# Dynamical system analysis of scalar field cosmology in coincident $f(Q)$
gravity

Sayantan Ghosh 0000-0002-3875-0849<EMAIL_ADDRESS>Department of
Mathematics, Birla Institute of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India. Raja Solanki 0000-0001-8849-7688
<EMAIL_ADDRESS>Department of Mathematics, Birla Institute of
Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India. P.K. Sahoo 0000-0003-2130-8832
<EMAIL_ADDRESS>Department of Mathematics, Birla Institute
of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India.

###### Abstract

In this article, we investigate scalar field cosmology in the coincident
$f(Q)$ gravity formalism. We calculate the motion equations of $f(Q)$ gravity
under the flat Friedmann-Lemaître-Robertson-Walker background in the presence
of a scalar field. We consider a non-linear $f(Q)$ model, particularly
$f(Q)=-Q+\alpha Q^{n}$, which is nothing but a polynomial correction to the
STEGR case. Further, we assumed two well-known specific forms of the potential
function, specifically the exponential from $V(\phi)=V_{0}e^{-\beta\phi}$ and
the power-law form $V(\phi)=V_{0}\phi^{-k}$. We employ some phase-space
variables and transform the cosmological field equations into an autonomous
system. We calculate the critical points of the corresponding autonomous
systems and examine their stability behaviors. We discuss the physical
significance corresponding to the exponential case for parameter values $n=2$
and $n=-1$ with $\beta=1$, and $n=-1$ with $\beta=\sqrt{3}$. Moreover, we
discuss the same corresponding to the power-law case for the parameter value
$n=-2$ and $k=0.16$. We also analyze the behavior of corresponding
cosmological parameters such as scalar field and dark energy density,
deceleration, and the effective equation of state parameter. Corresponding to
the exponential case, we find that the results obtained for the parameter
constraints in Case III is better among all three cases, and that represents
the evolution of the universe from a decelerated stiff era to an accelerated
de-Sitter era via matter-dominated epoch. Further, in the power-law case, we
find that all trajectories exhibit identical behavior, representing the
evolution of the universe from a decelerated stiff era to an accelerated de-
Sitter era. Lastly, we conclude that the exponential case shows better
evolution as compared to the power-law case.

Keywords: scalar field, $f(Q)$ gravity, dark energy, and autonomous system.

##  I Introduction

Modified gravity is a prominent route to enhance our understanding of the
evolution of the Universe, incorporating its early stages (inflation) and
later phases (dark energy) marked by accelerated expansion [1]. Additionally,
it may address potential observational conflicts [2]. These theories involve
constructing extensions and modifications of the General Relativity that
introduce additional degrees of freedom. These modifications can potentially
provide corrections at both the cosmological background and perturbation
levels, offering a more comprehensive description of the Universe’s behavior.
There exist various methods for constructing such gravitational modifications.
One different category of gravitational modifications emerges when utilizing
the equivalent formulation of gravity that incorporates non-metricity, which
is widely known as symmetric teleparallel gravity [3]. This approach employs a
generic affine connection with vanishing torsion and vanishing curvature with
respect to Levi-Civita connection, while relaxing the metricity condition with
respect to the generic connection. This formulation has recently been extended
to $f(Q)$ gravity [4]. This extension of the symmetric teleparallel formalism
has gained attention from the cosmology community as a potential avenue for
exploring new physics beyond the conventional $\Lambda$CDM cosmology.
Particular forms of the $f(Q)$ function have been demonstrated to alleviate
the $\sigma 8$ tension [5], whereas others enable a more accurate
representation of cosmological data [6, 7, 8]. Recently, some interesting
cosmological implications of the $f(Q)$ gravity in the different context have
been appeared, for instance, Black hole physics [9, 10], Neutrino physics
[11], Quantum cosmology [12, 13], Bouncing Cosmology [14], Inflation [15],
Phantom cosmology [16], Astrophysical objects [17, 18], Cosmological
perturbations [19], BBN constraints [20], and many others [21, 22, 23, 24,
25].

Scalar fields play a significant role in describing the physical properties of
the universe, particularly in the context of the inflationary scenario [26],
and they can offer explanations for the cosmic late-time acceleration. While
the $\Lambda$CDM model is highly consistent with observational data,
successfully describing structure formation, it has yet to quantify quantum
vacuum fluctuations effectively [27, 28]. This is the key inspiration for
proposing the dark energy candidate as an alternative to $\Lambda$. Various
examples include the quintessence field [29, 30], a quintom scalar field [31,
32], a phantom scalar field [33], and multi-scalar field models [34]. If one
assumes the cosmological constant $\Lambda$ is coming from a single (or
multiple) scalar field, then one can solve the cosmological constant problem,
as the scalar field can decay the vacuum energy to stop the exponential
increase. However, there exists a tension between the values of the Hubble
constant measured from early observations (such as Planck [35]) and estimated
via local observations (such as SH0ES [36]). One potential solution to this
tension involves assuming extensions beyond the $\Lambda$CDM. This article
explores scalar field cosmology in the modified symmetric teleparallel gravity
background, utilizing the dynamical system approach. The modified $f(Q)$
function is now responsible for the observed accelerating scenario, whereas
the quantum field theory prediction of the tiny value of $\Lambda$ ($\sim
10^{-120}$) cannot be described by such gravitational modification and thus
the addition of a scalar field take into account for quintessence scenario.

It is widely recognized that when investigating cosmological models,
introducing auxiliary variables allows for transforming cosmological equations
into an autonomous dynamical system [37]. This results in a system
$X^{\prime}=f(X)$, where $X$ represents the column vector consisting of
auxiliary variables and $f(X)$ denotes the vector field. The stability
analysis of the given autonomous involves several steps. Initially, critical
points (or equilibrium points) $X_{c}$ are identified by setting
$X^{\prime}=0$. Then, we consider linear perturbations around the critical
point $X_{c}$ as $X=X_{c}+P$, where $P$ represents the column vector of
perturbed auxiliary variables. As a result, one can have (up to first order)
$P^{\prime}=AP$, where $A$ is the matrix consisting of coefficients of the
perturbed equations. Finally, the eigenvalues of the coefficient matrix $A$
determine the stability behavior of each hyperbolic critical point. A critical
point $X_{c}$ is considered stable (unstable), or a saddle if the real parts
of the corresponding eigenvalues are negative (positive), or have real parts
with different signs. Several interesting outcomes in the context of modified
gravity utilizing dynamical system method can be found in references [38, 39,
40, 41, 42]. The present article is organized as follows. In Sec II, we
present the mathematical background of extended symmetric teleparallel gravity
with the Lagrangian density of a scalar field. In Sec III, we present
governing equations of motion under the FLRW background in the presence of a
scalar field. Further in Sec IV, we perform a detailed dynamical analysis of
two well-known scalar field potentials along with a non-linear $f(Q)$
cosmological background model. Finally in Sec V, we highlight our findings.

##  II $f(Q)$ gravity formulation with scalar field

We begin with the fact that a connection plays a vital role in the
transportation of tensors across a manifold. In the realm of general
relativity, which is built upon Riemannian geometry, the gravitational
interactions are ruled by a symmetric connection referred to as the Levi-
Civita connection. Nevertheless, a more generic connection comprises two
components: an antisymmetric part and another component exhibiting the non-
metricity condition. This extended affine connection can be expressed as
follows [43]:

$\Upsilon^{\alpha}_{\ \mu\nu}=\Gamma^{\alpha}_{\ \mu\nu}+K^{\alpha}_{\
\mu\nu}+L^{\alpha}_{\ \mu\nu},$ (1)

where the first term denotes the metric compatible Levi-Civita connection,

$\Gamma^{\alpha}_{\
\mu\nu}\equiv\frac{1}{2}g^{\alpha\lambda}(g_{\mu\lambda,\nu}+g_{\lambda\nu,\mu}-g_{\mu\nu,\lambda})$
(2)

The second one is the contortion tensor representing the antisymmetric part of
the affine connection, and that can estimate in terms of the torsion tensor
$T^{\alpha}_{\ \mu\nu}\equiv\Upsilon^{\alpha}_{\ \mu\nu}-\Upsilon^{\alpha}_{\
\nu\mu}$ as follows,

$K^{\alpha}_{\ \mu\nu}\equiv\frac{1}{2}(T^{\alpha}_{\ \mu\nu}+T_{\mu\ \nu}^{\
\alpha}+T_{\nu\ \mu}^{\ \alpha})$ (3)

, and the last one is the distortion tensor,

$L^{\alpha}_{\ \mu\nu}\equiv\frac{1}{2}(Q^{\alpha}_{\ \mu\nu}-Q_{\mu\ \nu}^{\
\alpha}-Q_{\nu\ \mu}^{\ \alpha})$ (4)

which is expressed in terms of non-metricity tensor,

$Q_{\alpha\mu\nu}\equiv\nabla_{\alpha}g_{\mu\nu}=\partial_{\alpha}g_{\mu\nu}-\Upsilon^{\beta}_{\,\,\,\alpha\mu}g_{\beta\nu}-\Upsilon^{\beta}_{\,\,\,\alpha\nu}g_{\mu\beta}$
(5)

In addition, we define the superpotential tensor as

$4P^{\lambda}\>_{\mu\nu}=-Q^{\lambda}\>_{\mu\nu}+2Q_{(\mu}\>^{\lambda}\>{}_{\nu)}+(Q^{\lambda}-\tilde{Q}^{\lambda})g_{\mu\nu}-\delta^{\lambda}_{(\mu}Q_{\nu)}.$
(6)

where $Q_{\alpha}=Q_{\alpha}\>^{\mu}\>{}_{\mu}$ and
$\tilde{Q}_{\alpha}=Q^{\mu}\>_{\alpha\mu}$ are non-metricity vectors. Now by
contracting the superpotential tensor with the non-metricity tensor, we obtain
non-metricity scalar in a more confine form,

$Q=-Q_{\lambda\mu\nu}P^{\lambda\mu\nu}.$ (7)

We know that the curvature tensor can be estimated as

$R^{\alpha}_{\>\beta\mu\nu}=2\partial_{[\mu}\Upsilon^{\alpha}_{\>\nu]\beta}+2\Upsilon^{\alpha}_{\>[\mu\mid\lambda\mid}\Upsilon^{\lambda}_{\nu]\beta}$
(8)

Now by using the affine connection (1), one can have

$R^{\alpha}_{\>\beta\mu\nu}=\mathring{R}^{\alpha}_{\>\beta\mu\nu}+\mathring{\nabla}_{\mu}X^{\alpha}_{\>\nu\beta}-\mathring{\nabla}_{\nu}X^{\alpha}_{\>\mu\beta}+X^{\alpha}_{\>\mu\rho}X^{\rho}_{\>\nu\beta}-X^{\alpha}_{\>\nu\rho}X^{\rho}_{\>\mu\beta}$
(9)

Here $\mathring{R}^{\alpha}_{\>\beta\mu\nu}$ and $\mathring{\nabla}$ are
described in terms of the Levi-Civita connection (2), and $X^{\alpha}_{\
\mu\nu}=K^{\alpha}_{\ \mu\nu}+L^{\alpha}_{\ \mu\nu}$. On applying suitable
contractions on curvature term and torsion free constraint $T^{\alpha}_{\
\mu\nu}=0$ in the equation (9), we have

$R=\mathring{R}-Q+\mathring{\nabla}_{\alpha}\left(Q^{\alpha}-\tilde{Q}^{\alpha}\right)$
(10)

where $\mathring{R}$ is the usual Ricci scalar evaluated in terms of the Levi-
Civita connection. Further on employing teleparallel constraint $R=0$, we
acquire curvature free teleparallel geometries, and hence relation (10)
becomes

$\mathring{R}=Q-\mathring{\nabla}_{\alpha}\left(Q^{\alpha}-\tilde{Q}^{\alpha}\right)$
(11)

The relation obtained in the equation (11) indicates that the Ricci scalar
curvature differs from the non-metricity scalar by a boundary term. The
equation (11) reveals that the gravity theory incorporates only non-metricity
scalar $Q$ in the action differs from the Einstein’s GR by a boundary term.
This indicates that STEGR presents an equivalent formulation to GR, and hence
the theory is known as symmetric teleparallel equivalent to GR [44].

