\begin{subarray}{c}\iota\in\mathcal{G}_{i}(n,q)\\\
\iota\cap\kappa\neq\emptyset\end{subarray}}\operatorname{proj}^{(i)}(c)(\iota)=0.$

On the other hand,

$\displaystyle\sum_{\begin{subarray}{c}\iota\in\mathcal{G}_{i}(n,q)\\\
\iota\cap\kappa\neq\emptyset\end{subarray}}\operatorname{proj}^{(i)}(c)(\iota)=\sum_{\begin{subarray}{c}\iota\in\mathcal{G}_{i}(n,q)\\\
\iota\cap\kappa\neq\emptyset\end{subarray}}\sum_{\begin{subarray}{c}\lambda\in\mathcal{G}_{j}(n,q)\\\
\iota\subseteq\lambda\end{subarray}}c(\lambda)=\sum_{\begin{subarray}{c}\lambda\in\mathcal{G}_{j}(n,q)\\\
\lambda\cap\kappa\neq\emptyset\end{subarray}}c(\lambda)=c\cdot\chi_{\kappa}^{(j)},$

where we used that

$|\left\\{\iota\in\mathcal{G}_{i}(n,q)\,\,||\,\,\iota\subseteq\lambda,\,\iota\cap\kappa\neq\emptyset\right\\}|\equiv\begin{cases}0&\text{if
}\lambda\cap\kappa=\emptyset\\\ 1&\text{if
}\lambda\cap\kappa\neq\emptyset\end{cases}\pmod{p},$

because the number of $i$-subspaces of a $j$-space, having nonempty
intersection with a given subspace, equals $1$ modulo $p$, see e.g. [BCN89,
Lemma 3.2].

Hence, $\operatorname{proj}^{(i)}(c)$ belongs to the dual code of $i$-spaces
and $k$-spaces of $\textnormal{PG}(n,q)$ if and only if
$c\cdot\chi_{\kappa}^{(j)}=0$ for each $k$-space $\kappa$, which is equivalent
to $c\in\mathcal{C}_{j,k}(n,q)^{\perp}$. ∎

######  Proposition 4.4.

Let $S$ denote the set of absolute lines of a symplectic polar space
$W(2n+1,q)$ embedded in $\textnormal{PG}(2n+1,q)$. Then
$\chi_{S}\in\mathcal{C}_{1,n}(2n+1,q)$ if and only if $q$ is even.

###### Proof.

First suppose that $q$ is even. Consider a hyperbolic quadric $Q^{+}(2n+1,q)$,
and an equivalence class $\mathcal{R}$ of the generators ($n$-spaces) of
$Q^{+}(2n+1,q)$. Consider the codeword

$c=\sum_{\lambda\in\mathcal{R}}\chi^{(1)}_{\lambda}$

of the code $\mathcal{C}_{1,n}(2n+1,q)$. Take a line $\ell$ in
$\textnormal{PG}(2n+1,q)$. Then $c(\ell)$ equals the number of generators in
$\mathcal{R}$ that intersect $\ell$ non-trivially. Suppose that $\ell$
intersects $Q^{+}(2n+1,q)$ in $m_{1}$ points. The number of generators of
$Q^{+}(2n+1,q)$ of $\mathcal{R}$ through a point $P$ equals 0 if $P\notin
Q^{+}(2n+1,q)$ and equals the number of generators of $\mathcal{R}$ in
$P^{\perp}$ otherwise, with $\perp$ the polarity associated to
$Q^{+}(2n+1,q)$. Since $P^{\perp}$ intersects $Q^{+}(2n+1,q)$ in a cone
$PQ^{+}(2n-1,q)$, this number equals the number of generators of one
equivalence class in $Q^{+}(2n-1,q)$, which is given by
$\prod_{i=1}^{n-1}(q^{i}+1)$ (where we use the convention that an empty
product equals $1$) by 2.3 (2) and (3). Let $m_{2}$ denote the number of
generators of $Q^{+}(2n+1,q)$ through $\ell$. Then

$c(\ell)=m_{1}\prod_{i=1}^{n-1}(q^{i}+1)-qm_{2}\equiv m_{1}\pmod{p}.$

Hence, $c$ is the characteristic vector of the singular lines of
$Q^{+}(2n+1,q)$. Since $q$ is even, these are the absolute lines with respect
to $\perp$, which must be symplectic, again by $q$ being even.

