Orbifold Modular GUT of Flavour

Francisco J. de Anda‡***E-mail<EMAIL_ADDRESS>Stephen F. King⋄†††E-mail:
<EMAIL_ADDRESS>
‡ Tepatitlán’s Institute for Theoretical Studies, C.P. 47600, Jalisco, México,

Dual CP Institute of High Energy Physics, C.P. 28045, Colima, México.

⋄ School of Physics and Astronomy, University of Southampton,

SO17 1BJ Southampton, United Kingdom.

We discuss an $SU(5)$ Grand Unified Theory (GUT) based on the 10d orbifold
$(\mathbb{T}^{2})^{3}/(\mathbb{Z}_{4}\times\mathbb{Z}_{2})$ plus three modular
$S_{4}$ groups with moduli at respective fixed points
$i,i+2,\omega=e^{2i\pi/3}$. The resulting model has hierarchical quark and
charged lepton mass matrices, arising from a double weighton mechanism, and
reproduces the highly predictive Littlest Seesaw Mechanism in the neutrino
sector. The down quark mass matrix has an upper triangular form, contributing
to CKM mixing, while the charged lepton mass matrix has a lower triangular
form with suppressed contributions to PMNS mixing. The orbifold yields
successful $SU(5)$ breaking with doublet-triplet splitting of the Higgs
multiplets.

##  1 Introduction

The flavour puzzle of the Standard Model (SM) remains one of the most
important questions in particle physics, both conceptually and pragmatically.
From a fundamental point of view, the origin of hierarchical quark and charged
lepton masses implies that the three families are somehow very different, even
though their gauge interactions are identical. Practically this results in a
plethora of unexplained parameters in the flavour sector of the SM, the more
so following the discovery of neutrino mass and mixing, and surely too many
for a fundamental theory. Recently there has been some progress in addressing
the flavour problem, using modular symmetry using bottom-up approaches [1]
(for a recent review see [2]). Although modular symmetry naturally arises from
orbifold models, such constructions have been relatively neglected in the
modular symmetry literature.

In a recent paper we developed a bottom-up approach to flavour models which
combines modular symmetry with orbifold constructions [3], especially on
orbifolds in 10d which can provide three modular groups and three moduli
fields in the low energy theory (below the compactification scales). Unlike 4d
models, in such 10d orbifold models the values of the moduli are not
completely free but are constrained geometry and symmetry. For example in the
orbifold example $(\mathbb{T}^{2})^{3}/(\mathbb{Z}_{4}\times\mathbb{Z}_{2})$,
the fixed points, two of the moduli are constrained by the orbifold to be at
$\tau=i$ (up to a discrete choice), while the third one is unconstrained by
geometry but fixed by stabilisation arguments to be at
$\tau=\omega=e^{2i\pi/3}$ [4]. The choice of the three moduli $\tau_{1}=i,\
\tau_{2}=i+2,\ \tau_{3}=\omega$ has recently been discussed in a 4d models of
modular symmetry based on $S_{4}^{3}$ [5, 6] as well as 10d orbifold models
also based on $S_{4}^{3}$ [3]. This choice is of phenomenological interest
since it leads to the highly predictive Littlest Modular Seesaw structure in
the neutrino sector [7, 8]. ‡‡‡The Littlest Seesaw model is based on
CSD($n\sim 3$) [9, 10, 11, 12, 13, 14, 15, 16], and was recently reviewed in
[17].

In the present paper we combine 10d orbifold Grand Unified Theories (GUTs)
with modular symmetry with the goal of providing a unified theory of flavour.
Orbifold GUTs [18, 19, 20, 21, 22, 23] have several advantages in terms of GUT
symmetry breaking and simplified Higgs doublet-triplet splitting. It is
therefore of interest to extend the above orbifold model of leptons [3], to
the case of orbifold GUTs, using the same orbifold and modular symmetry, in
order to provide a complete theory including quark mass and mixing. Here,
then, we shall present an $SU(5)$ GUT based on a 10d orbifold
$(\mathbb{T}^{2})^{3}/(\mathbb{Z}_{4}\times\mathbb{Z}_{2})$ with three modular
$S_{4}$ symmetries with moduli at the fixed points $i,i+2,\omega$, which can
successfully reproduce the Littlest Modular Seesaw as in the model of leptons
above. The model also attempts to explain the quark and charged lepton mass
hierarchies, using the weighton mechanism [24], where the down quark mass
matrix has an upper triangular form, contributing to CKM mixing, while the
charged lepton mass matrix has a lower triangular form with suppressed
contributions to PMNS mixing. This means that lepton mixing must originate
entirely from the neutrino sector, thereby preserving the predictions of the
Littlest Seesaw model. The same orbifold also yields successful $SU(5)$
breaking with doublet-triplet splitting of the Higgs multiplets. The approach
here follows an earlier work by two of us based on a 6d $SU(5)$ orbifold GUT
with $A_{4}$ modular symmetry [25] in which the resulting lepton sector had a
$\mu-\tau$ symmetry.

The layout of the remainder of the paper is as follows. In section 2 we
introduce the 10d orbifold of interest based on
$(\mathbb{T}^{2})^{3}/(\mathbb{Z}_{4}\times\mathbb{Z}_{2})$, and discuss the
resulting fixed points of interest, as well as the overlapping branes which
permit interactions between the matter fields located there. Section 3
describes an $SU(5)$ model based on this orbifold, including GUT breaking, a
discussion of fermion (matter) fields, Higgs doublet-triplet splitting, and
the 10d and resulting 4d Lagrangians. We also discuss the resulting quark and
lepton mass matrices and show that the model reproduces the Littlest seesaw
model predictions. Section 4 concludes the paper. In an Appendix we briefly
review modular symmetry and modular forms at level $N=4$ corresponding to
$S_{4}$.

##  2 The orbifold $(\mathbb{T}^{2})^{3}/(\mathbb{Z}_{4}\times\mathbb{Z}_{2})$

We assume a 10d spacetime where the 6 extra dimensions are factorisable into 3
tori, each defined by one complex coordinate $z_{i}$ with $i=1,2,3,$ and
compactified by the tori actions

$z_{i}\sim z_{i}+1,\ \ \ z_{i}\sim z_{i}+\tau_{i},$ (1)

where $\tau_{i}$ is a complex number to be fixed below.

The orbifold $(\mathbb{T}^{2})^{3}/\mathbb{Z}_{4}\times\mathbb{Z}_{2}$, which
is one from a restricted list of orbifolds which preserves supersymmetry [26],
as defined by the orbifolding actions

$\begin{split}\theta_{4}&:(x,z_{1},z_{2},z_{3})\sim(x,iz_{1},-iz_{2},z_{3}),\\\
\theta_{2}&:(x,z_{1},z_{2},z_{3})\sim(x,z_{1},-z_{2},-z_{3}).\end{split}$ (2)

Each of the orbifold operations $\theta_{2,4}$ can also be accompanied by a
gauge transformation $P_{2,4}\in SU(5)$ respectively that satisfy

$P_{2}^{2}=P_{4}^{4}=I,\ \ \ [P_{2},P_{4}]=0.$ (3)

We assume that each tori has an independent discrete modular symmetry $S_{4}$.

The lattice of the extra dimensions are defined by the $\tau_{i}$. These are
restricted geometrically by the orbifold action, as the orbifold action in Eq.
2 must be equivalent to some lattice translation as in Eq. 1 i.e. its action
over the lattice basis vectors $(1,\tau_{i})$ must be a linear combination of
the original lattice vectors, with integer coefficients. Therefore there must
exist integers $a_{1,2,3},b_{1,2,3},c_{1,2,3},d_{1,2,3}\in\mathbb{Z}$ such
that

$\begin{split}(i,i\tau_{1,2})&=(a_{1,2}+b_{1,2}\tau_{1,2},c_{1,2}+d\tau_{1,2}),\\\
(-1,-\tau_{3})&=(a_{3}+b_{3}\tau_{3},c_{3}+d\tau_{3}),\\\ \end{split}$ (4)

In the present example, solving Eq. 4 gives,

$\begin{split}\tau_{1,2}&=i+n_{1,2},\ \ \ |\ \ \ n_{1,2}\in\mathbb{Z},\\\
\tau_{3}&\in\mathbb{C}.\end{split}$ (5)

this fixes $\tau_{1,2}=i+n_{1,2}$ where $n_{1,2}=0,1,2,3$, as we are assuming
modular $S_{4}$ which constraints the available integers [3]. We choose
$n_{0}=0$ and $n_{2}=1$. The $\tau_{3}$ can be any complex number. We assume
that there is a remnant $S_{4}$ symmetry on the 4d branes of the $z_{3}$
torus, which then fixes $\tau_{3}=\omega$ [25, 27], which is energetically
favourable [4].

This way all the moduli $\tau_{i}$ are fixed geometrically and the lattice is
defined to be

$z_{i}\sim z_{i}+1,\ \ \ z_{i}\sim z_{i}+\tau_{i},\ \ \
\tau_{i}=\\{i,i+2,\omega\\},$ (6)

where $\omega=e^{2i\pi/3}$.

Each orbifold action in Eq. 2, leaves some invariant subspaces which are
called fixed branes.

We want a minimal model where all fields can behave as modular forms (with
different $\tau_{i}$ depending on their location) but can interact with each
other, we will only use the 6d branes

$\begin{split}\mathbb{T}^{2}_{A}&=(x,z_{1},0,0),\\\
\mathbb{T}^{2}_{B}&=(x,0,z_{2},0),\\\
\mathbb{T}^{2}_{C}&=(x,0,0,z_{3}),\end{split}$ (7)

where the $\mathbb{T}^{2}_{A}$ brane is left invariant by $\theta_{2}$, the
$\mathbb{T}^{2}_{B}$ brane is left invariant by $\theta_{2}\theta_{4}^{2}$ and
the $\mathbb{T}^{2}_{C}$ brane is left invariant by the action of
$\theta_{4}^{2}$. They all overlap with the origin brane, and this is where
all interactions will happen.

##  3 The model

We build an $SU(5)$ model in 10d with $S_{4}$ modular symmetry. We assume the
field content in Table 1, where each field is located in different branes
defined in Eq. 7.

Field | $SU(5)$ | $S_{4}^{A}$ | $S_{4}^{B}$ | $S_{4}^{C}$ |  $2k_{A}$ |  $2k_{B}$ |  $2k_{C}$ | Loc
---|---|---|---|---|---|---|---|---
$F$ | 5 | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{3}$ | 0 | 0 | $0$ | $\mathbb{T}^{2}_{C}$
$T_{1}$ | 10 | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | 0 | 0 |  $1$ | $\mathbb{T}^{2}_{C}$
$T_{2}$ | 10 | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | 0 | 0 |  $1/2$ | $\mathbb{T}^{2}_{C}$
$T_{3}$ | 10 | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | 0 | 0 |  $0$ | $\mathbb{T}^{2}_{C}$
$N_{a}^{c}$ | 1 | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | 0 | $-4$ | 0 | $\mathbb{T}^{2}_{B}$
$N_{s}^{c}$ | 1 | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $-2$ | 0 | 0 | $\mathbb{T}^{2}_{A}$
$H_{u}$ | 5 | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | 0 | 0 | 0 | Bulk
$H_{d}$ | $\overline{\textbf{5}}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | 0 | 0 | $1/2$ | Bulk
$H_{45}$ | 45 | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | 0 | 0 | $1/2$ | Bulk
$H_{\overline{45}}$ | $\overline{\textbf{45}}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | 0 | 0 | $0$ | Bulk
$\Phi_{BC}$ | 1 | $\mathbf{1}$ | $\mathbf{3}$ | $\mathbf{3}$ | 0 | 0 | $0$ | Bulk
$\Phi_{AC}$ | 1 | $\mathbf{3}$ | $\mathbf{1}$ | $\mathbf{3}$ | 0 | 0 | $0$ | Bulk
$\xi_{F}$ | 1 | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | 0 | 0 |  $-5/2$ | $\mathbb{T}^{2}_{C}$
$\xi_{T}$ | 1 | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | 0 | 0 |  $-1/2$ | $\mathbb{T}^{2}_{C}$

Yuk/Mass | $S_{4}^{A}$ | $S_{4}^{B}$ | $S_{4}^{C}$ |  $2k_{A}$ |  $2k_{B}$ |  $2k_{C}$
---|---|---|---|---|---|---
$Y_{e}(\tau_{3})$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{3}$ | 0 | 0 | $6$
$Y_{\mu}(\tau_{3})$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{3}$ | 0 | 0 | $4$
$Y_{\tau}(\tau_{3})$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{3}$ | 0 | 0 | $2$
$Y_{a}(\tau_{2})$ | $\mathbf{1}$ | $\mathbf{3}$ | $\mathbf{1}$ | 0 | $4$ | 0
$Y_{s}(\tau_{1})$ | $\mathbf{3}$ | $\mathbf{1}$ | $\mathbf{1}$ | $2$ | 0 | 0
$M_{a}(\tau_{2})$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | 0 | $8$ | 0
$M_{s}(\tau_{1})$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $4$ | 0 | 0

Table 1: Full list of the assumed fields of the model as well as they
localization. The ones in the bulk are 10d chiral superfields while the ones
in the defined branes are 6d chiral superfields. The modular forms in the
second table are fixed by the representation and weights of the fields. The
$H_{\overline{45}}$ is added to cancel anomalies and plays no other role in
the low energy effective model.

