11institutetext: IMCCE, Observatoire de Paris, PSL Research University, CNRS,
Sorbonne Université, Université de Lille, 75014 Paris, France 11email:
<EMAIL_ADDRESS>22institutetext: Department of Mathematics,
University of Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy

# The future large obliquity of Jupiter

Melaine Saillenfest 11 Giacomo Lari 22 Ariane Courtot 11

(Received 18 May 2020 / Accepted 3 June 2020)

###### Abstract

Aims. We aim to determine whether Jupiter’s obliquity is bound to remain
exceptionally small in the Solar System, or if it could grow in the future and
reach values comparable to those of the other giant planets.

Methods. The spin axis of Jupiter is subject to the gravitational torques from
its regular satellites and from the Sun. These torques evolve over time due to
the long-term variations of its orbit and to the migration of its satellites.
With numerical simulations, we explore the future evolution of Jupiter’s spin
axis for different values of its moment of inertia and for different migration
rates of its satellites. Analytical formulas show the location and properties
of all relevant resonances.

Results. Because of the migration of the Galilean satellites, Jupiter’s
obliquity is currently increasing, as it adiabatically follows the drift of a
secular spin-orbit resonance with the nodal precession mode of Uranus. Using
the current estimates of the migration rate of the satellites, the obliquity
of Jupiter can reach values ranging from $6^{\circ}$ to $37^{\circ}$ after $5$
Gyrs from now, according to the precise value of its polar moment of inertia.
A faster migration for the satellites would produce a larger increase in
obliquity, as long as the drift remains adiabatic.

Conclusions. Despite its peculiarly small current value, the obliquity of
Jupiter is no different from other obliquities in the Solar System: It is
equally sensitive to secular spin-orbit resonances and it will probably reach
comparable values in the future.

###### Key Words.:

celestial mechanics, Jupiter, secular dynamics, spin axis, obliquity

##  1 Introduction

The obliquity of a planet is the angle between its spin axis and the normal to
its orbit. A non-zero obliquity results in seasonal climate changes along the
planet’s orbit, as occurs on Earth. In the protoplanetary disc, giant planets
are expected to form with near-zero obliquities, while terrestrial planets
should exhibit more random values (see e.g. Ward & Hamilton, 2004; Rogoszinski
& Hamilton, 2020a). Yet, the planets of the Solar System all feature a large
variety of obliquities. The case of Mercury is special because the strong
tidal dissipation due to the proximity of the Sun now tightly maintains
Mercury’s obliquity to a near-zero value (see e.g. Correia & Laskar, 2010).
Excluding Mercury, Jupiter is by far the planet of the Solar System that has
the smallest obliquity (see Table 1). This small value seems to put Jupiter in
a different category, and it appears unclear why Jupiter should be the only
giant planet to indefinitely preserve its primordial obliquity.

Large obliquity changes can be produced by strong impacts. An impact with a
planetary-sized body is thought to have created the Moon and affected the spin
axis of the Earth, which has remained unchanged ever since (Canup & Asphaug,
2001; Li & Batygin, 2014b). Large-scale collisions have also probably
participated in increasing the obliquity of Uranus (Boué & Laskar, 2010;
Morbidelli et al., 2012; Rogoszinski & Hamilton, 2020a).

Table 1: Current obliquities of the planets of the Solar System.

$\begin{array}[]{rrcrr}\hline\cr\hline\cr&\text{obliquity}&&&\text{obliquity}\\\
\hline\cr\text{Mercury}&0.03^{\circ}&&\text{Jupiter}&3.12^{\circ}\\\
\text{Venus}&177.36^{\circ}&&\text{Saturn}&26.73^{\circ}\\\
\text{Earth}&23.45^{\circ}&&\text{Uranus}&97.86^{\circ}\\\
\text{Mars}&25.19^{\circ}&&\text{Neptune}&29.56^{\circ}\\\
\hline\cr\end{array}$

111Mercury’s obliquity is taken from Konopliv et al. (2020). Other values are
taken from Murray & Dermott (1999) who cite the compilation made by Yoder
(1995).

Apart from collisions, a well-known mechanism that can modify the obliquity of
a planet is a so-called “secular spin-orbit resonance”, that is, a near
commensurability between the frequency of precession of the spin axis and the
frequency of one (or several) harmonics appearing in the precession of the
orbit. This mechanism happens to be extremely common in planetary systems. The
overlap of such resonances produces a large chaotic region for the spin axis
of the terrestrial planets of the Solar System (see Laskar & Robutel, 1993).
This chaos probably had a strong influence on the early obliquity of Venus,
which was then driven to its current value by the solar tides combined with
its thick atmosphere (Correia & Laskar, 2001; Correia et al., 2003; Correia &
Laskar, 2003). The Moon currently protects the Earth from large chaotic
variations in its obliquity (Laskar et al., 1993; Li & Batygin, 2014a), but
due to tidal dissipation within the Earth-Moon system, the Earth will
eventually reach the chaotic region in a few gigayears from now (see Néron de
Surgy & Laskar, 1997). This chaotic zone also strongly affects the obliquity
of Mars, which still currently wanders between $0^{\circ}$ and more than
$60^{\circ}$ (Laskar et al., 2004a; Brasser & Walsh, 2011). As shown by
Millholland & Batygin (2019), secular spin-orbit resonances can also take
place very early in the history of a planet, that is, within the
protoplanetary disc itself. More generally, secular spin-orbit resonances are
thought to strongly affect the obliquity of exoplanets (see e.g. Atobe et al.,
2004; Brasser et al., 2014; Deitrick et al., 2018b, a; Shan & Li, 2018;
Millholland & Laughlin, 2018, 2019; Quarles et al., 2019; Saillenfest et al.,
2019; Kreyche et al., 2020).

For the giant planets of the Solar System, the secular spin-orbit resonances
are relatively thin today and well separated from each other. This is why it
is so difficult to explain the large obliquity of Uranus by a spin-orbit
coupling, now that the precession of Uranus’ spin axis is far from any first-
order resonances (see e.g. Boué & Laskar, 2010, Rogoszinski & Hamilton, 2020a,
b). Jupiter and Saturn, on the contrary, are located very close to strong
resonances: Jupiter is close to resonance with the nodal precession mode of
Uranus (Ward & Canup, 2006), and Saturn is close to resonance with the nodal
precession mode of Neptune (Ward & Hamilton, 2004; Hamilton & Ward, 2004; Boué
et al., 2009). Therefore, the dynamics of Jupiter’s spin axis seems to be
equally affected by secular spin-orbit resonances as other planets in the
Solar System. This was confirmed by Brasser & Lee (2015) and Vokrouhlický &
Nesvorný (2015), who show that models of the late planetary migration have to
be finely tuned to avoid overexciting Jupiter’s obliquity by spin-orbit
coupling, while tilting Saturn to its current orientation. In this regard, the
spin-axis dynamics of Jupiter does not appear to be special at all, in
contrast to its small obliquity value.

In this article, we aim to investigate the future long-term spin-axis dynamics
of Jupiter. In particular, we want to determine whether Jupiter’s obliquity is
bound to remain exceptionally small in the Solar System, or if it could grow
in the future and reach values comparable to those of the other planets.

The precession motion of a planet’s spin axis depends on the physical
properties of the planet (mass repartition and spin velocity), but also on
external torques applied to its equatorial bulge. These torques come from the
combined gravitational attraction of the Sun and of satellites (if it has
any). Since the orbit of Jupiter is stable over billions of years (Laskar,
1990), the direct torque from the Sun will not noticeably change in the
future. However, Jupiter’s satellites are known to migrate over time because
of tidal dissipation. The future long-term orbital evolution of the Galilean
satellites has been recently explored by Lari et al. (2020). The solutions
that they describe can therefore be used as a guide to study the future spin-
axis dynamics of Jupiter. Due to their much smaller masses, the other
satellites of Jupiter do not contribute noticeably to its spin-axis dynamics.

In Sect. 2, we describe our dynamical model and discuss the range of
acceptable values for the physical parameters of Jupiter, in particular its
polar moment of inertia. In Sect. 3, we present our results about the future
spin-axis dynamics of Jupiter: We explore the outcomes given by different
values of the poorly known physical parameters of Jupiter and by different
migration rates for its satellites. Our conclusions are summarised in Sect. 4.

##  2 Secular dynamics of the spin axis

###  2.1 Equations of motion

The spin-axis dynamics of an oblate planet subject to the lowest-order term of
the torque from the Sun is given for instance by Laskar & Robutel (1993) or
Néron de Surgy & Laskar (1997). Far from spin-orbit resonances, and due to the
weakness of the torque, the long-term evolution of the spin axis is accurately
described by the secular Hamiltonian function (i.e. averaged over rotational
and orbital motions). This Hamiltonian can be written

$\displaystyle\mathcal{H}(X,-\psi,t)$
$\displaystyle=-\frac{\alpha}{2}\frac{X^{2}}{\big{(}1-e(t)^{2}\big{)}^{3/2}}$
(1)
$\displaystyle-\sqrt{1-X^{2}}\big{(}\mathcal{A}(t)\sin\psi+\mathcal{B}(t)\cos\psi\big{)}$
$\displaystyle+2X\mathcal{C}(t),$

where the conjugate coordinates are $X$ (cosine of obliquity) and $-\psi$
(minus the precession angle). The Hamiltonian in Eq. (1) depends explicitly on
time $t$ through the orbital eccentricity $e$ and through the functions

$\left\\{\begin{aligned}
\mathcal{A}(t)&=\frac{2\big{(}\dot{q}+p\,\mathcal{C}(t)\big{)}}{\sqrt{1-p^{2}-q^{2}}}\,,\\\
\mathcal{B}(t)&=\frac{2\big{(}\dot{p}-q\,\mathcal{C}(t)\big{)}}{\sqrt{1-p^{2}-q^{2}}}\,,\\\
\end{aligned}\right.\quad\text{and}\quad\mathcal{C}(t)=q\dot{p}-p\dot{q}\,.$
(2)

In these expressions, $q=\eta\cos\Omega$ and $p=\eta\sin\Omega$, where
$\eta\equiv\sin(I/2)$, and $I$ and $\Omega$ are the orbital inclination and
the longitude of ascending node of the planet, respectively. If the orbit of
the planet is fixed in time, its obliquity is constant and its precession
angle $\psi$ circulates with constant angular velocity $\alpha
X/(1-e^{2})^{3/2}$. The quantity $\alpha$ is called the precession constant.
It depends on the spin rate of the planet and of its mass distribution,
through the formula:

$\alpha=\frac{3}{2}\frac{\mathcal{G}m_{\odot}}{\omega
a^{3}}\frac{J_{2}}{\lambda}\,,$ (3)

where $\mathcal{G}$ is the gravitational constant, $m_{\odot}$ is the mass of
the Sun, $\omega$ is the spin rate of the planet, $a$ is its semi-major axis,
$J_{2}$ is its second zonal gravity coefficient, and $\lambda$ is its
normalised polar moment of inertia. We retrieve the expression given for
instance by Néron de Surgy & Laskar (1997) by noting that

$J_{2}=\frac{2C-A-B}{2MR_{\mathrm{eq}}^{2}}\quad\text{and}\quad\lambda=\frac{C}{MR_{\mathrm{eq}}^{2}}\,,$
(4)

where $A$, $B$, and $C$ are the equatorial and polar moments of inertia of the
planet, $M$ is its mass, and $R_{\mathrm{eq}}$ is its equatorial radius.

The precession rate of the planet is increased if it possesses massive
satellites. If the satellites are far away from the planet, their equilibrium
orbital plane (called Laplace plane, see Tremaine et al., 2009) is close to
the orbital plane of the planet; therefore, far-away satellites increase the
torque exerted by the Sun on the equatorial bulge of the planet. If the
satellites are close to the planet, on the contrary, their equilibrium orbital
plane coincides with the equator of the planet and precesses with it as a
whole (Goldreich, 1965); therefore, close-in satellites artificially increase
the oblateness and the rotational angular momentum of the planet. In the
close-in satellite regime, an expression for the effective precession constant
has been derived by Ward (1975). As detailed by French et al. (1993), it
consists in replacing $J_{2}$ and $\lambda$ in Eq. (3) by the effective
values:

$J_{2}^{\prime}=J_{2}+\frac{1}{2}\sum_{k}\frac{m_{k}}{M}\frac{a_{k}^{2}}{R_{\mathrm{eq}}^{2}}\quad\text{and}\quad\lambda^{\prime}=\lambda+\sum_{k}\frac{m_{k}}{M}\frac{a_{k}^{2}}{R_{\mathrm{eq}}^{2}}\frac{n_{k}}{\omega}\,,$
(5)

where $m_{k}$, $a_{k}$, and $n_{k}$ are the mass, the semi-major axis, and the
mean motion of the $k$th satellite. In these expressions, the eccentricities
and inclinations of the satellites are neglected. This approximation has been
widely used in the literature. In the case of a single satellite, Boué &
Laskar (2006) have obtained a general expression for the precession rate of a
planet with an eccentric and inclined satellite, encompassing both the close-
in and far-away regimes. Using their article, we can verify that the Galilean
satellites are in the close-in regime. The Laplace plane of Callisto is
inclined today by less than $1^{\circ}$ with respect to Jupiter’s equator. The
small eccentricities and inclinations of the Galilean satellites would
contribute to $J_{2}^{\prime}$ and $\lambda^{\prime}$ with terms of order
$e_{k}^{2}$ and $\eta_{k}^{2}$, so even if $e_{k}$ increases up to $0.1$ (a
value found by Lari et al., 2020 in some cases) or if $I_{k}$ increases up to
$10^{\circ}$, the additional contribution to $J_{2}^{\prime}$ and
$\lambda^{\prime}$ would only be of order $10^{-4}$ and $10^{-6}$,
respectively. As we see below, this contribution is much smaller than our
uncertainty on the value of $\lambda$, allowing us to stick to the
approximation given by Eq. (5).