Now suppose that $q$ is odd. Let $\perp$ denote the polarity associated to
$W(2n+1,q)$. Take a 3-space $\Sigma$ of $W(2n+1,q)$ such that
$\Sigma\cap\Sigma^{\perp}=\emptyset$. Then the absolute subspaces of $\perp$
contained in $\Sigma$ are the lines and points of a $W(3,q)$ in $\Sigma$.
Consider a hyperbolic quadric $Q^{+}(3,q)$ in $\Sigma$ and let
$\mathcal{R}^{+}$ and $\mathcal{R}^{-}$ denote its two reguli. Consider the
function

$c:\mathcal{G}_{1}(2n+1,q)\to\mathbb{F}_{p}:\ell\mapsto\begin{cases}1&\text{if
}\ell\in\mathcal{R}^{+},\\\ -1&\text{if }\ell\in\mathcal{R}^{-},\\\
0&\text{otherwise.}\end{cases}$

Then $\operatorname{proj}^{(0)}(c)=\mathbf{0}$, hence
$c\in\mathcal{C}_{1,n}(2n+1,q)^{\perp}$ by 4.3. Therefore, if $\chi_{S}$ would
be in $\mathcal{C}_{1,n}(2n+1,q)$, then $\chi_{S}\cdot c=0$, which means that
$S$ contains the same number of lines of $\mathcal{R}^{+}$ and
$\mathcal{R}^{-}$ modulo $p$. Hence, in order to prove that
$\chi_{S}\notin\mathcal{C}_{1,n}(2n+1,q)$, it suffices to show that there
exists a hyperbolic quadric $Q^{+}(3,q)$ and a symplectic polar space $W(3,q)$
in $\Sigma\cong\textnormal{PG}(3,q)$ where the reguli of $Q^{+}(3,q)$ do not
contain the same number of lines of $W(3,q)$ modulo $p$.

To this end, consider the symplectic polar space of $\textnormal{PG}(3,q)$
defined by the bilinear form

$b(x,y)=x_{0}y_{1}-x_{1}y_{0}+x_{2}y_{3}-x_{3}y_{2}.$

From now on, let $\perp$ denote the polarity of the symplectic space $W(3,q)$
corresponding to $b$. A line through two distinct points of
$\textnormal{PG}(3,q)$, with coordinates $x=(x_{0},\dots,x_{3})$ and
$y=(y_{0},\dots,y_{3})$, is absolute if and only if $b(x,y)=0$. Now consider
the hyperbolic quadric with equation $x_{0}x_{1}=x_{2}x_{3}$. Its two reguli
are

$\displaystyle\mathcal{R}^{+}$
$\displaystyle=\left\\{\left\langle(\alpha,0,\beta,0),(0,\beta,0,\alpha)\right\rangle\,\,||\,\,\left\langle(\alpha,\beta)\right\rangle\in\textnormal{PG}(1,q)\right\\},$
$\displaystyle\mathcal{R}^{-}$
$\displaystyle=\left\\{\left\langle(\alpha,0,0,\beta),(0,\beta,\alpha,0)\right\rangle\,\,||\,\,\left\langle(\alpha,\beta)\right\rangle\in\textnormal{PG}(1,q)\right\\}.$

On the one hand,

$b\left((\alpha,0,\beta,0),(0,\beta,0,\alpha)\right)=2\alpha\beta.$

Since $q$ is odd, this means that $\mathcal{R}^{+}$ contains two absolute
lines. On the other hand

$b\left((\alpha,0,0,\beta),(0,\beta,\alpha,0)\right)=\alpha\beta-\beta\alpha=0,$

so all $q+1$ lines in $\mathcal{R}^{-}$ are absolute. Note that $q+1\equiv
1\not\equiv 2\pmod{p}$, which finishes the proof. ∎

Now we continue our investigation of the small weight codewords of $C(q)$.

######  Lemma 4.5.

Given $c\in C(q)$, there exists a scalar $\alpha\in\mathbb{F}_{p}$ with the
property that $c\cdot\chi_{S}=\alpha$ whenever $S$ is one of the following
types of sets:

  * $S$ equals the set of all lines in $\textnormal{PG}(3,q)$, i.e. $S=\mathcal{G}_{1}(3,q)$,

  * $S$ equals the set of lines in a plane of $\textnormal{PG}(3,q)$,

  * $S$ equals the set of lines through a point of $\textnormal{PG}(3,q)$,

  * $S$ equals the set of lines through a point in a plane of $\textnormal{PG}(3,q)$.