###  3.1 GUT breaking

In this model we will assume the $SU(5)$ breaking boundary conditions

$P_{2}=I,\ \ \ P_{4}=diag(1,1,1,-1,-1).$ (8)

The 10d vector superfield fulfils the boundary condition

$\begin{split}\mathcal{V}(x,z_{1},z_{2},z_{3})&=P_{4}\mathcal{V}(x,iz_{1},-iz_{2},z_{3})P_{4},\\\
\mathcal{V}(x,z_{1},z_{2},z_{3})&=\mathcal{V}(x,z_{1},-z_{2},-z_{3}).\end{split}$
(9)

As the 10d vector superfield decomposes into 4 4d superfields (1 vector and 3
left chiral superfields) $\mathcal{V}=\\{V,\phi_{1,2,3}\\}$ which fulfils the
conditions [28, 29, 30]

$\begin{array}[]{ll}V(x,z_{1},z_{2},z_{3})=P_{4}V(x,iz_{1},-iz_{2},z_{3})P_{4},&V(x,z_{1},z_{2},z_{3})=V(x,z_{1},-z_{2},-z_{3}),\\\
\phi_{1}(x,z_{1},z_{2},z_{3})=iP_{4}\phi_{1}(x,iz_{1},-iz_{2},z_{3})P_{4},&\phi_{1}(x,z_{1},z_{2},z_{3})=\phi_{1}(x,z_{1},-z_{2},-z_{3}),\\\
\phi_{2}(x,z_{1},z_{2},z_{3})=-iP_{4}\phi_{2}(x,iz_{1},-iz_{2},z_{3})P_{4},&\phi_{2}(x,z_{1},z_{2},z_{3})=-\phi_{2}(x,z_{1},-z_{2},-z_{3}),\\\
\phi_{3}(x,z_{1},z_{2},z_{3})=P_{4}\phi_{3}(x,iz_{1},-iz_{2},z_{3})P_{4},&\phi_{3}(x,z_{1},z_{2},z_{3})=-\phi_{3}(x,z_{1},-z_{2},-z_{3}),\end{array}$
(10)

where each 10d function decomposes into an infinite tower of KK modes. One can
easily find the zero modes by finding the solutions for the prior equations
when $z_{1}=z_{2}=z_{3}=0$. The only available zero modes are the SM gauge
vector superfields.

###  3.2 SM fermions

All the SM fermions are located in the 6d brane $\mathbb{T}_{C}$ and therefore
they are 6d chiral superfields which decompose as 2 4d chiral superfields
(left and right) $F=\\{F_{L},F_{R}\\}$. They comply with a single boundary
condition (since $\theta_{4}$ doesn’t act on the $\theta_{4}$ brane)

$F_{L}(x,z_{3})=F_{L}(x,-z_{3}),F_{R}(x,z_{3})=-F_{R}(x,-z_{3}),$ (11)

and therefore there is a single full left chiral superfield as a zero mode as
desired.

###  3.3 Higgs in the bulk and doublet-triplet splitting

The Higgs is to be located in the bulk, it decomposse into 4 4d chiral
superfields each fulfilling the boundary conditions (with a $-1$ charge under
$\theta_{4}$)

$\begin{array}[]{ll}H_{0}(x,z_{1},z_{2},z_{3})=-P_{4}H_{0}(x,iz_{1},-iz_{2},z_{3}),&H_{0}(x,z_{1},z_{2},z_{3})=H_{0}(x,z_{1},-z_{2},-z_{3}),\\\
H_{1}(x,z_{1},z_{2},z_{3})=-iP_{4}H_{1}(x,iz_{1},-iz_{2},z_{3}),&H_{1}(x,z_{1},z_{2},z_{3})=H_{1}(x,z_{1},-z_{2},-z_{3}),\\\
H_{2}(x,z_{1},z_{2},z_{3})=iP_{4}H_{2}(x,iz_{1},-iz_{2},z_{3}),&H_{2}(x,z_{1},z_{2},z_{3})=-H_{2}(x,z_{1},-z_{2},-z_{3}),\\\
H_{3}(x,z_{1},z_{2},z_{3})=-P_{4}H_{3}(x,iz_{1},-iz_{2},z_{3}),&H_{3}(x,z_{1},z_{2},z_{3})=-H_{3}(x,z_{1},-z_{2},-z_{3}),\end{array}$
(12)

which works leaves only a single doublet as a zero mode, solving the doublet
triplet splitting.

###  3.4 10d Lagrangian

The 10d lagrangian must have dimension $[\mathcal{L}]=10,$ where the 10d
scalars (like the one from $\Phi$ and $H$) have dimension $[\phi_{10}]=4$
while the 6d fermions (like any SM field) have dimension $[\psi_{6}]=5/2$.
Some interactions happen at a 6d brane and some at the origin 4d brane, which
are indicated by a Dirac $\delta$ function. There are no renormalizable
interactions and the superpotential will be written in terms of dimensionless
coupling constants divided by a common scale $\Lambda$.

The VEV $\braket{\Phi}\sim\delta^{i}_{j}$ breaks two $S_{4}$ groups into the
diagonal one [5], and its second power $\braket{\Phi}^{2}$ has non trivial
structure. The relevant SM Yukawa 10d lagrangian that fill up the entire mass
matrix structure of every fermion is

$\begin{split}\mathcal{L}_{10d}=&\mathcal{L}_{10d}^{(0)}+\mathcal{L}_{10d}^{(1)},\\\
\\\
\mathcal{L}_{10d}^{(0)}=&\left(\frac{y^{u}_{33}}{\Lambda^{5}}T_{3}T_{3}+\frac{y^{u}_{23}}{\Lambda^{7}}\xi_{T}T_{2}T_{3}\right)H_{u}\delta^{6}(z)\\\
&+\left(\frac{y^{u}_{22}}{\Lambda^{9}}\xi^{2}_{T}T_{2}T_{2}+\frac{y^{u}_{13}}{\Lambda^{9}}\xi_{T}^{2}T_{1}T_{3}+\frac{y^{u}_{12}}{\Lambda^{11}}(\xi^{3}_{T}+\xi_{F}^{3})T_{1}T_{2}\right)H_{u}\delta^{6}(z)\\\
&+\left(\frac{Y_{a}}{\Lambda^{9}}FN_{a}^{c}\Phi_{BC}+\frac{Y_{s}}{\Lambda^{9}}FN_{s}^{c}\Phi_{AC}\right)H_{u}\delta^{6}(z)\\\
&+\left(\frac{Y_{5\tau}}{\Lambda^{7}}\xi_{F}FT_{3}+\frac{Y_{5\tau}^{{}^{\prime}}}{\Lambda^{9}}\xi_{F}\xi_{T}FT_{2}+\frac{Y_{5\tau}^{{}^{\prime\prime}}}{\Lambda^{11}}\xi_{F}\xi_{T}^{2}FT_{1}\right)H_{5d}\delta^{6}(z)\\\
&+\left(\frac{Y_{5\mu}}{\Lambda^{9}}\xi_{F}^{2}FT_{2}+\frac{Y_{5\mu}^{{}^{\prime}}}{\Lambda^{11}}\xi_{F}^{2}\xi_{T}FT_{1}+\frac{Y_{5e}}{\Lambda^{11}}\xi_{F}^{3}FT_{1}\right)H_{5d}\delta^{6}(z)\\\
&\quad+(H_{5d}^{(0)}\to H_{45d}^{(0)}\ {\rm terms})\\\
&+\frac{M_{a}}{2}N_{a}^{c}N_{a}^{c}\delta^{2}(z_{1})\delta^{2}(z_{3})+\frac{M_{s}}{2}N_{s}^{c}N_{s}^{c}\delta^{2}(z_{2})\delta^{2}(z_{3}),\\\
\\\
\mathcal{L}_{10d}^{(1)}=&\left(\frac{y^{u}_{11}}{\Lambda^{13}}(\xi_{T}^{4}+\xi_{F}^{3}\xi_{T})T_{1}T_{1}+\frac{\tilde{y}^{u}_{33}}{\Lambda^{13}}T_{3}T_{3}\Phi^{2}_{AC,BC}\right)H_{u}\delta^{6}(z)\\\
&+\left(\frac{\tilde{Y}_{a}}{\Lambda^{13}}FN_{a}^{c}\Phi_{BC}^{2}+\frac{\tilde{Y}_{s}}{\Lambda^{13}}FN_{s}^{c}\Phi_{AC}^{2}\right)H_{u}\delta^{6}(z)\\\
&+\left(\frac{Y_{a}^{{}^{\prime}}}{\Lambda^{13}}\xi_{T,X}^{4}FN_{a}^{c}\Phi_{BC}Y_{\tau}+\frac{Y_{s}^{{}^{\prime}}}{\Lambda^{13}}\xi_{T,F}^{4}FN_{s}^{c}\Phi_{AC}Y_{\tau}\right)H_{u}\delta^{6}(z)\\\
&+\left(\frac{Y_{a}^{{}^{\prime\prime}}}{\Lambda^{21}}\xi_{T,X}^{8}FN_{a}^{c}\Phi_{BC}Y_{\mu}+\frac{Y_{s}^{{}^{\prime\prime}}}{\Lambda^{21}}\xi_{T,F}^{8}FN_{s}^{c}\Phi_{AC}Y_{\mu}\right)H_{u}\delta^{6}(z)\\\
&+\left(\frac{Y_{a}^{{}^{\prime\prime}}}{\Lambda^{29}}\xi_{T,X}^{12}FN_{a}^{c}\Phi_{BC}Y_{e}+\frac{Y_{s}^{{}^{\prime\prime}}}{\Lambda^{29}}\xi_{T,F}^{12}FN_{s}^{c}\Phi_{AC}Y_{e}\right)H_{u}\delta^{6}(z)\\\
&+\left(\frac{Y_{5\mu}^{{}^{\prime\prime}}}{\Lambda^{15}}\xi_{F}\xi_{F,T}^{4}FT_{3}+\frac{Y_{5e}^{{}^{\prime}}}{\Lambda^{23}}\xi_{F}\xi_{F,T}^{8}FT_{3}+\frac{Y_{5e}^{{}^{\prime\prime}}}{\Lambda^{17}}\xi_{F}^{2}\xi_{F,T}^{4}FT_{2}+\frac{\tilde{Y}_{\tau}}{\Lambda^{15}}\xi_{F}\Phi^{2}_{AC,BC})FT_{3}\right)H_{5}\delta^{6}(z)\\\
&\quad+(H_{5d}^{(0)}\to H_{45d}^{(0)}\ {\rm terms}),\\\ \end{split}$ (13)

where $\mathcal{L}_{10d}^{(1)}$ contains terms much more suppressed than
$\mathcal{L}_{10d}^{(0)}$.

As is mentioned in the appendix A, most singlet modular forms vanish.
Therefore powers of $\xi$ are necessary to allow up quark masses and therefore
they receive a non trivial structure.

We integrate the 6 extra dimensions and keep only the zero modes, where we
will assume, for simplicity, a single radius for the 3 different tori $\sim
R$. We will keep the same When integrating the extra dimensions $SU(5)$ is
broken but we will keep writing in terms of it for a simpler connection and
add a superscript $(0)$ to denote that we are talking about zero modes.

Let us assume that the $\Phi$ fields do get a VEV $\braket{\Phi}$ above
compactification but they do not have zero modes (they may have an $i$ charge
under $\theta_{4}$), therefore it is a KK mode which gets a VEV.

