11institutetext: Indraprastha Institute of Information Technology Delhi, India
11email<EMAIL_ADDRESS>22institutetext: Florida Atlantic
University, Boca Raton, Florida, USA
22email<EMAIL_ADDRESS>
# Antipodal two-weight rank metric codes††thanks: During the course of this
work, the first named author was partially supported by Grant 280731 from the
Research Council of Norway and currently, she is supported by a postdoctoral
researcher grant by iHub Anubhuti-IIITD Foundation (IHUB), IIIT Delhi. The
second named author is partially funded by the National Science Foundation
(NSF) grant CNS-1906360.

Rakhi Pratihar 11 Tovohery Hajatiana Randrianarisoa 22

###### Abstract

We consider the class of linear antipodal two-weight rank metric codes and
discuss their properties and characterization in terms of $t$-spreads. It is
shown that the dimension of such codes is $2$ and the minimum rank distance is
at least half of the length. We construct antipodal two-weight rank metric
codes from certain MRD codes. A complete classification of such codes is
obtained, when the minimum rank distance is equal to half of the length. As a
consequence of our construction of two-weight rank metric codes, we get some
explicit two-weight Hamming metric codes.

###### Keywords:

Rank metric codes Antipodal two-weight t-Spreads Subspreads

##  1 Introduction

Let $q$ be a prime power and let $\mathbb{F}_{q}$ be a finite field of size
$q$. For a positive integer $n$, the _Hamming weight_ of an element
$\mathbf{c}=(c_{1},\dots,c_{n})$ in $\mathbb{F}_{q}^{n}$ is defined by

$\mathsf{w}_{H}(\mathbf{c})=|\\{i:c_{i}\neq 0,1\leq i\leq n\\}|.$

The function “$\mathsf{w}_{H}$” induces a metric $\mathrm{d}_{H}$ on
$\mathbb{F}_{q}^{n}$ where
$\mathrm{d}_{H}(\mathbf{c},\mathbf{c}^{\prime})=\mathsf{w}_{H}(\mathbf{c}-\mathbf{c}^{\prime})$
for $\mathbf{c},\mathbf{c}^{\prime}$ in $\mathbb{F}_{q}^{n}$. An $[n,k,d]$
(linear) Hamming metric code $\mathcal{C}$ over $\mathbb{F}_{q}$ is an
$\mathbb{F}_{q}$-linear subspace of $\mathbb{F}_{q}^{n}$ of dimension $k$ such
that
$\min_{\mathbf{c}\in\mathcal{C}\backslash\\{\mathbf{0}\\}}\mathsf{w}_{H}(\mathbf{c})=d$.
A two-weight linear Hamming metric code over $\mathbb{F}_{q}$ is linear code
such that the weight distribution (i.e. the set of possible weights) of
$\mathcal{C}$ is of the form $\\{0,d,d^{\prime}\\}$ such that
$0<d<d^{\prime}$. The terminology is explained by the fact that there are only
two possible non-zero weights for the codewords. An $[n,k,d]$ linear Hamming
metric code over $\mathbb{F}_{q}$ is called _antipodal two-weight_ or ATW if
$d<n$ and any nonzero codewords have Hamming weight either $d$ or $n$ and
there is at least one codeword of Hamming weight $n$.

Linear codes with few distinct weights are important in coding theory, both
from practical and theoretical point of view. These classes of codes are for
example used in secret sharing schemes [13, 21] and authentication schemes
[7]. For codes with Hamming metric, the _constant weight codes_ i.e. codes
where all the nonzero codewords have the same weight and the _two-weight_
linear codes have been studied extensively. In [4], Bonisoli gives a
characterization of the constant weight codes. The class of two-weight linear
Hamming metric codes has been investigated by Delsarte [6]; see [5] for a
systematic exposition. A classification of the subclass of ATW linear codes
with Hamming metric is obtained in [11].

We are interested in the $q$-analogues of some of these results. For linear
rank metric codes, the class of constant rank weight codes are completely
classified in [16] following a geometric approach. It is proved that, up to
equivalence, there is only one non-degenerate constant rank weight code if the
code has dimension at least $2$. In this article we consider the class of ATW
rank metric codes. Using the classification of constant rank weight codes, we
show that the dimension of such codes is $2$ and the minimum distance $d$ must
be at least $\frac{n}{2}$. We provide an equivalent characterization of ATW
rank metric codes in terms of $t$-spreads, more precisely, as $t$-subspreads
of Desarguesian $t$-spreads induced by $q$-systems associated to ATW rank
metric codes. It is proved that, up to equivalence, there exists only one ATW
code for the case $d=\frac{n}{2}$ and a complete classification of such codes
is also obtained. For the case of $d>\frac{n}{2}$, we get partial results
regarding the classification. We construct ATW rank metric codes using MRD
codes of suitable parameters.

The article is organized as follows. In the next section, we collect some
preliminaries about notions such as rank metric codes and $t$-spreads. In
Section 3, we derive some properties of the ATW rank metric codes and
$t$-spreads and establish a relation between these two objects. Section 4
deals with classification and construction of ATW rank metric codes. In
Section 5, we provide a construction of two-weight Hamming metric codes from
two-weight rank metric codes. Finally, we conclude the paper with a few open
questions.

##  2 Preliminaries

Throughout, we use $\mathbf{a}\cdot\mathbf{b}$ to denote the usual dot product
of two vectors $\mathbf{a},\mathbf{b}\in\mathbb{F}_{q^{m}}^{n}$. For
$\mathbf{a}=(a_{1},\dots,a_{n})\in\mathbb{F}_{q^{m}}^{n}$, we denote by
$\left<a_{1},\dots,a_{n}\right>_{\mathbb{F}_{q}}$ the
$\mathbb{F}_{q}$-subspace of $\mathbb{F}_{q^{m}}$ generated by
$\\{a_{1},\dots,a_{n}\\}$.

###  2.1 Rank metric codes

Let $\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$ be a finite field extension of degree
$m$. We endow the space $\mathbb{F}_{q^{m}}^{n}$ with a norm, called
$\mathsf{rank}$, where
$\mathsf{rank}(\mathbf{c})=\dim_{\mathbb{F}_{q}}\left<c_{1},\dots,c_{n}\right>_{\mathbb{F}_{q}}$
for any $\mathbf{c}=(c_{1},\dots,c_{n})$ in $\mathbb{F}_{q^{m}}^{n}$. This
norm induces a metric on $\mathbb{F}_{q^{m}}^{n}$, which is defined by

$\mathrm{d}_{r}(\mathbf{c}_{1},\mathbf{c}_{2})=\mathsf{rank}(\mathbf{c}_{1}-\mathbf{c}_{2}),\text{
for }\mathbf{c}_{1},\mathbf{c}_{2}\in\mathbb{F}_{q^{m}}^{n}.$

######  Definition 1 (Rank metric codes)

Let $\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$ be a field extension of degree $m$. An
$[n,k,d]$ rank metric code $\mathcal{C}$ over
$\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$ is a $k$-dimensional
$\mathbb{F}_{q^{m}}$-subspace of $\mathbb{F}_{q^{m}}^{n}$ such that
$d=\min\\{\mathsf{rank}(\mathbf{c})\colon\mathbf{c}\in\mathcal{C}\backslash\\{\mathbf{0}\\}\\}$.
Here $d$ is called the minimum (rank) distance of the code.

When we are not concerned with the minimum distance, we simply call the codes
as $[n,k]$ rank metric codes. In [6], it was shown that for any $[n,k,d]$ rank
metric code over $\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$, $d\leq n-k+1$. Similar
to the Hamming metric case, this inequality is called the Singleton bound and
when the bound is attained i.e. $d=n-k+1$, the code is called a _maximum rank
distance_ (MRD) code. The first construction of MRD codes was independently
presented in [6, 8]. Later, many constructions of new classes of MRD codes
were obtained, e.g., in [9, 15, 18] to name a few.

For the purpose of classifying the rank metric codes, we need to have a notion
of equivalence of two codes. There are various notions of equivalence of two
rank metric codes. In this paper we consider the following notion defined in
[14].

###### Definition 2

[14] Two rank metric codes $\mathcal{C}$ and $\mathcal{C}^{\prime}$ over
$\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$ are called equivalent if
$\mathcal{C}=\alpha\mathcal{C}^{\prime}\mathbf{M}$ for some
$\alpha\in\mathbb{F}_{q^{m}}^{\times}$ and
$\mathbf{M}\in\mathbb{F}_{q}^{n\times n}$ invertible, where
$\mathcal{C}^{\prime}\mathbf{M}=\\{\mathbf{c}\mathbf{M}\colon\mathbf{c}\in\mathcal{C}^{\prime}\\}$.
It is equivalent to saying that any two generator matrices $\mathbf{G}$ and
$\mathbf{G}^{\prime}$, corresponding to $\mathcal{C}$ and
$\mathcal{C}^{\prime}$, respectively, satisfy the relation
$\mathbf{G}=\SS\mathbf{G}^{\prime}\mathbf{M}$, where
$\SS\in\mathbb{F}_{q^{m}}^{k\times k}$ is invertible.

A rank metric code $\mathcal{C}$ is called _non-degenerate_ if the columns of
its generator matrix $\mathbf{G}$ are linearly independent over
$\mathbb{F}_{q}$. Otherwise, $\mathcal{C}$ is called _degenerate_ and in this
case, $\mathcal{C}$ is equivalent to a code
$\\{(\mathbf{c}|\mathbf{0})\colon\mathbf{c}\in\mathcal{C}^{\prime}\\}$, where
$\mathcal{C}^{\prime}$ is non-degenerate of shorter length. Throughout this
paper, our code is assumed to be non-degenerate unless otherwise specified.