###  2.2 Orbital solution

The Hamiltonian given in Eq. (1) depends on the orbit of the planet and on its
temporal variations. In order to explore the long-term dynamics of Jupiter’s
spin axis, we need an orbital solution that is valid over billions of years.
This is well beyond the timespan covered by ephemerides. Luckily, the orbital
dynamics of the giant planets of the Solar System are almost integrable and
excellent solutions have been developed. We use the secular solution of Laskar
(1990) expanded in quasi-periodic series:

$\displaystyle z=e\exp(i\varpi)$
$\displaystyle=\sum_{k}E_{k}\exp(i\theta_{k})\,,$ (6)
$\displaystyle\zeta=\eta\exp(i\Omega)$
$\displaystyle=\sum_{k}S_{k}\exp(i\phi_{k})\,,$

where $\varpi$ is Jupiter’s longitude of perihelion. The amplitudes $E_{k}$
and $S_{k}$ are real constants, and the angles $\theta_{k}$ and $\phi_{k}$
evolve linearly over time $t$, with frequencies $\mu_{k}$ and $\nu_{k}$:

$\theta_{k}(t)=\mu_{k}\,t+\theta_{k}^{(0)}\hskip 14.22636pt\text{and}\hskip
14.22636pt\phi_{k}(t)=\nu_{k}\,t+\phi_{k}^{(0)}\,.$ (7)

The complete orbital solution of Laskar (1990) can be found in Appendix A for
amplitudes down to $10^{-8}$.

The series in Eq. (6) contain contributions from all the planets of the Solar
System. In the integrable approximation, the frequency of each term
corresponds to a unique combination of the fundamental frequencies of the
system, usually noted $g_{j}$ and $s_{j}$. In the limit of small masses, small
eccentricities and small inclinations (Lagrange-Laplace secular system), the
$z$ series only contains the frequencies $g_{j}$, while the $\zeta$ series
only contains the frequencies $s_{j}$ (see e.g. Murray & Dermott, 1999 or
Laskar et al., 2012). This is not the case in more realistic situations, as
recalled for instance by Kreyche et al. (2020) in the context of obliquity
dynamics. In planetary systems featuring mean-motion resonances, the spin axis
of a planet can be affected by shifted orbital precession frequencies
(Millholland & Laughlin, 2019) or by secondary resonances (Quillen et al.,
2017, 2018). However, this does not apply in the Solar System as it is today,
even when the existing near commensurabilities (like the “great Jupiter–Saturn
inequality”) are taken into account. Table 2 shows the combinations of
fundamental frequencies identified for the largest terms of Jupiter’s $\zeta$
series obtained by Laskar (1990).

Table 2: First twenty terms of Jupiter’s inclination and longitude of
ascending node in the J2000 equator and equinox reference frame.

$\begin{array}[]{rcrrr}\hline\cr\hline\cr k&\text{identification}&\nu_{k}\
(^{\prime\prime}\cdot\text{yr}^{-1})&S_{k}\times 10^{8}&\phi_{k}^{(0)}\
(^{\text{o}})\\\ \hline\cr 1&s_{5}&0.00000&1377467&107.59\\\
2&s_{6}&-26.33023&315119&307.29\\\ 3&s_{8}&-0.69189&58088&23.96\\\
4&s_{7}&-3.00557&48134&140.33\\\ 5&g_{5}-g_{6}+s_{7}&-26.97744&2308&222.98\\\
6&-g_{5}+g_{6}+s_{6}&-2.35835&1611&44.74\\\
7&2g_{6}-s_{6}&82.77163&1372&308.95\\\
8&g_{5}-g_{7}+s_{7}&-1.84625&1130&36.64\\\ 9&s_{1}&-5.61755&1075&168.70\\\
10&-g_{5}+g_{7}+s_{7}&-4.16482&946&51.54\\\
11&g_{5}+g_{6}-s_{6}&58.80017&804&32.90\\\
12&g_{5}-g_{6}+s_{6}&-50.30212&691&29.84\\\
13&2g_{5}-s_{6}&34.82788&636&114.12\\\
14&g_{7}-g_{8}+s_{7}&-0.58033&565&17.32\\\ 15&s_{2}&-7.07963&454&273.79\\\
16&-g_{5}+g_{7}+s_{6}&-27.48935&407&38.53\\\
17&g_{5}-g_{7}+s_{6}&-25.17116&385&35.94\\\
18&s_{1}+\gamma&-5.50098&383&162.89\\\
19&-g_{7}+g_{8}+s_{8}&-3.11725&321&326.97\\\
20&s_{2}+2\gamma&-6.84091&267&106.20\\\ \hline\cr\end{array}$

222Due to the secular resonance $(g_{1}-g_{5})-(s_{1}-s_{2})$, an additional
fundamental frequency $\gamma$ appears in terms 18 and 20 (see Laskar, 1990).

As explained by Saillenfest et al. (2019), at first order in the amplitudes
$S_{k}$ and $E_{k}$, secular spin-orbit resonant angles can only be of the
form $\sigma_{p}=\psi+\phi_{p}$, where $p$ is a given index in the $\zeta$
series. Resonances featuring terms of the $z$ series only appear at third
order and beyond. For the terrestrial planets of the Solar System, the $z$ and
$\zeta$ series converge very slowly, which implies that large resonances are
very numerous. These resonances overlap massively and produce wide chaotic
zones in the obliquity dynamics (see Laskar & Robutel, 1993; Néron de Surgy &
Laskar, 1997; Correia et al., 2003; Laskar et al., 2004a). The situation is
very different for the giant planets of the Solar System, for which the $z$
and $\zeta$ series converge quickly owing to the quasi-integrable nature of
their dynamics. Therefore, the secular spin-orbit resonances are small and
isolated from each other, and only first-order resonances play a substantial
role.

Figure 1 shows the location and width of every first-order resonance for the
spin-axis of Jupiter in an interval of precession constant $\alpha$ ranging
from $0^{\prime\prime}\cdot$yr-1 to $5^{\prime\prime}\cdot$yr-1. Because of
the chaotic dynamics of the Solar System (Laskar, 1989), the fundamental
frequencies related to the terrestrial planets (e.g. $s_{1}$, $s_{2}$, and
$\gamma$ appearing in Table 2) could vary substantially over billions of years
(Laskar, 1990). However, they only marginally contribute to Jupiter’s orbital
solution and none of them takes part in the resonances shown in Fig. 1. Our
secular orbital solution of Jupiter can therefore be considered valid over a
billion-year timescale.

Figure 1: Location and width of every first-order secular spin-orbit resonance
for Jupiter. Each resonant angle is of the form $\sigma_{p}=\psi+\phi_{p}$
where $\phi_{p}$ has frequency $\nu_{p}$ labelled on the graph according to
its index in the orbital series (see Table 2 and Appendix A). For a given
value of the precession constant $\alpha$, the interval of obliquity enclosed
by the separatrix is shown in pink, as computed using the exact formulas given
by Saillenfest et al. (2019). The green bar on the left shows Jupiter’s
current obliquity and the range for its precession constant considered in this
article, as detailed in Sects. 2.3 and 2.4.

###  2.3 Precession constant

As shown by the Hamiltonian function in Eq. (1), the precession constant
$\alpha$ is a key parameter of the spin-axis dynamics of a planet. Among the
physical parameters of Jupiter that enter into its expression (see Eq. 3), all
are very well constrained from observations except the normalised polar moment
of inertia $\lambda$.

While comparing the values of $\lambda$ given in the literature, one must be
careful about the normalisation used. Equation (4) explicitly requires a
normalisation using the equatorial radius $R_{\text{eq}}$, since it is linked
to the value of $J_{2}$. However, published values of the polar moment of
inertia are often normalised using the mean radius of Jupiter, which differs
from $R_{\text{eq}}$ by a factor of about $0.978$. This distinction seems to
have been missed by Ward & Canup (2006), who quote the nominal value given by
D. R. Williams in the _NASA Jupiter fact sheet_
333https://nssdc.gsfc.nasa.gov/planetary/factsheet/
jupiterfact.html as $0.254$, whereas it actually translates into
$\lambda=0.243$ when it is normalised using $R_{\text{eq}}$. Ward & Canup
(2006) also mention that “theoretical values [of $\lambda$] range from $0.255$
for the extreme of a constant-density core and massless envelope to $0.221$
for a constant-density envelope and point-mass core”. Unfortunately, these
numbers are taken from a conference talk given by W. B. Hubbard in 2005 so we
cannot check how they have been obtained. Since Eq. (3) is used, however, we
can assume that they have been properly normalised using $R_{\text{eq}}$.

As is shown in Fig. 1, the spin-axis of Jupiter is located very close to a
strong secular spin-orbit resonance. The corresponding term of the orbital
series is related to the precession mode of Uranus (term $k=4$ in Table 2),
and the resonant angle is $\sigma_{4}=\psi+\phi_{4}$. As noted by Ward & Canup
(2006), dissipative processes during the early planetary evolution are
expected to have forced Jupiter’s spin axis to spiral down towards the centre
of the resonance, called Cassini state 2. And indeed, the current value of
$\sigma_{4}$ is very close to zero, which has a low probability to happen if
Jupiter is far from Cassini state 2 because $\sigma_{4}$ would then circulate
between $0^{\circ}$ and $360^{\circ}$. In order to match Cassini state 2,
however, Jupiter’s normalised moment of inertia should be $\lambda\approx
0.2365$ (see Fig. 2). Since this value is not far from what is proposed in the
literature, this prompted Ward & Canup (2006) to consider this value as likely
for Jupiter.

Figure 2: Trajectory of Jupiter’s spin axis in the vicinity of resonance with
the fourth harmonics of $\zeta$ (see Table 2). Being farther away from
Jupiter’s precession frequency, the contribution of other harmonics can be
averaged; their mean contribution is included here up to third order in the
amplitudes (as in Eq. 17 of Saillenfest et al., 2019). Each trajectory
corresponds to a level curve of the Hamiltonian, which has only one degree of
freedom. The red dot shows the current location of Jupiter, and the black dot
shows Cassini state 2. The red curve is the current trajectory of Jupiter’s
spin axis for $\lambda=0.250$ (top) or $\lambda=0.237$ (bottom).

As noted by Le Maistre et al. (2016), the value of $\lambda\approx 0.2365$
corresponds to a massive core for Jupiter, and estimates obtained from models
of Jupiter’s interior structure are generally higher. Helled et al. (2011)
obtain values of $\lambda$ ranging from $0.251$ to $0.253$, that were
confirmed by Nettelmann et al. (2012). These values are consistent with the
range of $\lambda\in[0.221,0.255]$ quoted above. Other studies seem to agree
on even higher values: Wahl et al. (2017) and Ni (2018) present values of
$\lambda$ ranging between $0.2629$ and $0.2644$, compatible with the findings
of Hubbard & Marley (1989), Nettelmann et al. (2012), and Hubbard & Militzer
(2016). Finally, both low and high values are obtained by Vazan et al. (2016),
who give either $\lambda=0.247$ or $\lambda=0.262$ for three different models.
As explained by Le Maistre et al. (2016), however, all these values are model-
dependent and still a matter of debate. Hopefully, the _Juno_ mission will
provide direct observational constraints soon that will help us to determine
which models of Jupiter’s interior structure are the most relevant.

Here, instead of relying on one particular value of $\lambda$, we turn to the
exploration of the whole range of values given in the literature, namely
$\lambda\in[0.220,0.265]$. The rotation velocity of Jupiter is taken from
Archinal et al. (2018) and the other physical parameters are fixed to those
used by Lari et al. (2020) for consistency with the satellites’ orbital
evolution (see below). The corresponding value for the current precession
constant of Jupiter, computed from Eqs. (3) and (5), ranges from
$2.64^{\prime\prime}\cdot$yr-1 to $3.17^{\prime\prime}\cdot$yr-1. Given this
large range, using updated physical parameters (see e.g. Folkner et al., 2017;
Iess et al., 2018; Serra et al., 2019) would only slightly shift the value of
$\alpha$ within our exploration interval.

Because of tidal dissipation, satellites slowly migrate over time. This
produces a drift of the precession constant $\alpha$ on a timescale that is
much larger than the precession motion (i.e. the circulation of $\psi$). The
long-term spin-axis dynamics of a planet with migrating satellites is
described by the Hamiltonian in Eq. (1), but where $\alpha$ is a slowly-
varying function of time. In the Earth-Moon system, the outward migration of
the Moon produces a decrease of $\alpha$ that pushes the Earth towards a wide
chaotic region (see Néron de Surgy & Laskar, 1997). This decrease of $\alpha$
is due to the fact that the Moon is in the far-satellite regime (see Boué &
Laskar, 2006). The Galilean satellites, on the contrary, are in the close-
satellite regime, and their outward migration produces an increase of
$\alpha$, as shown by Eq. (5). This increase can be quantified using the long-
term orbital solution of Lari et al. (2020) depicted in Fig. 3 and
interpolating between data points. The result is presented in Fig. 4 for the
two extreme values of $\lambda$ considered in this article, as well as for the
value of $\lambda\approx 0.2365$ proposed by Ward & Canup (2006). Despite the
various outcomes of the dynamics described by Lari et al. (2020), the result
on the evolution of $\alpha$ is almost undistinguishable from one of their
simulations to another, even if the eccentricities of the satellites are taken
into account in Eq. (5). Indeed, the variation of $\alpha$ mostly depends on
the drift of the satellites’ semi-major axes, which is almost identical in
every simulation of Lari et al. (2020).

Since the rate of energy dissipation between Jupiter and its satellites is not
well known today, the timescale of the drift shown in Figs. 3 and 4 could
somewhat contract or expand. This point is further discussed in Sect. 3.
Moreover, other parameters in Eq. (3) probably slightly vary over billions of
years, such as the spin velocity of Jupiter or its oblateness. We consider
that the impact of their variations on the value of $\alpha$ is small and
contained within our exploration range.