Now, we present the action for $f(Q)$ gravity which is a generalization to
STEGR theory, in presence of a scalar field,

$\mathcal{S}=\int\frac{1}{2}\,f(Q)\sqrt{-g}\,d^{4}x+\int\mathcal{L}_{\phi}\,\sqrt{-g}\,d^{4}x\,,$
(12)

where $g=\text{det}(g_{\mu\nu})$, $f(Q)$ is an arbitrary function of the non-
metricity scalar $Q$, and $\mathcal{L}_{\phi}$ is the Lagrangian density of a
scalar field $\phi$ given by [45],

$\mathcal{L}_{\phi}=-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)$
(13)

Here $V(\phi)$ represents a potential for the field $\phi$. We obtain the
following governing field equation by varying the action (12) with respect to
the metric,

$\frac{2}{\sqrt{-g}}\nabla_{\lambda}(\sqrt{-g}f_{Q}P^{\lambda}\>_{\mu\nu})+\frac{1}{2}g_{\mu\nu}f+f_{Q}(P_{\mu\lambda\beta}Q_{\nu}\>^{\lambda\beta}-2Q_{\lambda\beta\mu}P^{\lambda\beta}\>_{\nu})=-T_{\mu\nu}^{\phi}$
(14)

where $f_{Q}=\frac{df}{dQ}$ and $T_{\mu\nu}^{\phi}$ represents the stress-
energy tensor of the scalar field given as

$T_{\mu\nu}^{\phi}=\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}g_{\mu\nu}g_{\alpha\beta}\partial^{\alpha}\phi\partial^{\beta}\phi-
g_{\mu\nu}V(\phi)$ (15)

Moreover, we obtain the following equation of motion for the scalar field i.e.
Klein-Gordon equation from the Euler-Lagrangian equation for the Lagrangian
density given by (13)

$\square\phi-V,_{\phi}=0$ (16)

Here $\square$ denotes the d’Alembertian and $V,_{\phi}=\frac{\partial
V}{\partial\phi}$. Further, on varying the action (12) with respect to the
connection (similar to Palatini prescription), we have,

$\nabla_{\mu}\nabla_{\nu}(\sqrt{-g}f_{Q}P^{\mu\nu}\>_{\lambda})=0$ (17)

##  III Equations of Motion

We begin with the following flat FLRW line element in order to probe the
cosmological implications under the assumption of spatial isotropy and
homogeneity of the universe,

$ds^{2}=-dt^{2}+a^{2}(t)[dx^{2}+dy^{2}+dz^{2}]$ (18)

Here $a(t)$ is a measure of expansion of the universe. Beginning with the
teleparallel constraint that corresponds to a flat geometry characterizing a
pure inertial connection, one can execute a gauge transformation parameterized
by $\Lambda^{\alpha}_{\mu}$ [46],

$\Upsilon^{\alpha}_{\>\mu\nu}=(\Lambda^{-1})^{\alpha}_{\>\>\beta}\partial_{[\mu}\Lambda^{\beta}_{\>\>\nu]}$
(19)

Consequently, one can express the generic affine connection in the following
manner, by utilizing the general element of $GL(4,\mathbb{R})$ characterized
by the transformation
$\Lambda^{\alpha}_{\>\>\mu}=\partial_{\mu}\zeta^{\alpha}$, where
$\zeta^{\alpha}$ is an arbitrary vector field,

$\Upsilon^{\alpha}_{\>\mu\nu}=\frac{\partial
x^{\alpha}}{\partial\zeta^{\rho}}\partial_{\mu}\partial_{\nu}\zeta^{\rho}$
(20)

This reveals the possibility of eliminating the connection through a
coordinate transformation. The coordinate transformation is responsible for
eliminating the connection (20) is termed gauge coincident. We utilize the
coincident gauge in the present manuscript. Hence, the non-metricity scalar
corresponds to the metric (18) becomes $Q=6H^{2}$.
The stress-energy tensor for the perfect fluid distribution reads as

$T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu\nu}$ (21)

where $u^{\mu}=(1,0,0,0)$ are components of the four velocities. On comparing
equation (21) and (15), we have,

$\rho=-\frac{1}{2}g_{\alpha\beta}\partial^{\alpha}\phi\partial^{\beta}\phi+V(\phi)$
(22)
$p=-\frac{1}{2}g_{\alpha\beta}\partial^{\alpha}\phi\partial^{\beta}\phi-V(\phi)$
(23)

As the scalar field considered here does not depend on the spatial
coordinates, we have following expressions for the energy density and pressure
component of the scalar field,

$\rho_{\phi}=\frac{1}{2}\dot{\phi}^{2}+V(\phi)$ (24)
$p_{\phi}=\frac{1}{2}\dot{\phi}^{2}-V(\phi)$ (25)

and the corresponding equation of state parameter can be written as,

$\omega_{\phi}=\frac{p_{\phi}}{\rho_{\phi}}=\frac{\frac{1}{2}\dot{\phi}^{2}-V(\phi)}{\frac{1}{2}\dot{\phi}^{2}+V(\phi)}$
(26)

Moreover, corresponding to the metric (18), the Klein-Gordon equation (16)
becomes,

$\ddot{\phi}+3H\dot{\phi}+V_{,\phi}=0$ (27)

We obtain the following Friedmann like equations governing the gravitational
interactions under the $f(Q)$ gravity background in the presence of scalar
field,

$3H^{2}=\frac{1}{2f_{Q}}\left(-\rho_{\phi}+\frac{f}{2}\right)$ (28)
$\dot{H}+3H^{2}+\frac{\dot{f_{Q}}}{f_{Q}}H=\frac{1}{2f_{Q}}\left(p_{\phi}+\frac{f}{2}\right)$
(29)

For the $f(Q)$ functional $f(Q)=-Q+\Psi(Q)$, we can rewrite the Friedmann
equations (28)-(29) as (where we can recover ordinary GR by putting $\Psi=0$)

$3H^{2}=\rho_{\phi}+\rho_{de}$ (30)
$\dot{H}=-\frac{1}{2}[\rho_{\phi}+p_{\phi}+\rho_{de}+p_{de}]$ (31)

where $\rho_{de}$ and $p_{de}$ represents the energy density and pressure of
the dark energy component evolving due to geometry of spacetime,

$\rho_{de}=-\frac{\Psi}{2}+Q\Psi_{Q}$ (32)
$p_{de}=-\rho_{de}-2\dot{H}\left(\Psi_{Q}+2Q\Psi_{QQ}\right)$ (33)

##  IV The Cosmological Model and Dynamical System Analysis

In this section, we attempt to examine some specific class of scalar field
potentials and $f(Q)$ functional forms, incorporating the phase-space
techniques. In order to do so, we transform motion equations of considered
cosmological settings, discussed in the previous section, into an autonomous
system with the help of phase-space variables. We define the following phase-
space variables,

$x^{2}=\frac{\dot{\phi}^{2}}{6H^{2}},\>\>y^{2}=\frac{V}{3H^{2}},\>\>\text{and}\>\>s^{2}=\Omega_{de}=\frac{\rho_{de}}{3H^{2}}$
(34)

Now using the fact that
$\Omega_{\phi}=\frac{\rho_{\phi}}{3H^{2}}=\frac{\dot{\phi}^{2}}{6H^{2}}+\frac{V}{3H^{2}}=x^{2}+y^{2}$
and the equation (30), we have following constraint,

$x^{2}+y^{2}+s^{2}=1$ (35)

Hence it follows that $\Omega_{de}=s^{2}=1-x^{2}-y^{2}=1-\Omega_{\phi}$ and
the constraint $0\leq x^{2}+y^{2}\leq 1$.

In order to obtain the close form of required autonomous system with general
potential, we define another phase-space variable as follows,

$\lambda=-\frac{V_{,\phi}}{V}$ (36)

We obtain the following dynamical system with respect to e-folding time
$N=ln(a)$,

$x^{\prime}=-3x-x\frac{\dot{H}}{H^{2}}+\sqrt{\frac{3}{2}}\lambda y^{2}$ (37)
$y^{\prime}=\frac{1}{2y}\left[\frac{\dot{V}}{3H^{2}}-y^{2}\frac{2\dot{H}}{H^{2}}\right]$
(38) $\lambda^{\prime}=-\sqrt{6}\lambda^{2}x(\Gamma-1)$ (39)

where $\Gamma=\frac{VV_{,\phi\phi}}{V_{,\phi}^{2}}$. Here using the equation
(31), we have

$\frac{\dot{H}}{H^{2}}=\frac{3x^{2}}{\left(\Psi_{Q}+2Q\Psi_{QQ}-1\right)}$
(40)

Note that the dynamical system (37)-(39) still not in its closed form, as it
depends on the choice of $f(Q)$ functional form. Further, it also appears that
we gain nothing from this extra phase-space variable $\lambda$, due to the
presence of a quantity $\Gamma$ in the given dynamical system, that directly
relies on the scalar field $\phi$. Nevertheless, considering that both
$\lambda$ and $\Gamma$ are functions dependent on the scalar field $\phi$,
there exists an explicit relation between them i.e. if the function
$\lambda(\phi)$ is invertible, allowing us to derive $\phi(\lambda)$, and thus
we have $\Gamma$ as a function of $\lambda$.

We consider a power-law functional form $\Psi(Q)=\alpha Q^{n}$ i.e.
$f(Q)=-Q+\alpha Q^{n}$, where $\alpha$ and $n$ are free parameters. The choice
of considered functional form is appropriate in order to close the obtained
dynamical system. Now, for the assumed $f(Q)$ function, the expression (40)
becomes

$\frac{\dot{H}}{H^{2}}=\frac{3x^{2}}{n\left(1-x^{2}-y^{2}\right)-1}$ (41)

Moreover, we have following expressions for two cosmological parameters that
plays a crucial role in characterizing the evolutionary phase of expansion of
the universe, namely deceleration and equation of state parameter,

$q=-1-\frac{3x^{2}}{n\left(1-x^{2}-y^{2}\right)-1}$ (42)
$\omega_{total}=-1-\frac{2x^{2}}{n\left(1-x^{2}-y^{2}\right)-1}$ (43)

Now in order to probe the cosmological implications of the considered scenario
we choose two specific forms of potential function, namely exponential and
power-law potentials, which are widely discussed in the literature in the
different cosmological contexts.

###  IV.1 Exponential potential

We assume the following form of exponential potential,

$V(\phi)=V_{0}e^{-\beta\phi}$ (44)

The assumed exponential potential represents the most basic illustration of
quintessence scenarios [47, 48], as it is a slowly varying scalar field and
could decay the cosmological constant $\Lambda$. Hence, it can be a suitable
candidate to address the cosmological constant problem. Such potential arises
naturally from the string theory compactifications discussed in [49]. Also,
this type of potential naturally arises from the Kaluza-Klein type
compactifications, and their effect on anisotropic product cosmology has been
investigated in the reference [50]. We note that the dynamical system analysis
of scale-invariant solutions of the exponential scalar field coupled with a
barotropic fluid system has been studied and has been shown that such a system
can have stable scale-invariant stable fixed point [37]. Now, by using
equation (44), the dynamical variable (36) becomes

$\lambda=\beta\>\>\text{with}\>\>\Gamma=1$ (45)

,and hence the system (37)-(39) reduces to the following autonomous system,

$x^{\prime}=-3x\left[1+\frac{x^{2}}{n\left(1-x^{2}-y^{2}\right)-1}\right]+\sqrt{\frac{3}{2}}\beta
y^{2}$ (46)
$y^{\prime}=-xy\left[\sqrt{\frac{3}{2}}\beta+\frac{3x}{n\left(1-x^{2}-y^{2}\right)-1}\right]$
(47)

The critical points and their physical behavior, corresponds to autonomous
system (46)-(47) are presented in the Table (1).