In [16], rank metric codes are described in terms of the geometric object
called $q$-systems. This geometric description offers the advantage of
studying the distance of a code by looking at the hyperplane sections of a
corresponding $q$-system. Next we review some of notions related to the
geometric description of rank metric codes relevant for our study.

######  Definition 3 ($q$-system)

Let $k,n$ be positive integers such that $k\leq n$. An $[n,k]$ $q$-system $X$
in $\mathbb{F}_{q^{m}}^{k}$ is an $\mathbb{F}_{q}$-subspace of
$\mathbb{F}_{q^{m}}^{k}$ of dimension $n$. If the set
$\\{\mathbf{c}_{i}\in\mathbb{F}_{q^{m}}^{k}\colon 1\leq i\leq n\\}$ forms an
$\mathbb{F}_{q}$-basis of $X$, then we write
$X=\left<\mathbf{c}_{1},\dots,\mathbf{c}_{n}\right>_{\mathbb{F}_{q}}$.

Let $\mathbf{G}$ be a generator matrix of a non-degenerate $[n,k]$ rank metric
code $\mathcal{C}$ over $\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$. The
$\mathbb{F}_{q}$-space generated by the columns of $\mathbf{G}$ is an $[n,k]$
$q$-system. As it was shown in [16, Theorem 2], this construction gives a one-
to-one correspondence between equivalence classes of $q$-systems and rank
metric codes. The key point of this approach is the geometric interpretation
of rank of a codeword which we explain below.

###### Lemma 1

For a vector $\mathbf{c}\in\mathbb{F}_{q^{m}}^{n}$,
$\mathsf{rank}(\mathbf{c})=n-\dim_{\mathbb{F}_{q}}\mathrm{Ker}\,\phi_{\mathbf{c}}$
where $\phi_{\mathbf{c}}$ is the $\mathbb{F}_{q}$-linear map defined as

$\displaystyle\phi_{\mathbf{c}}\colon\mathbb{F}_{q}^{n}$
$\displaystyle\longrightarrow\mathbb{F}_{q^{m}}$ $\displaystyle\mathbf{a}$
$\displaystyle\longmapsto\mathbf{c}\cdot\mathbf{a}.$

###### Proof

It is a straightforward consequence of the rank-nullity theorem. ∎

For any $\mathbf{x}\in\mathbb{F}_{q^{m}}^{k}\backslash\mathbf{0}$, we consider
the following $\mathbb{F}_{q^{m}}$-linear map

$\displaystyle\psi_{\mathbf{x}}\colon\mathbb{F}_{q^{m}}^{k}$
$\displaystyle\longrightarrow\mathbb{F}_{q^{m}}$ (1) $\displaystyle\mathbf{e}$
$\displaystyle\longmapsto\mathbf{x}\cdot\mathbf{e}.$

It is easy to see that kernel of $\psi_{\mathbf{x}}$ defines a unique
hyperplane, which we name as $H_{\mathbf{x}}$. In fact, any hyperplane of
$\mathbb{F}_{q^{m}}^{k}$ can be seen as kernel of $\phi_{\mathbf{x}}$ for some
$\mathbf{x}\in\mathbb{F}_{q^{m}}^{k}$.

Now let $\mathcal{C}$ be an $[n,k]$ rank metric code over
$\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$. Any codeword $\mathbf{c}\in\mathcal{C}$
is of the form $\mathbf{c}=\mathbf{x}_{\mathbf{c}}\mathbf{G}$ for some
$\mathbf{x}_{\mathbf{c}}\in\mathbb{F}_{q^{m}}^{k}$. Then $\mathbf{c}$ defines
a unique hyperplane
$H_{\mathbf{c}}=\mathrm{Ker}~{}(\psi_{\mathbf{x}_{\mathbf{c}}})$ of
$\mathbb{F}_{q^{m}}^{k}$. Conversely, from the discussion above, a hyperplane
$H$ of $\mathbb{F}_{q^{m}}^{k}$ corresponds to a codeword
$\mathbf{c}=\mathbf{x}\mathbf{G}$ defined by a vector
$\mathbf{x}\in\mathbb{F}_{q^{m}}^{k}$.

###### Lemma 2

Let $\mathcal{C}$ be an $[n,k]$ (non-degenerate) rank metric code over
$\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$ and let $X$ be the $[n,k]$ $q$-system
corresponding to a generator matrix $\mathbf{G}$ of $\mathcal{C}$. Then for
any $\mathbf{c}={\mathbf{x}_{\mathbf{c}}\mathbf{G}}\in\mathcal{C}$,

$\mathsf{rank}(\mathbf{c})=n-\dim_{\mathbb{F}_{q}}\left(X\cap
H_{\mathbf{c}}\right),\text{ where}$ (2)

$H_{\mathbf{c}}=\mathrm{Ker}~{}\psi_{\mathbf{x}_{\mathbf{c}}}$ is considered
as an $\mathbb{F}_{q}$-subspace of $\mathbb{F}_{q^{m}}^{k}$ of dimension
$m(k-1)$.

###### Proof

Following the definition of $\psi_{\mathbf{x}_{\mathbf{c}}}$ in Equation (1),
for any $\mathbf{e}\in\left(X\cap H_{\mathbf{c}}\right)$, we have
$\mathbf{x}_{\mathbf{c}}\cdot\mathbf{e}=0$. Moreover, $X$ being the $q$-system
corresponding to $\mathbf{G}$, we get $\mathbf{e}=\mathbf{G}\mathbf{a}^{T}$
for some $\mathbf{a}\in\mathbb{F}_{q}^{n}$. Note that $\mathbf{a}$ is unique
since $\mathcal{C}$ is non-degenerate. This implies that
$\mathbf{c}\cdot\mathbf{a}=0$. Therefore, $\mathbf{e}\in X\cap H_{\mathbf{c}}$
corresponds to a unique element
$\mathbf{a}\in\mathrm{Ker}~{}\phi_{\mathbf{c}}$. Also, the same argument
backward shows that any element of $\mathrm{Ker}\,\phi_{\mathbf{c}}$
corresponds to a unique element of $X\cap H_{\mathbf{c}}$. Hence $|X\cap
H_{\mathbf{c}}|=|\mathrm{Ker}\,\phi_{\mathbf{c}}|$ and the result follows from
Lemma 1. ∎

The geometric approach offers the advantage that if the $q$-system $X$
corresponding to the code $\mathcal{C}$ has nice properties, then it can be
simpler to study the intersection of $X$ with the hyperplanes of
$\mathbb{F}_{q^{m}}^{k}$. That makes it easier to study the weights of the
codewords of $\mathcal{C}$. Furthermore, if the hyperlanes
$\mathcal{H}_{\mathbf{c}}$ are replaced with subspaces of arbitrary dimension,
then it defines the “ _rank weight_ ” of subcodes of $\mathcal{C}$. And these
can be used to define the “generalized rank weights” of a rank metric code.
For more details on the geometric approach, one can look at [16]. Using the
geometric approach, the complete classification of the constant weight rank
metric codes has been obtained and we recall the result below.

###### Definition 4

[16, Definition 11] Let $X$ be the $\mathbb{F}_{q}$-vector space
$\mathbb{F}_{q^{m}}^{k}$ of dimension $mk$. A Hadamard rank metric code
$H_{1}(q,m,k)$ is an $[mk,k,m]$ linear code over
$\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$ with a generator matrix where the columns
consist of an $\mathbb{F}_{q}$-basis of $X=\mathbb{F}_{q^{m}}^{k}$.

Notice that Hadamard rank metric codes are unique up to equivalence. The
following theorem is the complete characterization of (linear) constant weight
rank metric codes i.e. all non-zero codewords have the same rank $d$ equal to
the minimum distance of the code.

###### Theorem 2.1

[16, Theorem 12] Let $\mathcal{C}$ be an $[n,k,d]$ non-degenerate constant
weight linear rank metric code over $\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$. We
have the following two cases:

  1. (a)

If $k=1$, then
$\mathcal{C}=\left<(a_{1},\dots,a_{n})\right>_{\mathbb{F}_{q^{m}}}$ with
$\mathsf{rank}(a_{1},\dots,a_{n})=d=n$, and

  2. (b)

if $k>1$, then $\mathcal{C}$ is an $[mk,k,m]$ Hadamard rank metric code
$H_{1}(q,m,k)$.

Theorem 2.1 gives a complete description of non-degenerate (and thus
degenerate) $[n,k,d]$ rank metric codes whose all non-zero codewords have the
same rank, that is equal to $d$. The next logical question is to consider
codes where there are two possible ranks for the non-zero codewords.

######  Definition 5 (Two-weight rank metric codes)

An $[n,k,d]$ rank metric code $\mathcal{C}$ over
$\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$ is called a two-weight rank metric code if
for any $\mathbf{c}\in\mathcal{C}$,
$\mathsf{rank}(\mathbf{c})\in\\{0,d,d^{\prime}\\}$ where $0<d<d^{\prime}$.