Figure 3: Typical evolution of the semi-major axes of the Galilean satellites
obtained by Lari et al. (2020). The values are expressed in unit of Jupiter’s
equatorial radius. The bump at about $1.8$ Gyrs is due to the capture of
Callisto into resonance. Figure 4: Evolution of the effective precession
constant of Jupiter due to the migration of its satellites. The top and bottom
curves correspond to the two extreme values of the normalised polar moment of
inertia $\lambda$ considered in this article. They appear into $\alpha$
through Eq. (3). The central curve corresponds to the value of $\lambda$ that
places Jupiter just near Cassini state 2 with the precession mode of Uranus
(Ward & Canup, 2006).

###  2.4 Initial conditions

The initial orientation of the spin axis is taken from the solution of
Archinal et al. (2018) averaged over short-period terms. At the level of
precision required by our exploratory study, the refined orientation obtained
by Durante et al. (2020) is undistinguishable from this nominal orientation.
With respect to Jupiter’s secular orbital solution (see Sect. 2.2), this gives
an obliquity $\varepsilon=3.120^{\circ}$ and a precession angle
$\psi=-137.223^{\circ}$ at time J2000. The uncertainty on these values is
extremely small compared to the range of $\alpha$ considered (see Sect. 2.3).
Since the uncertainty is smaller than the curve width of our figures, we do
not consider any error bar on the initial value of $\varepsilon$ and $\psi$.

##  3 Obliquity evolution with migrating satellites

For values of $\lambda$ finely sampled in our exploration interval, the spin
axis of Jupiter is numerically propagated forwards in time for $5$ Gyrs. By
virtue of trigonometric identities, moving Jupiter’s orbit one step forwards
in time using the quasi-periodic decomposition in Eq. (6) only amounts to
computing a few sums and products. The trajectories obtained are shown in Fig.
5 for a few values of $\lambda$. They are projected in the plane of the
obliquity and the precession constant of Jupiter, where we localise also the
centres and widths of all first-order secular spin-orbit resonances. See
Appendix B for further details about the geometry of the resonances.

Figure 5: Future evolution of Jupiter’s spin axis projected in the plane of
the obliquity and the precession constant $\alpha$. Each panel corresponds to
a value of the normalised polar moment of inertia of Jupiter
$\lambda=C/(MR_{\text{eq}}^{2})$ given in title. The green bar shows the
initial location of Jupiter’s spin axis according to our exploration interval
of $\lambda$; the central mark is the value proposed by Ward & Canup (2006).
The red curves show the centre of all first-order secular spin-orbit
resonances (Cassini state 2) and the coloured areas represent their widths
(same as Fig. 1). From bottom to top, the resonances are with $\phi_{6}$, with
$\phi_{4}$, and with $\phi_{19}$ (see Table 2). The black dots show the
numerical solutions obtained over a timespan of $5$ Gyrs from now; they evolve
from bottom to top. According to the exact migration rate of the Galilean
satellites, the timeline could somewhat contract or expand (see text).

For values of $\lambda$ smaller than about $0.228$, Jupiter starts outside of
the large resonance with $\phi_{4}$, and the increase of its precession
constant $\alpha$ pushes it even farther away over time. As shown by the
trajectory computed for $\lambda=0.227$, the crossing of the very thin
resonance with $\phi_{19}$ twists the trajectory a little, but this cannot
produce any large change of obliquity. Indeed the resonance with $\phi_{19}$
is not strong enough to capture Jupiter’s spin axis: It is crossed quickly as
$\alpha$ increases, and Fig. 6 shows that the libration period of
$\sigma_{19}=\psi+\phi_{19}$ is very long. This results in a non-adiabatic
crossing (see Appendix C for details). Consequently, no major obliquity
variation for Jupiter can be expected in the future if $\lambda<0.228$.
However, such small values of $\lambda$ seem to be ruled out by most models of
Jupiter’s interior (see Sect. 2.3).

Figure 6: Period of small oscillations about the resonance centre for a
resonance with $\phi_{4}$ or $\phi_{19}$. Even though complete closed-form
solutions exist (see Haponiak et al., 2020), the small-oscillation limit leads
to handier formulas, suitable for order-of-magnitude estimates. The resonant
angles are $\sigma_{4}=\psi+\phi_{4}$ and $\sigma_{19}=\psi+\phi_{19}$,
respectively. Dashed curves are used for oscillations about Cassini state 2
before the separatrix appears. The appearance of the separatrix is marked by a
blue dot.

For values of $\lambda$ larger than $0.228$, on the contrary, Jupiter is
currently located inside or below the large resonance with $\phi_{4}$. As
predicted, the value $\lambda=0.2365$ results in very small oscillations
around Cassini state 2. As its precession constant $\alpha$ slowly increases
with time, Jupiter is captured into the resonance and follows the drift of its
centre towards large obliquities. Indeed, the resonance with $\phi_{4}$ is
large, and the libration period of $\sigma_{4}=\psi+\phi_{4}$ is short
compared to the variation timescale of $\alpha$ (see Figs. 5 and 6). This
results in an adiabatic capture. The various possible outcomes of adiabatic
and non-adiabatic crossings of secular spin-orbit resonances have recently
been studied by Su & Lai (2020). However, the orbital motion is here not
limited to a single harmonic, and Appendix C shows that the separatrix of the
resonance is replaced by a chaotic “moat”. Properly speaking, the resonance
with $\phi_{4}$ becomes a “true resonance” only as soon as the separatrix
appears, that is, for $\alpha$ larger than $\alpha\approx
3.04^{\prime\prime}\cdot$yr-1 (see Appendix B). In the whole range of values
of $\lambda>0.228$ considered in this article, the spin axis of Jupiter is
initially located close enough to Cassini state 2 to invariably end up inside
the separatrix of the resonance when it appears. The capture probability is
therefore $100\%$. None of our simulation shows a release out of resonance or
a turn-off towards Cassini state 1, which could have been a possible outcome
if Jupiter’s spin axis was initially located farther away from Cassini state 2
or if the drift of $\alpha$ was not adiabatic444There is a typographical error
in Saillenfest et al. (2019): the list of the Cassini states given before Eq.
(22) should read (4,2,3,1) instead of (1,2,3,4) in order to match the
denomination introduced by Peale (1969).. Since in canonical coordinates the
resonance width increases for $\alpha$ growing up to
$4.244^{\prime\prime}\cdot$yr-1, no separatrix crossing can happen, even for a
large libration amplitude inside the resonance (e.g. for $\lambda=0.228$ in
Fig. 5). The maximum obliquity reached by Jupiter is therefore only limited by
the finite amount of time considered.

If the Galilean satellites migrate faster than shown in Fig. 3, the obliquity
reached in $5$ Gyrs would be larger than that presented in Fig. 5. The
migration rate of the satellites is not well known. According to Lari et al.
(2020), the long-term migration rate of the satellites varies by $\pm 15\%$
over the uncertainty range of the parameter $(k_{2}/Q)_{0,1}$ measured by
Lainey et al. (2009). This parameter quantifies the dissipation within Jupiter
at Io’s frequency. Figure 7 shows the maximum obliquity reached in $5$ Gyrs
for $\lambda$ sampled in our exploration interval and $(k_{2}/Q)_{0,1}$
sampled in its uncertainty range. We retrieve the discontinuity at
$\lambda\approx 0.228$ discussed before, below which only small obliquity
variations are possible. For $\lambda>0.228$, as expected, we see that a fast
migration and a small moment of inertia produce a fast increase of obliquity,
which reaches $37^{\circ}$ in $5$ Gyrs in the most favourable case of Fig. 7.
On the contrary, a slow migration and a large moment of inertia produce a slow
increase of obliquity, which barely reaches $6^{\circ}$ in $5$ Gyrs in the
most unfavourable case of Fig. 7.

Figure 7: Maximum obliquity reached by Jupiter after $5$ Gyrs from now as a
function of its normalised polar moment of inertia
$\lambda=C/(MR_{\text{eq}}^{2})$ (right vertical axis) and the dissipation
parameter of Jupiter at Io’s frequency (horizontal axis). The left vertical
axis shows the current precession constant $\alpha$ of Jupiter. Some level
curves are shown in red.

##  4 Discussion and conclusion

Prompted by the peculiarly small value of the current obliquity of Jupiter, we
studied the future long-term evolution of its spin axis under the influence of
its slowly migrating satellites.

Jupiter is located today near a strong secular spin-orbit resonance with the
nodal precession mode of Uranus (Ward & Canup, 2006). Because of this
resonance, the obliquity of Jupiter is found to be currently increasing,
provided that its normalised polar moment of inertia
$\lambda=C/(MR_{\text{eq}}^{2})$ is larger than about $0.228$. Such a small
value seems to be ruled out by models of Jupiter’s interior (see e.g. Helled
et al., 2011; Hubbard & Militzer, 2016; Wahl et al., 2017). For larger values
of $\lambda$, the migration of the Galilean satellites induces an adiabatic
drift of the precession constant $\alpha$ of Jupiter that pushes its spin axis
inside the resonance and forces it to follow the resonance centre towards high
obliquities. For the value $\lambda\approx 0.2365$ proposed by Ward & Canup
(2006), the obliquity can reach values as large as $30^{\circ}$ in the next
$5$ Gyrs. For the value $\lambda\approx 0.252$ obtained by Helled et al.
(2011), the obliquity reaches values ranging from about $17^{\circ}$ to
$23^{\circ}$. The increase is more modest for values close to $\lambda\approx
0.264$ found by other authors, for which the maximum value of the obliquity
ranges from about $6^{\circ}$ to $17^{\circ}$. Hence, our main conclusion is
that, contrary to Saturn, Jupiter did not have time to tilt much yet from its
primordial orientation, but it will in the future and possibly a lot.

The model of tidal dissipation applied by Lari et al. (2020) to the Galilean
satellites and used here to compute the drift of $\alpha$ is simplified. The
current migration rates of satellites in the Solar System have been proved to
be higher than previously thought (see Lainey et al., 2009, 2017). As
discussed by Lari et al. (2020), the migration of the Galilean satellites
could be even faster than considered here if ever one of the outer satellites
was pushed by a resonance with the frequency of an internal oscillation of
Jupiter (Fuller et al., 2016). This would result in a faster increase of
Jupiter’s obliquity. This increase would be halted, however, if the satellites
ever migrate so fast as to break the adiabaticity of the capture into secular
spin-orbit resonance. In this case, Jupiter would cross the resonance and exit
without following the drift of its centre (see e.g. Ward & Hamilton, 2004; Su
& Lai, 2020). Numerical experiments show that adiabaticity would be broken for
a migration more than $110$ times faster than currently estimated. Such an
extremely fast migration seems unlikely. Moreover, with such a fast migration,
Callisto and then Ganymede would soon go beyond the close-satellite regime
(Boué & Laskar, 2006): This would slow down the increase of $\alpha$ and
possibly restore the adiabaticity of its drift. Therefore, the future increase
of Jupiter’s obliquity appears to be a robust result.

The maximum obliquity that Jupiter will reach could be very large, but it
depends on the precise value of Jupiter’s polar moment of inertia and on the
precise migration rate of the Galilean satellites. We hope to obtain soon new
estimates for these two crucial parameters, in particular from the results of
the _Juno_ and JUICE missions.

###### Acknowledgements.

We thank Marco Fenucci for his help and his suggestions during the redaction
of our manuscript. We also thank the anonymous referee for her/his valuable
comments. G. L. acknowledges financial support from the Italian Space Agency
(ASI) through agreement 2017-40-H.0 in the context of the NASA _Juno_ mission.

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##  Appendix A Orbital solution for Jupiter

The secular orbital solution of Laskar (1990) is obtained by multiplying the
normalised proper modes $z_{i}^{\bullet}$ and $\zeta_{i}^{\bullet}$ (Tables VI
and VII of Laskar 1990) by the matrix $\tilde{S}$ corresponding to the linear
part of the solution (Table V of Laskar 1990). In the series obtained, the
terms with the same combination of frequencies are then merged together,
resulting in 56 terms in eccentricity and 60 terms in inclination. This forms
the secular part of the orbital solution of Jupiter, which is what is required
by our averaged model.

The orbital solution is expressed in the variables $z$ and $\zeta$ as
described in Eqs. (6) and (7). In Tables 3 and 4, we give the terms of the
solution in the J2000 ecliptic and equinox reference frame for amplitudes down
to $10^{-8}$.

Table 3: Quasi-periodic decomposition of Jupiter’s eccentricity and longitude
of perihelion (variable $z$).