###### Proof.

Let $\chi$ either be the characteristic vector of the lines of a symplectic
space $W(3,q)$, or a vector of the form $\chi_{\ell}$ for some line $\ell$ in
$\textnormal{PG}(3,q)$. Then for each of the sets $S$ listed above,
$\chi\cdot\chi_{S}=1$. The proof for the other codewords of $C(q)$ follows
from the linearity of the code $C(q)$. ∎

Note that if $\alpha\neq 0$ in the above lemma, then $c\cdot\chi_{S}\neq 0$
whenever $S$ is the set of lines through a point in a plane. Therefore, the
image of $\operatorname{supp}(c)$ under the Klein correspondence must be a
blocking set with respect to the lines of $Q^{+}(5,q)$. If
$\operatorname{wt}(c)$ is small, then we can apply 1.4 to obtain strong
structural information about $\operatorname{supp}(c)$. It remains to deal with
the case $\alpha=0$.

We will use a result in projective planes that is a special case of a result
in general combinatorial designs. For a proof, see [Sti13, Theorem 3.1] for
the projective planes and [Sti13, Theorem 4.1] or [Hae95, Corollary 5.3] for
the general result.

######  Result 4.6.

A set $S$ of lines in $\textnormal{PG}(2,q)$ covers at least
$\frac{(q+1)^{2}|S|}{q+|S|}$ points.

We need one more small lemma, and then we are ready to deal with the case
$\alpha=0$. Given a function $f$, and a subset $S$ of its domain, denote the
restriction of $f$ to $S$ by $f_{|S}$.

######  Lemma 4.7.

Suppose that $c\in C(q)$.

  * If $S$ equals the set of lines in a plane $\pi$ of $\textnormal{PG}(3,q)$, then $c_{|S}$ belongs to the code $\mathcal{C}_{1,0}(2,q)$ defined on the lines of $\pi$.

  * If $S$ equals the set of lines through a point $P$, then $c_{|S}$ belongs to the code of points and lines $\mathcal{C}_{0,1}(2,q)$ in the quotient geometry through $P$.

###### Proof.

Suppose that $S$ denotes the set of lines in the plane $\pi$. Then for each
line $\ell$, either $\ell\in\pi$ and $\chi^{(1)}_{\ell|S}=\mathbf{1}$, or
$\ell\cap\pi$ is some point $P$, and $\chi^{(1)}_{\ell|S}$ is the
characteristic vector of the lines through $P$ in $\pi$. Similarly, if $\chi$
is the characteristic vector of the lines of a symplectic space $W(3,q)$ with
polarity $\perp$, then $\chi_{|S}$ is the characteristic vector of the lines
through the point $\pi^{\perp}$ in $\pi$. In all cases, these vectors belong
to the code of $\mathcal{C}_{1,0}(2,q)$ defined on the lines of $\pi$. Then
this is valid for all $c\in C(q)$ by linearity.

The proof is analogous for $S$ equal to the set of lines through a point in
$\textnormal{PG}(3,q)$. ∎

######  Proposition 4.8.

Suppose that $q\geq 19$. Let $c\in C(q)$, with $c\cdot\mathbf{1}=0$. If
$c\neq\mathbf{0}$, then $\operatorname{wt}(c)\geq q^{3}+2q^{2}+q+3.$

###### Proof.

Assume that $c\cdot\mathbf{1}=0$ and $c\neq\mathbf{0}$. Let $\delta$ denote
$3$ if $q$ is prime and $12$ otherwise. Take a plane $\pi$ which is not
disjoint to $\operatorname{supp}(c)$ and let $S$ be the set of lines in $\pi$.
Consider the code $\mathcal{C}_{1,0}(2,q)$ of lines and points in $\pi$. Then
$c_{|S}\in\mathcal{C}_{1,0}(2,q)$ by 4.7. Now suppose that $\pi$ contains
fewer than $3q-\delta$ points of $\operatorname{supp}(c)$. By 2.2, $c_{|S}$ is
a linear combination of at most two characteristic vectors of points, i.e.
$c_{|S}=\alpha\chi_{P}^{(1)}+\beta\chi_{R}^{(1)}$ for some scalars
$\alpha,\beta\in\mathbb{F}_{p}$ and points $P,R\in\pi$. 4.5 implies that
$c\cdot\chi_{S}=c\cdot\mathbf{1}=0$. Since
$\alpha+\beta=c_{|S}\cdot\mathbf{1}=c\cdot\chi_{S}=0$, it follows that
$\alpha=-\beta$. Note that $\alpha,\beta\neq 0$ and $P\neq R$ since we assumed
that $c_{|S}\neq\mathbf{0}$.