It is shown in [1, Proposition 3.11] that, for an $[n,k,d]$ rank metric code
over $\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$, there exists a codeword with rank
equal to $\min\\{n,m\\}$. Therefore, for a two-weight rank metric code, there
are two possible weight distributions: $\\{0,d,m\\}$ or $\\{0,d,n\\}$. Now we
consider the particular class of two-weight rank metric codes that have a
weight distribution of the latter type i.e. $\\{0,d,n\\}$.

######  Definition 6 (Antipodal two-weight (ATW) rank metric codes)

An $[n,k,d]$ linear rank metric code over $\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$
with $d<n$ is called an _antipodal two-weight_ or ATW rank metric code if any
nonzero codeword has rank either $d$ or $n$ and there is at least one codeword
of full rank $n$.

The most straightforward examples of antipodal two-weight rank metric codes
are given by MRD codes of dimension $2$. To see this, let $\mathcal{C}$ be an
$[n,2,n-1]$ MRD codes over $\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$, where $n\leq
m$. Then, for any $\mathbf{c}\in\mathcal{C}$ such that
$\mathbf{c}\neq\mathbf{0}$, it is clear that
$n-1\leq\mathsf{rank}(\mathbf{c})\leq n$. Hence, the two dimensional MRD codes
are $[n,2,n-1]$ ATW rank metric codes. As, we explained earlier, there are
many constructions of MRD codes [8, 9, 15, 18] and those already provide
inequivalent classes of ATW rank metric codes of dimension $2$.

###  2.2 $t$-Spreads

Spreads are widely studied objects in finite geometry [2, 3, 10]. There are
several works which connect them to coding theory. For example, spreads when
considered as subspace codes are optimal subspace codes [12] i.e. they are the
largest given their parameters. In this work, we give a connection between
spreads and ATW rank metric codes. We start with a brief review of the basic
notions and relevant results related to spreads.

######  Definition 7 ($t$-Spreads)

A $t$-spread is a pair $(V,\Sigma)$ where $V$ is a vector space of dimension
$n$ over $\mathbb{F}_{q}$ and $\Sigma$ is a set of subspaces of $V$ of
dimension $t$ such that $\bigcup_{S\in\Sigma}S=V$ and for all $S_{1}\neq
S_{2}\in\Sigma$, $S_{1}\cap S_{2}=\\{\mathbf{0}\\}$. If the ambient space $V$
is clear from the context, we simply write $\Sigma$ to denote the $t$-spread.

From now onwards, whenever we write “spread”, we mean that $(V,\Sigma)$ is a
$t$-spread where $t=(\dim_{\mathbb{F}_{q}}V)/2$. It is well-known [17] that an
$\mathbb{F}_{q}$-space $V$ of dimension $n$ has a $t$-spread if and only if
$t$ divides $n$. Moreover, $|\Sigma|=\frac{q^{n}-1}{q^{t}-1}$.

Two $t$-spreads $\Sigma_{1},~{}\Sigma_{2}$ of $V$ are called equivalent if
there exists a collineation $\alpha$ of $\Gamma L(V)$ such that
$\Sigma_{1}^{\alpha}=\Sigma_{2}$.

Suppose $n=lt$. Any $r$-dimensional $\mathbb{F}_{q^{t}}$-subspace of
$\mathbb{F}_{q^{t}}^{l}$ is an $rt$-dimensional $\mathbb{F}_{q}$-subspace of
$\mathbb{F}_{q^{t}}^{l}$. Now, let $\mathcal{H}$ be the set of all one
dimensional subspaces of $\mathbb{F}_{q^{t}}^{l}$ and thus
$\\#\mathcal{H}=\frac{q^{n}-1}{q^{t}-1}$. These $1$-dimensional
$\mathbb{F}_{q^{t}}$-subspaces are considered as $t$-dimensional
$\mathbb{F}_{q}$-subspaces of
$\mathbb{F}_{q^{t}}^{l}\simeq\mathbb{F}_{q}^{tl}=\mathbb{F}_{q}^{n}$ and they
intersect trivially. Hence, up to isomorphism, $\mathcal{H}$ is a set of
$\frac{q^{n}-1}{q^{t}-1}$ non-intersecting $t$-dimensional subspaces of
$\mathbb{F}_{q}^{n}$. This gives the first example of $t$-spreads.

######  Definition 8 (Desarguesian $t$-spread)

Let $\mathcal{D}_{l,t,q}$ be the $t$-spread of $\mathbb{F}_{q^{t}}^{l}$
consisting of the $t$-dimensional subspaces of $\mathbb{F}_{q^{t}}^{l}$
defined by the $1$-dimensional $\mathbb{F}_{q^{t}}$-subspaces of
$\mathbb{F}_{q^{t}}^{l}$. A $t$-spread $\Sigma$ is called Desarguesian if
$\Sigma$ is equivalent to $\mathcal{D}_{l,t,q}$ for some $l,~{}t$ and $q$.

###### Definition 9

Let $V$ be an $\mathbb{F}_{q}$-vector space. Let $(V,\Sigma)$ be a $t$-spread
in $V$. For a subspace $W$ of $V$, the projection of $\Sigma$ onto $W$,
denoted as $\Sigma_{W}$, is defined as follows

$\Sigma_{W}=\\{S\cap W\colon S\in\Sigma\\}\backslash\\{\\{\mathbf{0}\\}\\}.$

The pair $(W,\Sigma_{W})$ is called a subsystem of $(V,\Sigma)$.

###### Remark 1

When the intersection is equal to the zero space $\\{\mathbf{0}\\}$, we remove
it from the subsystem. For any arbitrary subspace $W\subset V$, the projection
$\Sigma_{W}$ is not necessarily a $t^{\prime}$-spread, because the elements of
$\Sigma_{W}$ may not have the same dimension.

###### Definition 10

Let $(V,\Sigma)$ be a $t$-spread in $V$ and let $(W,\Sigma_{W})$ be a
subsystem of $(V,\Sigma)$. If $(W,\Sigma_{W})$ is itself $t^{\prime}$-spread
for some $t^{\prime}\leq t$, then it is called a $t^{\prime}$-subspread
induced by $(V,\Sigma)$ on $W$. If a $t^{\prime}$-subspread is a spread, then
we simply call it a subspread.

Thus the notion of $t$-subspreads generalizes the notion of subspreads of [2,
Definition 4.4.1] to arbitrary $t$. Later in Theorem 3.3, we will show how
$t$-subspreads are related to the main subject of this paper i.e. antipodal
two-weight rank metric codes.

##  3 Antipodal two-weight rank metric codes and $t$-spreads

In this section, we discuss properties of the _antipodal_ two-weight rank
metric codes and $t$-spreads and establish a relation between these two
objects.

###  3.1 Antipodal two-weight (ATW) rank metric codes

For an $[n,k,d]$ ATW rank metric code over
$\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$, we must have $n\leq m$ because it has a
codeword of full rank $n$. We first give a simple description of the form of
the generator matrix of an ATW rank metric code. As we progress, we will
refine this form.

###### Lemma 3

Let $\mathcal{C}$ be a non-degenerate $[n,k,d]$ ATW rank metric code over
$\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$. Then $\mathcal{C}$ is equivalent to a
rank metric code with a generator matrix of the form

$\mathbf{G}=\begin{pmatrix}\mathbf{c}_{1}&\mathbf{c}_{2}\\\
\mathbf{A}&\mathbf{0}\end{pmatrix},\text{ where}$

  1. (i)

$\mathbf{c}_{1}\in\mathbb{F}_{q^{m}}^{n-r}$,
$\mathbf{c}_{2}\in\mathbb{F}_{q^{m}}^{r}$ for some positive integer $r$ and
$\mathsf{rank}(\mathbf{c}_{1}|\mathbf{c}_{2})=n$.

  2. (ii)

$\mathbf{A}\in\mathbb{F}_{q^{m}}^{(k-1)\times(n-r)}$ is a generator matrix of
a non-degenerate constant weight $[n-r,k-1]$ rank metric code.

###### Proof

Since $\mathcal{C}$ is antipodal, there is a codeword $\mathbf{c}$ of full
rank $n$. After row reductions, we can have a generator matrix of the
following form

$\mathbf{G}^{\prime}=\begin{pmatrix}\mathbf{c}^{\prime}&\alpha\\\
\mathbf{A}^{\prime}&\mathbf{0}_{(k-1)\times 1}\end{pmatrix},$

where
$\mathbf{c}^{\prime}\in\mathbb{F}_{q^{m}}^{n-1},\;\alpha\in\mathbb{F}_{q^{m}}^{\times}$
such that $\mathbf{c}=(\mathbf{c}^{\prime}|\alpha)$ and
$\mathbf{A}^{\prime}\in\mathbb{F}_{q^{m}}^{(k-1)\times(n-1)}$. Since the
subcode $\mathcal{C}^{\prime}\subset\mathcal{C}$ generated by
$\left[\mathbf{A}^{\prime}|\mathbf{0}_{(k-1)\times 1}\right]$ can’t have a
codeword of rank $n$, then it must be a constant rank weight code with minimum
distance $d$. In case $\mathbf{A}^{\prime}$ generates a degenerate rank metric
code, then we can find some invertible matrix
$\mathbf{M}\in\mathbb{F}_{q}^{n\times n}$ such that

$\left[\mathbf{A}^{\prime}|\mathbf{0}_{(k-1)\times
1}\right]\mathbf{M}=\left[\mathbf{A}|\mathbf{0}_{(k-1)\times r}\right],$

where $1\leq r\leq n-1$ and
$\mathbf{A}\in\mathbb{F}_{q^{m}}^{(k-1)\times(n-r)}$ is a generator matrix of
a non-degenerate constant weight rank metric code. Therefore,
$\mathbf{G}:=\mathbf{G}^{\prime}\mathbf{M}$ has the desired form and the
codeword
$(\mathbf{c}_{1}|\mathbf{c}_{2})=(\mathbf{c}^{\prime}|\alpha)\mathbf{M}$ has
rank $n$ since $M$ is invertible. ∎

It is clear that any two-weight rank metric code cannot be of dimension $1$.
Such codes are constant weight rank metric codes. Now, following Theorem 2.1,
we can say that the matrix $\mathbf{A}$ in the decomposition of $G$ in Lemma 3
generates either a code of dimension $1$ or a Hadamard rank metric code. And
the later case leads to the following result.