$\begin{array}[]{rrrr}\hline\cr\hline\cr k&\mu_{k}\
(^{\prime\prime}\cdot\text{yr}^{-1})&E_{k}\times 10^{8}&\theta_{k}^{(0)}\
(^{\text{o}})\\\ \hline\cr 1&4.24882&4411915&30.67\\\
2&28.22069&1574994&308.11\\\ 3&3.08952&180018&121.36\\\
4&52.19257&51596&45.55\\\ 5&27.06140&18405&218.71\\\
6&29.37998&17762&217.54\\\ 7&28.86795&10743&32.64\\\ 8&27.57346&9436&43.74\\\
9&5.40817&6135&120.31\\\ 10&0.66708&5755&73.98\\\ 11&53.35188&4415&314.90\\\
12&76.16447&2441&143.03\\\ 13&51.03334&1354&316.29\\\ 14&7.45592&1354&20.24\\\
15&-19.72306&1083&293.24\\\ 16&4.89647&982&291.61\\\ 17&5.59644&941&290.35\\\
18&1.93168&767&198.10\\\ 19&3.60029&543&121.39\\\ 20&-56.90922&485&44.11\\\
21&2.97706&470&306.81\\\ 22&5.47449&354&95.01\\\ 23&17.91550&295&155.35\\\
24&5.71670&269&300.52\\\ 25&-20.88236&222&203.93\\\ 26&6.93423&173&349.25\\\
27&1.82121&161&150.50\\\ 28&5.35823&145&274.88\\\ 29&7.05595&139&178.82\\\
30&7.34103&114&27.85\\\ 31&17.36469&101&123.95\\\ 32&0.77840&99&65.10\\\
33&7.57299&80&191.47\\\ 34&5.99227&53&293.56\\\ 35&5.65485&51&219.22\\\
36&4.36906&49&40.82\\\ 37&5.23841&43&92.97\\\ 38&6.82468&37&14.53\\\
39&-0.49216&29&164.74\\\ 40&17.08266&28&179.38\\\ 41&16.81285&27&273.37\\\
42&7.20563&23&323.91\\\ 43&7.71663&15&273.52\\\ 44&19.01870&10&219.75\\\
45&17.15752&10&325.02\\\ 46&16.52731&6&131.91\\\ 47&17.63081&6&183.87\\\
48&17.81084&6&58.56\\\ 49&18.18553&5&57.27\\\ 50&17.47683&5&260.26\\\
51&17.72293&4&48.46\\\ 52&17.55234&4&197.65\\\ 53&18.01611&4&44.83\\\
54&16.26122&2&58.89\\\ 55&18.08627&2&356.17\\\ 56&18.46794&1&209.01\\\
\hline\cr\end{array}$

555This solution has been directly obtained from Laskar (1990) as explained in
the text. The phases $\theta_{k}^{(0)}$ are given at time J2000.

Table 4: Quasi-periodic decomposition of Jupiter’s inclination and longitude
of ascending node (variable $\zeta$).

$\begin{array}[]{rrrr}\hline\cr\hline\cr k&\nu_{k}\
(^{\prime\prime}\cdot\text{yr}^{-1})&S_{k}\times 10^{8}&\phi_{k}^{(0)}\
(^{\text{o}})\\\ \hline\cr 1&0.00000&1377467&107.59\\\
2&-26.33023&315119&307.29\\\ 3&-0.69189&58088&23.96\\\
4&-3.00557&48134&140.33\\\ 5&-26.97744&2308&222.98\\\ 6&-2.35835&1611&44.74\\\
7&82.77163&1372&308.95\\\ 8&-1.84625&1130&36.64\\\ 9&-5.61755&1075&168.70\\\
10&-4.16482&946&51.54\\\ 11&58.80017&804&32.90\\\ 12&-50.30212&691&29.84\\\
13&34.82788&636&114.12\\\ 14&-0.58033&565&17.32\\\ 15&-7.07963&454&273.79\\\
16&-27.48935&407&38.53\\\ 17&-25.17116&385&35.94\\\ 18&-5.50098&383&162.89\\\
19&-3.11725&321&326.97\\\ 20&-6.84091&267&106.20\\\ 21&-28.13656&256&134.07\\\
22&-7.19493&226&105.14\\\ 23&-6.96094&215&97.96\\\ 24&0.46547&162&286.88\\\
25&-17.74818&149&123.28\\\ 26&-7.33264&144&196.75\\\ 27&-5.85017&130&345.47\\\
28&11.50319&103&281.01\\\ 29&-5.21610&97&198.91\\\ 30&-5.37178&97&215.48\\\
31&-5.10025&94&15.38\\\ 32&0.57829&56&103.72\\\ 33&-5.96899&55&170.64\\\
34&-1.19906&53&133.26\\\ 35&-6.73842&47&44.50\\\ 36&-7.40536&44&233.35\\\
37&-7.48780&40&47.95\\\ 38&-6.15490&40&269.77\\\ 39&20.96631&40&57.78\\\
40&-6.56016&38&303.47\\\ 41&9.18847&32&1.15\\\ 42&-8.42342&32&211.21\\\
43&10.34389&32&190.85\\\ 44&-18.85115&23&240.06\\\ 45&-17.19656&17&334.19\\\
46&18.14984&15&291.19\\\ 47&-19.40256&14&207.96\\\ 48&-18.01114&11&242.09\\\
49&-17.66094&11&138.93\\\ 50&-17.83857&9&289.13\\\ 51&-17.54636&8&246.71\\\
52&-18.30007&6&267.05\\\ 53&-17.94404&5&212.26\\\ 54&-18.59563&5&98.11\\\
55&-19.13075&1&305.90\\\ \hline\cr\end{array}$

666This solution has been directly obtained from Laskar (1990) as explained in
the text. The phases $\phi_{k}^{(0)}$ are given at time J2000.

##  Appendix B Crossing the resonance with $\phi_{4}$

Figures 1 and 5 show the location and width of all first-order secular spin-
orbit resonances produced by Jupiter’s orbital solution (Appendix A). In
particular, Jupiter is located very close to the large resonance with
$\phi_{4}$, whose frequency is the nodal precession mode of Uranus. Figures 9
and 9 show the geometry of the resonance with $\phi_{4}$ for different values
of the precession constant $\alpha$ of Jupiter. These graphs can be understood
as horizontal sections of Fig. 5, where we can locate the centre of the
resonance (i.e. Cassini state 2) and the separatrix width. For easier
comparison, Figs. 5 and 9 share the same horizontal axis.

Figure 8: Level curves of the Hamiltonian function in the vicinity of
resonance with $\phi_{4}$. The resonant angle is $\sigma_{4}=\psi+\phi_{4}$.
Other terms are averaged and included up to the third order of their
amplitudes (see Saillenfest et al. 2019). Each panel corresponds to a
different value of the precession constant $\alpha$. Equilibrium points
(called Cassini states) are shown by black spots. The interior of the
resonance is coloured red and the separatrix is shown with a thick red curve.
The location and width of the resonance for continuous values of $\alpha$ can
be seen in Fig. 5. In order to avoid being misled by coordinate singularities,
Fig. 9 shows the same level curves in a different set of coordinates.

Figure 9: Same as Fig. 9, but using polar coordinates that are not singular
for an obliquity $\varepsilon=0^{\circ}$. The outer black circle corresponds
to an obliquity $\varepsilon=40^{\circ}$.

##  Appendix C Crossing the resonance with $\phi_{19}$

As a matter of fact, Jupiter’s orbital motion is not restricted to the
$\phi_{4}$ term. However, secular spin-orbit resonances with all other terms
(apart from $\phi_{19}$) are located very far from the location of Jupiter, so
that their effects average over time. The case of $\phi_{19}$ is special: even
though it is very thin, this resonance is not far from Jupiter’s location (see
the upper red curve in Fig. 5), which means that $\psi+\phi_{19}$ is a slow
angle that cannot be averaged out.

Instead of considering only $\phi_{4}$, as in Appendix B, a more rigorous
model of the long-term spin-axis dynamics of Jupiter consists in averaging the
Hamiltonian function over all angles except resonances with both $\phi_{4}$
and $\phi_{19}$. For a constant value of $\alpha$, this results in a two-
degree-of-freedom Hamiltonian system, in which the two angle coordinates are
$\sigma_{4}=\psi+\phi_{4}$ and $\sigma_{19}=\psi+\phi_{19}$. The dynamics can
then be studied using Poincaré surfaces of section. Figure 10 shows two
examples of sections. The lower island centred at $\sigma_{19}=0$ corresponds
to the thin resonance with $\phi_{19}$: As expected, it is completely
distorted as compared to the unperturbed separatrix (blue curve) due to the
proximity of the large resonance with $\phi_{4}$. It still persists, however,
as a set of periodic orbits. In contrast, the large resonance with $\phi_{4}$
is hardly affected at all by the $\phi_{19}$ term, which only transforms its
separatrix into a thin chaotic belt. In the left panel of Fig. 10, we can also
recognise Cassini state 1 with $\phi_{4}$ (for $\sigma_{4}=\pi$ and a small
obliquity), that is also visible in Fig. 9.

Figure 10: Poincaré surfaces of section showing the dynamics in the vicinity
of resonances with $\phi_{4}$ and $\phi_{19}$. Each graph corresponds to a
different value of $\alpha$ (see titles). The separatrices of the two
resonances taken separately are shown with coloured curves: red for $\phi_{4}$
and blue for $\phi_{19}$.

We investigated whether Jupiter could be trapped into the thin resonance with
$\phi_{19}$ and follow its resonance centre, but we found out that this can
never happen. On the one hand, the current phase of $\sigma_{19}$ is close to
$\pi$, so that even if $\lambda$ is finely tuned to place Jupiter right inside
the resonance, it ends up near the separatrix, leading to an unstable resonant
motion. On the other hand, as shown in Fig. 6, the libration period of
$\sigma_{19}$ is extremely large (the width and oscillation frequency both
scale as the square root of the amplitude $S_{19}$). This means that, as
$\alpha$ increases, the crossing of this resonance is not adiabatic. The
libration periods shown in Fig. 6 should be compared to the time needed for
$\alpha$ to go through the resonant region. According to Fig. 4, the mean
increase rate of $\alpha$ is $0.086^{\prime\prime}\cdot$yr${}^{-1}\cdot$Gyr-1
and according to Fig. 5, the resonances have a vertical width
$\Delta\alpha\approx 0.21^{\prime\prime}\cdot$yr-1 for the $\phi_{4}$
resonance and $\Delta\alpha\approx 0.01^{\prime\prime}\cdot$yr-1 for the
$\phi_{19}$ resonance (computed at the right separatrix when it appears).
Therefore, the time that would be needed for $\alpha$ to cross the $\phi_{4}$
resonance is $\Delta t\approx 2.5$ Gyrs, which corresponds to many oscillation
periods of $\sigma_{4}$ (about $100$): this is the adiabatic regime. On the
contrary, the time needed for $\alpha$ to cross the $\phi_{19}$ resonance is
$\Delta t\approx 0.1$ Gyrs, which corresponds to less than one oscillation
period of $\sigma_{19}$ (about $0.2$): this is the non-adiabatic regime. As a
result, a resonance capture with $\phi_{19}$ is extremely unlikely, even if
the orbital motion of Jupiter was restricted to its $19$th harmonic: Jupiter’s
spin axis enters the resonance and exits before $\sigma_{19}$ has time to
oscillate. With suitable initial conditions, Jupiter roughly follows the
resonance centre during the crossing, producing a bump in the obliquity
evolution (see Fig. 5 for $\lambda=0.227$), but nothing more can possibly
happen. This kind of non-adiabatic resonance crossing is described by Ward et
al. (1976), Laskar et al. (2004b), and Ward & Hamilton (2004) using Fresnel
integrals.


#  A new approach to construct minimal linear codes over $\mathbb{F}_{3}$

Wajid M. Shaikh  Rupali S. Jain  Associate Professor, School of Mathematical
Sciences, SRTMU Nanded  B. Surendranath Reddy  Assistant Professor, School of
Mathematical Sciences, SRTMU Nanded  Bhagyashri S. Patil  Sahar M. A. Maqbol
Department of Mathematics, Hodeidah University, P.O. 3114, Al-Hodeidah, Yemen.

###### Abstract

In this article, we present two new approaches to construct minimal linear
codes of dimension $n+1$ over $\mathbb{F}_{3}$ using characteristic and
ternary functions. We also obtain the weight distributions of these
constructed minimal linear codes. We further show that a specific class of
these codes violates Ashikhmin-Barg condition.

##  1 Introduction

Linear codes are fundamental in contemporary communication systems offering
vital error control capabilities, crucial for ensuring dependable data
transmission across a spectrum of applications from telecommunications to
digital broadcasting. Among these, minimal linear codes stand out as a
distinct class, finding significant utility in secret sharing, data storage,
and information processing systems within modern technology landscapes
[4],[8]. As such, the development and analysis of minimal linear codes have
been subject to extensive research efforts by scholars [5],[6],[7].

In 1998, Ashikhmin and Barg[1] gave a sufficient condition for linear code to
be minimal, which is stated below.

######  Lemma 1.1.

[1][Ashikhmin-Barg] Let $p$ be a prime, then a linear code $\mathscr{C}$ over
$\mathbb{F}_{p}$ is minimal if

$\frac{{wt}_{min}}{{wt}_{max}}>\frac{p-1}{p}$

where, ${wt}_{min}$ and ${wt}_{max}$ denote the minimum and maximum non-zero
Hamming weights for $\mathscr{C}$, respectively.

Many minimal linear codes can be constructed based on the lemma mentioned
above, however, the construction of minimal linear codes that violate the
Ashikhmin-Barg condition presents an intriguing research avenue. In the year
2018, Chang and Hyun[2] constructed an infinite family of minimal binary
linear codes, which do not satisfy Ashikhmin-Barg condition by using
mathematical objects called simplicial complexes and generic construction

$\mathscr{C}_{f}=\\{(\alpha f({\bf x})+{\bf w}\cdot{\bf x})_{\bf
x\in\mathbb{F}_{p}^{n*}}\ |\ \alpha\in\mathbb{F}_{p},\ {\bf
w}\in\mathbb{F}_{p}^{n}\\}$ (1.1)

The necessary and sufficient condition for binary linear code to be minimal
was given by Ding, Heng, and Zhou [9] in 2018, and they constructed three
classes of minimal binary linear codes. Heng, Z., Ding, C., & Zhou, Z. [3],
derived necessary and sufficient condition for ternary linear codes to be
minimal and constructed a class of minimal linear codes, which violating
Ashikhmin-Barg condition from Lloyd polynomial. Also several researcher,
construct minimal linear codes using different approaches [5],[6],[7]

In this paper, using the function defined in [9], we construct a class of
minimal linear codes over $\mathbb{F}_{3}$, which satisfies
$\frac{\text{wt}_{min}}{\text{wt}_{max}}\leq\frac{1}{3}<\frac{2}{3}$. Also we
define new ternary function and construct minimal linear codes over
$\mathbb{F}_{3}$, which violates Ashikhmin-Barg.

This paper is organised as follows: In Section 2, we give some basic
definitions and results, which are prerequisites. Section 4 contains class of
minimal linear codes over $\mathbb{F}_{3}$, using ternary functions. In
Section 5, we obtain a class of minimal linear codes, which violates
Ashikhmin-Barg condition using characteristic functions.