We conclude the following. For any plane $\pi$, one of three cases holds:

  * $\pi$ contains no lines of $\operatorname{supp}(c)$,

  * $\pi$ contains exactly $2q$ lines of $\operatorname{supp}(c)$, and there exist two distinct points $P,R$ in $\pi$ so that the lines of $\operatorname{supp}(c)$ contained in $\pi$ are exactly the lines of $\pi$ containing exactly one of the points $P$ and $R$,

  * $\pi$ contains at least $3q-\delta$ lines of $\operatorname{supp}(c)$.

Completely analogously, any point of $\textnormal{PG}(3,q)$ belongs to 0,
$2q$, or at least $3q-\delta$ lines of $\operatorname{supp}(c)$.

First consider the case where some plane $\pi$ of $\textnormal{PG}(3,q)$
contains $2q$ lines of $\operatorname{supp}(c)$, and let $P,R$ be as above.
Then every point of $\pi$ not on the line $\left\langle P,R\right\rangle$
belongs to two lines of $\operatorname{supp}(c)$ in $\pi$, hence, such a point
belongs to at least $2q-2$ lines of $\operatorname{supp}(c)$ outside of $\pi$.
Also, $P$ and $R$ lie on at least $2q$ lines of $\operatorname{supp}(c)$. This
yields

$\operatorname{wt}(c)\geq q^{2}(2q-2)+2\cdot
2q=2q^{3}-2q^{2}+4q>q^{3}+2q^{2}+q+3.$

If some point $P$ belongs to exactly $2q$ lines of $\operatorname{supp}(c)$,
we can apply a similar argument. Thus, the last case to consider is the case
where no plane or point of PG$(3,q)$ is incident with exactly $2q$ lines of
$\operatorname{supp}(c)$. Take a plane $\pi$ containing a line of
$\operatorname{supp}(c)$. Then $\pi$ contains at least $3q-\delta$ lines of
$\operatorname{supp}(c)$. By 4.6, these lines cover at least
$\frac{(q+1)^{2}(3q-\delta)}{4q-\delta}$ points of $\pi$. Each of these points
belongs to at least $3q-\delta-(q+1)=2q-\delta-1$ lines outside of $\pi$.
Therefore,

$\operatorname{wt}(c)\geq
3q-\delta+\frac{(q+1)^{2}(3q-\delta)}{4q-\delta}(2q-\delta-1).$

Plugging in that either $\delta=3$ and $q\geq 19$, or $\delta=12$ and $q\geq
25$ yields

$\operatorname{wt}(c)>q^{3}+2q^{2}+q+2.\qed$

######  Theorem 4.9.

Suppose that $q\geq 19$ and that $c$ is a non-zero codeword of $C(q)$ with
$\operatorname{wt}(c)\leq q^{3}+2q^{2}+q+1$. Then one of the following cases
holds.

  1. (1)

$\operatorname{wt}(c)=q^{3}+q^{2}+q+1$, and $c$ is a scalar multiple of the
characteristic vector of the lines of a symplectic space $W(3,q)$.

  2. (2)

$\operatorname{wt}(c)=q^{3}+2q^{2}+q+1$, and $c$ is a scalar multiple of
$\chi^{(1)}_{\ell}$ for some line $\ell$.

###### Proof.

Define $\alpha$ as $c\cdot\mathbf{1}$. By 4.8, $\alpha\neq 0$. As noted
before, 4.5 implies that $\operatorname{supp}(c)$ is equivalent to a blocking
set of $Q^{+}(5,q)$ under the Klein correspondence, whose size is at most
$q^{3}+2q^{2}+q+1$. By 1.4, there are two options:

  * $\operatorname{supp}(c)$ contains the set $S$ of lines of a symplectic space $W(3,q)$. In this case, let $\chi$ denote $\chi_{S}$. Since $S\subseteq\operatorname{supp}(c)$, $\operatorname{wt}(c-\alpha\chi)\leq\operatorname{wt}(c)$.

  * $\operatorname{supp}(c)$ contains the set $S$ of lines that intersect a given line $\ell$ of $\textnormal{PG}(3,q)$ exactly in a point. In this case, let $\chi$ denote $\chi^{(1)}_{\ell}$. Since $\ell$ is the only possible element of $\operatorname{supp}(\chi)\setminus\operatorname{supp}(c)$, $\operatorname{wt}(c-\alpha\chi)\leq\operatorname{wt}(c)+1$.