###### Theorem 3.1

There is no $[n,k,d]$ ATW rank metric code over
$\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$ of dimension $k\geq 3$.

###### Proof

Let $\mathcal{C}$ be an $[n,k,d]$ ATW rank metric code with of dimension
$k\geq 3$ and generator matrix $\mathbf{G}$. By Lemma 3, we can assume that

$\mathbf{G}=\begin{pmatrix}\mathbf{c}_{1}&\mathbf{c}_{2}\\\
\mathbf{A}&\mathbf{0}\end{pmatrix},$

where $\mathbf{A}$ is a generator matrix of a non-degenerate constant weight
rank metric code of dimension $k-1\geq 2$ and minimum distance $d$. Following
Theorem 2.1, $\mathbf{A}$ is a generator matrix of a Hadamard code
$H_{1}(q,m,k-1)$. But that implies $m=d<n\leq m$ which is a contradiction. ∎

So any ATW rank metric codes are only of dimension $2$. The next lemma gives a
restriction on the distance of the code.

###### Lemma 4

For any $[n,2,d]$ ATW rank metric code, the minimum distance $d$ is at least
$n/2$.

###### Proof

Let $\mathcal{C}$ be an $[n,2,d]$ ATW rank metric code. Following Lemma 3,
$\mathcal{C}$ is equivalent to a code $\mathcal{C}^{\prime}$ with a generator
matrix $\mathbf{G}=\begin{pmatrix}\mathbf{c}_{1}&\mathbf{c}_{2}\\\
\mathbf{c}_{3}&\mathbf{0}\end{pmatrix}$, where
$\mathbf{c}_{1},\mathbf{c}_{3}\in\mathbb{F}_{q^{m}}^{d}$,
$\mathbf{c}_{2}\in\mathbb{F}_{q^{m}}^{n-d}$, $\mathsf{rank}(\mathbf{c}_{3})=d$
and $\mathsf{rank}(\mathbf{c}_{1}|\mathbf{c}_{2})=n$. After row reduction,
$\mathbf{G}$ can be transformed into
$\mathbf{G}^{\prime}=\begin{pmatrix}\mathbf{c}_{1}^{\prime}&\mathbf{c}_{2}\\\
\mathbf{c}_{3}&\mathbf{0}\end{pmatrix}$, where the first entry of
$\mathbf{c}_{1}^{\prime}$ is zero and the two rows of the new generator matrix
$\mathbf{G}^{\prime}$ have rank $d$. The fact that
$\mathsf{rank}(\mathbf{c}_{1}|\mathbf{c}_{2})=n$ implies that $\mathbf{c}_{2}$
has full rank $n-d$. Since
$\mathsf{rank}(\mathbf{c}_{2})\leq\mathsf{rank}(\mathbf{c}_{1}^{\prime}|\mathbf{c}_{2})=d$,
then $d\geq n-d$ i.e. $d\geq n/2$. ∎

The following example gives an ATW rank metric code with $d=n/2$. Later we
show that any codes of the same parameters must be equivalent to this one.

###### Example 1

Let
$\mathbb{F}_{q^{d}}=\left<\alpha_{1},\dots,\alpha_{d}\right>_{\mathbb{F}_{q}}$
be an extension of degree $d$ over $\mathbb{F}_{q}$. Then, the matrix

$\mathbf{G}=\begin{pmatrix}0&0&\cdots&0&\alpha_{1}&\alpha_{2}&\cdots&\alpha_{d}\\\
\alpha_{1}&\alpha_{2}&\cdots&\alpha_{d}&0&0&\cdots&0\end{pmatrix}$

generates a non-degenerate ATW $[n=2d,2,d]$ rank metric code $\mathcal{C}$
over $\mathbb{F}/\mathbb{F}_{q}$, where $\mathbb{F}$ is any proper extension
of $\mathbb{F}_{q^{d}}$. For the proof, we know that all codewords of
$\mathcal{C}$ are of the form $(a_{1},a_{2})\mathbf{G}$, where
$a_{1},a_{2}\in\mathbb{F}$. Now, for any such
$(a_{1},a_{2})\in\mathbb{F}^{2}$, either $a_{2}a_{1}^{-1}\in\mathbb{F}_{q}$ or
$a_{2}a_{1}^{-1}\notin\mathbb{F}_{q}$. For the first case, the codeword is of
rank $d$, whereas the second case produces a codeword of rank $n$. Note that
we cannot take $\mathbb{F}=\mathbb{F}_{q^{d}}$, because then, there will be no
codeword of full rank in $\mathcal{C}$.

In fact this construction can be modified to provide constructions of two-
weight rank metric codes which are not necessarily antipodal. As we mentioned
in a paragraph after Definition 5, for non-antipodal two-weight rank metric
codes, the weight distribution is of the form $\\{0,d,m\\}$ where $m<n$. The
following example gives a construction of non-antipodal two-weight rank metric
codes.

###### Example 2

Let $d,n$ be positive integers such that $2d<n$. We have a sequence of finite
fields $\mathbb{F}_{q}\subset\mathbb{F}_{q^{d}}\subset\mathbb{F}_{q^{2d}}$.
Let $\\{\alpha_{1},\dots,\alpha_{d}\\}$ be an $\mathbb{F}_{q}$-basis of
$\mathbb{F}_{q^{d}}$. Then consider the rank metric code $\mathcal{C}$ over
$\mathbb{F}_{q^{2d}}/\mathbb{F}_{q}$, generated by the following $k\times dk$
matrix.

$\mathbf{G}=\left[\begin{array}[]{*{20}c}\alpha_{1}&\ldots&\alpha_{d}&0&\ldots&0&\ldots&\ldots&\ldots&0&\cdots&0\\\
0&\ldots&0&\alpha_{1}&\ldots&\alpha_{d}&\ldots&\ldots&\ldots&0&\ldots&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\ddots&\ddots&\vdots&\vdots&\vdots\\\
0&\ldots&0&0&\ldots&0&\ldots&\ldots&\ldots&\alpha_{1}&\ldots&\alpha_{d}\end{array}\right].$

The $i$-th row of the matrix $\mathbf{G}$ is chosen in such a way that the
i-th block (with $d$ entries) consists of $(\alpha_{1},\dots,\alpha_{d})$ and
the other blocks are made of zero vectors. Therefore, $\mathcal{C}$ has length
$n=kd$ and dimension $k>2$. Now, let us check the weights of the codewords of
$\mathcal{C}$. Suppose that $\mathbf{c}=\mathbf{x}\mathbf{G}$ where
$\mathbf{x}=(x_{1},\dots,x_{k})\in\mathbb{F}_{q^{2d}}^{k}$. There are two
possibilities:

  1. (i)

$\left<x_{1},\dots,x_{k}\right>_{\mathbb{F}_{q^{d}}}=\left<\beta\right>_{\mathbb{F}_{q^{d}}}$
for some $\beta\in\mathbb{F}_{q^{2d}}$.

  2. (ii)

$\dim_{\mathbb{F}_{q^{d}}}\left<x_{1},\dots,x_{k}\right>_{\mathbb{F}_{q^{d}}}>1$.

In the first case, $\mathsf{rank}\,\mathbf{c}=d$ if $\beta\neq 0$ and
$\mathsf{rank}\,\mathbf{c}=0$ if $\beta=0$. For the second case, without loss
of generality, suppose that
$\dim_{\mathbb{F}_{q^{d}}}\left<x_{1},x_{2}\right>_{\mathbb{F}_{q^{d}}}=2$.
Therefore,
$\mathsf{rank}(x_{1}\alpha_{1},\dots,x_{1}\alpha_{d},x_{2}\alpha_{1},\dots,x_{2}\alpha_{d})=2d$.
Thus

$2d=\mathsf{rank}(x_{1}\alpha_{1},\dots,x_{1}\alpha_{d},x_{2}\alpha_{1},\dots,x_{2}\alpha_{d})\leq\mathsf{rank}\,\mathbf{c}\leq
2d.$

The last inequality comes from the fact that the code is over
$\mathbb{F}_{q^{2d}}$ so that the rank of any codewords is at most $2d$.
Therefore $\mathsf{rank}\,\mathbf{c}=2d$.

Therefore, the code $\mathcal{C}$ generated by $\mathbf{G}$ is a $[kd,k,d]$
non-antipodal two-weight rank metric code over
$\mathbb{F}_{q^{2d}}/\mathbb{F}_{q}$ with weight distribution $\\{0,d,2d\\}$.

###  3.2 Some properties of $t$-spreads

Here we prove some properties of $t$-spreads that will be useful in latter
sections.