$\begin{split}\mathcal{L}_{4d}=&\mathcal{L}_{4d}^{(0)}+\mathcal{L}_{4d}^{(1)},\\\
\\\ \mathcal{L}_{4d}^{(0)}=&\left(\left[\frac{y^{u}_{33}}{(2\pi\Lambda
R)^{5}}\right]T_{3}^{(0)}T_{3}^{(0)}+\left[\frac{y^{u}_{23}}{(2\pi\Lambda
R)^{6}}\right]\frac{(\xi_{T}^{(0)})}{\Lambda}T_{2}^{(0)}T_{3}^{(0)}\right.\\\
&+\left.\left[\frac{y^{u}_{22}}{(2\pi\Lambda
R)^{7}}\right]\frac{(\xi_{T}^{(0)})^{2}}{\Lambda^{2}}T_{2}^{(0)}T_{2}^{(0)}+\left[\frac{y^{u}_{13}}{(2\pi\Lambda
R)^{7}}\right]\frac{(\xi_{T}^{(0)})^{2}}{\Lambda^{2}}T_{1}^{(0)}T_{3}^{(0)}+\left[\frac{y^{u}_{12}}{(2\pi\Lambda
R)^{8}}\right]\frac{(\xi_{T,F}^{(0)})^{3}}{\Lambda^{3}}T_{1}^{(0)}T_{2}^{(0)}\right)H_{u}^{(0)}\\\
&+\left(\left[\frac{Y_{a}\braket{\Phi_{BC}}}{(2\pi\Lambda
R)^{5}\Lambda^{4}}\right]F^{(0)}N_{a}^{c(0)}+\left[\frac{Y_{s}\braket{\Phi_{AC}}}{(2\pi\Lambda
R)^{5}\Lambda^{4}}\right]F^{(0)}N_{s}^{c(0)}\right)H_{u}^{(0)}\\\
&+\left(\left[\frac{Y_{5\tau}}{(2\pi\Lambda
R)^{6}}\right]\frac{(\xi_{F}^{(0)})}{\Lambda}F^{(0)}T_{3}^{(0)}+\left[\frac{Y_{5\tau}^{{}^{\prime}}}{(2\pi\Lambda
R)^{7}}\right]\frac{(\xi_{F}^{(0)})(\xi_{T}^{(0)})}{\Lambda^{2}}F^{(0)}T_{2}^{(0)}+\left[\frac{Y_{5\tau}^{{}^{\prime\prime}}}{(2\pi\Lambda
R)^{8}}\right]\frac{(\xi_{F}^{(0)})(\xi_{T}^{(0)})^{2}}{\Lambda^{3}}F^{(0)}T_{1}^{(0)}\right.\\\
&+\left.\left[\frac{Y_{5\mu}}{(2\pi\Lambda
R)^{7}}\right]\frac{(\xi_{F}^{(0)})^{2}}{\Lambda^{2}}F^{(0)}T_{2}^{(0)}+\left[\frac{Y_{5\mu}^{{}^{\prime}}}{(2\pi\Lambda
R)^{8}}\right]\frac{(\xi_{F}^{(0)})^{2}(\xi_{T}^{(0)})}{\Lambda^{3}}F^{(0)}T_{1}^{(0)}+\left[\frac{Y_{5e}}{(2\pi\Lambda
R)^{8}}\right]\frac{(\xi_{F}^{(0)})^{3}}{\Lambda^{3}}F^{(0)}T_{1}^{(0)}\right)H_{5d}^{(0)}\\\
&+(H_{5d}^{(0)}\to H_{45d}^{(0)}\ {\rm terms})\\\
&+\frac{1}{2}M_{a}N_{a}^{c(0)}N_{a}^{c(0)}+\frac{1}{2}M_{s}N_{s}^{c(0)}N_{s}^{c(0)},\\\
\\\ \mathcal{L}_{4d}^{(1)}=&\left(\left[\frac{y^{u}_{11}}{(2\pi\Lambda
R)^{9}}\right]\frac{(\xi_{F,T}^{(0)})^{3}(\xi_{T}^{(0)})}{\Lambda^{4}}T_{1}^{(0)}T_{1}^{(0)}+\left[\frac{\tilde{y}^{u}_{33}\braket{\Phi^{2}_{AC,BC}}^{i}_{i}}{(2\pi\Lambda
R)^{5}\Lambda^{8}}\right]T_{3}^{(0)}T_{3}^{(0)}\right)H_{u}^{(0)}\\\
&+\left(\left[\frac{\tilde{Y}_{a}\braket{\Phi_{BC}^{2}}}{(2\pi\Lambda
R)^{5}\Lambda^{8}}\right]F^{(0)}N_{a}^{c(0)}+\left[\frac{\tilde{Y}_{s}\braket{\Phi_{AC}^{2}}}{(2\pi\Lambda
R)^{5}\Lambda^{8}}\right]F^{(0)}N_{s}^{c(0)}\right.\\\
&+\left[\frac{Y_{a}^{{}^{\prime}}\braket{\Phi_{BC}}Y_{\tau}}{(2\pi\Lambda
R)^{9}\Lambda^{4}}\right]\frac{(\xi_{F,T}^{(0)})^{4}}{\Lambda^{4}}F^{(0)}N_{a}^{c(0)}+\left[\frac{Y_{s}^{{}^{\prime}}\braket{\Phi_{AC}}Y_{\tau}}{(2\pi\Lambda
R)^{9}\Lambda^{4}}\right]\frac{(\xi_{F,T}^{(0)})^{4}}{\Lambda^{4}}F^{(0)}N_{s}^{c(0)}\\\
&+\left[\frac{Y_{a}^{{}^{\prime\prime}}\braket{\Phi_{BC}}Y_{\mu}}{(2\pi\Lambda
R)^{13}\Lambda^{4}}\right]\frac{(\xi_{F,T}^{(0)})^{8}}{\Lambda^{8}}F^{(0)}N_{a}^{c(0)}+\left[\frac{Y_{s}^{{}^{\prime\prime}}\braket{\Phi_{AC}}Y_{\mu}}{(2\pi\Lambda
R)^{13}\Lambda^{4}}\right]\frac{(\xi_{F,T}^{(0)})^{8}}{\Lambda^{8}}F^{(0)}N_{s}^{c(0)}\\\
&\left.+\left[\frac{Y_{a}^{{}^{\prime\prime\prime}}\braket{\Phi_{BC}}Y_{e}}{(2\pi\Lambda
R)^{17}\Lambda^{4}}\right]\frac{(\xi_{F,T}^{(0)})^{8}}{\Lambda^{8}}F^{(0)}N_{a}^{c(0)}+\left[\frac{Y_{s}^{{}^{\prime\prime\prime}}\braket{\Phi_{AC}}Y_{e}}{(2\pi\Lambda
R)^{17}\Lambda^{4}}\right]\frac{(\xi_{F,T}^{(0)})^{8}}{\Lambda^{8}}F^{(0)}N_{s}^{c(0)}\right)H_{u}^{(0)}\\\
&+\left(\left[\frac{Y_{5\mu}^{{}^{\prime\prime}}}{(2\pi\Lambda
R)^{10}}\right]\frac{(\xi_{F}^{(0)})(\xi_{F,T}^{(0)})^{4}}{\Lambda^{5}}F^{(0)}T_{3}^{(0)}+\left[\frac{Y_{5e}^{{}^{\prime}}}{(2\pi\Lambda
R)^{14}}\right]\frac{(\xi_{F}^{(0)})(\xi_{F,T}^{(0)})^{8}}{\Lambda^{9}}F^{(0)}T_{3}^{(0)}\right.\\\
&\left.+\left[\frac{Y_{5e}^{{}^{\prime\prime}}}{(2\pi\Lambda
R)^{11}}\right]\frac{(\xi_{F}^{(0)})^{2}(\xi_{F,T}^{(0)})^{4}}{\Lambda^{6}}F^{(0)}T_{2}^{(0)}+\left[\frac{\tilde{Y}_{5\tau}^{{}^{\prime\prime}}\braket{\Phi^{2}_{AB,BC}}}{(2\pi\Lambda
R)^{6}\Lambda^{8}}\right]\frac{(\xi_{F}^{(0)})}{\Lambda}F^{(0)}T_{3}^{(0)}\right)H_{5d}^{(0)}\
\ \ +(H_{5d}^{(0)}\to H_{45d}^{(0)}\ {\rm terms})\end{split}$ (14)

Where everything inside brackets is a dimensionless coupling constant or an
effective modular form.

This lagrangian comes from a 10 dimensional one, which has implications on the
dimensional factors appearing. If this were a 4d lagrangian originally, we
would have the replacements

$\braket{\Phi}/\Lambda^{4}\to\braket{\Phi}/\Lambda,\ \ \ 2\pi R\Lambda\to 1.$
(15)

Note that we assume that the $\varphi$ obtains its VEV above compactification.
To enforce this, it does not have any zero modes and it is a KK mode which
obtains a VEV [31]. This implies it has dimensionality 4 and everytime it
appears it is suppressed by $\Lambda^{-4}$, making it much more suppressed
than the appearance of a 4d VEV like $\braket{\xi}$.

We assume that the dimensionless couplings reabsorb the factors $(2\pi\Lambda
R)\approx 1$. We define for simplicity

$\tilde{\xi}=\frac{\braket{\xi^{(0)}}}{\Lambda},\ \ \
\tilde{\Phi}=\frac{\braket{\Phi}}{\Lambda^{4}}\sim\tilde{\xi}^{4}\ll 1.$ (16)

These terms fill up the entire mass matrices, however we will ignore terms of

$O(\Lambda^{-13})<\frac{m_{u}}{m_{t}}\sim 10^{-5},$ (17)

which will become approximate texture zeroes in the mass matrices. Therefore
we ignore all the $\mathcal{L}^{(1)}$ terms. We can then simplify

$\begin{split}\mathcal{L}_{4d}^{(0)}=&\left(y^{u}_{33}T_{3}^{(0)}T_{3}^{(0)}+y^{u}_{23}\tilde{\xi}_{T}T_{2}^{(0)}T_{3}^{(0)}+y^{u}_{22}\tilde{\xi}_{T}^{2}T_{2}^{(0)}T_{2}^{(0)}+y^{u}_{13}\tilde{\xi}_{T}^{2}T_{1}^{(0)}T_{3}^{(0)}+y^{u}_{12}\tilde{\xi}_{T,F}^{3}T_{1}^{(0)}T_{2}^{(0)}\right)H_{u}^{(0)}\\\
&+\left(Y_{a}\tilde{\Phi}_{BC}F^{(0)}N_{a}^{c(0)}+Y_{s}\tilde{\Phi}_{AC}F^{(0)}N_{s}^{c(0)}\right)H_{u}^{(0)}\\\
&+\left(Y_{5\tau}\tilde{\xi}_{F}F^{(0)}T_{3}^{(0)}+Y_{5\tau}^{{}^{\prime}}\tilde{\xi}_{F}\tilde{\xi}_{T}F^{(0)}T_{2}^{(0)}+Y_{5\tau}^{{}^{\prime\prime}}\tilde{\xi}_{F}\tilde{\xi}_{T}^{2}F^{(0)}T_{1}^{(0)}\right.\\\
&\left.+Y_{5\mu}\tilde{\xi}_{F}^{2}F^{(0)}T_{2}^{(0)}+Y_{5\mu}^{{}^{\prime}}\tilde{\xi}_{F}^{2}\tilde{\xi}_{T}F^{(0)}T_{1}^{(0)}+Y_{5e}\tilde{\xi}_{F}^{3}F^{(0)}T_{1}^{(0)}\right)H_{5d}^{(0)}\
\ \ +(H_{5d}^{(0)}\to H_{45d}^{(0)}\ {\rm terms})\\\
&+\frac{1}{2}M_{a}N_{a}^{c(0)}N_{a}^{c(0)}+\frac{1}{2}M_{s}N_{s}^{c(0)}N_{s}^{c(0)}.\end{split}$
(18)

###  3.5 SM fermion mass matrices

The structure of the modular forms [7] is described in the Appendix A. Each
modular form has a different representation and weight, as shown in Table 1.
Furthermore, each modular form comes from different tori so that they are
defined by different moduli, giving each one different structure:

$\begin{split}Y_{a}&=Y_{\textbf{3}}^{(4)}(i+2)=y_{a}(0,1,-1)^{T},\\\
Y_{s}&=Y_{\textbf{3}}^{(2)}(i)=y_{s}(1,1+\sqrt{6},1-\sqrt{6})^{T},\\\
Y_{\tau}&=Y_{\textbf{3}}^{(2)}(\omega)=y_{\tau}(0,1,0)^{T},\\\
Y_{\mu}&=Y_{\textbf{3}}^{(4)}(\omega)=y_{\mu}(0,0,1)^{T},\\\
Y_{e}&=Y_{\textbf{3}I}^{(6)}(\omega)=y_{e}(1,0,0)^{T}.\end{split}$ (19)

we obtain the fermion masses. The $Y_{5e,5\mu,5\tau}$, $Y_{45e,45\mu,45\tau}$
and primed ones have the same modular form structure, which we simply write as
$Y_{e,\mu,\tau}$ respectively in Eq.19. The lower case $y$’s are arbitrary
complex dimensionless parameters. The primed, $5$ and $45$ subscript modular
forms indicate different $y$ complex parameter but the same flavour structure.

The symmetric up-quark mass matrix originates from the $TTH_{u}$ couplings in
Eq. 18,

$M_{u}=\left(\begin{array}[]{ccc}0&y_{12}^{u}\tilde{\xi}_{T,F}^{3}e^{i\phi_{u1}}&y_{13}^{u}\tilde{\xi}_{T}^{2}\\\
y_{12}^{u}\tilde{\xi}_{T,F}^{3}e^{i\phi_{u1}}&y_{22}^{u}\tilde{\xi}_{T}^{2}&y_{23}^{u}\tilde{\xi}_{T}e^{i\phi_{u2}}\\\
y_{13}^{u}\tilde{\xi}_{T}^{2}&y_{23}^{u}\tilde{\xi}_{T}e^{i\phi_{u2}}&y_{33}^{u}\end{array}\right)v_{u},$
(20)

where each $y$ is now an arbitrary real dimensionless constant. Phases can be
redefined so that there are 5 real parameters and 2 phases. This yields the
approximate up-type quark mass hierarchies,
$m_{u}\sim\tilde{\xi}_{T,F}^{4}v_{u}$, $m_{c}\sim\tilde{\xi}_{T}^{2}v_{u}$,
$m_{t}\sim v_{u}$.