Table 1: Table shows the critical points and their behavior corresponding to the model $f(Q)=-Q+\alpha Q^{n}$ with exponential potential $V(\phi)=V_{0}e^{-\beta\phi}$. Critical Points | Eigenvalues | Nature of critical point | $q$ | $\omega$
---|---|---|---|---
$(x_{c},y_{c})$ |  $\lambda_{1}$ and $\lambda_{2}$ |  |  |
$O(0,0)$ | $-3,0$ | Stable | $-1$ | $-1$
$A(-1,0)$ | $6-6n,3+\sqrt{\frac{3}{2}}\beta$ |  Stable for $(n>1\>\&\>\beta<-\sqrt{6})$ | $2$ | $1$
$B(1,0)$ | $6-6n,3-\sqrt{\frac{3}{2}}\beta$ |  Stable for $(n>1\>\&\>\beta>\sqrt{6})$ | $2$ | $1$
$C\left(\frac{\beta}{\sqrt{6}},\sqrt{1-\frac{\beta^{2}}{6}}\right)$ | $\frac{1}{2}(\beta^{2}-6),(1-n)\beta^{2}$ |  Stable for $(-\sqrt{6}<\beta<0\>\&\>n>1)$ or $(0<\beta<\sqrt{6}\>\&\>n>1)$ | $-1+\frac{\beta^{2}}{2}$ | $-1+\frac{\beta^{2}}{3}$
$D\left(\frac{\beta}{\sqrt{6}},-\sqrt{1-\frac{\beta^{2}}{6}}\right)$ | $\frac{1}{2}(\beta^{2}-6),(1-n)\beta^{2}$ |  Stable for $(-\sqrt{6}<\beta<0\>\&\>n>1)$ or $(0<\beta<\sqrt{6}\>\&\>n>1)$ | $-1+\frac{\beta^{2}}{2}$ | $-1+\frac{\beta^{2}}{3}$

We present the asymptotic behavior of some prominent cosmological parameters
such as the scale factor, the Hubble parameter, and the scalar field
corresponding to obtained set of critical points,

  * •

Corresponding to the case $x_{c}=0$ and $y_{c}=0$, we obtain the cosmic scalar
factor as $a(t)=e^{H_{0}(t-t_{0})}$, the Hubble parameter as
$H(t)=H(t_{0})=H_{0}$, and the scalar field $\phi(t)=\phi(t_{0})=\phi_{0}$,
where $t_{0}$ being the present time, and $a(t_{0})=1$ and $H_{0}$ are being
the present value of the scale factor and the Hubble parameter.

  * •

Corresponding to the case $x_{c}=\pm 1$ and $y_{c}=0$, we obtain the cosmic
scalar factor as $a(t)=\left[3H_{0}(t-t_{0})+1\right]^{\frac{1}{3}}$, the
Hubble parameter as $H(t)=\frac{H_{0}}{3H_{0}(t-t_{0})+1}$, and the scalar
field $\phi(t)=\phi_{0}\pm\frac{\sqrt{6}}{3}ln\left[3H_{0}(t-t_{0})+1\right]$.

  * •

Corresponding to the case $x_{c}=\frac{\beta}{\sqrt{6}}$ and
$y_{c}=\pm\sqrt{1-\frac{\beta^{2}}{6}}$, we obtain the cosmic scalar factor as
$a(t)=\left[\frac{\beta^{2}}{2}H_{0}(t-t_{0})+1\right]^{\frac{2}{\beta^{2}}}$,
the Hubble parameter as
$H(t)=\frac{H_{0}}{\frac{\beta^{2}}{2}H_{0}(t-t_{0})+1}$, and the scalar field
$\phi(t)=\phi_{0}+\left[\frac{\beta^{2}}{2}H_{0}(t-t_{0})+1\right]^{\frac{2}{\beta}}$.

The 2-D phase-space diagrams corresponding to autonomous system (46)-(47) for
some specific parameter values are presented below in the Figure (1). In
addition, the corresponding evolutionary profile of the scalar field and dark
energy density are presented in the Figure (2). Further in the Figure (3), we
present the corresponding evolutionary profile of the deceleration and the
effective equation of state parameter.

Figure 1: Phase-space plots for the case $n=2$ (left panel above) and $n=-1$
(right panel above), with $\beta=1$ , and for the case $n=-1$ with
$\beta=\sqrt{3}$ (below) corresponding to the exponential potential.

Figure 2: Evolutionary profile of the scalar field density and dark energy
density for the case $n=2$ (left panel) and $n=-1$ (middle panel), with
$\beta=1$, and for the case $n=-1$ with $\beta=\sqrt{3}$ (right panel)
corresponding to the exponential potential.

Figure 3: Evolutionary profile of the deceleration and the equation of state
parameter for the case $n=2$ (left panel) and $n=-1$ (middle panel), with
$\beta=1$, and for the case $n=-1$ with $\beta=\sqrt{3}$ (right panel)
corresponding to the exponential potential.

Now we discuss the physical significance and stability of the critical points
obtained for the specific parameter values as follows:

  * •

Case I ($n=2$ and $\beta=1$) : In this case, the obtained set of critical
points are $O_{1}(0,0)$, $A_{1}(-1,0)$, $B_{1}(1,0)$,
$C_{1}\left(\frac{1}{\sqrt{6}},\sqrt{\frac{5}{6}}\right)$, and
$D_{1}\left(\frac{1}{\sqrt{6}},-\sqrt{\frac{5}{6}}\right)$ with corresponding
eigenvalues $(\lambda_{1},\lambda_{2})=(-3,0)$,
$\left(-6,3+\sqrt{\frac{3}{2}}\right)$,
$\left(-6,3-\sqrt{\frac{3}{2}}\right)$, $\left(-\frac{5}{2},-1\right)$, and
$\left(-\frac{5}{2},-1\right)$ respectively. Moreover, the corresponding pair
of values $(q,\omega)$ are $(-1,-1)$, $(2,1)$, $(2,1)$,
$\left(-\frac{1}{2},-\frac{2}{3}\right)$, and
$\left(-\frac{1}{2},-\frac{2}{3}\right)$. Therefore critical points $O_{1}$,
$C_{1}$, and $D_{1}$ are stable, whereas points $A_{1}$ and $B_{1}$ are
saddle. It should also note that the one can neglect the flow of trajectories
below X-axis in the phase-space diagrams, since results are same as that of
the above X-axis due to the symmetry in positive and negative axes of Y.
Moreover, from the flow of phase-space trajectories in the Figure (1) (left
panel above), it is clear that we obtained a stable de-Sitter accelerated
universe (represented by $O_{1}$) without any transition epoch. The same is
reflected in the evolutionary profile of corresponding cosmological parameters
presented in the Figures (2) and (3) (left panel).

  * •

Case II ($n=-1$ and $\beta=1$) : In this case, the obtained set of critical
points are $O_{2}(0,0)$, $A_{2}(-1,0)$, $B_{2}(1,0)$,
$C_{2}\left(\frac{1}{\sqrt{6}},\sqrt{\frac{5}{6}}\right)$, and
$D_{2}\left(\frac{1}{\sqrt{6}},-\sqrt{\frac{5}{6}}\right)$ with corresponding
eigenvalues $(\lambda_{1},\lambda_{2})=(-3,0)$,
$\left(12,3+\sqrt{\frac{3}{2}}\right)$,
$\left(12,3-\sqrt{\frac{3}{2}}\right)$, $\left(-\frac{5}{2},2\right)$, and
$\left(-\frac{5}{2},2\right)$ respectively. Moreover, the corresponding pair
of values $(q,\omega)$ are $(-1,-1)$, $(2,1)$, $(2,1)$,
$\left(-\frac{1}{2},-\frac{2}{3}\right)$, and
$\left(-\frac{1}{2},-\frac{2}{3}\right)$. Therefore the critical point $O_{2}$
is stable, $A_{2}$ and $B_{2}$ are unstable, and points $C_{2}$ and $D_{2}$
are saddle. Further, from the flow of phase-space trajectories in the Figure
(1) (right panel above), we obtained evolutionary phase $A_{2}\rightarrow
O_{2}$ and $B_{2}\rightarrow O_{2}$, both representing the evolution of the
universe from stiff fluid dominated decelerated epoch to an accelerated de-
Sitter epoch. The same is reflected in the evolutionary profile of
corresponding cosmological parameters presented in the Figures (2) and (3)
(middle panel). The results obtained for the parameter constraints in Case II
is better than that of Case I.

  * •

Case III ($n=-1$ and $\beta=\sqrt{3}$) : In this case, the obtained set of
critical points are $O_{3}(0,0)$, $A_{3}(-1,0)$, $B_{3}(1,0)$,
$C_{3}\left(\frac{1}{\sqrt{2}},\sqrt{\frac{1}{2}}\right)$, and
$D_{3}\left(\frac{1}{\sqrt{2}},-\sqrt{\frac{1}{2}}\right)$ with corresponding
eigenvalues $(\lambda_{1},\lambda_{2})=(-3,0)$,
$\left(12,3+\frac{3}{\sqrt{2}}\right)$,
$\left(12,3-\frac{3}{\sqrt{2}}\right)$, $\left(-\frac{3}{2},6\right)$, and
$\left(-\frac{3}{2},6\right)$ respectively. Moreover, the corresponding pair
of values $(q,\omega)$ are $(-1,-1)$, $(2,1)$, $(2,1)$,
$\left(\frac{1}{2},0\right)$, and $\left(\frac{1}{2},0\right)$. Therefore the
critical point $O_{3}$ is stable, $A_{3}$ and $B_{3}$ are unstable, and points
$C_{3}$ and $D_{3}$ are saddle. Further, from the flow of phase-space
trajectories in the Figure (1) (below), we obtained evolutionary phase
$A_{3}\rightarrow O_{3}$, $B_{3}\rightarrow O_{3}$, and $A_{3}\rightarrow
C_{3}\rightarrow O_{3}$. The evolutionary trajectory $A_{3}\rightarrow
C_{3}\rightarrow O_{3}$ is quite interesting, representing the evolution of
the universe from a decelerated stiff era to an accelerated de-Sitter era via
matter dominated epoch. The same is reflected in the evolutionary profile of
corresponding cosmological parameters presented in the Figures (2) and (3)
(right panel). The results obtained for the parameter constraints in Case III
is better among all three cases.

###  IV.2 Power-law potential

We assume the following form of power-law potential,

$V(\phi)=V_{0}\phi^{-k}$ (48)

The power-law type scalar field potentials have already been investigated in
the context of both inflations [51] and to address the cosmological constant
problem [52]. Such a potential naturally occurs in the string theory, such as
law energy states of D-brains giving effective Tachyon field [53] and in Born-
Infeld inspired lagrangian [54]. Also, such potential has appeared in the
tachyon-dominated backgrounds to probe the cosmological perturbations [55].
Now, by using equation (48), the dynamical variable (36) becomes

$\lambda=\frac{k}{\phi}\>\>\text{with}\>\>\Gamma=\frac{k+1}{k}$ (49)

,and hence the system (37)-(39) reduces to the following autonomous system,

$x^{\prime}=-3x\left[1+\frac{x^{2}}{n\left(1-x^{2}-y^{2}\right)-1}\right]+\sqrt{\frac{3}{2}}\lambda
y^{2}$ (50) $y^{\prime}=\sqrt{\frac{3}{2}}\lambda
xy\left[1-\frac{\sqrt{6}x}{\lambda\\{n\left(1-x^{2}-y^{2}\right)-1\\}}\right]$
(51) $\lambda^{\prime}=-\frac{\sqrt{6}}{k}\lambda^{2}x$ (52)

Note that the autonomous system (50)-(52) is invariant under the
transformation $y\rightarrow-y$, and hence flow of trajectories obtained in
the negative region of y would be a copy of positive region. Moreover, the
system (50)-(52) is also invariant under simultaneous transformations
$x\rightarrow-x$ and $\lambda\rightarrow-\lambda$. Therefore, the physical
region of the given dynamical system is represented by the positive-y half
cylinder with infinite length from $\lambda=0$ to $\lambda=+\infty$. Hence, we
define the following phase-space variable in order to compactify the variable
$\lambda$,

$z=\frac{\lambda}{\lambda+1}\>\>\text{or}\>\>\lambda=\frac{z}{1-z}$ (53)

The new phase-space variable $z$ is bounded as $0\leq z\leq 1$ and follows
$z=0$ when $\lambda=0$ and $z=1$ when $\lambda\rightarrow+\infty$. Now, the
autonomous system (50)-(52) reduces to,

$x^{\prime}=-3x\left[1+\frac{x^{2}}{n\left(1-x^{2}-y^{2}\right)-1}\right]+\sqrt{\frac{3}{2}}\frac{z}{(1-z)}y^{2}$
(54)
$y^{\prime}=\sqrt{\frac{3}{2}}\frac{z}{(1-z)}xy\left[1-\frac{\sqrt{6}x(1-z)}{z\\{n\left(1-x^{2}-y^{2}\right)-1\\}}\right]$
(55) $z^{\prime}=-\frac{\sqrt{6}}{k}z^{2}x$ (56)

The critical points and their physical behavior, corresponds to autonomous
system (54)-(56) are presented in the Table (2).