###### Lemma 5

Let $(V,\Sigma)$ be a $t$-spread of $V$ where $V$ is a vector space over
$\mathbb{F}_{q}$ of dimension $n=tl$. For any integer $1<r<l$, if
$\\{S_{1},S_{2},\dots,S_{r}\\}$ are distinct subspaces in $\Sigma$ such that
$\dim_{\mathbb{F}_{q}}S_{1}+\dots+S_{r}=tr$, then there is an element
$S_{r+1}$ in $\Sigma$ such that
$\dim_{\mathbb{F}_{q}}S_{1}+\dots+S_{r+1}=t(r+1)$.

###### Proof

By contradiction, suppose that for any $S_{j}\neq S_{i}$, $i=1,\dots,r$,
$\dim_{\mathbb{F}_{q}}S_{1}+\dots+S_{r}+S_{j}<t(r+1)$. Hence
$S_{j}\cap(S_{1}+\dots+S_{r})$ contains a non-zero vector. Since the elements
of a $t$-spread pairwise intersect trivially, each space in
$\Sigma\backslash\\{S_{1},\dots,S_{r}\\}$ contains a distinct non-zero vector
in $S_{1}+\dots+S_{r}$. Therefore,
$\\#(S_{1}+\dots+S_{r})\backslash\\{\mathbf{0}\\}\geq\frac{q^{n}-1}{q^{t}-1}-r$.
Hence,

$\displaystyle q^{tr}-1\geq\frac{q^{n}-1}{q^{t}-1}-r$
$\displaystyle\Rightarrow$ $\displaystyle q^{t(r+1)}-q^{tr}\geq
q^{n}-1-(r-1)(q^{t}-1)$ $\displaystyle\Rightarrow$ $\displaystyle
q^{n}-q^{tr}\geq q^{n}-1-(r-1)(q^{t}-1)~{}(\text{As }q^{n}\geq
q^{t(r+1)},\text{for }r+1\leq l)$ $\displaystyle\Rightarrow$ $\displaystyle
q^{tr}-1\leq(r-1)(q^{t}-1)<(r-1)q^{t}$ $\displaystyle\Rightarrow$
$\displaystyle q^{tr}-1<rq^{t}-q^{t}+(q^{t}-1)$ $\displaystyle\Rightarrow$
$\displaystyle q^{tr}-1<rq^{t}-1.$

This implies that $q^{tr}<rq^{t}$. Since $t\geq 1$ and $r\geq 2$, this leads
to a contradiction as $q^{tr}\geq rq^{t}$ and hence the Lemma is proved. ∎

###### Theorem 3.2

Suppose that $V$ is a vector space of dimension $n$ over $\mathbb{F}_{q}$ and
let $n=tl$. Let $\Sigma$ be a $t$-spread of $V$. Then there are
$S_{1},\dots,S_{l}$ in $\Sigma$ such that $V=S_{1}\oplus
S_{2}\oplus\dots\oplus S_{l}$.

###### Proof

Choose $S_{1}$ and $S_{2}$ as any elements of $\Sigma$. By the definition of
$t$-spreads, $S_{1}+S_{2}$ is a direct sum. The previous lemma says that we
can increase it to $S_{1}\oplus S_{2}\oplus S_{3}$ and applying the lemma
repetitively, we get the result. ∎

###### Remark 2

The above property can be easily seen when the spread is Desarguesian. We can
choose the $S_{i}$’s to be the $\mathbb{F}_{q}$-subspaces defined by the
elements of a basis of $\mathbb{F}_{q^{t}}^{l}$ over $\mathbb{F}_{q^{t}}$.

###  3.3 Relation between ATW codes and $t$-spreads

We now give a relation between $(n-d)$-spreads and $[n,2,d]$ ATW rank metric
codes for $d\geq n/2$. The following lemma is obtained by using the same
method as used in classifying constant rank weight codes in [16].

###### Lemma 6

Let $\mathcal{C}$ be an $[n,2,d]$ ATW rank metric code over
$\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$ with a generator matrix $\mathbf{G}$ and
let $X$ be the $q$-system corresponding to $\mathbf{G}$. If $\mathcal{H}$ is
the set of all hyperplanes of $\mathbb{F}_{q^{m}}^{2}$, then we can partition
$\mathcal{H}$ into $\mathcal{H}_{1}\sqcup\mathcal{H}_{2}$, where

$\mathcal{H}_{1}=\\{H\in\mathcal{H}\colon\dim_{\mathbb{F}_{q}}H\cap
X=n-d\\},\text{ and
}\mathcal{H}_{2}=\\{H\in\mathcal{H}\colon\dim_{\mathbb{F}_{q}}H\cap X=0\\}.$
(3)

###### Proof

Let $\mathbf{c}\in\mathcal{C}$ be a nonzero codeword and let $H_{\mathbf{c}}$
be the corresponding hyperplane i.e.
$H_{\mathbf{c}}=\mathrm{Ker}~{}\psi_{\mathbf{x}_{\mathbf{c}}}$ in Lemma (1).
Then Lemma 2 implies that $\mathbb{F}_{q}$-dimension of $H_{\mathbf{c}}\cap X$
is $n-d$ if $\mathsf{rank}(\mathbf{c})=d$ and $0$ otherwise. Since any
hyperplane of $\mathbb{F}_{q^{m}}^{2}$ corresponds to some codewords, we get
the desired partition of $\mathcal{H}$ by considering $\mathcal{H}_{1}$ (resp.
$\mathcal{H}_{2}$) to be the set of hyperplanes corresponding to codewords of
rank $d$ (resp. $n$). ∎

###### Theorem 3.3

Let $\mathcal{C}$ be an $[n,2,d]$ ATW rank metric code over
$\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$ with a generator matrix $\mathbf{G}$ and
let $X$ be the $q$-system corresponding to $\mathbf{G}$. Let $\mathcal{H}$ be
the set of all hyperplanes of $\mathbb{F}_{q^{m}}^{2}$ and
$\mathcal{H}_{1}=\\{H\in\mathcal{H}\colon\dim_{\mathbb{F}_{q}}H\cap X=n-d\\}$.
Then $|\mathcal{H}_{1}|=\frac{q^{n}-1}{q^{n-d}-1}$ and thus the set
$\Sigma=\\{H\cap X:H\in\mathcal{H}_{1}\\}$ defines an $(n-d)$-spread
$(X,\Sigma)$ of $X$.

###### Proof

Lemma 6 implies $\mathcal{H}=\mathcal{H}_{1}\sqcup\mathcal{H}_{2}$ where
$\mathcal{H}_{2}$ as in equation (3). Note that the hyperplanes have pairwise
trivial intersection. Therefore we can partition the set
$\mathbb{F}_{q^{m}}^{2}\backslash\\{\mathbf{0}\\}$ as

$\mathbb{F}_{q^{m}}^{2}\backslash\\{\mathbf{0}\\}=\left(\bigsqcup_{H\in\mathcal{H}_{1}}H\backslash\\{\mathbf{0}\\}\right)\bigsqcup\left(\bigsqcup_{H\in\mathcal{H}_{2}}H\backslash\\{\mathbf{0}\\}\right),$
(4)

where $H\backslash\\{\mathbf{0}\\}$ is the set of non-zero elements of $H$.
Define a valuation $v$ on $\mathbb{F}_{q^{m}}^{2}$ by

$v(\mathbf{a})=\begin{cases}1~{}\text{ if }\mathbf{a}\in X,\\\ 0~{}\text{
otherwise},\end{cases}$

and by an abuse of notation, we also define $v(S)=\sum_{\mathbf{a}\in
S}v(\mathbf{a})$ for any subset $S$ of $\mathbb{F}_{q^{m}}^{2}$. Since all the
unions in equation (4) are disjoint, then

$v(\mathbb{F}_{q^{m}}^{2}\backslash\\{\mathbf{0}\\})=\sum_{H\in\mathcal{H}_{1}}v\left(H\backslash\\{\mathbf{0}\\}\right)+\sum_{H\in\mathcal{H}_{2}}v\left(H\backslash\\{\mathbf{0}\\}\right),$

Now $v(H\backslash\\{\mathbf{0}\\})=q^{n-d}-1$ for $H\in\mathcal{H}_{1}$, and
$v(H\backslash\\{\mathbf{0}\\})=0$ if $H\in\mathcal{H}_{2}$. Furthermore
$v(\mathbb{F}_{q^{m}}^{2}\backslash\\{\mathbf{0}\\})=v(X\backslash\\{\mathbf{0}\\})$.
Therefore,

$q^{n}-1=|\mathcal{H}_{1}|(q^{n-d}-1).$ (5)

The pairwise trivial intersection of elements of $\Sigma$ implies that
$\Sigma$ is an $(n-d)$-spread. ∎

###### Corollary 1

If $\mathcal{C}$ is an $[n,2,d]$ ATW rank metric code, then $n-d$ divides $n$.

###### Proof

Theorem 3.3 implies that a $q$-system $X$ has an $(n-d)$-spread $\Sigma$ such
that $|\Sigma|=\frac{q^{n}-1}{q^{n-d}-1}$ and thus $n-d$ divides $n$. ∎

Corollary 1 recovers the result of Lemma 4.