The $H_{45}$ breaks the charged lepton and down quark degeneracy. The
$Y_{5e,5\mu,5\tau}$, $Y_{45e,45\mu,45\tau}$ and primed ones have the same
modular form structure but different overall complex constants multiplying
them. Therefore both are diagonal mass matrices but the actual masses are
determined after Higgs mixing [32]§§§The presence of both $H_{{45}}$ and
$H_{\overline{45}}$ allows the mass term $M_{45}H_{{45}}H_{\overline{45}}$
which ensures that all the components of these Higgs fields are heavy, apart
from the Higgs doublet component of $H_{\overline{45}}$ that mixes with the
Higgs doublet contained in $H_{5d}$, to produce the light linear combination
identified as the physical Higgs doublet $H_{d}$. In this way, the Higgs
doublet-triplet splitting mechanism discussed earlier is sufficient to ensure
one light physical combination of down-type Higgs doublets which we identify
as $H_{d}$.,

$\begin{array}[]{ll}y_{e11}v_{d}=y_{5e}v_{d5}-3y_{5e}v_{d45},&y_{d11}v_{d}=y_{5d}v_{d}+y_{45d}v_{d45},\\\
y_{e22}v_{d}=y_{5\mu}v_{d5}-3y_{5\mu}v_{d45},&y_{d22}v_{d}=y_{5s}v_{d5}+y_{45s}v_{d45},\\\
y_{e33}v_{d}=y_{5\tau}v_{d5}-3y_{5\tau}v_{d45},&y_{d33}v_{d}=y_{5b}v_{d5}+y_{45b}v_{d45},\\\
y_{e21}v_{d}=y^{\prime}_{5\mu}v_{d5}-3y^{\prime}_{5\mu}v_{d45},&y_{d12}v_{d}=y^{\prime}_{5s}v_{d5}+y^{\prime}_{45s}v_{d45},\\\
y_{e32}v_{d}=y^{\prime}_{5\tau}v_{d5}-3y^{\prime}_{5\tau}v_{d45},&y_{d23}v_{d}=y^{\prime}_{5b}v_{d5}+y^{\prime}_{45b}v_{d45},\\\
y_{e31}v_{d}=y^{\prime\prime}_{5\tau}v_{d5}-3y^{\prime\prime}_{5\tau}v_{d45},&y_{d13}v_{d}=y^{\prime\prime}_{5b}v_{d5}+y^{\prime\prime}_{45b}v_{d45},\\\
\end{array}$ (21)

where $v_{d}$ is an effective down Higgs VEV.

The triangular down-quark and charged lepton mass matrices originate from the
$FTH_{d}$ couplings in Eq. 18,

$M_{d}=\left(\begin{array}[]{ccc}y_{d11}\tilde{\xi}_{F}^{3}&y_{d12}\tilde{\xi}_{F}^{2}\tilde{\xi}_{T}&y_{d13}\tilde{\xi}_{F}\tilde{\xi}_{T}^{2}\\\
0&y_{d22}\tilde{\xi}_{F}^{2}&y_{d23}\tilde{\xi}_{F}\tilde{\xi}_{T}e^{i\phi_{d2}}\\\
0&0&y_{d33}\tilde{\xi}_{F}\end{array}\right)v_{d},$ (22)
$M_{e}=\left(\begin{array}[]{ccc}y_{e11}\tilde{\xi}_{F}^{3}&0&0\\\
y_{e21}\tilde{\xi}_{F}^{2}\tilde{\xi}_{T}&y_{e22}\tilde{\xi}_{F}^{2}&0\\\
y_{e31}\tilde{\xi}_{F}\tilde{\xi}_{T}^{2}&y_{e32}\tilde{\xi}_{F}\tilde{\xi}_{T}e^{i\phi_{d1}}&y_{e33}\tilde{\xi}_{F}\end{array}\right)v_{d},$
(23)

where each matrix has 6 real parameters and 1 phase. These yield the the
approximate down-type quark and charged lepton mass hierarchies, $m_{d}\sim
m_{e}\sim\tilde{\xi}_{F}^{3}v_{d}$, and $m_{s}\sim
m_{\mu}\sim\tilde{\xi}_{F}^{2}v_{d}$, and $m_{b}\sim
m_{\tau}\sim\tilde{\xi}_{F}v_{d}$. We have written each mass matrix in LR
convention so that, upon diagonalisation, $M_{d}$ will yield left-handed
mixing angles arising from the upper-right off-diagonal terms, while $M_{e}$
will yield approximately zero left-handed mixing angles (with non-zero right-
handed mixing angles from the lower-left off-diagonal terms). This means that
$M_{d}$ (as well as $M_{u}$) will both contribute approximately equally to the
CKM mixing angles, while $M_{e}$ will not contribute appreciably to the PMNS
mixing angles.

Finally the neutrino Dirac and Majorana mass matrices from $FNH_{u}$ and
$N^{c}N^{c}$ terms are

$M_{D}=\left(\begin{array}[]{cc}0&y_{s}\tilde{\Phi}_{AC}\\\
y_{a}\tilde{\Phi}_{BC}&y_{s}\tilde{\Phi}_{AC}(1-\sqrt{6})\\\
-y_{a}\tilde{\Phi}_{BC}&y_{s}\tilde{\Phi}_{AC}(1+\sqrt{6})\end{array}\right)v_{u},\
\ \ M_{N}=\left(\begin{array}[]{cc}M_{a}&0\\\ 0&M_{s}\end{array}\right),$ (24)

which has the structure of a type-I seesaw mechanism which generates effective
mass matrix for the light neutrinos:

$\footnotesize m_{\nu}=M_{D}\cdot M_{R}^{-1}\cdot
M_{D}^{T}=v_{u}^{2}\begin{pmatrix}\dfrac{(y_{s}\tilde{\Phi}_{AC})^{2}}{M_{s}}&\dfrac{(y_{s}\tilde{\Phi}_{AC})^{2}(2-n)}{M_{s}}&\dfrac{(y_{s}\tilde{\Phi}_{AC})^{2}n}{M_{s}}\\\\[12.0pt]
.&\dfrac{(y_{a}\tilde{\Phi}_{BC})^{2}}{M_{a}}+\dfrac{(y_{s}\tilde{\Phi}_{AC})^{2}(2-n)^{2}}{M_{s}}&-\dfrac{(y_{a}\tilde{\Phi}_{BC})^{2}}{M_{a}}+\dfrac{(y_{s}\tilde{\Phi}_{AC})^{2}n(2-n)}{M_{s}}\\\\[12.0pt]
.&.&\dfrac{(y_{a}\tilde{\Phi}_{BC})^{2}}{M_{a}}+\dfrac{(y_{s}\tilde{\Phi}_{AC})^{2}n^{2}}{M_{s}}\end{pmatrix},$
(25)

where $n=1+\sqrt{6}\approx 3.45$. This can be redefined in terms of 3
independent physical parameters

$m_{\nu}=m_{a}\left(\begin{matrix}0&0&0&\\\ 0&1&-1\\\
0&-1&1\end{matrix}\right)+m_{b}e^{i\eta}\left(\begin{matrix}1&(2-n)&n&\\\
(2-n)&(2-n)^{2}&n(2-n)\\\ n&n(2-n)&n^{2}\end{matrix}\right),$ (26)

where

$m_{a}=\left|\frac{v_{u}^{2}(y_{a}\tilde{\Phi}_{BC})^{2}}{M_{a}}\right|,\ \ \
m_{b}=\left|\frac{v_{u}^{2}(y_{s}\tilde{\Phi}_{AC})^{2}}{M_{s}}\right|$ (27)

which corresponds to flipped CSD($n$) with $n=1+\sqrt{6}\approx 3.45$ in the
notation of ref. [17]. Therefore the model has only these three parameters for
the whole neutrino sector, where the PMNS mixing parameters do not receive any
appreciable contribution from the charged lepton sector as mentioned above and
discussed further below. This results in a highly predictive flipped
CSD($1+\sqrt{6}$) setup [3, 17] with an excellent fit to neutrino oscillation
parameters involving three real input parameters to determine the three
neutrino masses and the six parameters of the PMNS matrix, where one neutrino
mass and one Majorana phase are predicted to be zero.

|  |  NuFit 5.2 $\pm 1\sigma$ | Natural Fit | Unrestricted fit
---|---|---|---|---
| $\theta_{12}/^{\circ}$ | $33.41_{-0.72}^{+0.75}$ | 34.34 | $34.32$
| $\theta_{23}/^{\circ}$ | $49.1_{-1.3}^{+1.0}$ | 48.31 | $48.95$
| $\theta_{13}/^{\circ}$ | $8.54_{-0.12}^{+0.11}$ | 8.54 | $8.54$
| $\delta/^{\circ}$ | $197_{-25}^{+42}$ | 284 | $253$
| $\dfrac{\Delta m^{2}_{21}}{10^{-5}~{}\text{eV}^{2}}$ | $7.41_{-0.20}^{+0.21}$ | 7.42 | $7.41$
| $\dfrac{\Delta m^{2}_{3\ell}}{10^{-3}~{}\text{eV}^{2}}$ | $+2.511_{-0.021}^{+0.028}$ | 2.510 | $2.511$
| $\dfrac{m_{a}}{10^{-3}~{}{\rm eV}}$ |  | $31.47$ | $31.47$
| $\dfrac{m_{b}}{10^{-3}~{}{\rm eV}}$ |  | 2.32 | $-2.28$
| $\eta/\pi$ |  | $1.24$ | $0.26$
| $y_{e33}$ |  | $0.46$ | $-0.06$
| $y_{e22}$ |  | $0.13$ | $0.89$
| $y_{e11}$ |  | $0.003$ | $0.003$
| $y_{e32}$ |  | $<0.7$ | $1.03$
| $y_{e31}$ |  | $<0.7$ | $0.04$
| $y_{e21}$ |  | $<0.7$ | $0.05$
| $\phi_{d1}/\pi$ |  | $0-2$ | $1.33$
| $\chi^{2}$ |  | $6.3$ | $1.56$
Table 2: Normal Ordering NuFit 5.2 values [33] for the neutrino observables,
and the best fit point from the model. Note that the predictions for the PMNS
mixing parameters for the natural fit are practically identical to that for
the Littlest Seesaw Model in which the charged lepton mass matrix is diagonal.
Assumed values for the fit: $\tan\beta=10$, $\tilde{\xi}_{F}=0.22$, and
$\tilde{\xi}_{T}=0.41$.

One may ask how the fit changes due to the off-diagonal charged lepton mass
matrix parameters in the lower-left of the mass matrix $M_{e}$ in Eq. 23. To
address this question we show two fits in Table 2. First we assume that the
off-diagonal terms in $M_{e}$ are smaller than the diagonal ones on the same
row, which is a natural choice, since they are relatively suppressed by small
expansion parameters. The results show that this generates quite a good for a
wide range of off-diagonal terms, which supports the assertion that such terms
are practically irrelevant. If we relax this condition, we can slightly
improve the fit when they become larger than the diagonal ones, for instance
by significantly increasing $y_{e32}$ relative to $y_{e33}$. We can see that
even in this less natural case, the predictions emerging from the fit are
similar to the first case. We conclude that the predictions for the PMNS
parameters are quite independent of the parameters appearing in the charged
lepton mass matrix. This means that the model maintains very similar
predictions to the Littlest Modular Seesaw model where the charged lepton mass
matrix is diagonal and neutrino masses and PMNS parameters are all determined
from just the three input parameters in Eq. 26. The present model in addition
also accounts for the quark mass hierarchies and CKM mixing parameters, as
well as the charged lepton mass hierarchies, in terms of two small expansion
parameters arising from the weighton fields. However the charged fermion
masses also depend on undetermined coefficients and so are not precisely
predicted. Similarly the CKM angles, although generically predicted to be
small, suppressed by powers of weighton expansion parameters, receive
contributions from both the up and down quark sectors, and are not predicted
or constrained. For this reason we do not show fits for the quark masses and
CKM mixing parameters.

##  4 Conclusion

In this paper we have discussed 10d orbifold GUTS based on modular symmetry as
candidates for a theory of Flavour. To illustrate the approach, we presented
an $SU(5)$ 10d orbifold theory, based on $S_{4}^{3}$ modular symmetry, capable
of giving quark and charged lepton mass hierarchies, using the weighton
mechanism. The assumed
$(\mathbb{T}^{2})^{3}/(\mathbb{Z}_{4}\times\mathbb{Z}_{2})$ orbifold yields
three $S_{4}$ modular groups and three moduli at the fixed points
$i,i+2,\omega=e^{2i\pi/3}$, which is a stable configuration. Using these fixed
points, the model reproduces the highly predictive Littlest Seesaw Mechanism
in the neutrino sector to very good approximation. The off-diagonal lower
triangular entries in the charged lepton mass matrix yield only very small
contributions to the left-handed PMNS angles, since the upper triangular
entries are zero, a feature that we have verified numerically. Although the
model is very predictive in the neutrino and PMNS sectors, it does not make
any predictions for CKM parameters. The same orbifold also yields successful
$SU(5)$ breaking with doublet-triplet splitting of the Higgs multiplets.

## Acknowledgements

SFK acknowledges the STFC Consolidated Grant ST/L000296/1 and the European
Union’s Horizon 2020 Research and Innovation programme under Marie Sklodowska-
Curie grant agreement HIDDeN European ITN project (H2020-MSCA-
ITN-2019//860881-HIDDeN).

##  Appendix A $S_{4}$ Modular forms

With only two extra dimensions the single complex modulus $\tau$ has a modular
symmetry $\overline{\Gamma}=SL(2,\mathbb{Z})$ The group $\overline{\Gamma}$ is
the group of linear transformations which acts on the modulus $\tau$ as
follows,

$\tau\rightarrow\gamma\tau=\frac{a\tau+b}{c\tau+d},~{}~{}\text{with}~{}~{}a,b,c,d\in\mathbb{Z},~{}~{}ad-
bc=1,~{}~{}~{}\mathop{\mathrm{Im}}\tau>0\,.$ (28)

The modular group $\overline{\Gamma}$ can be generated by $S$ and $T$

$S:\tau\mapsto-\frac{1}{\tau},~{}~{}~{}~{}\quad T:\tau\mapsto\tau+1.$ (29)

From the infinite modular group the finite subgroup
$\Gamma_{N}=PSL(2,\mathbb{Z})/\Gamma(N)$ may be obtained. A crucial element of
the modular invariance is the modular form $f(\tau)$ of weight $k$ and level
$N$. The modular form $f(\tau)$ is a holomorphic function of the modulus
$\tau$ and it is required to transform under the action of
$\overline{\Gamma}(N)$ as,

$f\left(\frac{a\tau+b}{c\tau+d}\right)=(c\tau+d)^{k}f(\tau)$ (30)

The modular forms of level $N=4$ have been constructed in [34, 35] .