##  2 Preliminaries

In this section, we provide some definitions and results that are necessary
prior knowledge for understanding this article. We denote a finite field with
$p$ elements by $\mathbb{F}_{p}$. A $[n,k,d]$ linear code over
$\mathbb{F}_{3}$ is a $k$-dimensional subspace of $\mathbb{F}_{3}^{n}$, where
$d$ is the minimum Hamming distance. Let $A_{i}$ denote the count of the
codewords having weight $i$ in $\mathscr{C}$. The weight enumerator of a code
$\mathscr{C}$ is defined as $1+A_{1}z+A_{2}z+\cdots+A_{n}z^{n}$, and the
sequence $(1,A_{1},A_{2},\ldots,A_{n})$ is called the weight distribution of a
code $\mathscr{C}$. If there are $t$ non-zero $A_{i}$, then $\mathscr{C}$ is
called the $t$-weight code.

######  Definition 2.1.

[10] The $support$ $(\text{Supp}({\bf c}))$ of a codeword ${\bf c}$ is the set
of the coordinates at which ${c}$ is non-zero. i.e.

$\text{Supp}({\bf c})=\\{i\ |\ c_{i}\neq 0,\ 1\leq i\leq n\\}$

Note that, $\text{wt}({\bf c})=|\text{Supp}({\bf c})|$, where $\text{wt}({\bf
x})$ is the Hamming weight of ${\bf c}$.

######  Definition 2.2.

For ${\bf w}_{1},{\bf w}_{2}\in\mathbb{F}_{p}^{n}$, if $\text{Supp}({\bf
w}_{1})\subseteq\text{Supp}({\bf w}_{2})$, then we say ${\bf w}_{2}$ covers
${\bf w}_{1}$ and denoted by ${\bf w}_{2}\preceq{\bf w}_{1}$.

######  Definition 2.3.

A codeword ${\bf w}$ is minimal if ${\bf w}$ covers only the codeword of the
form $\alpha{\bf w}$, where $\alpha\in\mathbb{F}_{p}^{*}$. A code
$\mathscr{C}$ is called minimal if every codeword in $\mathscr{C}$ is minimal.

######  Definition 2.4.

A ternary function $f$ is a function from $\mathbb{F}_{3}^{n}$ to
$\mathbb{F}_{3}$. The Walsh transform for a ternary function is given by

$\widehat{f}({\bf w})=\sum_{{\bf x}\in\mathbb{F}_{3}^{n}}\zeta_{3}^{f({\bf
x})-{\bf w}\cdot{\bf x}}$

where, $\zeta_{3}$ is a cube root of unity, ${\bf w}\in\mathbb{F}_{3}^{n}$ and
${\bf w}\cdot{\bf x}$ is the standard inner product of ${\bf w}$ and ${\bf
x}$.

######  Lemma 2.5.

[3] For any ${\bf w}_{1},{\bf w}_{2}\in\mathbb{F}_{q}^{n}$, ${\bf
w}_{1}\preceq{\bf w}_{2}$ if and only if

$\sum_{{c\in\mathbb{F}_{q}^{*}}}\text{wt}({\bf w}_{2}+c{\bf
w}_{1})=(q-1)\text{wt}({\bf w}_{2})-\text{wt}({\bf w}_{1})$

######  Theorem 2.6.

[3] Assume that $f({\bf x})\neq{\bf w}\cdot{\bf x}$ for any ${\bf
w}\in\mathbb{F}_{3}^{n}$. The linear code $\mathscr{C}_{f}$ in 1.1 has length
$3^{n}-1$ and dimension $n+1$. In addition, the weight distribution of
$\mathscr{C}_{f}$ is given by the following multiset union:

$\displaystyle\left\\{\ 2\left(3^{n-1}-\frac{\text{Re}(\widehat{f}({\bf
w}))}{3}\right)\ |\ \alpha=-1,{\bf w}\in\mathbb{F}_{3}^{n}\right\\}\cup$
$\displaystyle\left\\{\ 2\left(3^{n-1}-\frac{\text{Re}(\widehat{f}({-\bf
w}))}{3}\right)\ |\ \alpha=1,{\bf
w}\in\mathbb{F}_{3}^{n}\right\\}\cup\\{3^{n}-3^{n-1}\ |\ \alpha=0,{\bf
w}\in\mathbb{F}_{3}^{n}\\}$

######  Theorem 2.7.

[3] Let $\mathscr{C}_{f}$ be the ternary code in 1.1. Assume that $f({\bf
x})\neq{\bf w}\cdot{\bf x}$ for any ${\bf w}\in\mathbb{F}_{3}^{n}$. Then
$\mathscr{C}_{f}$ is a minimal $[3^{n}-1,n+1]$ code if and only if

$\displaystyle\text{Re}(\widehat{f}({\bf w}_{1}))+\text{Re}(\widehat{f}({\bf
w}_{2}))+\text{Re}(\widehat{f}({\bf w}_{3}))$ $\displaystyle\neq 3^{n}$

and

$\displaystyle\text{Re}(\widehat{f}({\bf w}_{1}))+\text{Re}(\widehat{f}({\bf
w}_{2}))-2\text{Re}(\widehat{f}({\bf w}_{3}))$ $\displaystyle\neq 3^{n}$

for all pairwise distinct ${\bf w}_{1},{\bf w}_{2}$ and ${\bf w}_{3}$ with
${\bf w}_{1}+{\bf w}_{2}+{\bf w}_{3}={\bf 0}$.

##  3 Construction of minimal linear codes over $\mathbb{F}_{3}$ from ternary
functions

In this section, first we define ternary functions using partial spread and
using the ternary functions we construct minimal linear codes over
$\mathbb{F}_{3}$, which violates Ashikhmin-Barg condition. Throughout this
paper, we assume that $n$ be a positive even integer and $t=\frac{n}{2}$. A
partial spread of order $\nu$ in $\mathbb{F}_{3}^{n}$ is a collection of $\nu$
disjoint $t$-dimensional subspaces $W_{1},W_{2},\ldots,W_{\nu}$ of
$\mathbb{F}_{3}^{n}$. Also, observe that $\nu\leq 3^{t}+1$.
Let $f_{ij}:\mathbb{F}_{3}^{n}\rightarrow\mathbb{F}_{3}$ be ternary function
given by

$f_{ij}({\bf x})=\begin{cases}1,&\text{ if }{\bf x}\in
W_{i}\setminus\\{0\\}\\\ 2,&\text{ if }{\bf x}\in W_{j}\setminus\\{0\\}\\\
0,&\text{Otherwise}\end{cases}$

where, $i\neq j$ and $1\leq i,j\leq 3^{t}+1$. Note that $2f_{ij}=f_{ji}$.
For $1\leq s\leq\frac{3^{t}+1}{2}$ and
$\mathcal{A}=\\{i_{1},i_{2},\ldots,i_{s},i_{s+1},\ldots,i_{2s}\\}$, we define
ternary function $f$ as follows

$f({\bf x})=\sum_{k=1}^{s}f_{i_{k}i_{s+k}}({\bf x})$ (3.1)

Using the function $f$ defined above, we will construct minimal linear code
$\mathscr{C}_{f}$ over $\mathbb{F}_{3}$. For construction of minimal linear
code $\mathscr{C}_{f}$, we need the values of $\text{Re}(\widehat{f}({\bf
x}))$. The following lemma help us to find these values.

######  Theorem 3.1.

Let $\mathbb{F}_{p}^{n}$ be an $n$-dimensional vector space over a finite
field $\mathbb{F}_{p}$ and $W$ be a $m$-dimensional subspace of
$\mathbb{F}_{p}^{n}$. If ${\bf y}\notin W^{\perp}$, then
$|\mathcal{S}_{k}|=p^{m-1}$, for all $k\in\mathbb{F}_{p}$, where
$\mathcal{S}_{k}=\\{{\bf x}\in W\ |\ {\bf y}\cdot{\bf x}=k\\}$.

###### Proof.

We prove this theorem by mathematical induction on $m$.
If $m=1$, then we have $W=\text{span}(\\{{\bf w}\\})$. Since ${\bf y}\notin
W^{\perp}$, we have ${\bf y}\cdot{\bf w}\neq 0$, which implies ${\bf
y}\cdot{\bf w}=k$, for some $k\in\mathbb{F}^{*}_{p}$. From this we get ${\bf
x}\cdot\alpha{\bf w}=\alpha k$, for all $\alpha\in\mathbb{F}_{p}$. Hence the
result is true for $m=1$.
Assume that result is true for $m-1$ and we prove the result for $m$.
Let $W$ be a $m$-dimensional subspace of $\mathbb{F}_{p}^{n}$. Then
$W=\text{span}(\\{{\bf w_{1}},{\bf w_{2}},\ldots,{\bf w_{m}}\\})$. Since ${\bf
y}\notin W^{\perp}$, we have ${\bf y}\cdot{\bf w}_{i}\neq 0$, for some $1\leq
i\leq m$. Without loss of generality, consider the case ${\bf y}\cdot{\bf
w}_{i}\neq 0$, for some $1\leq i\leq m-1$ and ${\bf y}\cdot{\bf w}_{m}=k$, for
some $k\in\mathbb{F}_{p}$. Now $W^{\prime}=\text{span}(\\{{\bf w}_{1},{\bf
w}_{2},\ldots,{\bf w}_{m-1}\\})$ is $(m-1)$-dimensional subspace of
$\mathbb{F}_{p}^{n}$. Hence, the result is true for $W^{\prime}$, from this we
have,
$\mathcal{S}^{\prime}_{0},\mathcal{S}^{\prime}_{1},\ldots,\mathcal{S}^{\prime}_{p-1}$
be a collection of subsets of $W^{\prime}$, with
$|\mathcal{S}^{\prime}_{i}|=p^{m-2}$, for all $0\leq i\leq p-1$ such that
${\bf y}\cdot{\bf x}=i$, for all ${\bf x}\in\mathcal{S}^{\prime}_{i}$ and
$0\leq i\leq p-1$. Note that
$\mathcal{S}^{\prime}_{0},\mathcal{S}^{\prime}_{1},\ldots,\mathcal{S}^{\prime}_{p-1}$
is also subsets of $W$. Now, consider the sets ${\bf
w}_{m}+\mathcal{S}^{\prime}_{0},{\bf
w}_{m}+\mathcal{S}^{\prime}_{1},\ldots,{\bf w}_{m}+\mathcal{S}^{\prime}_{p-1}$
be a collection of subsets of $W$, with $|{\bf
w}_{m}+\mathcal{S}^{\prime}_{i}|=p^{m-2}$ and ${\bf y}\cdot({\bf w}_{m}+{\bf
x})=k+i$, for all $0\leq i\leq p-1$ and for all ${\bf
x}\in\mathcal{S}^{\prime}_{i}$. Also, observe that
$\mathcal{S}^{\prime}_{i}\neq{\bf w}_{m}+\mathcal{S}^{\prime}_{j}$, for $0\leq
i,j\leq p-1$. Similarly, $2{\bf w}_{m}+\mathcal{S}^{\prime}_{0},2{\bf
w}_{m}+\mathcal{S}^{\prime}_{1},\ldots,2{\bf
w}_{m}+\mathcal{S}^{\prime}_{p-1}$ be a collection of subsets of $W$, with
$|2{\bf w}_{m}+\mathcal{S}^{\prime}_{i}|=p^{m-2}$ and ${\bf y}\cdot(2{\bf
w}_{m}+{\bf x})=2k+i$, for all $0\leq i\leq p-1$ and for all ${\bf
x}\in\mathcal{S}^{\prime}_{i}$. Continuing in this manner, we get
$\mathcal{S}_{0},\mathcal{S}_{1},\ldots,\mathcal{S}_{p-1}$ be a subset of $W$,
with $|\mathcal{S}_{i}|=p^{m-1}$ such that ${\bf y}\cdot{\bf x}=i$, for ${\bf
x}\in\mathcal{S}_{i}$. Hence, the result. ∎

######  Remark 3.2.

Note that, if $W_{1}$ and $W_{2}$ are disjoint subspace of a finite
dimensional vector space $V$ over a field $\mathbb{F}_{p}$ such that

$\text{dim}(W_{1})+\text{dim}(W_{2})=\text{dim}(V)$

then $W^{\perp}_{1}$ and $W^{\perp}_{2}$ are also disjoint subspaces of $V$.

Now following theorem give the weight distribution of $\mathscr{C}_{f}$ and
the values of Walsh transform.

######  Theorem 3.3.

Let $f$ be a ternary function defined in 3.1. Then $\mathscr{C}_{f}$ is a
$[3^{n}-1,n+1]$ linear code over $\mathbb{F}_{3}$ with weight distribution
given in Table 1.

Weight of ${\bf w}$ |  Multiplicity $A_{{\bf w}}$
---|---
0 | 1
$2s(3^{t}-1)$ | 2
$3^{n}-3^{n-1}$ | $3^{n}-1$
$3^{n}-3^{n-1}-2s$ | $4s(3^{t}-1)$
$3^{n}-3^{n-1}+3^{t}-2s$ | $2(3^{t}+1-2s)(3^{t}-1)$
Table 1: Weight Distribution of $\mathscr{C}_{f}$

###### Proof.

Note that $f({\bf x})\neq{\bf v}\cdot{\bf x}$, for all ${\bf
v}\in\mathbb{F}_{3}^{n}$. Hence, from Theorem 2.6, we have $\mathscr{C}_{f}$
is a $[3^{n}-1,n+1]$ linear code over $\mathbb{F}_{3}$. Now, we find the
weight distribution of the linear code $\mathscr{C}_{f}$. For any ${\bf
w}\in\mathbb{F}_{3}^{n}$, we have three cases

  1. (I)

${\bf w}={\bf 0}$

  2. (II)

${\bf w}\notin W_{i}^{\perp}$, for all $i\in\mathcal{A}$.