###### Corollary 2

The weight distribution of an $[n,2,d]$ ATW code over $\mathbb{F}_{q^{m}}$ is

$\left(1,\,(q^{m}-1)\frac{q^{n}-1}{q^{n-d}-1},\,(q^{2m}-1)-(q^{m}-1)\frac{q^{n}-1}{q^{n-d}-1}\right).$

###### Proof

Following Equation (4), we have

$q^{2m}-1=(q^{m}-1)|\mathcal{H}_{1}|+(q^{m}-1)|\mathcal{H}_{2}|.$

It is clear that the codewords of rank weight $d$ correspond to the
hyperplanes in $\mathcal{H}_{1}$ and those of rank weight $n$ correspond to
the hyperplanes in $\mathcal{H}_{2}$, where $\mathcal{H}_{1},H_{2}$ are as
described in Theorem 3.3. From Equation (5) it follows that the number of
codewords of rank weight $d$ is equal to
$(q^{m}-1)|\mathcal{H}_{1}|=(q^{m}-1)\frac{q^{n}-1}{q^{n-d}-1}$. Since the
number of hyperplanes of $\mathbb{F}_{q^{m}}^{2}$ is $|\mathcal{H}|=q^{m}+1$,
and since $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ give a partition of
$\mathcal{H}$, the number of codewords of rank weight $n$ is
$(q^{m}-1)|\mathcal{H}_{2}|=(q^{m}-1)(q^{m}+1-\frac{q^{n}-1}{q^{n-d}-1})$. ∎

###### Theorem 3.4

Let $\mathcal{C}$ be an $[n,2,d]$ ATW rank metric code. Suppose that
$n=l(n-d)$, then $\mathcal{C}$ is equivalent to a rank metric code with
generator matrix $\mathbf{G}=[\mathbf{G}_{1}|\dots|\mathbf{G}_{l}]$, where,
for $1\leq i\leq l$, the columns of each $\mathbf{G}_{i}$’s belong to a
hyperplane $H_{i}$ and the $H_{i}$’s are pairwise distinct.

###### Proof

Theorem 3.3 says that $X$ admits an $(n-d)$-spread. Then we choose the columns
of $\mathbf{G}$ from the decomposition in Theorem 3.2, where each block
$\mathbf{G}_{i}$ consists of a $\mathbb{F}_{q}$-basis of $H_{i}\cap X$ for
$H_{i}\in\mathcal{H}_{1}$. ∎

We have shown that ATW rank metric codes define $t$-spreads in the q-system
$X$. The following theorem shows that these $t$-spreads satisfy a certain
property.

###### Theorem 3.5

Let $\mathcal{C}$ be an $[n,2,d]$ rank metric code over
$\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$ generated by $\mathbf{G}$ with the
corresponding $q$-system $X$. Let $(\mathbb{F}_{q^{m}}^{2},\Delta)$ be a
Desarguesian $m$-spread. The following assertions are equivalent.

  1. (i)

$\mathcal{C}$ is an ATW rank metric code.

  2. (ii)

$(\mathbb{F}_{q^{m}}^{2},\Delta)$ induces an $(n-d)$-spread $(X,\Delta_{X})$
of $X$.

###### Proof

  * $\eqref{thm:6-i}\Rightarrow\eqref{thm:6-ii}$: Since $\mathcal{C}$ is an ATW rank metric code, following Theorem 3.3 we know that $\Delta_{X}=\\{X\cap H:H\in\mathcal{H}_{1}\\}$ is an $(n-d)$-spread of $X$. Also, from Definition 10, it follows that $(X,\Delta_{X})$ is an $(n-d)$-subspread of $(\mathbb{F}_{q^{m}}^{2},\Delta)$.

  * $\eqref{thm:6-ii}\Rightarrow\eqref{thm:6-i}$: Suppose that $(X,\Delta_{X})$ is an $(n-d)$-subspread of $(\mathbb{F}_{q^{m}}^{2},\Delta)$. Let $\mathcal{H}$ be the set of all hyperplanes in $\mathbb{F}_{q^{m}}^{2}$. Then from Definition 8 and Definition 10, it follows that an element of $\mathcal{H}$ intersects $X$ in an $\mathbb{F}_{q}$-space of dimension either $n-d$ or $0$. From Lemma 2, we have that for any $H\in\mathcal{H}$, $H=H_{\mathbf{c}}$ for some $\mathbf{c}\in\mathcal{C}$ and conversely, any $\mathbf{c}\in\mathcal{C}$ defines a hyperplane $H_{\mathbf{c}}$ (two linearly dependent codewords define the same hyperplane). Then Equation (2) in Lemma 2 implies that the only possible rank for any non-zero codewords are $d$ and $n$. ∎

In the following result, we give a condition for any general $(n-d)$-spread of
$\mathbb{F}_{q}^{n}$ to induce $[n,2,d]$ ATW rank metric codes.

###### Corollary 3

Let $n,m,d$ be positive integers with $d\leq n\leq m$ and let $\Sigma$ be an
$(n-d)$-spread of $\mathbb{F}_{q}^{n}$. Let $X\subseteq\mathbb{F}_{q^{m}}^{2}$
be a $q$-system and suppose that $\mathbf{G}$ is a $(2\times n)$-matrix whose
columns consist of an $\mathbb{F}_{q}$-basis of $X$. If for any $S\in\Sigma$,
$\mathbf{G}S^{T}=\\{\mathbf{G}\mathbf{x}^{T}\colon\mathbf{x}\in S\\}$ is
contained in a hyperplane $H$ of $\mathbb{F}_{q^{m}}^{2}$ and each different
$S$ correspond to different $H$, then $\mathbf{G}$ generates an ATW rank
metric code.

##  4 Constructions of ATW rank metric codes

In Example 1 of Section 3.1 we have seen a construction of ATW rank metric
codes. We construct ATW rank metric codes for more general parameters using
suitable MRD codes.

###### Theorem 4.1

Let $d,l,m,n$ be positive integers such that $d\leq n\leq m$, $(n-d)|m$, and
$n=l(n-d)$. Suppose $\mathcal{C}$ is a non-degenerate $[l,2,l-1]$ MRD code
over $\mathbb{F}_{q^{m}}/\mathbb{F}_{q^{n-d}}$ with a generator matrix
$\mathbf{G}$. Fix a basis $\\{a_{1},\ldots,a_{n-d}\\}$ of
$\mathbb{F}_{q^{n-d}}$ over $\mathbb{F}_{q}$ and let $\tilde{\mathbf{G}}$ be
the $2\times l(n-d)$ block matrix
$(\tilde{\mathbf{G}}_{1}|\ldots|\tilde{\mathbf{G}}_{l})$ where the $(n-d)$
columns of $\tilde{\mathbf{G}}_{i}$ are given by
$\\{a_{j}\mathbf{G}_{i},j=1,2,\ldots,n-d\\}$ with $\mathbf{G}_{i}$ being the
$i$-th column of $\mathbf{G}$. Then $\tilde{\mathbf{G}}$ generates an
$[n,2,d]$ ATW rank metric code $\tilde{\mathcal{C}}$ over
$\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$.

###### Proof

We claim that if the codeword $c=(c_{1},c_{2})\mathbf{G}$ of $\mathcal{C}$ has
rank weight $l$ (respectively, $l-1$), then the codeword
$\tilde{c}=(c_{1},c_{2})\tilde{\mathbf{G}}$ of $\tilde{\mathcal{C}}$ has rank
weight $l(n-d)$ (respectively, $(l-1)(n-d)$). It is easy to see that if the
claim is proved then $\tilde{\mathcal{C}}$ is an ATW $[n,2,d]$ code over
$\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$.

Proof of the claim: Let $X$ (resp. $\tilde{X}$) be the $q$-system
corresponding to the generator matrix $\mathbf{G}$ (resp.
$\tilde{\mathbf{G}}$). It is crucial to note that the $\mathbb{F}_{q}$-space
$\tilde{X}$ and the $\mathbb{F}_{q^{n-d}}$-space $X$ are same as a set. Now
let the codeword $c=(c_{1},c_{2})\mathbf{G}$ of $\mathcal{C}$ has rank weight
$l$. Thus from equation (1), we get $\dim_{\mathbb{F}_{q^{n-d}}}H_{c}\cap X=0$
and hence, $\dim_{\mathbb{F}_{q}}H_{c}\cap\tilde{X}=0$. Therefore,
$rank(\tilde{c})=n-\dim_{\mathbb{F}_{q}}H_{c}\cap\tilde{X}=n$. Similarly, for
the case when $rank(c)=l-1$ we have $\dim_{\mathbb{F}_{q^{n-d}}}H_{c}\cap X=1$
(follows from equation (1)). Then
$\dim_{\mathbb{F}_{q}}H_{c}\cap\tilde{X}=\dim_{\mathbb{F}_{q}}H_{c}\cap{X}=n-d$
and therefore, $rank(\tilde{c})=n-(n-d)=d$. ∎

The above theorem says that if $X$ is a $q^{n-d}$-system corresponding to an
$[l,2,l-1]$ MRD code over $\mathbb{F}_{q^{m}}/\mathbb{F}_{q^{n-d}}$, then $X$
is a $q$-system corresponding to an ATW $[n,2,d]$ code $\tilde{\mathcal{C}}$
over $\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$. We call the code
$\tilde{\mathcal{C}}$ the _ATW rank metric code induced_ by the MRD rank
metric code $\mathcal{C}$.