The associated finite modular group $\Gamma_{4}$ with generators $S$ and $T$
which fulfill the following rations

$S^{2}=(ST)^{3}=(TS)^{3}=T^{4}=1\,.$ (31)

The modular forms are built from the functions $Y_{i}(\tau)$ with $i=1,...,5$
[35]. The ones we use are:

$\begin{split}Y_{\textbf{3}}^{(2)}=&\left(\begin{array}[]{c}Y_{3}\\\ Y_{4}\\\
Y_{5}\end{array}\right),\\\
Y_{\textbf{3}}^{(4)}=&\left(\begin{array}[]{c}Y_{1}Y_{4}+Y_{2}Y_{5}\\\
Y_{2}Y_{3}+Y_{1}Y_{5}\\\ Y_{1}Y_{3}+Y_{2}Y_{4}\end{array}\right),\\\
Y_{\textbf{3}I}^{(6)}=&\left(\begin{array}[]{c}2Y_{1}Y_{2}Y_{3}\\\
2Y_{1}Y_{2}Y_{4}\\\ 2Y_{1}Y_{2}Y_{5}\end{array}\right),\\\
Y_{\textbf{3}II}^{(6)}=&\left(\begin{array}[]{c}Y_{2}^{2}Y_{4}+Y_{1}^{2}Y_{5}\\\
Y_{1}^{2}Y_{3}+Y_{2}^{2}Y_{5}\\\
Y_{2}^{2}Y_{3}+Y_{1}^{2}Y_{4}\end{array}\right),\\\
Y_{\textbf{1}}^{(0)}=&1,\\\ Y_{\textbf{1}}^{(2)}=&0,\\\
Y_{\textbf{1}}^{(4)}=&2Y_{1}Y_{2},\\\
Y_{\textbf{1}}^{(6)}=&Y_{1}^{3}+Y_{2}^{3},\\\
Y_{\textbf{1}}^{(8)}=&Y_{1}^{2}Y_{2}^{2},\\\
Y_{\textbf{1}}^{(10)}=&Y_{2}Y_{1}^{4}+Y_{2}^{4}Y_{1},\\\ \end{split}$ (32)

Where at $\tau=\omega$ we have

$Y_{1}(\omega)=Y_{3}(\omega)=Y_{5}(\omega)=0,\ \ \ Y_{2}(\omega)=\alpha,\ \ \
Y_{4}(\omega)=\alpha(1-\omega^{2}),$ (33)

where $\alpha\approx 1.79288$.

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11footnotetext:  Department of Mathematics and Computing, Indian Institute of
Technology (Indian School of Mines), Dhanbad 826 004, India.

# A Class of few-Lee weight $\mathbb{Z}_{2}[u]$-linear codes using simplicial
complexes and minimal codes via Gray map

Pramod Kumar Kewat1 and Nilay Kumar Mondal1,∗<EMAIL_ADDRESS>(P.K.
Kewat), [corresponding<EMAIL_ADDRESS>(N.K. Mondal)

###### Abstract.

Recently some mixed alphabet rings are involved in constructing few-Lee weight
additive codes with optimal or minimal Gray images using suitable defining
sets or down-sets. Inspired by these works, we choose the mixed alphabet ring
$\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$ to construct a special class of linear code
$C_{L}$ over $\mathbb{Z}_{2}[u]$ with $u^{2}=0$ by employing simplicial
complexes generated by a single maximal element. We show that $C_{L}$ has few-
Lee weights by determining the Lee weight distribution of $C_{L}$.
Theoretically, this shows that we may employ simplicial complexes to obatin
few-weight codes even in the case of mixed alphabet rings. We show that the
Gray image of $C_{L}$ is self-orthogonal and we have an infinite family of
minimal codes over $\mathbb{Z}_{2}$ via Gray map, which can be used to secret
sharing schemes.

###### Key words and phrases:

Few-Lee weight codes, Mixed alphabet ring, Simplicial complexes, Minimal
codes.

###### 1991 Mathematics Subject Classification:

94B05, 94A62

##  1\. Introduction

Constructing few-weight codes over finite fields is a long standing area of
interest in algebraic coding theory. It has key applications in graph theory,
finite geometry and combinatorial designs (see [3]). The authors, in [15]
first introduced few-Lee weight codes over the finite ring
$\mathbb{F}_{2}+u\mathbb{F}_{2}$ with $u^{2}=0$ and as an application they
found some infinite families of optimal (with respect to the Griesmer bound)
few-weight codes over $\mathbb{F}_{2}$ through Gray map. Apart from this,
using a famous characterization of minimal codes, namely the Ashikhmin-Barg
lemma, they showed that some of the Gray images are also minimal over
$\mathbb{F}_{2}$, which have direct applications in secret sharing schemes and
authentication codes (see [1], [4], [7], [23]). Actually finding minimal codes
over finite fields is itself a central topic in algebraic coding theory and in
[15] the authors succesfully clubbed these two important topics together.
After that lots of researchers have constructed several classes of few-Lee
weight codes over different finite chain rings of characteristic $p$ having
optimal or minimal codes as appliactions. In particular for $p=2$, see [11],
[12], [14], [16], [18], [22]. The researchers mainly used suitable defining
sets, simplicial complexes or down-sets (for $p>2$ only) to construct such
codes (see [5], [6], [21]). Very recently the authors in [13] first studied
few-Lee weight codes over a non-chain ring
$\mathbb{F}_{2}+u\mathbb{F}_{2}+v\mathbb{F}_{2}+uv\mathbb{F}_{2}$. The
authors, in [8] constructed one-Lee weight additive codes over the mixed
alphabet ring $\mathbb{Z}_{2}\mathbb{Z}_{4}$. In [17], the authors studied
one-Lee weight and two-Lee weight codes over
$\mathbb{Z}_{2}\mathbb{Z}_{2}[u,v]$ with $u^{2}=0,v^{2}=0$ and $uv=vu$. The
authors, in [9] generalized [8] and studied the construction of one-Lee weight
and two-Lee weight $\mathbb{Z}_{2}\mathbb{Z}_{4}[u]$-additive codes. In [19],
the authors constructed few-Lee weight additive codes over
$\mathbb{Z}_{p}\mathbb{Z}_{p}[u]$ with $u^{2}=0$ having good codes as Gray
images by using a suitable defining set
$D=(D_{0}^{e},D_{0}^{e}+u\mathbb{F}_{p^{m}})\subseteq\mathbb{F}_{p^{m}}\times(\mathbb{F}_{p^{m}}+u\mathbb{F}_{p^{m}})$,
where $D_{0}^{e}$ is the cyclotomic class of order e in $\mathbb{F}_{p^{m}}$.
After that the authors, in [20], raised a natural question that whether we can
construct such few-Lee weight codes with minimal or optimal Gray images over
mixed alphabet rings by employing down-sets or simplicial complexes and they
provided a partial answer to that by using the mixed alphabet ring
$\mathbb{Z}_{p}\mathbb{Z}_{p}[u]$ with $u^{2}=u$ and $p>2$ to study some few-
Lee weight additive codes by using the set
$L=\\{t_{1},ut_{2}+(1-u)t_{3}~{}|~{}t_{i}\in\Delta_{i}^{c}\\}\subseteq(\mathbb{Z}_{p}^{m}\times\mathbb{Z}_{p}^{m}+u\mathbb{Z}_{p}^{m})$
consisting of down-sets $\Delta_{i}\subseteq\mathbb{Z}_{p}^{m}$ generated by a
single element. In this paper, we choose the mixed alphabet ring
$\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$ and defined an inner product on
$\mathbb{Z}_{2}^{m}\mathbb{Z}_{2}^{m}[u]$ to construct a class of linear codes
over $\mathbb{Z}_{2}[u]$ with $u^{2}=0$ using the set
$L=\\{t_{1},t_{2}+ut_{3}~{}|~{}t_{i}\in\Delta_{i}^{c}\\}\subseteq(\mathbb{Z}_{2}^{m}\times\mathbb{Z}_{2}^{m}+u\mathbb{Z}_{2}^{m})$
consisting of simplicial complexes $\Delta_{i}\subseteq\mathbb{Z}_{2}^{m}$
generated by a single element. Further we compute the Lee weight distribution
for this class of codes and find it to be of few-Lee weight. To the best of
our knowledge, this is the first time simplicial complexes are being used in
the case of mixed alphabet rings and we must note that down-sets (a subclass
of simplicial complexes) can be used only in the case $p>2$. So it gives the
complete answer to the question raised in [20]. We have also studied their
Gray images and show that the Gray image of $C_{L}$ is self-orthogonal.
Moreover, we have an infinite family of binary minimal codes with the help of
Ashikhmin-Barg lemma. This paper is organized as follows. Section 2 covers the
basic definitions and related prerequisites. In Section 3, we compute the Lee
weight distributions of the linear code $C_{L}$ over $\mathbb{Z}_{2}[u]$ as
defined in Section 2. As an application, we also find an infinite family of
minimal codes over $\mathbb{Z}_{2}$ as Gray images of $C_{L}$ in Section 4.
Finally we give an overview of this material in Section 5.

##  2\. Preliminaries

###  2.1. Rings and simplicial complexes

Let $\mathbb{Z}_{2}$ be the ring of integers modulo $2$ and
$\mathbb{Z}_{2}[u]=\mathbb{Z}_{2}+u\mathbb{Z}_{2},~{}u^{2}=0$. We define
$\mathcal{R}=\mathbb{Z}_{2}\mathbb{Z}_{2}[u]=\\{(x,y+uz)|x,y,z\in\mathbb{Z}_{2}\\}$
and $\mathcal{R}^{m}=\\{(p,q+ur)|p,q,r\in\mathbb{Z}_{2}^{m}\\}$, where $m$ is
any positive integer strictly greater than 1. Since $\mathbb{Z}_{2}$ is a
subring of $\mathbb{Z}_{2}[u]$, it is easy to observe that $\mathcal{R}^{m}$
is a $\mathbb{Z}_{2}[u]$-module under component-wise addition and
$\mathbb{Z}_{2}[u]$-scalar multiplication defined as:
$(p_{1},q_{1}+ur_{1})+(p_{2},q_{2}+ur_{2})=((p_{1}+p_{2}),(q_{1}+q_{2})+u(r_{1}+r_{2}))$
and $(y+uz)(p,q+ur)=(yp,yq+u(yr+zq))$. For any vector
$w\in\mathbb{Z}_{2}^{m}$, the support of $w$ denoted by $supp(w)$ is defined
as the set of all nonzero coordinate positions of $w$. Clearly, $w\mapsto
supp(w)$ is a bijection between $\mathbb{Z}_{2}^{m}$ and the power set of
$[m]=\\{1,\ldots,m\\}$ i.e., $2^{[m]}$. Let
$\Delta\subseteq\mathbb{Z}_{2}^{m}$ such that if $w_{1}\in\Delta$ and
$supp(w_{2})\subseteq supp(w_{1})$ imply $w_{2}\in\Delta$, then $\Delta$ is
called a simplicial complex. For a vector $w\in\Delta$, if there does not
exist any vector $v\in\Delta$ such that $supp(w)\subseteq supp(v)$, then we
call $w$ is a maximal element of $\Delta$. Clearly a simplical complex
$\Delta$ may admit more than one maximal elements. Here we consider the
simplicial complexes with a single maximal element. The simplicial complex
$\Delta_{supp(w)}$ generated by a single maximal element $w$ is defined to be
the set of all subsets of $supp(w)\subseteq[m]$ and is of size
$2^{|supp(w)|}$, where $|\cdot|$ denotes the size of a set.