  3. (III)

${\bf w}\in W_{i}^{\perp}\setminus\\{{\bf 0}\\}$, for some $i\in\mathcal{A}$.

Case I: If ${\bf w}=0$, then from the definition of $f_{ij}$ and $f$, we have

$\displaystyle\widehat{f}({\bf 0})$ $\displaystyle=\sum_{{\bf
x}\in\mathbb{F}_{3}^{n}}\zeta_{3}^{\sum_{k=1}^{s}f_{i_{k}i_{s+k}}({\bf
x})-{\bf 0}\cdot{\bf x}}$ $\displaystyle=1+\sum_{{\bf x}\in
W_{i_{1}}\setminus\\{0\\}}\zeta_{3}^{1}+\sum_{{\bf x}\in
W_{i_{s+1}}\setminus\\{0\\}}\zeta_{3}^{2}+\cdots+\sum_{{\bf x}\in
W_{i_{s}}\setminus\\{0\\}}\zeta_{3}^{1}+\sum_{{\bf x}\in
W_{i_{2s}}\setminus\\{0\\}}\zeta_{3}^{2}+\sum_{{\bf
x}\notin\bigcup_{k=1}^{2s}W_{i_{k}}}\zeta_{3}^{0}$

Using Theorem 3.1, we get

$\displaystyle\widehat{f}({\bf 0})$
$\displaystyle=1+\left(3^{t}-1\right)\zeta_{3}+\left(3^{t}-1\right)\zeta_{3}^{2}+\cdots+\left(3^{t}-1\right)\zeta_{3}+\left(3^{t-1}-1\right)\zeta_{3}^{2}+(3^{t}+1-2s)\left(3^{t}-1\right)$
$\displaystyle=1+s(3^{t}-1)(\zeta_{3}+\zeta_{3}^{2})+(3^{t}+1-2s)\left(3^{t}-1\right)$
$\displaystyle=3^{n}-3s(3^{t}-1)$

Case II: If ${\bf w}\notin W_{i}^{\perp}$, for all $i\in\mathcal{A}$, then
${\bf w}\in W^{\perp}_{j}$, for some $j\notin\mathcal{A}$. Without loss of
generality assume that ${\bf w}\in W^{\perp}_{i_{2s+1}}$. We have

$\displaystyle\widehat{f}({\bf w})$ $\displaystyle=\sum_{{\bf
x}\in\mathbb{F}_{3}^{n}}\zeta_{3}^{\sum_{k=1}^{s}f_{i_{k}i_{s+k}}({\bf
x})-{\bf w}\cdot{\bf x}}$ $\displaystyle=1+\sum_{{\bf x}\in
W_{i_{1}}\setminus\\{0\\}}\zeta_{3}^{1-{\bf w}\cdot{\bf x}}+\sum_{{\bf x}\in
W_{i_{s+1}}\setminus\\{0\\}}\zeta_{3}^{2-{\bf w}\cdot{\bf x}}\cdots+\sum_{{\bf
x}\in W_{i_{s}}\setminus\\{0\\}}\zeta_{3}^{1-{\bf w}\cdot{\bf x}}$
$\displaystyle+\sum_{{\bf x}\in W_{i_{2s}}\setminus\\{0\\}}\zeta_{3}^{2-{\bf
w}\cdot{\bf x}}+\sum_{{\bf x}\in
W_{i_{2s+1}}\setminus\\{0\\}}\zeta_{3}^{0}+\sum_{{\bf
x}\notin\bigcup_{k=1}^{2s+1}W_{i_{k}}}\zeta_{3}^{-{\bf w}\cdot{\bf x}}$

From Theorem 3.1 and Remark 3.2, we get

$\displaystyle\widehat{f}({\bf w})$
$\displaystyle=1+\left(3^{t-1}\zeta_{3}^{2}+(3^{t-1}-1)\zeta_{3}+3^{t-1}\right)+\left((3^{t-1}-1)\zeta_{3}^{2}+3^{t-1}\zeta_{3}+3^{t-1}\right)+\cdots$
$\displaystyle+\left(3^{t-1}\zeta_{3}^{2}+(3^{t-1}-1)\zeta_{3}+3^{t-1}\right)+\left((3^{t-1}-1)\zeta_{3}^{2}+3^{t-1}\zeta_{3}+3^{t-1}\right)$
$\displaystyle+(3^{t}-1)+(3^{t}+1-(2s+1))\left(3^{t-1}\zeta_{3}^{2}+3^{t-1}\zeta_{3}+(3^{t-1}-1)\right)$
$\displaystyle=1-\zeta_{3}-\zeta_{3}^{2}-\cdots-\zeta_{3}-\zeta_{3}^{2}+(3^{t}-1)+(3^{t}+1-(2s+1))\left(3^{t-1}\zeta_{3}^{2}+3^{t-1}\zeta_{3}+(3^{t-1}-1)\right)$
$\displaystyle=3s$

Case III: Let ${\bf w}\in W^{\perp}_{i}\setminus\\{{\bf 0}\\}$ for some
$i\in\mathcal{A}$. Suppose ${\bf w}\in W^{\perp}_{i}\setminus\\{{\bf 0}\\}$,
for some $i\in\\{i_{1},i_{2},\ldots,i_{s}\\}$. Without loss of generality
assume that ${\bf w}\in W^{\perp}_{i_{1}}\setminus\\{{\bf 0}\\}$.
We have

$\displaystyle\widehat{f}({\bf w})$ $\displaystyle=\sum_{{\bf
x}\in\mathbb{F}_{3}^{n}}\zeta_{3}^{\sum_{k=1}^{s}f_{i_{k}i_{s+k}}({\bf
x})-{\bf w}\cdot{\bf x}}$ $\displaystyle=1+\sum_{{\bf x}\in
W_{i_{1}}\setminus\\{0\\}}\zeta_{3}^{1}+\sum_{{\bf x}\in
W_{i_{s+1}}\setminus\\{0\\}}\zeta_{3}^{2-{\bf w}\cdot{\bf x}}\cdots+\sum_{{\bf
x}\in W_{i_{s}}\setminus\\{0\\}}\zeta_{3}^{1-{\bf w}\cdot{\bf x}}$
$\displaystyle+\sum_{{\bf x}\in W_{i_{2s}}\setminus\\{0\\}}\zeta_{3}^{2-{\bf
w}\cdot{\bf x}}+\sum_{{\bf
x}\notin\bigcup_{k=1}^{2s}W_{i_{k}}}\zeta_{3}^{1-{\bf w}\cdot{\bf x}}$
$\displaystyle=1+(3^{t}-1)\zeta_{3}+\left((3^{t-1}-1)\zeta_{3}^{2}+3^{t-1}\zeta_{3}+3^{t-1}\right)+\cdots$
$\displaystyle+\left(3^{t-1}\zeta_{3}^{2}+(3^{t-1}-1)\zeta_{3}+3^{t-1}\right)+\left((3^{t-1}-1)\zeta_{3}^{2}+3^{t-1}\zeta_{3}+3^{t-1}\right)$
$\displaystyle+(3^{t}+1-2s)\left(3^{t-1}\zeta_{3}^{2}+3^{t-1}\zeta_{3}+(3^{t-1}-1)\right)$
$\displaystyle=3^{t}(\zeta_{3}^{2}-1)+3s$

Similarly, if ${\bf w}\in W^{\perp}_{i}$, for some
$i\in\\{i_{s+1},i_{s+2},\ldots,i_{2s}\\}$, then we get,
$\widehat{f}({\bf w})=3^{t}(\zeta_{3}-1)+3s$. Hence, we get

$\text{Re}(\widehat{f}({\bf w}))=\begin{cases}3^{n}-3s(3^{t}-1),&\text{ if
}{\bf w}={\bf 0}\\\ 3s,&\text{ if }{\bf w}\notin W^{\perp}_{i},\text{ for all
}i\in\mathcal{A}\\\ -\frac{3^{t+1}}{2}+3s,&\text{ if }{\bf w}\in
W^{\perp}_{i}\setminus\\{0\\},\text{ for some }i\in\mathcal{A}\end{cases}$

The weight distribution of $\mathscr{C}_{f}$ follows from Theorem 2.6. ∎

We now prove the following two lemmas, which are crucial in proving
$\mathscr{C}_{f}$ is minimal.

######  Lemma 3.4.

Let $n\geq 6$ and $f$ be a ternary function defined in 3.1. If
$s\neq\frac{3^{t}+1}{2}$, then

$\text{Re}(\widehat{f}({\bf w}_{1}))+\text{Re}(\widehat{f}({\bf
w}_{2}))+\text{Re}(\widehat{f}({\bf w}_{3}))\neq 3^{n}$

for all pairwise distinct ${\bf w}_{1},{\bf w}_{2},{\bf
w}_{3}\in\mathbb{F}_{3}^{n}$ such that ${\bf w}_{1}+{\bf w}_{2}+{\bf
w}_{3}={\bf 0}$.

###### Proof.

For different choices of ${\bf w}_{1},{\bf w}_{2}$ and ${\bf w}_{3}$, we have
the following cases:
Case I: For ${\bf w}_{i}={\bf 0}$, for some $i$. Without loss of generality
assume that ${\bf w}_{1}={\bf 0}$, then ${\bf w}_{2}=-{\bf w}_{3}$. Now,
consider the following subcases:
(a) If ${\bf w}_{2}\notin W^{\perp}_{i}$, for all $i\in\mathcal{A}$, then

$\displaystyle\text{Re}(\widehat{f}({\bf 0}))+\text{Re}(\widehat{f}({\bf
w}_{2}))+\text{Re}(\widehat{f}({\bf w}_{3}))=3^{n}-3s(3^{t}-1)+3s+3s\neq
3^{n}\text{ as }s\neq 0$

(b) If ${\bf w}_{2}\in W^{\perp}_{i}$, for some $i\in\mathcal{A}$, then

$\displaystyle\text{Re}(\widehat{f}({\bf 0}))+\text{Re}(\widehat{f}({\bf
w}_{2}))+\text{Re}(\widehat{f}({\bf
w}_{3}))=3^{n}-3s(3^{t}-1)-\frac{3^{t+1}}{2}+3s-\frac{3^{t+1}}{2}+3s\neq
3^{n}$ $\displaystyle\text{ as }s\neq 3^{t-1}(s+1)$

Case II: If ${\bf w}_{i}\neq 0$, for all $i\in\\{1,2,3\\}$, then consider the
following subcases:
(a) If ${\bf w}_{1},{\bf w}_{2},{\bf w}_{3}\notin W^{\perp}_{i}$, for all
$i\in\mathcal{A}$, then

$\displaystyle\text{Re}(\widehat{f}({\bf w}_{1}))+\text{Re}(\widehat{f}({\bf
w}_{2}))+\text{Re}(\widehat{f}({\bf w}_{3}))=3s+3s+3s=9s\neq 3^{n}(\text{as
}s\leq 3^{t}+1)$

(b) If ${\bf w}_{1},{\bf w}_{2},{\bf w}_{3}\in W^{\perp}_{i}$, for some
$i\in\mathcal{A}$, then

$\displaystyle\text{Re}(\widehat{f}({\bf w}_{1}))+\text{Re}(\widehat{f}({\bf
w}_{2}))+\text{Re}(\widehat{f}({\bf
w}_{3}))=\frac{-3^{t+1}}{2}+3s-\frac{3^{t+1}}{2}+3s-\frac{3^{t}+1}{2}+3s\neq
3^{n}$ $\displaystyle(\text{as }s\neq 3^{n-2}+\frac{3^{t}}{2})$

(c) If ${\bf w_{1}}\in W^{\perp}_{i}$, for some $i\in\mathcal{A}$ and ${\bf
w_{2}},{\bf w_{3}}\notin W_{i}$, for all $i\in\mathcal{A}$, then

$\displaystyle\text{Re}(\widehat{f}({\bf w}_{1}))+\text{Re}(\widehat{f}({\bf
w}_{2}))+\text{Re}(\widehat{f}({\bf w}_{3}))=-\frac{3^{t+1}}{2}+3s+3s+3s\neq
3^{n}(\text{as }s\leq 3^{t}+1)$

All other cases follows in similar way.
Hence, we get $\text{Re}(\widehat{f}({\bf w}_{1}))+\text{Re}(\widehat{f}({\bf
w}_{2}))+\text{Re}(\widehat{f}({\bf w}_{3}))\neq 3^{n}$, for all pairwise
distinct ${\bf w}_{1},{\bf w}_{2},{\bf w}_{3}\in\mathbb{F}_{3}^{n}$ such that
${\bf w}_{1}+{\bf w}_{2}+{\bf w}_{3}={\bf 0}$. ∎

######  Lemma 3.5.

Let $n\geq 6$ and $f$ be a ternary function defined in 3.1. If
$s\neq\frac{3^{t}+1}{2}$, then

$\text{Re}(\widehat{f}({\bf w}_{1}))+\text{Re}(\widehat{f}({\bf
w}_{2}))-2\text{Re}(\widehat{f}({\bf w}_{3}))\neq 3^{n}$

for all pairwise distinct ${\bf w}_{1},{\bf w}_{2},{\bf
w}_{3}\in\mathbb{F}_{3}^{n}$ such that ${\bf w}_{1}+{\bf w}_{2}+{\bf
w}_{3}={\bf 0}$.