In both cases, we find a codeword $c-\alpha\chi$ of $C(q)$ with
$(c-\alpha\chi)\cdot\mathbf{1}=0$ and
$\operatorname{wt}(c-\alpha\chi)\leq\operatorname{wt}(c)+1<q^{3}+2q^{2}+q+3$.
By 4.8, this implies that $c=\alpha\chi$. ∎

Note that 1.2 follows directly from 4.9 and 4.4

### Acknowledgements.

To obtain the results of this article, we needed to improve 1.3 of Metsch on
blocking sets with respect to the lines of the hyperbolic quadric $Q^{+}(5,q)$
[Met00]. We thank Metsch for allowing us to repeat large parts of his proofs
in Section 3 of this article to obtain the improvement. We are also grateful
to Mrinmoy Datta for helpful discussions.

Robin Simoens is supported by the Research Foundation Flanders (FWO) through
the grant 11PG724N.

## References

  * [AD23] S. Adriaensen and L. Denaux.  Small weight codewords of projective geometric codes II, 2023.  arXiv:2309.00490.
  * [BCN89] A. Brouwer, A. Cohen, and A. Neumaier.  Distance-Regular Graphs.  Springer-Verlag, Berlin, 1989.
  * [BG04] L. D. Baumert and D. M. Gordon.  On the existence of cyclic difference sets with small parameters.  In High primes and misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams, volume 41 of Fields Inst. Commun., pages 61–68. Amer. Math. Soc., Providence, RI, 2004.
  * [BI02] B. Bagchi and S. P. Inamdar.  Projective geometric codes.  J. Combin. Theory Ser. A, 99(1):128–142, 2002.
  * [Bru70] A. Bruen.  Baer subplanes and blocking sets.  Bulletin of the American Mathematical Society, 76:342–344, 1970\.
  * [Cam15] P. Cameron.  Projective and polar spaces, 2015.  https://cameroncounts.files.wordpress.com/2015/04/pps1.pdf.
  * [DGM70] P. Delsarte, J.-M. Goethals, and F. J. MacWilliams.  On generalized Reed-Muller codes and their relatives.  Information and Control, 16:403–442, 1970.
  * [Hae95] W. H. Haemers.  Interlacing eigenvalues and graphs.  Linear Algebra Appl., 226/228:593–616, 1995.
  * [Ham68] N. Hamada.  The rank of the incidence matrix of points and $d$-flats in finite geometries.  J. Sci. Hiroshima Univ. Ser. A-I Math., 32:381–396, 1968.
  * [HT16] J. W. P. Hirschfeld and J. A. Thas.  General Galois geometries.  Springer Monographs in Mathematics. Springer, London, 2016.
  * [LSVdV10] M. Lavrauw, L. Storme, and G. Van de Voorde.  Linear codes from projective spaces.  In Error-correcting codes, finite geometries and cryptography, volume 523 of Contemp. Math., pages 185–202. Amer. Math. Soc., Providence, RI, 2010.
  * [Met00] K. Metsch.  On blocking sets of quadrics.  Journal of Geometry, 67(1-2):188–207, 2000.
  * [PT09] Stanley E. Payne and Joseph A. Thas.  Finite generalized quadrangles.  EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), Zürich, second edition, 2009.
  * [Sin04] P. Sin.  The $p$-rank of the incidence matrix of intersecting linear subspaces.  Des. Codes Cryptogr., 31(3):213–220, 2004.
  * [Sti13] D. R. Stinson.  Nonincident points and blocks in designs.  Discrete Math., 313(4):447–452, 2013.
  * [SW18] T. Szőnyi and Zs. Weiner.  Stability of $k\bmod p$ multisets and small weight codewords of the code generated by the lines of PG(2, $q$).  J. Combin. Theory Ser. A, 157:321–333, 2018.
  * [vL99] J. H. van Lint.  Introduction to coding theory, volume 86 of Graduate Texts in Mathematics.  Springer-Verlag, Berlin, third edition, 1999.
  * [Wel66] E. J. Weldon, Jr.  Difference-set cyclic codes.  Bell System Tech. J., 45:1045–1055, 1966.

Sam Adriaensen
Department of Mathematics and Data Science
Vrije Universiteit Brussel
Pleinlaan 2, 1050 Elsene, Belgium
E-mail address<EMAIL_ADDRESS>

Robin Simoens
Department of Mathematics: Analysis, Logic and Discrete Mathematics
Ghent University
Krijgslaan 281, 9000 Gent, Belgium
Department of Mathematics
Universitat Politècnica de Catalunya
C. Pau Gargallo 14, 08028 Barcelona, Spain
E-mail address<EMAIL_ADDRESS>

Leo Storme
Department of Mathematics: Analysis, Logic and Discrete Mathematics
Ghent University
Krijgslaan 281, 9000 Gent, Belgium
E-mail address<EMAIL_ADDRESS>