Table 2: Table shows the critical points and their behavior corresponding to the model $f(Q)=-Q+\alpha Q^{n}$ with potential $V(\phi)=V_{0}\phi^{-k}$. Critical Points $(x_{c},y_{c},z_{c})$ |  Eigenvalues ($\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$) | Nature of critical point | $q$ | $\omega$
---|---|---|---|---
$O^{\prime}(0,0,0)$ | $(-3,0,0)$ | Stable | $-1$ | $-1$
$A^{\prime}(0,y,0)$ | $(-3,0,0)$ | Stable | $-1$ | $-1$
$B^{\prime}(0,0,z)$ | $(0,0,-3)$ | Stable | $-1$ | $-1$
$C^{\prime}(1,0,0)$ | $(3,0,6-6n)$ |  Unstable for $n\leq 1$ and N.H. for $n>1$ | $2$ | $1$
$D^{\prime}(-1,0,0)$ | $(3,0,6-6n)$ |  Unstable for $n\leq 1$ and N.H. for $n>1$ | $2$ | $1$

We present the asymptotic behavior of some prominent cosmological parameters
such as the scale factor, the Hubble parameter, and the scalar field
corresponding to obtained set of critical points,

  * •

Corresponding to the case $x_{c}=0$, we obtain the cosmic scalar factor as
$a(t)=e^{H_{0}(t-t_{0})}$, the Hubble parameter as $H(t)=H(t_{0})=H_{0}$, and
the scalar field $\phi(t)=\phi(t_{0})=\phi_{0}$, where $t_{0}$ being the
present time, and $a(t_{0})=1$ and $H_{0}$ are being the present value of the
scale factor and the Hubble parameter.

  * •

Corresponding to the case $x_{c}=\pm 1$ and $y_{c}=0$, we obtain the cosmic
scalar factor as $a(t)=\left[3H_{0}(t-t_{0})+1\right]^{\frac{1}{3}}$, the
Hubble parameter as $H(t)=\frac{H_{0}}{3H_{0}(t-t_{0})+1}$, and the scalar
field $\phi(t)=\phi_{0}\pm\frac{\sqrt{6}}{3}ln\left[3H_{0}(t-t_{0})+1\right]$.

The 3-D phase-space diagram plotted for a set of solutions to the autonomous
system (54)-(56) utilizing numerical approach for specific parameter value is
presented in the Figure (4). Further, the corresponding evolutionary profile
of the scalar field and dark energy density with the deceleration and the
effective equation of state parameter are presented in the Figure (5).

Figure 4: The 3-D phase-space trajectories plotted for a set of solutions to
the autonomous system for the parameter value $n=-2$ and $k=0.16$
corresponding to the power-law potential.

Figure 5: Evolutionary profile of scalar field density, dark energy density,
deceleration, and the equation of state parameter for the parameter value
$n=-2$ and $k=0.16$ corresponding to the power-law potential.

Now we discuss the physical significance and stability of the critical points
obtained for the specific parameter value $n=-2$ and $k=0.16$. In this case,
the obtained set of critical points are $O^{\prime}(0,0,0)$,
$A^{\prime}(0,y,0)$ where $0\leq y\leq 1$, $B^{\prime}(0,0,z)$ where $0\leq
z\leq 1$, $C^{\prime}(1,0,0)$, and $D^{\prime}(-1,0,0)$ with corresponding
eigenvalues $(\lambda_{1},\lambda_{2},\lambda_{3})=(-3,0,0)$, $(-3,0,0)$,
$(0,0,-3)$, $(3,0,18)$, and $(3,0,18)$ respectively. Moreover, the
corresponding pair of values $(q,\omega)$ are $(-1,-1)$, $(-1,-1)$, $(-1,-1)$,
$(2,1)$, and $(2,1)$. Therefore critical points $O^{\prime}$, $A^{\prime}$,
and $B^{\prime}$ are stable, whereas points $C^{\prime}$ and $D^{\prime}$ are
unstable. Further, from the flow of trajectories in the 3-D phase-diagram
presented in the Figure (4), we obtained evolutionary phase
$C^{\prime}\rightarrow O^{\prime}$, $D^{\prime}\rightarrow O^{\prime}$,
$C^{\prime}\rightarrow A^{\prime}$, $D^{\prime}\rightarrow A^{\prime}$, and
$D^{\prime}\rightarrow B^{\prime}$. All the presented flow of trajectories
exhibits identical behavior, and representing the evolution of the universe
from a decelerated stiff era to an accelerated de-Sitter era. The same is
reflected in the evolutionary profile of corresponding cosmological parameters
presented in the Figure (5). The left panel of the Figure (5) indicates that
the scalar field density vanishes with the expansion of the universe, whereas
the dark energy density dominates completely at the late-times. The right
panel of the Figure (5) indicates the transition from decelerated epoch to
accelerated epoch of the universe in the recent past, and that end up with a
de-Sitter universe at late-times.

##  V Conclusion

In this manuscript, we studied scalar field cosmology in the coincident $f(Q)$
gravity formalism. Modified gravity with non-metricity has been extensively
investigated recently in different contexts such as late-time observational
constraints, black holes, wormholes, and dynamical analysis, whereas the
scalar field cosmology has been prominent in describing inflation as well as
late-time acceleration. The modified $f(Q)$ function is now responsible for
observed accelerating expansion, whereas the quantum field theory prediction
of a tiny value of $\Lambda$ cannot be described by such a modified gravity
scenario, and hence the addition of a scalar field take into account for
quintessence scenario. In section II, we presented the mathematical
formulation of $f(Q)$ gravity with the Lagrangian density of the scalar field.
Further, in section III, we presented Friedmann-like equations of $f(Q)$
gravity ruling the gravitational interactions under the FLRW background in the
presence of a scalar field. In section IV, we begin with a non-linear $f(Q)$
functional, specifically $f(Q)=-Q+\Psi(Q)=-Q+\alpha Q^{n}$, where $\alpha$ and
$n$ are free model parameters. The assumed functional form is a polynomial
correction to the STEGR case and has great significance in early and late-time
cosmology. The considered $f(Q)$ function with $n>1$ can potentially apply to
the inflationary scenario, whereas the case $n<1$ corrects the late-time
cosmology. Further, we have defined phase-space variables (presented in
equations (34) and (36)), and then we expressed the deceleration and the
effective equation of state parameter in terms of phase-space variables. Now
to probe the cosmological implications of the considered scenario, we assumed
two specific forms of the potential function, specifically the exponential one
$V(\phi)=V_{0}e^{-\beta\phi}$ and the power-law $V(\phi)=V_{0}\phi^{-k}$,
which are widely discussed in the literature. The corresponding critical
points and their behaviors for the obtained autonomous systems are presented
in Table (1) and (2). Moreover, for both cases, the asymptotic behavior of the
scale factor, the Hubble parameter, and the scalar field corresponding to the
obtained set of critical points have been presented. In addition, we presented
the 2-D phase-space diagrams corresponding to the exponential case for some
parameter values, specifically, $n=2$ and $n=-1$ with $\beta=1$ , and $n=-1$
with $\beta=\sqrt{3}$, in the Figure (1). The behavior of corresponding
cosmological parameters such as scalar field and dark energy density,
deceleration, and the effective equation of state parameter is presented in
Figure (2) and (3). Then we have discussed the physical significance and
stability of the obtained critical points corresponding to all three cases. We
found that the results obtained for the parameter constraints in Case III are
better among all three cases. We obtained evolutionary phase $A_{3}\rightarrow
C_{3}\rightarrow O_{3}$ representing the universe’s evolution from a
decelerated stiff era to an accelerated de-Sitter era via matter-dominated
epoch. Further, we have presented the 3-D phase-space diagram plotted for a
set of solutions to the autonomous system corresponding to the power-law case
for the parameter value $n=-2$ and $k=0.16$ (see Figure (4)) and the behavior
of corresponding cosmological parameters presented in the Figure (5). In this
case, all trajectories exhibit identical behavior, representing the universe’s
evolution from a decelerated stiff era to an accelerated de-Sitter era. Thus,
we can conclude that the exponential case shows better evolution than the
power-law case, and hence the present study successfully describes the
different cosmological epochs of the universe.

## Data availability

There are no new data associated with this article.

## Acknowledgments

SG acknowledges Council of Scientific and Industrial Research (CSIR),
Government of India, New Delhi, for junior research fellowship (File
no.09/1026(13105)/2022-EMR-I). RS acknowledges University Grants Commission
(UGC), New Delhi, India, for awarding a Senior Research Fellowship (UGC-Ref.
No.: 191620096030). PKS acknowledges the Science and Engineering Research
Board, Department of Science and Technology, Government of India for financial
support to carry out the Research project No.: CRG/2022/001847 and
Transilvania University of Brasov for Transilvania Fellowship for Visiting
Professors.

## References

  * [1] E. N. Saridakis et al., Modified Gravity and Cosmology: An Update by the CANTATA Network, Springer (2021).
  * [2] E. Abdalla et al., JHEAp 34, 49-211 (2022).
  * [3] J. M. Nester and H. J. Yo, Chin. J. Phys. 37, 113 (1999).
  * [4] J. B. Jiménez, L. Heisenberg, and T. Koivisto, Phys. Rev. D 98, 044048 (2018).
  * [5] B. J. Barros, T. Barreiro, T. Koivisto, and N. J. Nunes, Phys. Dark Univ. 30, 100616 (2020).
  * [6] F. K. Anagnostopoulos, S. Basilakos, E. N. Saridakis, Phys. Lett. B 822, 136634 (2021).
  * [7] O. Sokoliuk, S. Arora, S. Praharaj, A. Baransky, and P. K. Sahoo, MNRAS 522, 252-267 (2023).
  * [8] J. Ferreira, T. Barreiro, J. Mimoso, and N. J. Nunes, Phys. Rev. D 105, 123531 (2022).
  * [9] J. T. S. S. Junior and M. E. Rodrigues, Eur. Phys. J. C 83, 475 (2023).
  * [10] F. D’Ambrosio, S. D. B. Fell, L. Heisenberg, and S. Kuhn, Phys. Rev. D 105, 024042 (2022).
  * [11] L. Atayde and N. Frusciante, arXiv, arXiv:2306.03015 (2023).
  * [12] F. Bajardi and S. Capozziello, Eur. Phys. J. C 83, 531 (2023).
  * [13] N. Dimakis, A. Paliathanasis, and T. Christodoulakis, Class. Quantum Grav. 38, 225003 (2021).
  * [14] G. N. Gadbail, A. Kolhatkar, S. Mandal, and P. K. Sahoo, arXiv, arXiv:2304.10245 (2023).
  * [15] S. Capozziello and M. Shokri, Phys. Dark Univ. 37, 101113 (2022).
  * [16] A. Lymperis, JCAP 11, 018 (2022).
  * [17] S. Pradhan, S. Mandal, and P. K. Sahoo, Chin. Phys. C 47, 055103 (2023).
  * [18] Z. Hassan, S. Ghosh, P. K. Sahoo, and K. Bamba, Eur. Phys. J. C 82, 1116 (2022).
  * [19] S. Sahlu and E. Tsegaye, arXiv, arXiv:2206.02517 (2022).
  * [20] F. K. Anagnostopoulos, V. Gakis, E. N. Saridakis, and S. Basilakos, Eur. Phys. J. C 83, 58 (2023).
  * [21] G. Subramaniam, A. De, T. H. Loo, and Y. K. Goh, Fortschr. Phys. 2023, 2300038 (2023).
  * [22] G. Subramaniam, A. De, T. H. Loo, and Y. K. Goh, Phys. Dark Univ. 41, 101243 (2023).
  * [23] N. Dimakis, M. Roumeliotis, A. Paliathanasis, P. S. Apostolopoulos, and T. Christodoulakis, Phys. Rev. D 106, 123516 (2022).
  * [24] M. Hohmann, Phys. Rev. D 104, 124077 (2021).
  * [25] W. Khyllep, A. Paliathanasis, and J. Dutta, Phys. Rev. D 103, 103521 (2021).
  * [26] A. H. Guth, Phys. Rev. D 23, 347-356 (1981).
  * [27] Y. B. Zeldovich, Sov. phys., Usp. 11, (1968).
  * [28] S. Weinberg, Rev. Mod. Phys. 61, 1 (1989).
  * [29] B. Ratra and P. J. E. Peebles, Phys. Rev. D 37, 3406 (1988).
  * [30] C. Rubano and P. Scudellaro, Gen. Rel. Grav 34, 307-328 (2002).
  * [31] Y. F. Cai, E. N. Saridakis, M. R. Setare, and J. Q. Xia, Phys. Rept. 493, 1-60 (2010).
  * [32] Z. K. Guo, Y. S. Piao, X. M. Zhang, and Y. Z. Zhang, Phys. Lett. B 608, 177-182 (2005).
  * [33] L. A. Urena-Lopez, JCAP 09, 013 (2005).
  * [34] E. Elizalde, S. Nojiri, and S. D. Odintsov, Phys. Rev. D 70, 043539 (2004).
  * [35] N. Aghanim et. al, A & bA 641, A6 (2020).
  * [36] A. G. Riess, S. Casertano, W. Yuan, L. M. Macri, and D. Scolnic, Astrophys. J. 876, 85 (2019).
  * [37] E. J. Copeland, A. R. Liddle, and D. Wands, Phys. Rev. D 57, 4686-4690 (1998).
  * [38] H. Shabani, Avik De, T. H. Loo, arXiv, arxiv:2304.02949 (2023).
  * [39] W. Khyllep, J. Dutta, E. N. Saridakis, and K. Yesmakhanova, Phys. Rev. D 107, 044022 (2023).
  * [40] L. K. Duchaniya, S. V. Lohakare, B. Mishra, and S. K. Tripathy, Eur. Phys. J. C 82, 448 (2022).
  * [41] S. A. Kadam, B. Mishra, and J. L. Said, Eur. Phys. J. C 82, 680 (2022).
  * [42] H. Shabani and M. Farhoudi, Phys. Rev. D 88, 044048 (2013).
  * [43] J. B. Jiménez, L. Heisenberg, and T. S. Koivisto, Universe 5, 173 (2019).
  * [44] F. D’Ambrosio, L. Heisenberg, and S. Kuhn, Class. Quantum Grav. 39, 025013 (2021).
  * [45] S. Bahamonde, C. G. Bohmer, S. Carloni, E. J. Copeland, W. Fang, and N. Tamanini, Phys. Rept. 775, 1-122 (2018).
  * [46] J. B. Jiménez, L. Heisenberg, T. S. Koivisto, and S. Pekar, Phys. Rev. D 101, 103507 (2020).
  * [47] L. A. Salo and J. Haro, Eur. Phys. J. C  81, 105 (2021).
  * [48] T. Barreiro, E. J. Copeland, and N. J. Nunes, Phys. Rev. D 61, 127301 (2000).
  * [49] D. Baumann and L. McAllister, Inflation and String Theory (Cambridge Monographs on Mathematical Physics, Cambridge University Press 2015).
  * [50] J. Yearsley and J. D Barrow, Class. Quantum Grav. 13, 2693 (1996).
  * [51] B. Ratra and P. J. E. Peebles, Phys. Rev. D 37, 3406 (1988).
  * [52] P. J. E. Peebles and B. Ratra, Astrophys. J. Lett. 325, L17 (1988).
  * [53] A. Sen, Mod. Phys. Lett. A. 17, 1797-1804 (2002).
  * [54] W. Fang, H. Q. Lu and Z. G. Huang, Int J Mod Phys A. 22, 2173-2195 (2007).
  * [55] L. R. Abramo and F. Finelli, Phys. Lett. B 575, 165-171 (2003).