###### Corollary 4

Let $d,l,m,n,$ be positive integers such that $d\leq n\leq m$, $n=l(n-d)$, and
$(n-d)|m$. Let $X$ be a $q$-system corresponding to an $[n,2,d]$ ATW code
$\tilde{\mathcal{C}}$ over $\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$. Then
$\tilde{\mathcal{C}}$ is induced by an $[l,2,l-1]$ MRD code $\mathcal{C}$ over
$\mathbb{F}_{q^{m}}/\mathbb{F}_{q^{n-d}}$ if and only if $X$ is also an
$\mathbb{F}_{q^{n-d}}$-space.

It is natural to ask if all ATW rank metric codes are induced by MRD codes as
shown in the previous construction. The answer is affirmative when $d=n/2$.

###### Theorem 4.2

Let $\mathcal{C}$ be an $[n,2,d=n/2]$ rank metric code over
$\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$. Then $\mathcal{C}$ is an ATW rank metric
code if and only if $d$ divides $m$ and $\mathcal{C}$ is equivalent to a code
with generator matrix

$\begin{pmatrix}0&0&\dots&0&1&\alpha_{2}&\dots&\alpha_{d}\\\
1&\alpha_{2}&\dots&\alpha_{d}&0&0&\dots&0\end{pmatrix},\text{ such that
}\left<1,\alpha_{2},\dots,\alpha_{d}\right>_{\mathbb{F}_{q}}=\mathbb{F}_{q^{d}}.$

###### Proof

Following Theorem 3.4, we know that $\mathcal{C}$ has a generator matrix of
the form $\mathbf{G}=[\mathbf{G}_{1}|\mathbf{G}_{2}]$ where the columns of
$\mathbf{G}_{1}$ (resp. $\mathbf{G}_{2}$) belong to a hyperplane $H_{1}$
(resp. $H_{2}$). For $i=1,2$, let $\mathbf{x}_{i}\in\mathbb{F}_{q^{m}}^{2}$
such that $H_{i}=\mathrm{Ker}~{}\psi_{\mathbf{c}_{i}}$ where
$\mathbf{c}_{i}=\mathbf{x}_{i}\mathbf{G}$ as in Equation (1). Then
$\mathcal{C}$ has another generator matrix
$\mathbf{G}^{\prime}=\begin{pmatrix}\mathbf{x}_{1}\\\
\mathbf{x}_{2}\end{pmatrix}\mathbf{G}=\begin{pmatrix}\mathbf{0}&\mathbf{e}_{1}\\\
\mathbf{e}_{2}&\mathbf{0}\end{pmatrix},$ where $\mathbf{e}_{1}$ and
$\mathbf{e}_{2}$ both have rank $d$.

Without loss of generality we can assume that
$\mathbf{e}_{1}=(1,\alpha_{2}\dots,\alpha_{d})$ and
$\mathbf{e}_{2}=(1,\beta_{2},\dots,\beta_{d})$ where
$\alpha_{i},\beta_{i}\in\mathbb{F}_{q^{m}}^{\times}$. Take the codeword
$(\mathbf{e}_{2}|\mathbf{e}_{1})$, because the first and second part both
contain $1$, then $0<\mathsf{rank}(\mathbf{e}_{2}|\mathbf{e}_{1})<n$ and
therefore, $\mathsf{rank}(\mathbf{e}_{2}|\mathbf{e}_{1})=d$. Thus the
$\beta_{i}$’s are $\mathbb{F}_{q}$-linear combination of the
$\alpha_{i}^{\prime}s$ and $1$. Hence, $\mathcal{C}$ is equivalent to a code
with generator matrix $\overline{\mathbf{G}}$ of the form
$\overline{\mathbf{G}}=\begin{pmatrix}\mathbf{0}&\mathbf{e}_{1}\\\
\mathbf{e}_{1}&\mathbf{0}\end{pmatrix}.$ For $i=2,\dots,d$, let
$\mathbf{c}^{(i)}=(\mathbf{e}_{1}|\alpha_{i}\mathbf{e}_{1})$. Because the
first and second part of $\mathbf{c}^{(i)}$ contain $\alpha_{i}$, then
$0<\mathsf{rank}(\mathbf{c}^{(i)})<n$ and thus
$\mathsf{rank}(\mathbf{c}^{(i)})=d$. This implies that
$\alpha_{i}\alpha_{j}\in\left<1,\alpha_{2},\dots,\alpha_{d}\right>_{\mathbb{F}_{q}}$
for all $2\leq i,j\leq d$. So the space
$\left<1,\alpha_{2},\dots,\alpha_{d}\right>_{\mathbb{F}_{q}}$ is in fact a
subfield, say $K$, of $\mathbb{F}_{q^{m}}$ with $[K:\mathbb{F}_{q}]=d$ and
hence $d\mid m$. In this case, $\mathcal{C}$ is a $[2d,2,d]$ rank metric code
equivalent to a code with generator matrix

$\mathbf{G}"=\begin{pmatrix}0&0&\dots&0&1&\alpha_{2}&\dots&\alpha_{d}\\\
1&\alpha_{2}&\dots&\alpha_{d}&0&0&\dots&0\end{pmatrix},\text{ such that
}\left<1,\alpha_{2},\dots,\alpha_{d}\right>_{\mathbb{F}_{q}}=\mathbb{F}_{q^{d}}.$

For the converse, it follows from the arguments in Example 1 that
$\mathbf{G}"$ generates a non-degenerate ATW rank metric codes. ∎

For $d=n/2$, we showed that the ATW rank metric code essentially corresponds
to a Desarguesian spread $(\mathbb{F}_{q^{n/2}},\Delta)$. This gives an
interesting property of Desarguesian subspreads.

###### Corollary 5

Let $(\mathbb{F}_{q^{m}}^{2},\Delta)$ be a Desarguesian spread. Let
$X\subseteq\mathbb{F}_{q^{m}}^{2}$ be an $\mathbb{F}_{q}$-space of dimension
$n$ and suppose that $(\mathbb{F}_{q^{m}}^{2},\Delta)$ induces a subspread
i.e. $n/2$-subspread, $(X,\Delta_{X})$ of $X$, then $(X,\Delta_{X})$ is a
Desarguesian spread.

###### Proof

Since $(X,\Delta_{X})$ is a subspread, then by definition, the elements of
$\Delta_{X}$ has dimension $n/2$. Since $(X,\Delta_{X})$ is a subspread
induced by a Desarguesian spread, following Theorem 3.5, $X$ defines an
$[n,2,n/2]$ ATW rank metric code. Now Theorem 4.2 gives a generator matrix of
the ATW code, from which it is clear that each element $S$ of $\Delta_{X}$ is
an $\mathbb{F}_{q^{n/2}}$ vector space. So the induced subspread
$(X,\Delta_{X})$ is a Desarguesian spread of $X$. ∎

##  5 Two-weight Hamming metric codes from two-weight rank metric codes

As mentioned in the Introduction, there are already existing constructions of
two-weight Hamming metric codes. In this section, as a small appendix to this
work, we provide constructions of other two-weight Hamming metric codes via
the expansion of rank metric codes as presented in [1]. To do this, it is
better to also use the geometric approach for Hamming metric codes as
introduced in [19]. We only consider _non-degenerate_ Hamming metric codes
i.e. no generator matrix of the code has a zero column.

######  Definition 11 (Projective system)

Let $\mathbb{P}=\mathbb{P}(\mathbb{F}_{q}^{k})$ be a projective space of
dimension $k-1$ over a finite field $\mathbb{F}_{q}$. A projective $[n,k]_{q}$
system is a finite unordered family $\mathcal{P}$ of points of $\mathbb{P}$,
which does not lie in a (projective) hyperplane and such that
$|\mathcal{P}|=n$ (counted with multiplicity).

######  Definition 12 (Projective system associated to a code)

Let $\mathcal{C}$ be an $[n,k]$ non-degenerate Hamming metric code over
$\mathbb{F}_{q}$ and let $\mathbf{G}\in\mathbb{F}_{q}^{k\times n}$ be a
generator matrix of $\mathcal{C}$. A projective system
$\mathcal{P}(\mathcal{C})$ associated to $\mathcal{C}$ is the projective
$[n,k]_{q}$ system in $\mathbb{P}(\mathbb{F}_{q}^{k})$ consisting of the
columns of $\mathbf{G}$ (with multiplicity).

On the other hand, given a projective system $\mathcal{P}$ in
$\mathbb{P}(\mathbb{F}_{q}^{k})$ of size $n$, one can consider a $k\times n$
matrix $\mathbf{G}$ with the elements in the projective system $\mathcal{P}$
as its columns. This $k\times n$ matrix $\mathbf{G}$ generates an $[n,k]$ code
$\mathcal{C}(\mathcal{P})$ over $\mathbb{F}_{q}$. In [19, Chapter 1.1],
Tsfasman and Vladut showed that, up to some notion of equivalences, there is a
one-to-one correspondence between linear codes with Hamming metric and
projective systems.

As it was shown in [19], if $\mathbf{c}=\mathbf{x}_{\mathbf{c}}\mathbf{G}$ is
a codeword of $\mathcal{C}$, then the Hamming weight of $\mathbf{c}$ is given
by

$\mathsf{w}_{H}(\mathbf{c})=n-|\mathcal{P}\cap\mathcal{H}_{\mathbf{c}}|,$

where the cardinality on the right-hand side is counted with multiplicity and
$\mathcal{H}_{\mathbf{c}}=\mathrm{Ker}~{}\psi_{\mathbf{x}_{\mathbf{c}}}$ for
$\psi_{\mathbf{x}_{\mathbf{c}}}$ as defined in Equation (1). This is in fact
the analogue of Lemma 2 and it is the key to the construction of Hamming
metric codes from rank metric codes which we explain next.