###  2.2. Lee weight, inner product and code

A Gray map $\phi:\mathbb{Z}_{2}[u]\rightarrow\mathbb{Z}_{2}^{2}$ is defined as
$y+uz\mapsto(z,y+z)$, where $y,z\in\mathbb{Z}_{2}$. We can naturally extend
the map $\phi$ to
$\Phi:\mathbb{Z}_{2}^{m}+u\mathbb{Z}_{2}^{m}\rightarrow\mathbb{Z}_{2}^{2m}$ as
$\Phi(q+ur)=(r,q+r)$, where $q,r\in\mathbb{Z}_{2}^{m}$. For a vector
$w\in\mathbb{Z}_{2}^{m}$, the Hamming weight of $w$ denoted by $wt_{H}(w)$ is
defined as $wt_{H}(w)=|supp(w)|$. The Lee weight of a vector
$w^{\prime}=(q+ur)\in\mathbb{Z}_{2}^{m}+u\mathbb{Z}_{2}^{m}$ denoted by
$wt_{L}(a)$ is defined as its Hamming weight under the Gray map $\Phi$, i.e.
$wt_{L}(w^{\prime})=wt_{L}(q+ur)=wt_{H}(r)+wt_{H}(q+r)$. The Lee distance of
$w^{\prime}_{1},w^{\prime}_{2}\in\mathbb{Z}_{2}^{m}+u\mathbb{Z}_{2}^{m}$ is
defined as $wt_{L}(w^{\prime}_{1}-w^{\prime}_{2})$. It is quite easy to check
that $\Phi$ is an isometry between
$(\mathbb{Z}_{2}^{m}+u\mathbb{Z}_{2}^{m},d_{L})$ and
$(\mathbb{Z}_{2}^{2m},d_{H})$. A linear code $C$ of length $m$ over a finite
commutative ring $R$ is an $R$-submodule of $R^{m}$. The Euclidean inner
product of vectors
$a=(a_{1},a_{2},\ldots,a_{m}),b=(b_{1},b_{2},\ldots,b_{m})\in R^{m}$ is
$\langle a,b\rangle=\sum\limits_{i=1}^{m}a_{i}b_{i}\in R$. Let
$w_{1},w_{2}\in\mathbb{Z}_{2}^{m}$ and
$w^{\prime}_{1},w^{\prime}_{2}\in\mathbb{Z}_{2}^{m}+u\mathbb{Z}_{2}^{m}$. Now
we define the inner product of two vectors $(w_{1},w^{\prime}_{1})$ and
$(w_{2},w^{\prime}_{2})$ in $\mathcal{R}^{m}$ as
$((w_{1},w^{\prime}_{1})\cdot(w_{2},w^{\prime}_{2}))=u\langle
w_{1},w_{2}\rangle+\langle w^{\prime}_{1},w^{\prime}_{2}\rangle$. Let
$\Delta_{1},\Delta_{2},\Delta_{3}\subseteq\mathbb{Z}_{2}^{m}$ be three
simplicial complexes generated by a single maximal elements having support
$D,E,F\subseteq[m]$ respectively and they are not equal to
$\mathbb{Z}_{2}^{m}$. From now on we denote $\Delta_{1},\Delta_{2},\Delta_{3}$
as $\Delta_{D},\Delta_{E},\Delta_{F}$ throughout this paper. We set
$L=\\{l=(t_{1},t_{2}+ut_{3})|t_{1}\in\Delta_{D}^{c},t_{2}\in\Delta_{E}^{c},t_{3}\in\Delta_{F}^{c}\\}\subseteq\mathcal{R}^{m}$,
where $\Delta^{c}=\mathbb{Z}_{2}^{m}\setminus\Delta$. We define
$C_{L}=\\{c_{a}=((a\cdot l))_{l\in L}|a\in\mathcal{R}^{m}\\}$. It is easy to
check that $C_{L}$ is a code over $\mathbb{Z}_{2}[u]$ of length $|L|$. The
code $C_{L}$ is said to be a $t$-Lee weight code if all of its codewords have
only $t$ different nonzero Lee weights.

###  2.3. Generating function and characteristic function

Let $X$ be a subset of $\mathbb{Z}_{2}^{m}$. Chang and Hyun ([5]) introduced
the following $m$-varriable generating function associated with the set $X$:

$\mathcal{H}_{X}(x_{1},x_{2},\ldots,x_{m})=\sum\limits_{u\in
X}\prod\limits_{i=1}^{m}(x_{i})^{u_{i}}\in\mathbb{Z}[x_{1},x_{2},\ldots,x_{m}],~{}\text{where}~{}u=(u_{1},u_{2},\ldots,u_{m})\in\mathbb{Z}_{2}^{m}.$
(1)

######  Lemma 2.1.

[5, Lemma 2] Let $\Delta\subseteq\mathbb{Z}_{2}^{m}$ be a simplicial complex.
Then the following results hold.

  * (1)

$\mathcal{H}_{\Delta}(x_{1},x_{2},\ldots,x_{m})+\mathcal{H}_{\Delta^{c}}(x_{1},x_{2},\ldots,x_{m})=\mathcal{H}_{\mathbb{Z}_{2}^{m}}(x_{1},x_{2},\ldots,x_{m})=\prod\limits_{i\in[m]}(1+x_{i})$,

  * (2)

$\mathcal{H}_{\Delta_{S}}(x_{1},x_{2},\ldots,x_{m})=\prod\limits_{i\in
S}(1+x_{i})$, where $\Delta_{S}$ is a simplicial complex generated by a single
maximal elemnt having support $S\subseteq[m]$.

Suppose $X$ and $Y$ are two subsets of $[m]$. We define a characteristic
function $\chi:2^{[m]}\times 2^{[m]}\rightarrow\\{0,1\\}$ as $\chi(X|Y)=1$ if
and only if $X\cap Y=\emptyset$ and zero otherwise.

######  Lemma 2.2.

[22, Lemma 3.1] Let $D,E\subseteq[m]$. Then

  * (1)

$|\\{X\subseteq[m]~{}|~{}\emptyset\neq X,\chi(X|D)=1\\}|=2^{m-|D|}-1$ and
$|\\{X\subseteq[m]~{}|~{}\chi(X|D)=0\\}|=2^{m}-2^{m-|D|}$.

  * (2)

$|\\{X\subseteq[m]~{}|~{}\emptyset\neq X,\chi(X|D)\chi(X|E)=1\\}|=2^{m-|D\cup
E|}-1$ and $|\\{X\subseteq[m]~{}|~{}\chi(X|D)\chi(X|E)=0\\}|=2^{m}-2^{m-|D\cup
E|}$.

  * (3)

Define $D\oplus E=(D\cup E)\setminus(D\cap E)$. Let
$T_{i}=|\\{(X,Y)~{}|~{}\emptyset\neq X,Y\subseteq[m],X\neq
Y,\chi(Y|E)=1,\chi(X|D)+\chi(X\oplus Y|D)=i\\}|$, where $i\in\\{0,1,2\\}$.
Then $T_{0}=2^{m}(2^{m-|E|}-1)+2^{m-|D|}(1+2^{m-|D\cup E|}-2^{m+1-|E|})$,
$T_{1}=2(2^{m-|D|}-1)(2^{m-|E|}-2^{m-|D\cup E|})$, and
$T_{2}=(2^{m-|D|}-2)(2^{m-|D\cup E|}-1)$.

##  3\. Main Result

Let $a=(p,q+ur),l=(t_{1},t_{2}+ut_{3})$, where $a\in\mathcal{R}^{m}$ and $l\in
L$. The Lee weigt of the codeword $c_{a}$ of $C_{L}$ is

$\displaystyle wt_{L}(c_{a})$
$\displaystyle=wt_{L}(((p,q+ur)\cdot(t_{1},t_{2}+ut_{3}))_{t_{1},t_{2},t_{3}})$
$\displaystyle=wt_{L}((u\langle p,t_{1}\rangle+\langle
q+ur,t_{2}+ut_{3}\rangle)_{t_{1},t_{2},t_{3}})$
$\displaystyle=wt_{L}((qt_{2}+u(pt_{1}+qt_{3}+rt_{2}))_{t_{1},t_{2},t_{3}})$
$\displaystyle=wt_{H}((pt_{1}+qt_{3}+rt_{2})_{t_{1},t_{2},t_{3}})+wt_{H}((pt_{1}+(q+r)t_{2}+qt_{3})_{t_{1},t_{2},t_{3}})$
$\displaystyle=|L|-\frac{1}{2}\sum\limits_{y\in\mathbb{Z}_{2}}\sum\limits_{t_{1}\in\Delta_{D}^{c}}\sum\limits_{t_{2}\in\Delta_{E}^{c}}\sum\limits_{t_{3}\in\Delta_{F}^{c}}(-1)^{(pt_{1}+qt_{3}+rt_{2})y}$
$\displaystyle~{}+|L|-\frac{1}{2}\sum\limits_{y\in\mathbb{Z}_{2}}\sum\limits_{t_{1}\in\Delta_{D}^{c}}\sum\limits_{t_{2}\in\Delta_{E}^{c}}\sum\limits_{t_{3}\in\Delta_{F}^{c}}(-1)^{(pt_{1}+(q+r)t_{2}+qt_{3})y}$
$\displaystyle=|L|-\frac{1}{2}\sum\limits_{t_{1}\in\Delta_{D}^{c}}(-1)^{pt_{1}}\sum\limits_{t_{2}\in\Delta_{E}^{c}}(-1)^{rt_{2}}\sum\limits_{t_{3}\in\Delta_{F}^{c}}(-1)^{qt_{3}}$
$\displaystyle~{}-\frac{1}{2}\sum\limits_{t_{1}\in\Delta_{D}^{c}}(-1)^{pt_{1}}\sum\limits_{t_{2}\in\Delta_{E}^{c}}(-1)^{(q+r)t_{2}}\sum\limits_{t_{3}\in\Delta_{F}^{c}}(-1)^{qt_{3}}.$

Using Equation (1), it is easy to observe that for $X=\Delta_{D}$ and
$x_{i}=(-1)^{p_{i}}$, where $\Delta_{D}$ is defined as above and
$p=(p_{1},p_{2},\ldots,p_{m})\in\mathbb{Z}_{2}^{m}$, we have
$\mathcal{H}_{\Delta_{D}}((-1)^{p_{1}},(-1)^{p_{2}},\ldots,(-1)^{p_{m}})=\sum\limits_{t\in\Delta_{D}}\prod\limits_{i=1}^{m}(-1)^{p_{i}t_{i}}=\sum\limits_{t\in\Delta_{D}}(-1)^{pt}$,
where $t=(t_{1},t_{2},\ldots,t_{m})\in\Delta_{D}\subseteq\mathbb{Z}_{2}^{m}$.
So, by Lemma 2.1,
$\sum\limits_{t\in\Delta_{D}^{c}}(-1)^{pt}=\mathcal{H}_{\Delta_{D}^{c}}((-1)^{p_{1}},(-1)^{p_{2}},\ldots,(-1)^{p_{m}})=\prod\limits_{i\in[m]}(1+(-1)^{p_{i}})-\mathcal{H}_{\Delta_{D}}((-1)^{p_{1}},(-1)^{p_{2}},\ldots,(-1)^{p_{m}})$
$=2^{m}\delta_{0,p}-\mathcal{H}_{\Delta_{D}}((-1)^{p_{1}},(-1)^{p_{2}},\ldots,(-1)^{p_{m}})$,
where $\delta_{i,j}$ is the Kronecker delta function i.e. $\delta_{i,j}=1$ if
and only if $i=j$ and zero otherwise. Also we have
$\mathcal{H}_{\Delta_{D}}((-1)^{p_{1}},(-1)^{p_{2}},\ldots,(-1)^{p_{m}})=\prod\limits_{i\in
D}(1+(-1)^{p_{i}})=2^{|D|}\chi(supp(p)|D)$, where $\chi$ is the characteristic
function on the set $2^{[m]}\times 2^{[m]}$ defined in Section 2. So, further
we can write

$\displaystyle wt_{L}(c_{a})=$
$\displaystyle|L|-\frac{1}{2}[(2^{m}\delta_{0,p}-2^{|D|}\chi(supp(p)|D))(2^{m}\delta_{0,r}-2^{|E|}\chi(supp(r)|E))$
$\displaystyle(2^{m}\delta_{0,q}-2^{|F|}\chi(supp(q)|F))]-\frac{1}{2}[(2^{m}\delta_{0,p}-2^{|D|}\chi(supp(p)|D))$
$\displaystyle(2^{m}\delta_{0,(q+r)}-2^{|E|}\chi(supp(q+r)|E))(2^{m}\delta_{0,q}-2^{|F|}\chi(supp(q)|F))].$
(2)

######  Lemma 3.1.

Let $D,E,F\subseteq[m]$. Then

  * (1)

$|\\{X\subseteq[m]~{}|~{}\chi(X|D)=0,\chi(X|E)=0\\}|=2^{m}-(2^{m-|D\cup
E|})[(2^{|D|-|D\cap E|})+(2^{|E|-|D\cap E|})-1]$ and
$|\\{X\subseteq[m]~{}|~{}\chi(X|D)=0,\chi(X|E)=1\\}|=(2^{m-|D\cup
E|})(2^{|D|-|D\cap E|}-1)$ and
$|\\{X\subseteq[m]~{}|~{}\chi(X|D)=1,\chi(X|E)=0\\}|=(2^{m-|D\cup
E|})(2^{|E|-|D\cap E|}-1)$.

  * (2)

Let $T^{\prime}_{i}=|\\{(X,Y)~{}|~{}\emptyset\neq X,Y\subseteq[m],X\neq
Y,\chi(Y|E)=0,\chi(X|D)+\chi(X\oplus Y|D)=i\\}|$, where $i\in\\{0,1,2\\}$.
Then $\sum\limits_{i=1}^{i=3}T^{\prime}_{i}=(2^{m}-2^{m-|E|})(2^{m}-2)$.

###### Proof.

  * (1)

Let us consider the set $\\{X\subseteq[m]~{}|~{}\chi(X|D)=0,\chi(X|E)=1\\}$.
Since $X\cap D\neq\emptyset$, $X$ has $(2^{|D|}-1)$ choices but also $X\cap
E=\emptyset$, so the choices redeuces to $(2^{|D|-|D\cap E|}-1)$ and for each
of these choices we have $(2^{m-|D\cup E|})$ possibilities as $X\subseteq[m]$.
So all total we have
$|\\{X\subseteq[m]~{}|~{}\chi(X|D)=0,\chi(X|E)=1\\}|=(2^{m-|D\cup
E|})(2^{|D|-|D\cap E|}-1)$. Using similar arguments we have
$|\\{X\subseteq[m]~{}|~{}\chi(X|D)=1,\chi(X|E)=0\\}|=(2^{m-|D\cup
E|})(2^{|E|-|D\cap E|}-1)$. Now it is easy to observe that
$\\{X\subseteq[m]~{}|~{}\chi(X|D)\chi(X|E)=0\\}=\\{X\subseteq[m]~{}|~{}\chi(X|D)=0,\chi(X|E)=0\\}\sqcup\\{X\subseteq[m]~{}|~{}\chi(X|D)=0,\chi(X|E)=1\\}\sqcup\\{X\subseteq[m]~{}|~{}\chi(X|D)=1,\chi(X|E)=0\\}$.
So using the above two results and (2) of lemma 2.2, we have
$|\\{X\subseteq[m]~{}|~{}\chi(X|D)=0,\chi(X|E)=0\\}|=2^{m}-(2^{m-|D\cup
E|})[(2^{|D|-|D\cap E|})+(2^{|E|-|D\cap E|})-1]$.