# Two-weight rank-metric codes

Ferdinando Zullo1, Olga Polverino1, Paolo Santonastaso1 and John Sheekey2

Corresponding author: F. Zullo (email<EMAIL_ADDRESS>1Dipartimento di Matematica e Fisica, Università degli
Studi della Campania “Luigi Vanvitelli", Caserta, Italy  2School of
Mathematics and Statistics,
University College Dublin, Dublin, Ireland

###### Abstract

Two-weight linear codes are linear codes in which any nonzero codeword can
have only two possible distinct weights. Those in the Hamming metric have
proven to be very interesting for their connections with authentication codes,
association schemes, strongly regular graphs, and secret sharing schemes. In
this paper, we characterize two-weight codes in the rank metric, answering a
recent question posed by Pratihar and Randrianarisoa.

##  I Introduction

Rank-metric codes have garnered significant attention in recent decades, owing
to their applications and intriguing connections with various mathematical
objects, such as in linear random network coding, see [5]. Nevertheless, the
origin of rank-metric codes traces its roots to Delsarte [14] in 1978, and
later to Gabidulin in [15] and Roth in [24].

Most of the focus on the study of rank-metric codes has regarded the linear
case, that is ${\mathbb{F}}_{q^{m}}$-subspaces of ${\mathbb{F}}_{q^{m}}^{n}$,
where ${\mathbb{F}}_{q^{m}}^{n}$ is equipped with the rank distance, defined
as follows: for any
$x=(x_{1},\ldots,x_{n}),y=(y_{1},\ldots,y_{n})\in{\mathbb{F}}_{q^{m}}^{n}$
then

$d(x,y)=w(x-y)=\dim_{\mathbb{F}_{q}}(\langle
x_{1}-y_{1},\ldots,x_{n}-y_{n}\rangle_{\mathbb{F}_{q}}),$

where $w(x-y)$ denotes the rank weight of $x-y$.

In this paper, a rank-metric code is any ${\mathbb{F}}_{q^{m}}$-subspace of
${\mathbb{F}}_{q^{m}}^{n}$. Let
$\operatorname{\mathcal{C}}\subseteq{\mathbb{F}}_{q^{m}}^{n}$ be a rank-metric
code. We will write that $\operatorname{\mathcal{C}}$ is an
$[n,k,d]_{q^{m}/q}$ code if $k$ is the ${\mathbb{F}}_{q^{m}}$-dimension of
$\operatorname{\mathcal{C}}$ and $d$ is its minimum distance, that is

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

When the minimum distance is not known or is irrelevant, we simply write
$[n,k]_{q^{m}/q}$.

It is possible to prove a Singleton-like bound for a rank-metric code.

######  Theorem I.1.

(see [14]) Let $\operatorname{\mathcal{C}}\subseteq{\mathbb{F}}_{q^{m}}^{n}$
be an $[n,k,d]_{q^{m}/q}$ code. Then

$mk\leq\max\\{m,n\\}(\min\\{n,m\\}-d+1).$ (1)

An $[n,k,d]_{q^{m}/q}$ code is called Maximum Rank Distance code (or shortly
MRD code) if its parameters attains the bound (1). A family of MRD codes has
been shown in [14, 15, 24].

In the classical Hamming-metric setting, well-studied codes are those such
that the weight of its nonzero elements can only assume two distinct values;
see the well celebrated paper by Calderbank and Kantor [10]. These codes
present a wide variety of properties, and their geometry can be very
different, although some classification results have been proved; see e.g.
[16].

In this paper we analyze rank-metric codes $\operatorname{\mathcal{C}}$ which
are two-weight codes, that is the rank of its nonzero elements can only assume
two distinct values. An example of two-weight code in the rank metric is given
by MRD codes in ${\mathbb{F}}_{q^{m}}^{n}$ with $n\leq m$ and with minimum
distance $n-1$. The study of two-weight rank-metric codes was initiated by
Pratihar and Randrianarisoa in [22], considering _antipodal_ two-weight rank-
metric codes $\operatorname{\mathcal{C}}$. An $[n,k]_{q^{m}/q}$ code is an
antipodal code if there exists a codeword of weight $n$. They ask whether or
not all of two-weight antipodal codes _arise_ from MRD codes. Using a
geometric description of rank-metric codes developed in [23] and also in [25]
and by a clever characterization in terms of spreads, they prove a complete
classification of antipodal two-weight rank-metric codes when the minimum rank
distance is equal to half of the length. More precisely, they proved that such
a code is always induced by an MRD code over an extension of $\mathbb{F}_{q}$
(see [22] and [21]). This proved that in this case, two-weight codes in the
rank metric have a strong geometric structure.

In this paper we give a more general geometric characterisation of antipodal
two-weight codes, proving that, when $q$ is large enough, all the two-weight
rank-metric codes arise from well-studied combinatorial objects known as
_scattered spaces_. Indeed, in Section IV, we prove that this is the case for
all (not necessarily antipodal) two-weight codes. Moreover, using the lower
bound on the dimension of scattered subspaces and their existence results, we
are able to give a lower bound on the length of a two-weight code and a range
of values for its length in which we can guarantee the existence of such
codes.

##  II Preliminaries

In this section we briefly recall some definitions and results we will need in
the paper.

###  II-A Linear sets

Let $V$ be a $k$-dimensional vector space over ${\mathbb{F}}_{q^{m}}$ and let
$\Lambda=\mathrm{PG}(V,{\mathbb{F}}_{q^{m}})=\mathrm{PG}(k-1,q^{m})$. Recall
that, if $U$ is an $\mathbb{F}_{q}$-subspace of $V$ of dimension $n$, then the
set of points

$L_{U}=\\{\langle{u}\rangle_{\mathbb{F}_{q^{m}}}:{u}\in
U\setminus\\{{0}\\}\\}\subseteq\Lambda$

is said to be an $\mathbb{F}_{q}$-linear set of rank $n$. Let $P=\langle
v\rangle_{{\mathbb{F}}_{q^{m}}}$ be a point in $\Lambda$. The weight of $P$ in
$L_{U}$ is defined as

$w_{U}(P)=\dim_{\mathbb{F}_{q}}(U\cap\langle
v\rangle_{{\mathbb{F}}_{q^{m}}}).$

If $N_{i}$ denotes the number of points of $\Lambda$ having weight
$i\in\\{0,\ldots,n\\}$ in $L_{U}$, the following relations hold:

$|L_{U}|\leq\frac{q^{n}-1}{q-1},$ (2) $|L_{U}|=N_{1}+\ldots+N_{n},$ (3)
$N_{1}+N_{2}(q+1)+\ldots+N_{n}(q^{n-1}+\ldots+q+1)=q^{n-1}+\ldots+q+1.$ (4)

Furthermore, $L_{U}$ is called scattered if it has the maximum number
$\frac{q^{n}-1}{q-1}$ of points, or equivalently, if all points of $L_{U}$
have weight one. Moreover, for a positive integer $e\mid m$, with $e<m$, we
say that an ${\mathbb{F}}_{q^{e}}$-subspace $U$ of $V$ is an
${\mathbb{F}}_{q^{e}}$-scattered subspace of $V$ if

$\dim_{{\mathbb{F}}_{q^{e}}}(U\cap\langle v\rangle_{{\mathbb{F}}_{q^{m}}})\leq
1,$

for any $v\in V$. Blokhuis and Lavrauw provided the following bound on the
rank of a scattered liner set.

######  Theorem II.1.

(see [7]) Let $L_{U}$ be a scattered $\mathbb{F}_{q}$-linear set of rank $n$
in $\mathrm{PG}(k-1,q^{m})$, then

$n\leq\frac{mk}{2}.$

A scattered $\mathbb{F}_{q}$-linear set of rank $km/2$ in
$\mathrm{PG}(k-1,q^{m})$ is said to be a maximum scattered and $U$ is said to
be a maximum scattered $\mathbb{F}_{q}$-subspace as well.

In the next result we summarize what is known on the existence of maximum
scattered linear sets/subspaces.

######  Theorem II.2.

(see [7, 3, 4, 12]) If $mk$ is even then there exist maximum scattered
subspaces in ${\mathbb{F}}_{q^{m}}^{k}$.

If $L_{U}$ is a scattered $\mathbb{F}_{q}$-linear set for which $L_{U}$ is not
contained in another scattered $\mathbb{F}_{q}$-linear set is said to be
maximally scattered and $U$ is called a maximally scattered subspace. Clearly,
maximum scattered linear sets are maximally scattered, but the converse does
not hold in general; see e.g. [18].

Now, we recall the notion of the dual of a linear set. Let $\sigma\colon
V\times V\rightarrow{\mathbb{F}}_{q^{m}}$ be a nondegenerate reflexive
bilinear form on the $k$-dimensional ${\mathbb{F}}_{q^{m}}$-vector space $V$
and consider

$\begin{array}[]{cccc}\sigma^{\prime}:&V\times
V&\longrightarrow&{\mathbb{F}}_{q}\\\
&(x,y)&\longmapsto&\mathrm{Tr}_{q^{m}/q}(\sigma(x,y)).\end{array}$

So, $\sigma^{\prime}$ is a nondegenerate reflexive bilinear form on $V$ seen
as an $\mathbb{F}_{q}$-vector space of dimension $km$. Then we may consider
$\perp$ and $\perp^{\prime}$ as the orthogonal complement maps defined by
$\sigma$ and $\sigma^{\prime}$. For an ${\mathbb{F}}_{q}$-linear set in
$\mathrm{PG}(V,{\mathbb{F}}_{q^{m}})$ of rank $n$, the
${\mathbb{F}}_{q}$-linear set $L_{U^{\perp^{\prime}}}$ in
$\mathrm{PG}(V,{\mathbb{F}}_{q^{m}})$ of rank $km-n$ is, called the dual
linear set of $L_{U}$ with respect to $\sigma$. The definition of the dual
linear set does not depend on the choice of $\sigma$; see e.g. [20,
Proposition 2.5].