###### Definition 13

Let $X\subset\mathbb{F}_{q^{m}}^{k}$ be a $q$-system and let $\sim$ be the
equivalence relation on $X$ given by $\mathbf{x}\sim\mathbf{y}$ for
$\mathbf{x},\mathbf{y}\in X$ if and only if $\mathbf{x}=\alpha\mathbf{y}$ for
some $\alpha\in\mathbb{F}_{q}^{\times}$. The projective system associated to
$X$ is the set
$\mathcal{P}(X)=(X/\sim)\backslash\\{\mathbf{0}\\}\subset\mathbb{P}(\mathbb{F}_{q^{m}}^{k})$.

It follows easily from Definition 13 that given an $[n,k]$ $q$-system $X$ in
$\mathbb{F}_{q^{m}}^{k}$, the associated projective system $\mathcal{P}(X)$ is
an $\left[\frac{q^{n}-1}{q-1},k\right]$ projective system in
$\mathbb{P}(\mathbb{F}_{q^{m}}^{k})$.

###### Definition 14

[1] Let $\mathcal{C}$ be an $[n,k]$ rank metric code over
$\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$ corresponding to a $q$-system
$X\subset\mathbb{F}_{q^{m}}^{k}$ of dimension $n$ over $\mathbb{F}_{q}$. A
Hamming metric code $\mathfrak{H}(\mathcal{C})$ associated to $\mathcal{C}$ is
a $\left[\frac{q^{n}-1}{q-1},k\right]$ Hamming metric code over
$\mathbb{F}_{q^{m}}$ corresponding to the projective system $\mathcal{P}(X)$.

For a rank metric code $\mathcal{C}$, the Hamming weights of the codewords in
an associated Hamming metric code $\mathfrak{H}(\mathcal{C})$ are determined
by the rank weights of the codewords of $\mathcal{C}$.

###### Lemma 7

[1, Sec 4.2] Let $\mathcal{C}$ be an $[n,k]$ rank metric code over
$\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$ corresponding to a $q$-system
$X\subset\mathbb{F}_{q^{m}}^{k}$ with generator matrix
$\mathbf{G}\in\mathbb{F}_{q^{m}}^{k\times n}$. Suppose that
$\mathfrak{H}(\mathcal{C})$ is a Hamming metric code associated to
$\mathcal{C}$ with generator matrix
$\mathbf{G}^{\prime}\in\mathbb{F}_{q^{m}}^{k\times\left(\frac{q^{n}-1}{q-1}\right)}$
and let $\mathbf{x}\in\mathbb{F}_{q^{m}}^{k}$. If
$\mathsf{rank}(\mathbf{x}\mathbf{G})=t$, then
$\mathsf{w}_{H}(\mathbf{x}\mathbf{G}^{\prime})=\frac{q^{n}-q^{n-t}}{q-1}$.

From Lemma 7, a rank weight in the rank metric code $\mathcal{C}$ corresponds
to a unique Hamming weight in the associated Hamming metric code
$\mathfrak{H}(\mathcal{C})$. Therefore we have the following theorem.

###### Theorem 5.1

Let $\mathcal{C}$ be an $[n,k,d]$ rank metric code over the extension
$\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$. If $\mathcal{C}$ is a two-weight rank
metric code, then the associated Hamming code $\mathfrak{H}(\mathcal{C})$ is a
$\left[\frac{q^{n}-1}{q-1},k,\frac{q^{n}-q^{n-d}}{q-1}\right]$ two-weight
Hamming metric code. Moreover, if $\mathcal{C}$ is an antipodal two-weight
rank metric code, then $\mathfrak{H}(\mathcal{C})$ is an antipodal two-weight
Hamming metric code.

###### Proof

Following Lemma 7, a codeword in $\mathcal{C}$ of rank $t$ corresponds to a
codeword of $\mathfrak{H}(\mathcal{C})$ with Hamming weight
$\frac{q^{n}-q^{n-t}}{q-1}$. Hence if the weight distribution of $\mathcal{C}$
is $\\{0,d,d^{\prime}\\}$, then the weight distribution of
$\mathfrak{H}(\mathcal{C})$ is
$\left\\{0,\frac{q^{n}-q^{n-d}}{q-1},\frac{q^{n}-q^{n-d^{\prime}}}{q-1}\right\\}$.

As a consequence of the previous theorem, Example 1 and Theorem 4.1 provide
constructions of antipodal two-weight Hamming metric codes. Furthermore,
Example 2 provides constructions of non-antipodal two-weight Hamming metric
codes.

## Open questions

We end this paper with some open questions. Theorem 4.1 gives a construction
of $[n,2,d]$ ATW rank metric codes over $\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$
from suitable MRD codes over $\mathbb{F}_{q^{m}}/\mathbb{F}_{q^{n-d}}$. In
fact, for the case $d=n/2$ this is the only way of constructing ATW rank
metric codes as proved in Theorem 4.2. So it is natural to ask if this happens
for the case when $d>n/2$? In other words, are there any $[n,2,d]$ ATW rank
metric code which are not induced by MRD codes? Equivalently, in terms of
$t$-spreads, one may ask if a $t$-subspread of a Desarguesian spread is again
a Desarguesian $t$-spread. In other words, is there a non-Desarguesian
t-subspread of a Desarguesian spread?

The main focus of this work was the study of antipodal two-weight rank metric
codes. However, one may also want to explore the classification and
characterization of non-antipodal two-weight rank metric codes.

## References

  * [1] G.N. Alfarano, M. Borello, A. Neri, and A. Ravagnani, _Linear cutting blocking sets and minimal codes in the rank metric_ , J. Combin. Theory Ser. A 192 (2022), paper No. 105658.
  * [2] M. Biliotti, V. Jha, and N.L. Johnson, _Foundations of translation planes_ , Monographs and Textbooks in Pure and Applied Mathematics, 243, Marcel Dekker, Inc., New York, 2001, xvi+542 pp.
  * [3] M, Biliotti, V. Jha, and N.L. Johnson, _Handbook of finite translation planes_ , Pure and Applied Mathematics (Boca Raton), 289, Chapman and Hall/CRC, 2007, xxii+861 pp.
  * [4] A. Bonisoli, _Every equidistant linear code is a sequence of dual Hamming codes_ , Ars Combin., 18 (1983), pp. 181–186.
  * [5] R. Calderbank and W.M. Kantor, _The geometry of two-weight codes_ , Bulletin of the London Mathematical Society, 18 (1986), no. 2, Wiley Online Library, pp. 97–122.
  * [6] P. Delsarte, Bilinear forms over a finite field, with applications to coding theory, J. Combin. Theory Ser. A, 25 (1978), no. 3, pp. 226–241.
  * [7] C. Ding and X. Wang, _A coding theory construction of new systematic authentication codes_ , Theor. Comput. Sci., 330(1):81–99 (2005).
  * [8] E. M. Gabidulin, _Theory of codes with maximum rank distance_ , Probl. Inf. Transm. 21 (1985), 1-12.
  * [9] E.M. Gabidulin, A. Kshevetskiy, _The new construction of rank codes_ , Proceedings, ISIT 2005.
  * [10] J. Hirschfeld, _Projective geometries over finite fields_ , Oxford Mathematical Monographs, Second edition, The Clarendon Press, Oxford University Press, New York, 1998, xii+555 pp.
  * [11] D. Jungnickel and V. D. Tonchev, _The classification of antipodal two-weight linear codes_ , Finite Fields Appl. 50 (2018), pp. 372–381.
  * [12] F. Manganiello, E. Gorla, and J. Rosenthal, _Spread codes and spread decoding in network coding_ , 2008 IEEE International Symposium on Information Theory, 2008.
  * [13] J. L. Massey, _Minimal codewords and secret sharing_ , in Proc. 6th Joint Swedish-Russian Workshop on Information Theory, Mölle, Sweden (1993), pp. 276–-279.
  * [14] K. Morrison, _Equivalence for rank metric and matrix codes and automorphism groups of Gabidulin codes_ , IEEE Trans. Inf. Theory, 60 (2014), no. 11, pp. 7035–7046.
  * [15] R. Pratihar and T.H. Randrianarisoa, _Constructions of optimal rank metric codes from automorphisms of rational function fields_ , Adv. Math. Commun., 2022.
  * [16] T.H. Randrianarisoa, _A geometric approach to rank metric codes and a classification of constant weight codes_ , Des. Codes Cryptogr., 88 (2020), no. 7, pp. 1331–1348.
  * [17] B. Segre, _Teoria di Galois, fibrazioni proiettive e geometrie non Desarguesiane_ , Annali di Matematica Pura ed Applicata (4), 64 (1964), pp. 1–76.
  * [18] J. Sheekey, _A new family of linear maximum rank distance codes_ , Adv. Math. Commun. 10 (2016), no. 3, pp. 475 –488.
  * [19] M. Tsfasman and S.G. Vladut, _Algebraic-geometric codes_ (Vol. 58), Springer Science & Business Media (1991).
  * [20] H.N. Ward, _An introduction to divisible codes_ , Des. Codes Cryptogr. 17 (1999), Springer, no. 1–3, pp. 73–79.
  * [21] J. Yuan and C. Ding, _Secret sharing schemes from two-weight codes_ , in Proc. R. C. Bose Centenary Symp. Discrete Mathematics and Applications, India, (2002).