###### Proof.

For different choices of ${\bf w}_{1},{\bf w}_{2}$ and ${\bf w}_{3}$, we have
following cases:
Case I: If ${\bf w}_{1}={\bf 0}$, then ${\bf w}_{2}=-{\bf w}_{3}$. Now
consider the following subcases:
(a) If ${\bf w}_{2}\notin W^{\perp}_{i}$, for all $i\in\mathcal{A}$, then

$\displaystyle\text{Re}(\widehat{f}({\bf 0}))+\text{Re}(\widehat{f}({\bf
w}_{2}))-2\text{Re}(\widehat{f}({\bf w}_{3}))=3^{n}-3s(3^{t}-1)+3s-2(3s)\neq
3^{n}(\text{ as }s\neq 0)$

(b) If ${\bf w}_{2}\in W^{\perp}_{i}\setminus\\{{\bf 0}\\}$, for some
$i\in\mathcal{A}$, then

$\displaystyle\text{Re}(\widehat{f}({\bf 0}))+\text{Re}(\widehat{f}({\bf
w}_{2}))-2\text{Re}(\widehat{f}({\bf
w}_{3}))=3^{n}-3s(3^{t}-1)-\frac{3^{t+1}}{2}+3s-2\left(-\frac{3^{t+1}}{2}+3s\right)\neq
3^{n}$ $\displaystyle(\text{ as }s\neq\frac{1}{2})$

Case II: If ${\bf w}_{3}={\bf 0}$, then ${\bf w}_{1}=-{\bf w}_{2}$. Now
consider the following subcases:
(a) If ${\bf w}_{1}\notin W^{\perp}_{i}$, for all $i\in\mathcal{A}$, then

$\displaystyle\text{Re}(\widehat{f}({\bf w}_{1}))+\text{Re}(\widehat{f}({\bf
w}_{2}))-2\text{Re}(\widehat{f}({\bf
0}))=3s+3s-2\left(3^{n}-3s(3^{t}-1)\right)\neq 3^{n}(\text{ as
}s\neq\frac{3^{t}}{2})$

(b) If ${\bf w}_{1}\in W^{\perp}_{i}\setminus\\{{\bf 0}\\}$, for some
$i\in\mathcal{A}$, then

$\displaystyle\text{Re}(\widehat{f}({\bf w}_{1}))+\text{Re}(\widehat{f}({\bf
w}_{2}))-2\text{Re}(\widehat{f}({\bf
0}))=-\frac{3^{t+1}}{2}+3s-\frac{3^{t+1}}{2}+3s-2\left(3^{n}-3s(3^{t}-1)\right)\neq
3^{n}$ $\displaystyle(\text{as }s\neq\frac{3^{t}+1}{2})$

Case III: If ${\bf w}_{i}\neq{\bf 0}$, for $i\in\\{1,2,3\\}$ then consider the
following subcases:
(a) If ${\bf w}_{1},{\bf w}_{2},{\bf w}_{3}\notin W^{\perp}_{i}$, for all
$i\in\mathcal{A}$, then

$\displaystyle\text{Re}(\widehat{f}({\bf w}_{1}))+\text{Re}(\widehat{f}({\bf
w}_{2}))-2\text{Re}(\widehat{f}({\bf w}_{3}))=3s+3s-2(3s)=0\neq 3^{n}$

(b) If ${\bf w}_{1},{\bf w}_{2}\notin W^{\perp}_{i}$, for all
$i\in\mathcal{A}$ and ${\bf w}_{3}\in W^{\perp}_{i}\setminus\\{{\bf 0}\\}$,
for some $i\in\mathcal{A}$, then

$\displaystyle\text{Re}(\widehat{f}({\bf w}_{1}))+\text{Re}(\widehat{f}({\bf
w}_{2}))-2\text{Re}(\widehat{f}({\bf w}_{3}))\neq 3^{n}\Leftrightarrow
3s+3s-2\left(-\frac{3^{t+1}}{2}+3s\right)=3^{t+1}\neq 3^{n}$

(c) If ${\bf w}_{1}\notin W^{\perp}_{i}$, for all $i\in\mathcal{A}$ and ${\bf
w}_{2},{\bf w}_{3}\in W^{\perp}_{i}$, for some $i\in\mathcal{A}$, then

$\displaystyle\text{Re}(\widehat{f}({\bf w}_{1}))+\text{Re}(\widehat{f}({\bf
w}_{2}))-2\text{Re}(\widehat{f}({\bf
w}_{3}))=3s-\frac{3^{t+1}}{2}+3s-2\left(-\frac{3^{t+1}}{2}+3s\right)=\frac{3^{t+1}}{2}\neq
3^{n}$

All other cases follows in similar way.
Hence, we get $\text{Re}(\widehat{f}({\bf w}_{1}))+\text{Re}(\widehat{f}({\bf
w}_{2}))-2\text{Re}(\widehat{f}({\bf w}_{3}))\neq 3^{n}$, for all pairwise
distinct ${\bf w}_{1},{\bf w}_{2},{\bf w}_{3}\in\mathbb{F}_{3}^{n}$ such that
${\bf w}_{1}+{\bf w}_{2}+{\bf w}_{3}={\bf 0}$. ∎

######  Theorem 3.6.

Let $n\geq 6$ and $f$ be a ternary function defined above. If
$s\neq\frac{3^{t}+1}{2}$, then $\mathscr{C}_{f}$ is a $[2^{n}-1,n+1]$ minimal
linear code. Moreover, if $s\leq 3^{t-2}$, then
$\frac{\text{wt}_{min}}{\text{wt}_{max}}\leq\frac{1}{3}<\frac{2}{3}$.

###### Proof.

From Lemma 3.4 and Lemma 3.5, $\mathscr{C}_{f}$ satisfies the necessary and
sufficient condition of the Theorem 2.7. Hence $\mathscr{C}_{f}$ is minimal.
If $s\leq 3^{t-2}$, then $\text{wt}_{min}=2s(3^{t}-1)$ and
$\text{wt}_{max}=3^{n}-3^{n-1}+3^{t}-2s$.
Then

$\displaystyle\frac{\text{wt}_{min}}{\text{wt}_{max}}$
$\displaystyle=\frac{2s(3^{t}-1)}{3^{n}-3^{n-1}+3^{t}-2s}$
$\displaystyle\leq\frac{2\cdot 3^{t-2}(3^{t}-1)}{2\cdot 3^{2t-1}+3^{t}-2\cdot
3^{t-2}}$ $\displaystyle=\frac{2(3^{t}-1)}{2\cdot
3^{t+1}+7}\leq\frac{1}{3}<\frac{2}{3}$

Hence, the result. ∎

##  4 Construction of minimal linear codes over $\mathbb{F}_{3}$ from
characteristic functions

In this section, we construct minimal linear codes over $\mathbb{F}_{3}$,
using the function define in [9]. First we define characteristic functions
using partial spread and using these functions, we construct minimal linear
codes over $\mathbb{F}_{3}$.
Consider a partial spread of order $\nu$ in $\mathbb{F}_{3}^{n}$ consisting of
$\nu$ disjoint $t$-dimensional subspaces $W_{1},W_{2},\ldots,W_{\nu}$ of
$\mathbb{F}_{3}^{n}$.
Let $f_{i}:\mathbb{F}_{3}^{n}\rightarrow\mathbb{F}_{3}$ be a characteristic
function given by

$\displaystyle f_{i}({\bf x})=\begin{cases}1,&\text{ if }{\bf x}\in
W_{i}\setminus\\{0\\}\\\ 0&\text{Otherwise}\end{cases}$

For $1\leq s\leq 3^{t}+1$ and $\mathcal{A}=\\{i_{1},i_{2},\ldots,i_{s}\\}$,
define the characteristic function $f$ as

$f({\bf x})=\sum_{i\in\mathcal{A}}f_{i}({\bf x})$ (4.1)

We now give the weight distribution of $\mathscr{C}_{f}$ and values of the
Walsh transform.

######  Theorem 4.1.

Let $f$ be a characteristic function defined in 4.1. Then $\mathscr{C}_{f}$ is
a $[3^{n}-1,n+1]$ linear code over $\mathbb{F}_{3}$ with weight distribution
given in Table 2.

Weight $w$ |  Multiplicity $A_{w}$
---|---
0 | 1
$s(3^{t}-1)$ | 2
$3^{n}-3^{n-1}$ | $3^{n}-1$
$3^{n}-3^{n-1}-s$ | $2s(3^{t}-1)$
$3^{n}-3^{n-1}+3^{t}-s$ | $2(3^{t}+1-s)(3^{t}-1)$
Table 2: Weight distribution of $\mathscr{C}_{f}$

###### Proof.

Note that $f({\bf x})\neq{\bf v}\cdot{\bf x}$, for all ${\bf
v}\in\mathbb{F}_{3}^{n}$. Hence, from Theorem 2.6, we have $\mathscr{C}_{f}$
is a $[3^{n}-1,n+1]$ linear code over $\mathbb{F}_{3}$. For any ${\bf
w}\in\mathbb{F}_{3}^{n}$, we have three cases

  1. (I)

${\bf w}={\bf 0}$

  2. (II)

${\bf w}\notin W_{i}^{\perp}$, for all $i\in\mathcal{A}$.

  3. (III)

${\bf w}\in W_{i}^{\perp}\setminus\\{{\bf 0}\\}$, for some $i\in\mathcal{A}$.

Case I: If ${\bf w}={\bf 0}$, then we have

$\displaystyle\widehat{f}({\bf 0})$ $\displaystyle=\sum_{{\bf
x}\in\mathbb{F}_{3}}\zeta_{3}^{\sum_{i\in\mathcal{A}}f_{i}({\bf x})-{\bf
0}\cdot{\bf x}}$ $\displaystyle=1+\sum_{{\bf x}\in
W_{i_{1}}\setminus\\{0\\}}\zeta_{3}^{1}+\sum_{{\bf x}\in
W_{i_{2}}\setminus\\{0\\}}\zeta_{3}^{1}+\cdots+\sum_{{\bf x}\in
W_{i_{s}}\setminus\\{0\\}}\zeta_{3}^{1}+\sum_{{\bf
x}\notin\bigcup_{k=1}^{s}W_{i_{k}}}\zeta_{3}^{0}$
$\displaystyle=1+\left(3^{t}-1\right)\zeta_{3}+\cdots+\left(3^{t}-1\right)\zeta_{3}+(3^{t}+1-s)\left(3^{t}-1\right)$
$\displaystyle=3^{n}-3^{t}s+s+s(3^{t}-1)\zeta_{3}$ (4.2)

Case II: If ${\bf w}\notin W_{i}^{\perp}$, for all $i\in\mathcal{A}$ then
${\bf w}\in W^{\perp}_{j}$, for some $j\notin\mathcal{A}$. Without loss of
generality, assume that ${\bf x}\in W_{i_{s+1}}^{\perp}$. We have

$\displaystyle\widehat{f}({\bf w})$ $\displaystyle=\sum_{{\bf
x}\in\mathbb{F}_{3}^{n}}\zeta_{3}^{\sum_{i\in\mathcal{A}}f_{i}({\bf
x})-w\cdot{\bf x}}$ $\displaystyle=1+\sum_{{\bf x}\in
W_{i_{1}}\setminus\\{0\\}}\zeta_{3}^{1-{\bf w}\cdot{\bf x}}+\cdots+\sum_{{\bf
x}\in W_{i_{s}}\setminus\\{0\\}}\zeta_{3}^{1-{\bf w}\cdot{\bf x}}+\sum_{{\bf
x}\in W_{i_{s+1}}\setminus\\{0\\}}\zeta_{3}^{0}+\sum_{{\bf
x}\notin\bigcup_{k=1}^{s+1}W_{i_{k}}}\zeta_{3}^{-{\bf w}\cdot{\bf x}}$
$\displaystyle=\left(3^{t-1}+(3^{t-1}-1)\zeta_{3}+3^{t-1}\zeta_{3}^{2}\right)+\cdots+\left(3^{t-1}+(3^{t-1}-1)\zeta_{3}+3^{t-1}\zeta_{3}^{2}\right)$
$\displaystyle+(3^{t}-1)+\left(3^{t}+1-(s+1)\right)\left((3^{t-1}-1)+3^{t-1}\zeta_{3}+3^{t-1}\zeta_{3}^{2}\right)$
$\displaystyle=-s\zeta_{3}+s$ (4.3)

Case III: For ${\bf w}\in W_{i}^{\perp}\setminus\\{{\bf 0}\\}$, for some
$i\in\\{i_{1},i_{2},\ldots,i_{s}\\}$. Without loss of generality assume that
${\bf w}\in W_{i_{1}}^{\perp}\setminus\\{{\bf 0}\\}$. We have

$\displaystyle\widehat{f}({\bf w})$ $\displaystyle=\sum_{{\bf
x}\in\mathbb{F}_{3}^{n}}\zeta_{3}^{\sum_{i\in\mathcal{A}}f_{i}({\bf
x})-w\cdot{\bf x}}$ $\displaystyle=1+\sum_{{\bf x}\in
W_{i_{1}}\setminus\\{0\\}}\zeta_{3}^{1}+\cdots+\sum_{{\bf x}\in
W_{i_{s}}\setminus\\{0\\}}\zeta_{3}^{1-{\bf w}\cdot{\bf x}}+\sum_{{\bf
x}\notin\bigcup_{k=1}^{s}W_{i_{k}}}\zeta_{3}^{-{\bf w}\cdot{\bf x}}$
$\displaystyle=(3^{t}-1)\zeta_{3}+\cdots+\left(3^{t-1}+(3^{t-1}-1)\zeta_{3}+3^{t-1}\zeta_{3}^{2}\right)$
$\displaystyle+\left(3^{t}+1-s\right)\left((3^{t-1}-1)+3^{t-1}\zeta_{3}+3^{t-1}\zeta_{3}^{2}\right)$
$\displaystyle=(3^{t}-1)\zeta_{3}-(s-1)\zeta_{3}-(3^{t}+1-s)$ (4.4)

Hence, from 4.1,4.1 and 4.1, we have

$\text{Re}(\widehat{f}({\bf
w}))=\begin{cases}3^{n}-\frac{3}{2}s(3^{t}-1),&\text{ if }{\bf w}={\bf 0}\\\
\frac{3}{2}s,&\text{ if }{\bf w}\notin W^{\perp}_{i},\text{ for all
}i\in\mathcal{A}\\\ -\frac{3^{t+1}}{2}+\frac{3}{2}s,&\text{ if }{\bf w}\in
W^{\perp}_{i}\setminus\\{0\\},\text{ for some }i\in\mathcal{A}\end{cases}$

Note that, $\text{Re}(\widehat{f}({\bf w}))=\text{Re}(2\widehat{f}({\bf w}))$.
The weight distribution of $\mathscr{C}_{f}$ follow from Theorem 2.6 ∎

Now, we prove the following two results, which help us to prove
$\mathscr{C}_{f}$ is a minimal linear code.