Moreover, we have the following relation between the weight of a subspace with
respect to a linear set and the weight of its polar space with respect to the
dual linear set.

######  Proposition II.3.

(see [20, Property 2.6]) Let $L_{U}\subseteq\mathrm{PG}(k-1,q^{m})$ be a
linear set of rank $n$. Let $\Omega_{s}=\mathrm{PG}(W,{\mathbb{F}}_{q^{m}})$
be an $s$-dimensional projective space of $\mathrm{PG}(k-1,q^{m})$. Then

$w_{U^{\perp^{\prime}}}(\Omega_{s}^{\tau})=w_{U}(\Omega_{s})+km-n-(s+1)m,$

i.e.

$\dim_{\mathbb{F}_{q}}(U^{\perp^{\prime}}\cap
W^{\perp})=\dim_{\mathbb{F}_{q}}(U\cap W)+\dim_{\mathbb{F}_{q}}(V)$
$-\dim_{\mathbb{F}_{q}}(U)-\dim_{\mathbb{F}_{q}}(W).$

We refer to [19] and [20] for comprehensive references on linear sets.

###  II-B Rank-metric codes and $q$-systems

Now, we recall the definition of equivalence between rank-metric codes in
${\mathbb{F}}_{q^{m}}^{n}$. An ${\mathbb{F}}_{q^{m}}$-linear isometry $\phi$
of ${\mathbb{F}}_{q^{m}}^{n}$ is an ${\mathbb{F}}_{q^{m}}$-linear map of
${\mathbb{F}}_{q^{m}}^{n}$ that preserves the rank weight, i.e.
$w(x)=w(\phi(x))$, for every $x\in{\mathbb{F}}_{q^{m}}^{n}$, or equivalently
$d(x,y)=d(\phi(x),\phi(y))$, for every $x,y\in{\mathbb{F}}_{q^{m}}^{n}$. It is
known that the group of ${\mathbb{F}}_{q^{m}}$-linear isometries of
${\mathbb{F}}_{q^{m}}^{n}$ equipped with rank distance is generated by the
(nonzero) scalar multiplications of ${\mathbb{F}}_{q^{m}}$ and the general
linear group $\mathrm{GL}(n,{\mathbb{F}}_{q})$, see e.g. [6]. So we say that
two rank-metric codes
$\operatorname{\mathcal{C}},\operatorname{\mathcal{C}}^{\prime}\subseteq{\mathbb{F}}_{q^{m}}^{n}$
are (linearly) equivalent if there exists an isometry $\phi$ such that
$\phi(\operatorname{\mathcal{C}})=\operatorname{\mathcal{C}}^{\prime}$.
Clearly, when studying equivalence of $[n,k]_{q^{m}/q}$ codes the action of
${\mathbb{F}}_{q^{m}}^{*}$ is trivial. This means that two $[n,k]_{q^{m}/q}$
codes $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{C}}^{\prime}$
are equivalent if and only if there exists $A\in\mathrm{GL}(n,q)$ such that
$\operatorname{\mathcal{C}}^{\prime}=\operatorname{\mathcal{C}}A=\\{vA:v\in\operatorname{\mathcal{C}}\\}$.
We will consider codes that are _nondegenerate_.

######  Definition II.4.

An $[n,k]_{q^{m}/q}$ code $\operatorname{\mathcal{C}}$ is said to be
nondegenerate if the columns of any generator matrix of
$\operatorname{\mathcal{C}}$ are $\mathbb{F}_{q}$-linearly independent.

The geometric counterpart of rank-metric are the systems.

######  Definition II.5.

An $[n,k,d]_{q^{m}/q}$ system $U$ is an ${\mathbb{F}}_{q}$-subspace of
${\mathbb{F}}_{q^{m}}^{k}$ of dimension $n$, such that $\langle
U\rangle_{{\mathbb{F}}_{q^{m}}}={\mathbb{F}}_{q^{m}}^{k}$ and

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

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

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

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

######  Theorem II.6.

(see [23]) Let $\operatorname{\mathcal{C}}$ be an $[n,k,d]_{q^{m}/q}$ rank-
metric code and let $G$ be an its generator matrix. Let
$U\subseteq{\mathbb{F}}_{q^{m}}^{k}$ be the ${\mathbb{F}}_{q}$-span of the
columns of $G$. The rank weight of an element
$xG\in\operatorname{\mathcal{C}}$, with $x\in{\mathbb{F}}_{q^{m}}^{k}$ is

$w(xG)=n-\dim_{\mathbb{F}_{q}}(U\cap x^{\perp}),$ (5)

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

$d=n-\max\left\\{\dim_{\mathbb{F}_{q}}(U\cap H)\colon H\mbox{ is an
}{\mathbb{F}}_{q^{m}}\mbox{-hyperpl. of }{\mathbb{F}}_{q^{m}}^{k}\right\\}.$
(6)

Thanks to Theorem II.6, it can be proved that there is a one-to-one
correspondence between equivalence classes of $[n,k,d]_{q^{m}/q}$ codes and
equivalence classes of $[n,k,d]_{q^{m}/q}$ systems; see [2, 23].

Finally, we recall that by using the dual of a system associated to a code, it
is possible to define a new code as follows, see [9].

######  Definition II.7.

Let $\operatorname{\mathcal{C}}$ be an $[n,k,d]_{q^{m}/q}$ and let $U$ be a
system associated with $\operatorname{\mathcal{C}}$. Assume that
$\dim_{{\mathbb{F}}_{q}}(U\cap\langle v\rangle_{{\mathbb{F}}_{q^{m}}})<m$, for
any $v\in{\mathbb{F}}_{q^{m}}^{k}$. Then a geometric dual of
$\operatorname{\mathcal{C}}$ (with respect to $\perp^{\prime}$) is defined as
$\operatorname{\mathcal{C}}^{\prime}$, where
$\operatorname{\mathcal{C}}^{\prime}$ is any code associated with the system
$U^{\perp^{\prime}}$.

Note that the geometric dual of an $[n,k]_{q^{m}/q}$ code is an
$[mk-n,k]_{q^{m}/q}$ code.

##  III Auxiliary results on linear sets

The problem of determining bounds on the size of a linear set is a hard
problem. The following is a lower bound on the number of points of a linear
set admitting a point of weight one (which generalizes [8, Lemma 2.2]).

######  Theorem III.1.

(see [13, Theorem 1.2]) Let $L_{U}$ be an ${\mathbb{F}}_{q}$-linear set in
$\mathrm{PG}(1,q^{m})$ of rank $n$, such that there exists $P\in L_{U}$ with
$w_{U}(P)=1$. Then

$|L_{U}|\geq q^{n-1}+1.$

We also refer to [1] for more bounds in higher dimensional projective spaces.
We also recall the following recent result, which regards the linearity of a
linear set.

######  Theorem III.2.

(see [11, Theorem 2]) Let $L_{U}$ be an ${\mathbb{F}}_{q}$-linear set of rank
$n$ in $\mathrm{PG}(k-1,q^{m})$ with $n\leq(k-1)m$ and $m\leq q$. Let
$e=\min\\{w_{U}(P)\colon P\in L_{U}\\}$ and suppose $e>1$. Then there exists a
positive integer $t\geq e$ such that $t\lvert m$ and $L_{U}=L_{W}$ where
$W=\langle U\rangle_{{\mathbb{F}}_{q^{t}}}$.

In the following we use the above mentioned results to prove that if an
$\mathbb{F}_{q}$-subspace $U$ of ${\mathbb{F}}_{q^{m}}^{2}$ meets every one-
dimensional ${\mathbb{F}}_{q^{m}}$-subspace of ${\mathbb{F}}_{q^{m}}^{2}$
either trivially or in an $\mathbb{F}_{q}$-subspace of dimension $e>0$, then
$e\mid m$ and $U$ is ${\mathbb{F}}_{q^{e}}$-linear, as well.

######  Corollary III.3.

Assume that $q\geq m$. Let $U$ be an $n$-dimensional $\mathbb{F}_{q}$-subspace
of a $k$-dimensional vector space $V$ over ${\mathbb{F}}_{q^{m}}$. Suppose
that $n\leq(k-1)m$, $\langle U\rangle_{{\mathbb{F}}_{q^{m}}}=V$ and

$\dim_{\mathbb{F}_{q}}(U\cap\langle
w\rangle_{{\mathbb{F}}_{q^{m}}})\in\\{0,e\\},$

for any $w\in V$, then $e\mid n$, $e\mid m$ and $U$ is an
${\mathbb{F}}_{q^{e}}$-subspace of dimension $n/e$. In other words, $U$ is an
${\mathbb{F}}_{q^{e}}$-scattered subspace of $V$.

###### Proof.

The assertion is clear if $e=1$. So suppose that $1<e<m$. First, assume that
$k=2$. Let consider the $\mathbb{F}_{q}$-linear set $L_{U}$ which, by the
assumption on $U$, has size $\frac{q^{n}-1}{q^{e}-1}$ and so $e\mid n$ with
$e<n$. By Theorem III.2, it follows that there exists a positive integer
$t\geq e$ such that $t\lvert m$ and such that $L_{U}=L_{W}$, where $W=\langle
U\rangle_{{\mathbb{F}}_{q^{t}}}$. Denote by
$n^{\prime}=\dim_{{\mathbb{F}}_{q}}(W)=tg$ and note that $n^{\prime}\geq n$
and $t<n^{\prime}$ (i.e. $g>1$). We may also assume that there exists a point
$P=\langle u\rangle_{{\mathbb{F}}_{q^{m}}}\in\mathrm{PG}(1,q^{m})$ such that

$w_{W}(P)=\dim_{{\mathbb{F}}_{q^{t}}}(W\cap\langle
u\rangle_{{\mathbb{F}}_{q^{m}}})=1,$

where $L_{W}$ is seen as an ${\mathbb{F}}_{q^{t}}$-linear set, otherwise we
may apply Theorem III.2 again to $L_{W}$. In particular, applying Theorem
III.1 and the bound (2) we obtain

$(q^{t})^{g-1}+1\leq\frac{q^{n}-1}{q^{e}-1}\leq\frac{q^{gt}-1}{q^{t}-1}.$ (7)

From the former bound in (7) we obtain

$q^{t(g-1)+e}+q^{e}\leq q^{n}+q^{t(g-1)},$

which implies that $n\geq t(g-1)+e$, that is

$n-e\geq n^{\prime}-t,$ (8)

and also

$n+t\geq n^{\prime}+e\geq n^{\prime}+2.$ (9)

By the latter bound in (7) we obtain

$q^{n+t}+q^{e}+q^{n^{\prime}}\leq q^{n^{\prime}+e}+q^{t}+q^{n}.$ (10)

Taking into account (9) and that $n^{\prime}+e\geq\max\\{n^{\prime}+2,t\\}$,
it follows that $n^{\prime}+e\geq n+t$, that, together with (8), implies that

$n-e=n^{\prime}-t.$

As a consequence, since by (9) $n\geq t(g-1)+2\geq t$, (10) reads as

$q^{e}+q^{n^{\prime}}\leq q^{t}+q^{n},$

implies that $n^{\prime}\leq n$ and hence the equality and $t=e$. Since $U$ is
contained in $W$ and have the same dimension, we have $U=W$ and hence $U$ is
an ${\mathbb{F}}_{q^{e}}$-subspace. So the assertion is proved for $k=2$. Now,
assume that $k>2$. For any two vectors $u,v\in U$ which are not proportional
over ${\mathbb{F}}_{q^{m}}$ and consider

$U^{\prime}_{u,v}=\langle u,v\rangle_{{\mathbb{F}}_{q^{m}}}\cap U.$

Then $U^{\prime}_{u,v}$ is an $\mathbb{F}_{q}$-subspace with the property that

$\dim_{{\mathbb{F}}_{q}}(U^{\prime}_{u,v}\cap\langle
w\rangle_{{\mathbb{F}}_{q^{m}}})\in\\{0,e\\},$

for any nonzero $w\in\langle u,v\rangle_{{\mathbb{F}}_{q^{m}}}$. Applying the
case $k=2$ to $U^{\prime}_{u,v}$, from which we obtain that $e\mid m$ and
$U^{\prime}_{u,v}$ is an ${\mathbb{F}}_{q^{e}}$-subspace of $V$ for every
${\mathbb{F}}_{q^{m}}$-linearly independent $u,v\in U$ and hence $U$ is an
${\mathbb{F}}_{q^{e}}$-subspace as well. ∎

The above result cannot be extended to ${\mathbb{F}}_{q}$-subspaces meeting
every one-dimensional ${\mathbb{F}}_{q^{m}}$-subspaces in
$\mathbb{F}_{q}$-subspaces of dimension a multiple of a fixed integer $e$.