  * (2)

It is easy to observe that
$\sum\limits_{i=1}^{i=3}T^{\prime}_{i}=|\\{(X,Y)~{}|~{}\emptyset\neq
X,Y\subseteq[m],X\neq Y,\chi(Y|E)=0\\}|$. Hence the result is clear as $Y$ has
$(2^{m}-2^{m-|E|})$ choices (using (1) of Lemma 2.2) and $X$ has $(2^{m}-2)$
as $X\neq\emptyset$ and $X\neq Y$.

∎

######  Theorem 3.2.

Let $\Delta_{D},\Delta_{E},\Delta_{F}\subseteq\mathbb{Z}_{2}^{m}$ are three
simplicial complexes with single maximal elements having support
$D,E,F\subseteq[m]$ respectively. Let
$L=\\{(t_{1},t_{2}+ut_{3})|t_{1}\in\Delta_{D}^{c},t_{2}\in\Delta_{E}^{c},t_{3}\in\Delta_{F}^{c}\\}$,
where $\Delta^{c}=\mathbb{Z}_{2}^{m}\setminus\Delta$. Then the code $C_{L}$ of
length $(2^{m}-2^{|D|})(2^{m}-2^{|E|})(2^{m}-2^{|F|})$ (as defined in
Subsection 2.2) is linear over $\mathbb{Z}_{2}[u]$. The Lee weight
distribution of $C_{L}$ is given in Table 1 and the size of $C_{L}$ is given
by $|C_{L}|=\begin{cases}2^{3m-1}&if~{}|D|=|E|=m-1,\\\
2^{3m}&otherwise.\end{cases}$

###### Proof.

The length of $C_{L}$ is quite straightforward. To show $C_{L}$ is linear over
$\mathbb{Z}_{2}[u]$, let us consider the the map $f$ between two
$\mathbb{Z}_{2}[u]$-module $\mathcal{R}^{m}$ and $\mathbb{Z}_{2}[u]^{|L|}$
defined as

$\displaystyle f:$
$\displaystyle\mathcal{R}^{m}\rightarrow\mathbb{Z}_{2}[u]^{|L|}$
$\displaystyle a\mapsto((a\cdot l))_{l\in L}$

Clearly, $Im(f)=C_{L}$ and to prove $C_{L}$ is $\mathbb{Z}_{2}[u]$-linear, it
is enough to show that $f$ is a $\mathbb{Z}_{2}[u]$-module homomorphism. Let
$a_{1}=(p_{1},q_{1}+ur_{1})$ and $a_{2}=(p_{2},q_{2}+ur_{2})$. Then
$((a_{1}\cdot l))=(q_{1}t_{2}+u(p_{1}t_{1}+q_{1}t_{3}+r_{1}t_{2}))$ and
$((a_{2}\cdot l))=(q_{2}t_{2}+u(p_{2}t_{1}+q_{2}t_{3}+r_{2}t_{2}))$ and
$(((a_{1}+a_{2})\cdot
l))=((q_{1}+q_{2})t_{2}+u((p_{1}+p_{2})t_{1}+(q_{1}+q_{2})t_{3}+(r_{1}+r_{2})t_{2}))=((a_{1}\cdot
l))+((a_{2}\cdot l))$ (as $\mathbb{Z}_{2}[u]$ is a ring under component-wise
addition). So, $(((a_{1}+a_{2})\cdot l))_{l\in L}=((a_{1}\cdot l))_{l\in
L}+((a_{2}\cdot l))_{l\in L}$, i.e., $f(a_{1}+a_{2})=f(a_{1})+f(a_{2})$.
Again, let $\alpha=(y+uz)\in\mathbb{Z}_{2}[u]$. Then $((\alpha a_{1}\cdot
l))=(((yp_{1},yq_{1}+u(yr_{1}+zq_{1}))\cdot l))=u\langle
yp_{1},t_{1}\rangle+\langle
yq_{1}+u(yr_{1}+zq_{1}),t_{2}+ut_{3}\rangle=\langle
yq_{1},t_{2}\rangle+u(\langle yq_{1},t_{3}\rangle+\langle
yr_{1},t_{2}\rangle+\langle zq_{1},t_{2}\rangle+\langle
yp_{1},t_{1}\rangle)=y\langle q_{1},t_{2}\rangle+u(y\langle
q_{1},t_{3}\rangle+y\langle r_{1},t_{2}\rangle+z\langle
q_{1},t_{2}\rangle+y\langle p_{1},t_{1}\rangle)=(y+uz)(\langle
q_{1},t_{2}\rangle+u(\langle q_{1},t_{3}\rangle+\langle
r_{1},t_{2}\rangle+\langle p_{1},t_{1}\rangle))=\alpha((a_{1}\cdot l))$. So,
$((\alpha a_{1}\cdot l))_{l\in L}=\alpha((a_{1}\cdot l))_{l\in L}$ i.e,
$f(\alpha a_{1})=\alpha f(a_{1})$. So, $C_{L}$ is linear over
$\mathbb{Z}_{2}[u]$ and $C_{L}\cong\frac{\mathcal{R}^{m}}{Ker(f)}$ with
$|C_{L}|=\frac{|\mathcal{R}^{m}|}{|Ker(f)|}=\frac{2^{3m}}{|Ker(f)|}$. Now,
first we compute the Lee weight distribution of $C_{L}$ and then we compute
the size of $C_{L}$. To compute the Lee weight distribution of $C_{L}$, we
have $supp(q+r)=supp(q)\oplus supp(r)$ as decsribed in Lemma 2.2. Let
$f_{i}~{}1\leq i\leq 57$ be the frequency of $wt_{L}(c_{a})$. Next we divide
the proof in the following cases and calculate $f_{i}$ using Equation (3),
Lemma 2.2 and Lemma 3.1.

  * (1)

$p=q=r=0$, $wt_{L}(c_{a})=0$ and $f_{1}=1$.

  * (2)

$p=q=0,r\neq 0$,
$wt_{L}(c_{a})=|L|-[(2^{m}-2^{|D|})(-2^{|E|}\chi(supp(r)|E))(2^{m}-2^{|F|})]$.

    * –

If $\chi(supp(r)|E)=0$, then $wt_{L}(c_{a})=|L|$ and $f_{2}=2^{m}-2^{m-|E|}$.

    * –

If $\chi(supp(r)|E)=1$, then
$wt_{L}(c_{a})=|L|+[(2^{m}-2^{|D|})(2^{|E|})(2^{m}-2^{|F|})]$ and
$f_{3}=2^{m-|E|}-1$.

  * (3)

$p=0,q\neq 0,r=0$,
$wt_{L}(c_{a})=|L|-\frac{1}{2}[(2^{m}-2^{|D|})(2^{m}-2^{|E|})(-2^{|F|}\chi(supp(q)|F))]-\frac{1}{2}[(2^{m}-2^{|D|})(-2^{|E|}\chi(supp(q)|E))(-2^{|F|}\chi(supp(q)|F))]$.

    * –

If $\chi(supp(q)|E)=0$, $\chi(supp(q)|F)=0$, then $wt_{L}(c_{a})=|L|$ and
$f_{4}=2^{m}-(2^{m-|E\cup F|})$ $[(2^{|E|-|E\cap F|})+(2^{|F|-|E\cap F|})-1]$.

    * –

If $\chi(supp(q)|E)=1$, $\chi(supp(q)|F)=0$, then $wt_{L}(c_{a})=|L|$ and
$f_{5}=(2^{m-|E\cup F|})$ $(2^{|F|-|E\cap F|}-1)$.

    * –

If $\chi(supp(q)|E)=0$, $\chi(supp(q)|F)=1$, then
$wt_{L}(c_{a})=|L|+\frac{1}{2}[(2^{m}-2^{|D|})(2^{m}-2^{|E|})(2^{|F|})]$ and
$f_{6}=(2^{m-|E\cup F|})(2^{|E|-|E\cap F|}-1)$.

    * –

If $\chi(supp(q)|E)=1$, $\chi(supp(q)|F)=1$,
$wt_{L}(c_{a})=|L|+\frac{1}{2}[(2^{m}-2^{|D|})(2^{m}-2^{|E|})(2^{|F|})]-\frac{1}{2}[(2^{m}-2^{|D|})(2^{|E|})(2^{|F|})]$
and $f_{7}=2^{m-|E\cup F|}-1$.

  * (4)

$p=0,q\neq 0,r\neq 0$,
$wt_{L}(c_{a})=|L|-\frac{1}{2}[(2^{m}-2^{|D|})(-2^{|E|}\chi(supp(r)|E))(-2^{|F|}\chi(supp(q)|F))]-\frac{1}{2}[(2^{m}-2^{|D|})(2^{m}\delta_{0,(q+r)}-2^{|E|}\chi(supp(q+r)|E))(-2^{|F|}\chi(supp(q)|F))]$.

    * –

Let $q\neq r$,
$wt_{L}(c_{a})=|L|-\frac{1}{2}[(2^{m}-2^{|D|})(-2^{|E|}\chi(supp(r)|E))(-2^{|F|}\chi(supp(q)|F))]-\frac{1}{2}[(2^{m}-2^{|D|})(-2^{|E|}\chi(supp(q+r)|E))(-2^{|F|}\chi(supp(q)|F))]$.

      * *

If $\chi(supp(q)|F)=0$, $\chi(supp(r)|E)=0$, $\chi(supp(q+r)|E)=0$, then
$wt_{L}(c_{a})=|L|$ and let $f_{8}$ be the frequency in this case.

      * *

If $\chi(supp(q)|F)=0$, $\chi(supp(r)|E)=0$, $\chi(supp(q+r)|E)=1$, then
$wt_{L}(c_{a})=|L|$ and let $f_{9}$ be the frequency in this case.

      * *

If $\chi(supp(q)|F)=0$, $\chi(supp(r)|E)=1$, $\chi(supp(q+r)|E)=0$, then
$wt_{L}(c_{a})=|L|$ and let $f_{10}$ be the frequency in this case.

      * *

If $\chi(supp(q)|F)=0$, $\chi(supp(r)|E)=1$, $\chi(supp(q+r)|E)=1$, then
$wt_{L}(c_{a})=|L|$ and let $f_{11}$ be the frequency in this case. Now,
$\sum\limits_{i=8}^{11}f_{i}=(2^{m}-2^{m-|F|})(2^{m}-2)$.

      * *

If $\chi(supp(q)|F)=1$, $\chi(supp(r)|E)=0$, $\chi(supp(q+r)|E)=0$, then
$wt_{L}(c_{a})=|L|$ and $f_{12}=2^{m}(2^{m-|F|}-1)+2^{m-|E|}(1+2^{m-|E\cup
F|}-2^{m+1-|F|})$.

      * *

If $\chi(supp(q)|F)=1$, $\chi(supp(r)|E)=0$, $\chi(supp(q+r)|E)=1$, then
$wt_{L}(c_{a})=|L|-\frac{1}{2}[(2^{m}-2^{|D|})(2^{|E|})(2^{|F|})]$ and let
$f_{13}$ be the frequency in this case.

      * *

If $\chi(supp(q)|F)=1$, $\chi(supp(r)|E)=1$, $\chi(supp(q+r)|E)=0$, then
$wt_{L}(c_{a})=|L|-\frac{1}{2}[(2^{m}-2^{|D|})(2^{|E|})(2^{|F|})]$ and let
$f_{14}$ be the frequency in this case. Now,
$f_{13}+f_{14}=2(2^{m-|E|}-1)(2^{m-|F|}-2^{m-|E\cup F|})$.

      * *

If $\chi(supp(q)|F)=1$, $\chi(supp(r)|E)=1$, $\chi(supp(q+r)|E)=1$, then
$wt_{L}(c_{a})=|L|-\frac{1}{2}[(2^{m}-2^{|D|})(2^{|E|})(2^{|F|})]-\frac{1}{2}[(2^{m}-2^{|D|})(2^{|E|})(2^{|F|})]$
and $f_{15}=(2^{m-|E|}-2)(2^{m-|E\cup F|}-1)$.

    * –

Let $q=r$,
$wt_{L}(c_{a})=|L|-\frac{1}{2}[(2^{m}-2^{|D|})(-2^{|E|}\chi(supp(r)|E))(-2^{|F|}\chi(supp(q)|F))]-\frac{1}{2}[(2^{m}-2^{|D|})(2^{m}-2^{|E|})(-2^{|F|}\chi(supp(q)|F))]$.

      * *

If $\chi(supp(r)|E)=0$, $\chi(supp(q)|F)=0$, then $wt_{L}(c_{a})=|L|$ and
$f_{16}=2^{m}-(2^{m-|E\cup F|})[(2^{|E|-|E\cap F|})+(2^{|F|-|E\cap F|})-1]$.

      * *

If $\chi(supp(r)|E)=1$, $\chi(supp(q)|F)=0$, then $wt_{L}(c_{a})=|L|$ and
$f_{17}=(2^{m-|E\cup F|})$ $(2^{|F|-|E\cap F|}-1)$.

      * *

If $\chi(supp(r)|E)=0$, $\chi(supp(q)|F)=1$, then
$wt_{L}(c_{a})=|L|+\frac{1}{2}[(2^{m}-2^{|D|})(2^{m}-2^{|E|})(2^{|F|})]$ and
$f_{18}=(2^{m-|E\cup F|})(2^{|E|-|E\cap F|}-1)$.