######  Lemma 4.2.

Let $n\geq 6$ and $f$ be a characteristic function defined in 4.1. If
$s\notin\\{1,3^{t},3^{t}+1\\}$, then

$\text{Re}(\widehat{f}({\bf w}_{1}))+\text{Re}(\widehat{f}({\bf
w}_{2}))+\text{Re}(\widehat{f}({\bf w}_{3}))\neq 3^{n}$

for all pairwise distinct ${\bf w}_{1},{\bf w}_{2},{\bf
w}_{3}\in\mathbb{F}_{3}^{n}$ such that ${\bf w}_{1}+{\bf w}_{2}+{\bf
w}_{3}={\bf 0}$.

###### Proof.

For different choices of ${\bf w}_{1},{\bf w}_{2}$ and ${\bf w}_{3}$, we have
following subcases:
Case I: For ${\bf w}_{i}={\bf 0}$, for some $i$. Without loss of generality
assume that ${\bf w}_{1}={\bf 0}$, then ${\bf w}_{2}=-{\bf w}_{3}$. Now
consdier the following subcases:
(a) If ${\bf w}_{2}\notin W^{\perp}_{i}$, for all $i\in\mathcal{A}$, then

$\displaystyle\text{Re}(\widehat{f}({\bf 0}))+\text{Re}(\widehat{f}({\bf
w}_{2}))+\text{Re}(\widehat{f}({\bf
w}_{3}))=3^{n}-\frac{3}{2}s(3^{t}-1)+\frac{3}{2}s+\frac{3}{2}s\neq
3^{n}(\text{since }s\neq 0)$

(b) If ${\bf w}_{2}\in W^{\perp}_{i}\setminus\\{{\bf 0}\\}$, for some
$i\in\mathcal{A}$, then

$\displaystyle\text{Re}(\widehat{f}({\bf 0}))+\text{Re}(\widehat{f}({\bf
w}_{2}))+\text{Re}(\widehat{f}({\bf
w}_{3}))=3^{n}-\frac{3}{2}s(3^{t}-1)-\frac{3^{t+1}}{2}+\frac{3}{2}s-\frac{3^{t+1}}{2}+\frac{3}{2}s\neq
3^{n}$ $\displaystyle(\text{since }s\neq 3^{t-1}(s+2))$

Case II: If ${\bf w}_{i}\neq 0$, for $i=1,2,3$, then consider the following
subcases:
(a) If ${\bf w}_{1},{\bf w}_{2},{\bf w}_{3}\notin W^{\perp}_{i}$, for all
$i\in\mathcal{A}$, then

$\displaystyle\text{Re}(\widehat{f}({\bf w}_{1}))+\text{Re}(\widehat{f}({\bf
w}_{2}))+\text{Re}(\widehat{f}({\bf
w}_{3}))=\frac{3}{2}s+\frac{3}{2}s+\frac{3}{2}s\neq 3^{n}=\frac{9}{2}s\neq
3^{n}$

(b) If ${\bf w}_{1},{\bf w}_{2},{\bf w}_{3}\in W^{\perp}_{i}\setminus\\{{\bf
0}\\}$, for some $i\in\mathcal{A}$, then

$\displaystyle\text{Re}(\widehat{f}({\bf w}_{1}))+\text{Re}(\widehat{f}({\bf
w}_{2}))+\text{Re}(\widehat{f}({\bf
w}_{3}))=-\frac{3^{t+1}}{2}+\frac{3}{2}s-\frac{3^{t+1}}{2}+\frac{3}{2}s-\frac{3^{t+1}}{2}+\frac{3}{2}s$
$\displaystyle=-\frac{3^{t+2}}{2}+\frac{9}{2}s\neq 3^{n}(\text{since }s\leq
3^{t}+1)$

(c) If ${\bf w_{1}}\in W^{\perp}_{i}\setminus\\{{\bf 0}\\}$, for some
$i\in\mathcal{A}$ and ${\bf w_{2}},{\bf w_{3}}\notin W^{\perp}_{i}$, for all
$i\in\mathcal{A}$, then

$\displaystyle\text{Re}(\widehat{f}({\bf w}_{1}))+\text{Re}(\widehat{f}({\bf
w}_{2}))+\text{Re}(\widehat{f}({\bf
w}_{3}))=-\frac{3^{t+1}}{2}+\frac{3}{2}s+\frac{3}{2}s+\frac{3}{2}s=-\frac{3^{t+1}}{2}+\frac{9}{2}s\neq
3^{n}$ $\displaystyle(\text{as }s\leq 3^{t}+1)$

All other cases follows in similar way.
Hence, we get $\text{Re}(\widehat{f}({\bf w}_{1}))+\text{Re}(\widehat{f}({\bf
w}_{2}))+\text{Re}(\widehat{f}({\bf w}_{3}))\neq 3^{n}$, for all pairwise
distinct ${\bf w}_{1},{\bf w}_{2},{\bf w}_{3}\in\mathbb{F}_{3}^{n}$ such that
${\bf w}_{1}+{\bf w}_{2}+{\bf w}_{3}={\bf 0}$. ∎

######  Lemma 4.3.

Let $n\geq 6$ and $f$ be a characteristic function defined in 4.1. If
$s\notin\\{1,3^{t},3^{t}+1\\}$, then

$\text{Re}(\widehat{f}({\bf w}_{1}))+\text{Re}(\widehat{f}({\bf
w}_{2}))-2\text{Re}(\widehat{f}({\bf w}_{3}))\neq 3^{n}$

for all pairwise distinct ${\bf w}_{1},{\bf w}_{2},{\bf
w}_{3}\in\mathbb{F}_{3}^{n}$ such that ${\bf w}_{1}+{\bf w}_{2}+{\bf
w}_{3}={\bf 0}$.

###### Proof.

For different choices of ${\bf w}_{1},{\bf w}_{2}$ and ${\bf w}_{3}$, we have
following subcases:
Case I: If ${\bf w}_{1}={\bf 0}$, then ${\bf w}_{2}=-{\bf w}_{3}$. Now
consider the following subcases:
(a) If ${\bf w}_{2}\notin W^{\perp}_{i}$, for all $i\in\mathcal{A}$, then

$\displaystyle\text{Re}(\widehat{f}({\bf 0}))+\text{Re}(\widehat{f}({\bf
w}_{2}))-2\text{Re}(\widehat{f}({\bf
w}_{3}))=-\frac{3}{2}s(3^{t}-1)+\frac{3}{2}s-2\left(\frac{3}{2}s\right)\neq
3^{n}(\text{as }s\neq 0)$

(b) If ${\bf w}_{2}\in W^{\perp}_{i}\setminus\\{{\bf 0}\\}$, for some
$i\in\mathcal{A}$, then

$\displaystyle\text{Re}(\widehat{f}({\bf 0}))+\text{Re}(\widehat{f}({\bf
w}_{2}))-2\text{Re}(\widehat{f}({\bf
w}_{3}))=3^{n}-\frac{3}{2}s(3^{t}-1)-\frac{3^{t+1}}{2}+\frac{3}{2}s-2\left(-\frac{3^{t+1}}{2}+\frac{3}{2}s\right)$
$\displaystyle=3^{n}-\frac{3^{t+1}}{2}(s-1)\neq 3^{n}(\text{since }s\neq 1)$

Case II: If ${\bf w}_{3}={\bf 0}$, then ${\bf w}_{1}=-{\bf w}_{2}$. Now
consider the following subcases:
(a) If ${\bf w}_{1}\notin W^{\perp}_{i}$, for all $i\in\mathcal{A}$, then

$\displaystyle\text{Re}(\widehat{f}({\bf w}_{1}))+\text{Re}(\widehat{f}({\bf
w}_{2}))-2\text{Re}(\widehat{f}({\bf
0}))=\frac{3}{2}s+\frac{3}{2}s-2\left(3^{n}-\frac{3}{2}s(3^{t}-1)\right)$
$\displaystyle=-2\cdot 3^{n}+3^{t+1}s\neq 3^{n}(\text{since }s\neq 3^{t})$

(b) If ${\bf w}_{1}\in W^{\perp}_{i}\setminus\\{{\bf 0}\\}$, for some
$i\in\mathcal{A}$, then

$\displaystyle\text{Re}(\widehat{f}({\bf w}_{1}))+\text{Re}(\widehat{f}({\bf
w}_{2}))-2\text{Re}(\widehat{f}({\bf
0}))=-\frac{3^{t+1}}{2}+\frac{3}{2}s-\frac{3^{t+1}}{2}+\frac{3}{2}s-2\left(3^{n}-\frac{3}{2}s(3^{t}-1)\right)$
$\displaystyle=-2\cdot 3^{n}-3^{t+1}+3^{t+1}s\neq 3^{n}(\text{since }s\neq
3^{t}+1)$

Case III: If ${\bf w}_{i}\neq{\bf 0}$, for all $i\in\\{1,2,3\\}$ then consider
the following subcases:
(a) If ${\bf w}_{1},{\bf w}_{2},{\bf w}_{3}\notin W^{\perp}_{i}$, for all
$i\in\mathcal{A}$, then

$\displaystyle\text{Re}(\widehat{f}({\bf w}_{1}))+\text{Re}(\widehat{f}({\bf
w}_{2}))-2\text{Re}(\widehat{f}({\bf
w}_{3}))=\frac{3}{2}s+\frac{3}{2}s-2\left(\frac{3}{2}s\right)=0\neq 3^{n}$

(b) If ${\bf w}_{1},{\bf w}_{2}\notin W^{\perp}_{i}$, for all
$i\in\mathcal{A}$ and ${\bf w}_{3}\in W^{\perp}_{i}\setminus\\{{\bf 0}\\}$,
for some $i\in\mathcal{A}$, then

$\displaystyle\text{Re}(\widehat{f}({\bf w}_{1}))+\text{Re}(\widehat{f}({\bf
w}_{2}))-2\text{Re}(\widehat{f}({\bf
w}_{3}))=\frac{3}{2}s+\frac{3}{2}s-2\left(-\frac{3^{t+1}}{2}+\frac{3}{2}s\right)=3^{t+1}\neq
3^{n}$

(c) If ${\bf w}_{1}\notin W^{\perp}_{i}$, for all $i\in\mathcal{A}$ and ${\bf
w}_{2},{\bf w}_{3}\in W_{i}^{\perp}\setminus\\{{\bf 0}\\}$, for some
$i\in\mathcal{A}$, then

$\displaystyle\text{Re}(\widehat{f}({\bf w}_{1}))+\text{Re}(\widehat{f}({\bf
w}_{2}))-2\text{Re}(\widehat{f}({\bf
w}_{3}))=\frac{3}{2}s-\frac{3^{t+1}}{2}+\frac{3}{2}s-2\left(-\frac{3^{t+1}}{2}+\frac{3}{2}s\right)=\frac{3^{t+1}}{2}\neq
3^{n}$

All other cases follows in similar way.
Hence, we get $\text{Re}(\widehat{f}({\bf w}_{1}))+\text{Re}(\widehat{f}({\bf
w}_{2}))-2\text{Re}(\widehat{f}({\bf w}_{3}))\neq 3^{n}$, for all pairwise
distinct ${\bf w}_{1},{\bf w}_{2},{\bf w}_{3}\in\mathbb{F}_{3}^{n}$ such that
${\bf w}_{1}+{\bf w}_{2}+{\bf w}_{3}={\bf 0}$. ∎

Now, we provide a class of minimal linear codes violating Ashikhmin-Barg
condition.

######  Theorem 4.4.

Let $n\geq 6$ and $f$ be a characteristic function defined in 4.1. If
$s\notin\\{1,3^{t},3^{t}+1\\}$, then $\mathscr{C}_{f}$ is a $[2^{n}-1,n+1]$
minimal linear code. Moreover, if $s\leq 3^{t-2}$, then
$\frac{\text{wt}_{min}}{\text{wt}_{max}}\leq\frac{1}{3}<\frac{2}{3}$.

###### Proof.

From Lemma 4.2 and 4.3, $\mathscr{C}_{f}$ satisfies conditions in Theorem 2.7.
Hence, $\mathscr{C}_{f}$ is minimal.
If $s\leq 3^{t-2}$, then $\text{wt}_{min}=s(3^{t}-1)$ and
$\text{wt}_{max}=3^{n}-3^{n-1}+3^{t}-s$.

$\displaystyle\text{Therefore,}\quad\frac{\text{wt}_{min}}{\text{wt}_{max}}$
$\displaystyle=\frac{s(3^{t}-1)}{3^{n}-3^{n-1}+3^{t}-s}$
$\displaystyle\leq\frac{3^{t-2}(3^{t}-1)}{2\cdot 3^{2t-1}+3^{t}-3^{t-2}}$
$\displaystyle=\frac{(3^{t}-1)}{2\cdot 3^{t+1}+8}\leq\frac{1}{3}<\frac{2}{3}$

Hence, the result. ∎

##  5 Conclusion

In this paper, we constructed two classes of ternary minimal linear codes of
dimension $n+1$. One class is constructed using characteristic function over
partial spread and another class is constructed using ternary function on
partial spread. We also obtained weight distribution for these minimal linear
codes. We further prove that a family of these constructed minimal linear
codes violates Ashikhmin-Barg condition. Using the functions defined in this
paper, constructing minimal linear codes over $\mathbb{F}_{3}$ of dimension
greater than $n+1$ can be a further scope of the study.

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