######  Remark III.4.

Let $\lambda\in{\mathbb{F}}_{q^{4}}$ such that
${\mathbb{F}}_{q}(\lambda)={\mathbb{F}}_{q^{4}}$. Let $S_{1}=\langle
1,\lambda\rangle_{{\mathbb{F}}_{q}}$. Let $S_{2}$ be a $3$-dimensional
${\mathbb{F}}_{q^{4}}$-subspace of ${\mathbb{F}}_{q^{16}}$ and consider

$U=S_{1}\times S_{2}=\\{(s_{1},s_{2}):s_{1}\in S_{1},s_{2}\in
S_{2}\\}\subseteq{\mathbb{F}}_{q^{16}}^{2}.$

Then $U$ is a $14$-dimensional ${\mathbb{F}}_{q}$-subspace of
${\mathbb{F}}_{q^{16}}^{2}$ and note that $U$ cannot be also an
${\mathbb{F}}_{q^{i}}$-subspace for some $i>1$. Moreover,

$\dim_{\mathbb{F}_{q}}(U\cap\langle
w\rangle_{{\mathbb{F}}_{q^{16}}})\in\\{0,2,12\\},$

for every $w\in{\mathbb{F}}_{q^{16}}^{2}$. Therefore, $U$ is an
$\mathbb{F}_{q}$-subspace meeting every one-dimensional
${\mathbb{F}}_{q^{16}}$-subspace of ${\mathbb{F}}_{q^{16}}^{2}$ in a subspace
of dimension a multiple of $2$ but $U$ is strictly $\mathbb{F}_{q}$-linear.

##  IV Two-weight linear rank-metric codes

In this section we completely describe two-weight codes. A way to construct
two-weight codes is via scattered subspaces.

######  Construction IV.1.

Let $U$ be an ${\mathbb{F}}_{q^{e}}$-scattered subspace in
${\mathbb{F}}_{q^{m}}^{k}$ such that $\dim_{{\mathbb{F}}_{q}}(U)=n$ and
$\langle U\rangle_{{\mathbb{F}}_{q^{m}}}={\mathbb{F}}_{q^{m}}^{k}$. A code
$\operatorname{\mathcal{C}}$ having $U$ as an its associated system is said to
be an ${\mathbb{F}}_{q^{e}}$-scattered code.

Note that, when $k=2$, an ${\mathbb{F}}_{q^{e}}$-scattered code
$\operatorname{\mathcal{C}}$ is an $[n,2,n-e]_{q^{m}/q}$ code with nonzero
weights $n-e$ and $n$. In addition, when $e=1$, $\operatorname{\mathcal{C}}$
is MRD for any value of $n\leq m$. The above construction can be extended via
duality as follows.

######  Construction IV.2.

Let $U$ be an ${\mathbb{F}}_{q^{e}}$-scattered subspace in
${\mathbb{F}}_{q^{m}}^{k}$ such that $\dim_{{\mathbb{F}}_{q}}(U)=n$ and
$\langle U\rangle_{{\mathbb{F}}_{q^{m}}}={\mathbb{F}}_{q^{m}}^{k}$. Consider
$U^{\perp^{\prime}}$ the dual of $U$. Then
$\dim_{{\mathbb{F}}_{q}}(U^{\perp^{\prime}})=km-n$, $\langle
U^{\perp^{\prime}}\rangle_{{\mathbb{F}}_{q^{m}}}={\mathbb{F}}_{q^{m}}^{k}$ and
by Proposition II.3

$\dim_{\mathbb{F}_{q}}(U^{\perp^{\prime}}\cap H)\in\\{(k-1)m-n,(k-1)m-n+e\\}.$

Consider $\operatorname{\mathcal{C}}$ the rank-metric code associated with
$U^{\perp^{\prime}}$, then $\operatorname{\mathcal{C}}$ is a nondegenerate
$[km-n,k,m-e]_{q^{m}/q}$ code with nonzero weights $m-e$ and $m$. In other
words, the geometric dual of an ${\mathbb{F}}_{q^{e}}$-scattered
$[n,k]_{q^{m}/q}$ code is an $[km-n,k,m-e]_{q^{m}/q}$ with nonzero weights
$m-e$ and $m$.

######  Remark IV.3.

For an ${\mathbb{F}}_{q^{e}}$-scattered subspace $U$ in
${\mathbb{F}}_{q^{m}}^{k}$ such that $\dim_{{\mathbb{F}}_{q}}(U)=n$ and
$\langle U\rangle_{{\mathbb{F}}_{q^{m}}}={\mathbb{F}}_{q^{m}}^{k}$, it results
that $\langle
U^{\perp^{\prime}}\rangle_{{\mathbb{F}}_{q^{m}}}={\mathbb{F}}_{q^{m}}^{k}$.
Indeed, suppose that $U^{\perp^{\prime}}$ is contained in a hyperplane of
${\mathbb{F}}_{q^{m}}^{k}$, i.e. $\langle
U^{\perp^{\prime}}\rangle_{{\mathbb{F}}_{q^{m}}}\subseteq H$. This means that
$H^{\perp}\subseteq U$ and so $\dim_{{\mathbb{F}}_{q}}(H^{\perp}\cap U)=m>e$,
a contradiction.

In Construction IV.2, it can happen that the obtained codes are antipodal, for
instance when $m=n$ and $k=2$.

######  Remark IV.4.

The $[km-n,k,m-e]_{q^{m}/q}$ code $\operatorname{\mathcal{C}}$ from
Construction IV.2 is an MRD code if and only if

$|\operatorname{\mathcal{C}}|=q^{\max\\{m,km-n\\}(\min\\{m,km-n\\}-d+1)},$

which can be read as

$km=(km-n)(e+1),$ (11)

since $n\leq km/2$ via the bound of Theorem II.1. Again, by II.1, (11) implies
that

$km=(km-n)(e+1)\geq\frac{km}{2}(e+1),$

from which $e=1$. Hence, from (11) it follows that $n=\frac{km}{2}$.

We will start by characterizing the antipodal two weight codes.

Our approach now is to use the geometric correspondence between rank-metric
codes and systems of ${\mathbb{F}}_{q^{m}}^{2}$. We will start by recalling
some results for the antipodal case from [22].

######  Proposition IV.5.

Let $\operatorname{\mathcal{C}}$ be a nondegenerate antipodal two-weight
$[n,k,d]_{q^{m}/q}$ code. Then $n\leq m$.

######  Theorem IV.6.

(see [22, Theorem 2]) Let $\operatorname{\mathcal{C}}$ be a nondegenerate
$[n,k,d]_{q^{m}/q}$ code. If $\operatorname{\mathcal{C}}$ is an antipodal two-
weight code then $k=2$.

######  Lemma IV.7.

(see [22, Corollary 1]) Let $\operatorname{\mathcal{C}}$ be a nondegenerate
$[n,2,d]_{q^{m}/q}$ code. If $\operatorname{\mathcal{C}}$ is an antipodal two-
weight code then $n-d\mid n$.

Now, we are able to characterize antipodal codes, extending [22].

######  Theorem IV.8.

Assume that $q\geq m.$ Let $\operatorname{\mathcal{C}}$ be a nondegenerate
antipodal two-weight $[n,2,d]_{q^{m}/q}$ code with $n=\ell(n-d)$. Then
$n-d\mid m$ and $\operatorname{\mathcal{C}}$ is an
${\mathbb{F}}_{q^{n-d}}$-scattered code.

###### Proof.

Let $U\subseteq\mathbb{F}_{q^{m}}^{2}$ be a system associated with the code
$\operatorname{\mathcal{C}}$ and

$\dim_{\mathbb{F}_{q}}(U\cap\langle
v\rangle_{\mathbb{F}_{q^{m}}}^{\perp})\in\\{0,n-d\\},$

for any $v\in\mathbb{F}_{q^{m}}^{2}$. Note that
$\dim_{\mathbb{F}_{q}}(U)=\ell(n-d)$. By Corollary III.3, it follows that
$n-d$ divides $m$ and $U$ is an ${\mathbb{F}}_{q^{n-d}}$-subspace of
${\mathbb{F}}_{q^{m}}^{2}$ with the property that

$\dim_{{\mathbb{F}}_{q^{n-d}}}(U\cap\langle
w\rangle_{{\mathbb{F}}_{q^{m}}})\leq 1,$

for any $w\in{\mathbb{F}}_{q^{m}}^{2}$ and hence by Theorem II.6 the assertion
follows. ∎

This answers to the question posed by Pratihar and Randrianarisoa in [22],
proving that every antipodal two-weight code is induced by an MRD code over an
extension field of ${\mathbb{F}}_{q}$.

Moreover, this result allows us to completely characterize
${\mathbb{F}}_{q^{m}}$-linear rank-metric codes with two distinct weights.

######  Remark IV.9.

Note that, since a nondegenerate $[n,k]_{q^{m}/q}$ code always has at least
one codeword having rank weight equal to $\min\\{m,n\\}$ (see [2, Proposition
3.11]), a nondegenerate two-weight $[n,k,d]_{q^{m}/q}$ code admits nonzero
weights $d$ and $\min\\{m,n\\}$.

Let us start with the case $k=2$.

######  Corollary IV.10.

Assume that $q\geq m$. Let $\operatorname{\mathcal{C}}$ be a nondegenerate
two-weight $[n,2,d]_{q^{m}/q}$ code. Suppose that $\operatorname{\mathcal{C}}$
is not antipodal. Then $m-d\mid m$ and $\operatorname{\mathcal{C}}$ is a
geometric dual of an ${\mathbb{F}}_{q^{m-d}}$-scattered
$[2m-n,2,m-n+d]_{q^{m}/q}$ code.

###### Proof.

Let $U$ be the system associated with $\operatorname{\mathcal{C}}$. Let $d$
and $d^{\prime}$ be the two nonzero weights of $\operatorname{\mathcal{C}}$.
By Remark IV.9, we know that $d^{\prime}=m$ and $m<n$, since
$\operatorname{\mathcal{C}}$ is not antipodal. Then by Theorem II.6 we obtain
that

$\dim_{\mathbb{F}_{q}}(U\cap\langle
v\rangle_{{\mathbb{F}}_{q^{m}}})\in\\{n-d,n-m\\},$

and $n-d,n-m\neq 0$. The dual $U^{\perp^{\prime}}$ of $U$ has dimension
$2m-n<m$ and

$\dim_{\mathbb{F}_{q}}(U^{\perp^{\prime}}\cap\langle
v\rangle_{{\mathbb{F}}_{q^{m}}})\in\\{0,m-d\\}.$

We can then use Corollary III.3, obtaining that $U^{\perp^{\prime}}$ is
${\mathbb{F}}_{q^{m-d}}$-linear, i.e. $U^{\perp^{\prime}}$ is a scattered
${\mathbb{F}}_{q^{m-d}}$-subspace. ∎

We can now extend the above result to the remaining dimensions.

######  Theorem IV.11.

Assume that $q\geq m$. Let $\operatorname{\mathcal{C}}$ be a nondegenerate
two-weight $[n,k,d]_{q^{m}/q}$ code with $k>2$. Then $m-d\mid m$, $m-d\mid n$
and a system $U$ associated with $\operatorname{\mathcal{C}}$ is the dual of
an ${\mathbb{F}}_{q^{m-d}}$-scattered subspace in ${\mathbb{F}}_{q^{m}}^{k}$.
In particular, $n\geq\frac{mk}{2}$. Moreover,

$\frac{km}{2(m-d)}\leq\frac{n}{m-d}\leq\frac{mk}{m-d}-1.$

###### Proof.

By Remark IV.9, we know that the nonzero weights of
$\operatorname{\mathcal{C}}$ are $d$ and $m=\min\\{m,n\\}$, and $m<n$, since
$\operatorname{\mathcal{C}}$ is not antipodal (otherwise $k=2$). Let
$U\subseteq\mathbb{F}_{q^{m}}^{k}$ be a system associated with the code
$\operatorname{\mathcal{C}}$. Then by Theorem II.6 we obtain that
$\dim_{\mathbb{F}_{q}}(U\cap H)\in\\{n-m,n-d\\}$, for any hyperplane of
${\mathbb{F}}_{q^{m}}^{k}$. By Proposition II.3, it follows that

$\dim_{\mathbb{F}_{q}}(U^{\perp^{\prime}}\cap H^{\perp})\in\\{0,m-d\\},$

or in other words,

$\dim_{{\mathbb{F}}_{q}}(U^{\perp^{\prime}}\cap\langle
w\rangle_{{\mathbb{F}}_{q^{m}}})\in\\{0,m-d\\},$

for any $w\in{\mathbb{F}}_{q^{m}}^{k}$. Let $S=\langle
U^{\perp^{\prime}}\rangle_{{\mathbb{F}}_{q^{m}}}\subseteq{\mathbb{F}}_{q^{m}}^{k}$
and let $\dim_{{\mathbb{F}}_{q^{m}}}(S)=h$. If
$\dim_{{\mathbb{F}}_{q}}(U^{\perp^{\prime}})>(h-1)m$ then

$\dim_{\mathbb{F}_{q}}(U^{\perp^{\prime}}\cap\langle
w\rangle_{{\mathbb{F}}_{q^{m}}})=m-d$

for any $w\in S\setminus\\{\underline{0}\\}$, and hence

$(q^{m-d}-1)\frac{q^{hm}-1}{q^{m}-1}=q^{km-n}-1$

which gives $d=0$, a contradiction. This means that
$\dim_{{\mathbb{F}}_{q}}(U^{\perp^{\prime}})\leq(h-1)m$ and hence by Corollary
III.3 we get, $m-d\mid m$, $m-d\mid(km-n)$ and $U^{\perp^{\prime}}$ is an
${\mathbb{F}}_{q^{m-d}}$-scattered subspace of dimension $\frac{km-n}{m-d}$ of
$S\subseteq{\mathbb{F}}_{q^{m}}^{k}$. In particular, applying the bound of
Theorem II.1 to $U^{\perp^{\prime}}$, we get
$\frac{km-n}{m-d}\leq\frac{hm}{2(m-d)}$, and hence $n\geq\frac{mk}{2}$.
Finally, since $\operatorname{\mathcal{C}}$ is a two-weight code, $n<km$ and
hence $n\leq km-(m-d)$. ∎

In the above theorem, we classified all the two-weight rank-metric codes as
those codes arising from the dual of scattered subspaces over extensions of
$\mathbb{F}_{q}$. In other terms, such codes arise from a rank-metric code in
$\mathbb{F}_{q^{\prime m^{\prime}}}^{n^{\prime}}$, where $q^{\prime}=q^{m-d}$,
$m^{\prime}=m/(m-d)$ and $n^{\prime}=n/(m-d)$, see [21].

We will now show that, under certain assumptions, two-weight rank-metric codes
exists for any length satisfying the above bound.

######  Theorem IV.12.

Let $m,k$ and $d$ be positive integers such that $d<m$, $(m-d)\mid m$ and
$mk/(m-d)$ is even, then for any $q$ and for any $n$ multiple of $m-d$
satisfying

$\frac{km}{2(m-d)}\leq\frac{n}{m-d}\leq\frac{mk}{m-d}-1,$

there exists a nondegenerate two-weight $[n,k,d]_{q^{m}/q}$ code
$\operatorname{\mathcal{C}}$.

###### Proof.

Because of Theorem II.2, there exists at least one scattered
${\mathbb{F}}_{q^{m-d}}$-subspace $W$ with dimension $t$ in
${\mathbb{F}}_{(q^{m-d})^{\frac{m}{m-d}}}^{k}$ for any

$1\leq t\leq mk/2(m-d).$

Note that, since $d<m$ and $W$ is a scattered
${\mathbb{F}}_{q^{m-d}}$-subspace, $W$ does not contain any
${\mathbb{F}}_{q^{m}}$-subspace of ${\mathbb{F}}_{q^{m}}^{k}$. This implies
that any code associated with $U=W^{\perp^{\prime}}$ is a non-degenerate two-
weight $[n,k,d]_{q^{m}/q}$ code with $n=km-t(m-d)$. ∎

Another natural question regards whether or not the _puncturing_ of a two-
weight rank-metric code preserves the property of having exactly two weights.
As we will see in the next result, this property strongly depends on whether
or not the subspace is _maximally scattered_.

######  Remark IV.13.

The puncturing operation has been defined for rank-metric codes of matrices by
multiplying every codeword by a fixed full-rank (not necessarily square)
matrix. In order to have the analogue definition in vector terms, we need to
consider a matrix $A\in{\mathbb{F}}_{q}^{n\times n^{\prime}}$, with
$n^{\prime}\leq n$ of rank $n^{\prime}$. The punctured code of
$\mathcal{C}\subseteq{\mathbb{F}}_{q^{m}}^{n}$ via the matrix $A$ is

$\mathcal{C}^{\prime}=\\{(c_{1},\ldots,c_{n})A\colon(c_{1},\ldots,c_{n})\in\operatorname{\mathcal{C}}\\}.$

Note that if $\mathcal{C}$ is ${\mathbb{F}}_{q^{m}}$-linear, then
$\mathcal{C}^{\prime}$ is ${\mathbb{F}}_{q^{m}}$-linear as well. Moreover, if
$\mathcal{C}$ and $\mathcal{C}^{\prime}$ are have the same dimension, then the
system associated with $\mathcal{C}^{\prime}$ is contained in the system
associated with $\mathcal{C}$.

######  Theorem IV.14.

Assume that $q\geq m$. Let $\operatorname{\mathcal{C}}$ be a nondegenerate
$[n,k,d]_{q^{m}/q}$ code with $d=m-1$. Let $U$ be an its associated system. If
$U^{\perp^{\prime}}$ is maximally scattered then a punctured code of
$\operatorname{\mathcal{C}}$ which has dimension $k$ is either a two-weight
code such that an its associated system is the dual of an
${\mathbb{F}}_{q^{e}}$-scattered subspace, with $e>1$, or it has more than
three weights.

###### Proof.

Let $W$ be the system of a punctured code of $\operatorname{\mathcal{C}}$. In
particular, this means that $W$ is contained in $U$ and hence
$W^{\perp^{\prime}}\supset U^{\perp^{\prime}}$. Since $U^{\perp^{\prime}}$ is
maximally scattered, then $W^{\perp^{\prime}}$ is not scattered over
$\mathbb{F}_{q}$. So, either $W^{\perp^{\prime}}$ is a
${\mathbb{F}}_{q^{e}}$-scattered subspace (for some positive integer $e$) or
it is not scattered. This implies that the code associated with $W$ either it
is a two-weight code associated with a scattered subspace over a proper
extension of $\mathbb{F}_{q}$ or it has more than three nonzero weights. ∎

Apart from the maximum scattered subspaces (which give MRD codes), very few
examples of maximally scattered $\mathbb{F}_{q}$-subspaces are known, see [17,
Example 3.2].

## Acknowledgements

The research was supported by the project COMBINE of “VALERE: VAnviteLli pEr
la RicErca" of the University of Campania “Luigi Vanvitelli” and was partially
supported by by the INdAM - GNSAGA Project _Tensors over finite fields and
their applications_ , number E53C23001670001. This research was also supported
by Bando Galileo 2024 – G24-216 and by the project “The combinatorics of
minimal codes and security aspects”, Bando Cassini.

## References

  * [1] S. Adriaensen and P. Santonastaso.  On the minimum size of linear sets.  arXiv preprint arXiv:2301.13001, 2023.
  * [2] G. N. Alfarano, M. Borello, A. Neri, and A. Ravagnani.  Linear cutting blocking sets and minimal codes in the rank metric.  Journal of Combinatorial Theory, Series A, 192:105658, 2022.
  * [3] S. Ball, A. Blokhuis, and M. Lavrauw.  Linear $(q+1)$-fold blocking sets in PG$(2,q^{4})$.  Finite Fields and Their Applications, 6(4):294–301, 2000.
  * [4] D. Bartoli, M. Giulietti, G. Marino, and O. Polverino.  Maximum scattered linear sets and complete caps in Galois spaces.  Combinatorica, 38(2):255–278, 2018.
  * [5] H. Bartz, L. Holzbaur, H. Liu, S. Puchinger, J. Renner, and A. Wachter-Zeh.  Rank-metric codes and their applications.  Foundations and Trends® in Communications and Information Theory, 19(3):390–546, 2022.
  * [6] T. P. Berger.  Isometries for rank distance and permutation group of Gabidulin codes.  IEEE Transactions on Information Theory, 49(11):3016–3019, 2003\.
  * [7] A. Blokhuis and M. Lavrauw.  Scattered spaces with respect to a spread in PG$(n,q)$.  Geometriae Dedicata, 81(1):231–243, 2000.
  * [8] G. Bonoli and O. Polverino.  $\mathbb{F}_{q}$-linear blocking sets in PG$(2,q^{4})$.  Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial, 2(1):35–56, 2005.
  * [9] M. Borello and F. Zullo.  Geometric dual and sum-rank minimal codes.  arXiv preprint arXiv:2303.07288, 2023.
  * [10] R. Calderbank and W. M. Kantor.  The geometry of two-weight codes.  Bulletin of the London Mathematical Society, 18(2):97–122, 1986\.
  * [11] B. Csajbók, G. Marino, and V. Pepe.  On the maximum field of linearity of linear sets.  arXiv preprint arXiv:2306.07488, 2023.
  * [12] B. Csajbók, G. Marino, O. Polverino, and F. Zullo.  Maximum scattered linear sets and MRD-codes.  Journal of Algebraic Combinatorics, 46(3):517–531, 2017.
  * [13] J. De Beule and G. Van de Voorde.  The minimum size of a linear set.  Journal of Combinatorial Theory, Series A, 164:109–124, 2019.
  * [14] P. Delsarte.  Bilinear forms over a finite field, with applications to coding theory.  Journal of Combinatorial Theory, Series A, 25(3):226–241, 1978\.
  * [15] E. M. Gabidulin.  Theory of codes with maximum rank distance.  Problemy Peredachi Informatsii, 21(1):3–16, 1985.
  * [16] D. Jungnickel and V. D. Tonchev.  The classification of antipodal two-weight linear codes.  Finite Fields and Their Applications, 50:372–381, 2018.
  * [17] M. Lavrauw.  Scattered spaces in Galois geometry.  Contemporary developments in finite fields and applications, pages 195–216, 2016.
  * [18] M. Lavrauw and G. Van de Voorde.  Scattered linear sets and pseudoreguli.  The electronic journal of combinatorics, 20(1), 2013.
  * [19] M. Lavrauw and G. Van de Voorde.  Field reduction and linear sets in finite geometry.  Topics in finite fields, 632:271–293, 2015.
  * [20] O. Polverino.  Linear sets in finite projective spaces.  Discrete mathematics, 310(22):3096–3107, 2010.
  * [21] O. Polverino, P. Santonastaso, J. Sheekey, and F. Zullo.  Divisible linear rank metric codes.  IEEE Transactions on Information Theory, 69(7):4528 – 4536, 2023\.
  * [22] R. Pratihar and T. H. Randrianarisoa.  Antipodal two-weight rank metric codes.  Designs, Codes and Cryptography, 92(3):753–769, 2024.
  * [23] T. H. Randrianarisoa.  A geometric approach to rank metric codes and a classification of constant weight codes.  Designs, Codes and Cryptography, 88:1331–1348, 2020.
  * [24] R. M. Roth.  Maximum-rank array codes and their application to crisscross error correction.  IEEE Transactions on Information Theory, 37(2):328–336, 1991.
  * [25] J. Sheekey.  (scattered) linear sets are to rank-metric codes as arcs are to Hamming-metric codes.  in: M. Greferath, C. Hollanti, J. Rosenthal (Eds.), Oberwolfach Report No. 13/2019, 2019.