      * *

If $\chi(supp(r)|E)=1$, $\chi(supp(q)|F)=1$,
$wt_{L}(c_{a})=|L|-\frac{1}{2}[(2^{m}-2^{|D|})(2^{|E|})(2^{|F|})]+\frac{1}{2}[(2^{m}-2^{|D|})(2^{m}-2^{|E|})(2^{|F|})]$
and $f_{19}=2^{m-|E\cup F|}-1$.

  * (5)

$p\neq 0,q=r=0$,
$wt_{L}(c_{a})=|L|+[(2^{|D|}\chi(supp(p)|D))(2^{m}-2^{|E|})(2^{m}-2^{|F|})]$.

    * –

If $\chi(supp(p)|D)=0$, then $wt_{L}(c_{a})=|L|$ and $f_{20}=2^{m}-2^{m-|D|}$.

    * –

If $\chi(supp(p)|D)=1$, then
$wt_{L}(c_{a})=|L|+[(2^{|D|})(2^{m}-2^{|E|})(2^{m}-2^{|F|})]$ and
$f_{21}=2^{m-|D|}-1$.

  * (6)

$p\neq 0,q=0,r\neq 0$,
$wt_{L}(c_{a})=|L|-\frac{1}{2}[(-2^{|D|}\chi(supp(p)|D))(-2^{|E|}\chi(supp(r)|E))(2^{m}-2^{|F|})]-\frac{1}{2}[(-2^{|D|}\chi(supp(p)|D))(-2^{|E|}\chi(supp(r)|E))(2^{m}-2^{|F|})]$.

    * –

If $\chi(supp(p)|D)=0$, $\chi(supp(r)|E)=0$, then $wt_{L}(c_{a})=|L|$ and
$f_{22}=(2^{m}-2^{m-|D|})(2^{m}-2^{m-|E|})$.

    * –

If If $\chi(supp(p)|D)=1$, $\chi(supp(r)|E)=0$, then $wt_{L}(c_{a})=|L|$ and
$f_{23}=(2^{m-|D|}-1)(2^{m}-2^{m-|E|})$.

    * –

If $\chi(supp(p)|D)=0$, $\chi(supp(r)|E)=1$, then $wt_{L}(c_{a})=|L|$ and
$f_{24}=(2^{m}-2^{m-|D|})$ $(2^{m-|E|}-1)$.

    * –

If $\chi(supp(p)|D)=1$, $\chi(supp(r)|E)=1$,
$wt_{L}(c_{a})=|L|-[(2^{|D|})(2^{|E|})(2^{m}-2^{|F|})]$ and
$f_{25}=(2^{m-|D|}-1)(2^{m-|E|}-1)$.

  * (7)

$p\neq 0,q\neq 0,r=0$,
$wt_{L}(c_{a})=|L|-\frac{1}{2}[(-2^{|D|}\chi(supp(p)|D))(2^{m}-2^{|E|})(-2^{|F|}\chi(supp(q)|F))]-\frac{1}{2}[(-2^{|D|}\chi(supp(p)|D))(-2^{|E|}\chi(supp(q)|E))(-2^{|F|}\chi(supp(q)|F))]$.

    * –

If $\chi(supp(p)|D)=0$, $\chi(supp(q)|F)=0$, $\chi(supp(q)|E)=0$, then
$wt_{L}(c_{a})=|L|$ and let $f_{26}$ be the frequency in this case.

    * –

If $\chi(supp(p)|D)=0$, $\chi(supp(q)|F)=0$, $\chi(supp(q)|E)=1$, then
$wt_{L}(c_{a})=|L|$ and let $f_{27}$ be the frequency in this case.

    * –

If $\chi(supp(p)|D)=0$, $\chi(supp(q)|F)=1$, $\chi(supp(q)|E)=0$, then
$wt_{L}(c_{a})=|L|$ and let $f_{28}$ be the frequency in this case.

    * –

If $\chi(supp(p)|D)=0$, $\chi(supp(q)|F)=1$, $\chi(supp(q)|E)=1$, then
$wt_{L}(c_{a})=|L|$ and let $f_{29}$ be the frequency in this case. Now,
$\sum\limits_{i=26}^{29}f_{i}=(2^{m}-2^{m-|D|})(2^{m}-1)$.

    * –

If $\chi(supp(p)|D)=1$, $\chi(supp(q)|F)=0$, $\chi(supp(q)|E)=0$, then
$wt_{L}(c_{a})=|L|$ and $f_{30}=(2^{m-|D|}-1)[2^{m}-(2^{m-|E\cup
F|})\\{(2^{|E|-|E\cap F|})+(2^{|F|-|E\cap F|})-1\\}]$.

    * –

If $\chi(supp(p)|D)=1$, $\chi(supp(q)|F)=0$, $\chi(supp(q)|E)=1$, then
$wt_{L}(c_{a})=|L|$ and $f_{31}=(2^{m-|D|}-1)[(2^{m-|E\cup F|})(2^{|F|-|E\cap
F|}-1)]$.

    * –

If $\chi(supp(p)|D)=1$, $\chi(supp(q)|F)=1$, $\chi(supp(q)|E)=0$, then
$wt_{L}(c_{a})=|L|-\frac{1}{2}[(2^{|D|})(2^{m}-2^{|E|})(2^{|F|})]$ and
$f_{32}=(2^{m-|D|}-1)[(2^{m-|E\cup F|})(2^{|E|-|E\cap F|}-1)]$.

    * –

If $\chi(supp(p)|D)=1$, $\chi(supp(q)|F)=1$, $\chi(supp(q)|E)=1$, then
$wt_{L}(c_{a})=|L|-\frac{1}{2}[(2^{|D|})(2^{m}-2^{|E|})(2^{|F|})]+\frac{1}{2}[(2^{|D|})(2^{|E|})(2^{|F|})]$
and $f_{33}=(2^{m-|D|}-1)[2^{m-|E\cup F|}-1]$.

  * (8)

$p\neq 0,q\neq 0,r\neq 0$,
$wt_{L}(c_{a})=|L|-\frac{1}{2}[(-2^{|D|}\chi(supp(p)|D))(-2^{|E|}\chi(supp(r)|E))(-2^{|F|}\\\
\chi(supp(q)|F))]-\frac{1}{2}[(-2^{|D|}\chi(supp(p)|D))(2^{m}\delta_{0,(q+r)}-2^{|E|}\chi(supp(q+r)|E))(-2^{|F|}\chi(supp(q)|F))]$.

    * –

Let $q\neq r$,
$wt_{L}(c_{a})=|L|-\frac{1}{2}[(-2^{|D|}\chi(supp(p)|D))(-2^{|E|}\chi(supp(r)|E))\\\
(-2^{|F|}\chi(supp(q)|F))]-\frac{1}{2}[(-2^{|D|}\chi(supp(p)|D))(-2^{|E|}\chi(supp(q+r)|E))(-2^{|F|}\chi(supp(q)|F))]$.

      * *

If $\chi(supp(p)|D)=0$, $\chi(supp(q)|F)=0$, $\chi(supp(r)|E)=0$,
$\chi(supp(q+r)|E)=0$, then $wt_{L}(c_{a})=|L|$ and let $f_{34}$ be the
frequency in this case.

      * *

If $\chi(supp(p)|D)=0$, $\chi(supp(q)|F)=0$, $\chi(supp(r)|E)=0$,
$\chi(supp(q+r)|E)=1$, then $wt_{L}(c_{a})=|L|$ and let $f_{35}$ be the
frequency in this case.

      * *

If $\chi(supp(p)|D)=0$, $\chi(supp(q)|F)=0$, $\chi(supp(r)|E)=1$,
$\chi(supp(q+r)|E)=0$, then $wt_{L}(c_{a})=|L|$ and let $f_{36}$ be the
frequency in this case.

      * *

If $\chi(supp(p)|D)=0$, $\chi(supp(q)|F)=0$, $\chi(supp(r)|E)=1$,
$\chi(supp(q+r)|E)=1$, then $wt_{L}(c_{a})=|L|$ and let $f_{37}$ be the
frequency in this case.

      * *

If $\chi(supp(p)|D)=0$, $\chi(supp(q)|F)=1$, $\chi(supp(r)|E)=0$,
$\chi(supp(q+r)|E)=0$, then $wt_{L}(c_{a})=|L|$ and let $f_{38}$ be the
frequency in this case.

      * *

If $\chi(supp(p)|D)=0$, $\chi(supp(q)|F)=1$, $\chi(supp(r)|E)=0$,
$\chi(supp(q+r)|E)=1$, then $wt_{L}(c_{a})=|L|$ and let $f_{39}$ be the
frequency in this case.

      * *

If $\chi(supp(p)|D)=0$, $\chi(supp(q)|F)=1$, $\chi(supp(r)|E)=1$,
$\chi(supp(q+r)|E)=0$, then $wt_{L}(c_{a})=|L|$ and let $f_{40}$ be the
frequency in this case.

      * *

If $\chi(supp(p)|D)=0$, $\chi(supp(q)|F)=1$, $\chi(supp(r)|E)=1$,
$\chi(supp(q+r)|E)=1$, then $wt_{L}(c_{a})=|L|$ and let $f_{41}$ be the
frequency in this case. Now,
$\sum\limits_{i=34}^{41}f_{i}=(2^{m}-2^{m-|D|})(2^{m}-1)(2^{m}-2)$.

      * *

If $\chi(supp(p)|D)=1$, $\chi(supp(q)|F)=0$, $\chi(supp(r)|E)=0$,
$\chi(supp(q+r)|E)=0$, then $wt_{L}(c_{a})=|L|$ and let $f_{42}$ be the
frequency in this case.

      * *

If $\chi(supp(p)|D)=1$, $\chi(supp(q)|F)=0$, $\chi(supp(r)|E)=0$,
$\chi(supp(q+r)|E)=1$, then $wt_{L}(c_{a})=|L|$ and let $f_{43}$ be the
frequency in this case.

      * *

If $\chi(supp(p)|D)=1$, $\chi(supp(q)|F)=0$, $\chi(supp(r)|E)=1$,
$\chi(supp(q+r)|E)=0$, then $wt_{L}(c_{a})=|L|$ and let $f_{44}$ be the
frequency in this case.

      * *

If $\chi(supp(p)|D)=1$, $\chi(supp(q)|F)=0$, $\chi(supp(r)|E)=1$,
$\chi(supp(q+r)|E)=1$, then $wt_{L}(c_{a})=|L|$ and let $f_{45}$ be the
frequency in this case. Now,
$\sum\limits_{i=42}^{45}f_{i}=(2^{m-|D|}-1)[(2^{m}-2^{m-|F|})(2^{m}-2)]$.

      * *

If $\chi(supp(p)|D)=1$, $\chi(supp(q)|F)=1$, $\chi(supp(r)|E)=0$,
$\chi(supp(q+r)|E)=0$, then $wt_{L}(c_{a})=|L|$ and
$f_{46}=(2^{m-|D|}-1)[2^{m}(2^{m-|F|}-1)+2^{m-|E|}(1+2^{m-|E\cup
F|}-2^{m+1-|F|})]$.

      * *

If $\chi(supp(p)|D)=1$, $\chi(supp(q)|F)=1$, $\chi(supp(r)|E)=0$,
$\chi(supp(q+r)|E)=1$, then
$wt_{L}(c_{a})=|L|+\frac{1}{2}[(2^{|D|})(2^{|E|})(2^{|F|})]$ and let $f_{47}$
be the frequency in this case.

      * *

If $\chi(supp(p)|D)=1$, $\chi(supp(q)|F)=1$, $\chi(supp(r)|E)=1$,
$\chi(supp(q+r)|E)=0$, then
$wt_{L}(c_{a})=|L|+\frac{1}{2}[(2^{|D|})(2^{|E|})(2^{|F|})]$ and let $f_{48}$
be the frequency in this case. Now
$f_{47}+f_{48}=(2^{m-|D|}-1)[2(2^{m-|E|}-1)(2^{m-|F|}-2^{m-|E\cup F|})]$.

      * *

If $\chi(supp(p)|D)=1$, $\chi(supp(q)|F)=1$, $\chi(supp(r)|E)=1$,
$\chi(supp(q+r)|E)=1$, then $wt_{L}(c_{a})=|L|+[(2^{|D|})(2^{|E|})(2^{|F|})]$
and $f_{49}=(2^{m-|D|}-1)[(2^{m-|E|}-2)(2^{m-|E\cup F|}-1)]$.

    * –

Let $q=r$,
$wt_{L}(c_{a})=|L|-\frac{1}{2}[(-2^{|D|}\chi(supp(p)|D))(-2^{|E|}\chi(supp(r)|E))\\\
(-2^{|F|}\chi(supp(q)|F))]-\frac{1}{2}[(-2^{|D|}\chi(supp(p)|D))(2^{m}-2^{|E|})(-2^{|F|}\chi(supp(q)|F))]$.

      * *

If $\chi(supp(p)|D)=0$, $\chi(supp(q)|F)=0$, $\chi(supp(r)|E)=0$, then
$wt_{L}(c_{a})=|L|$ and let $f_{50}$ be the frequency in this case.

      * *

If $\chi(supp(p)|D)=0$, $\chi(supp(q)|F)=0$, $\chi(supp(r)|E)=1$, then
$wt_{L}(c_{a})=|L|$ and let $f_{51}$ be the frequency in this case.

      * *

If $\chi(supp(p)|D)=0$, $\chi(supp(q)|F)=1$, $\chi(supp(r)|E)=0$, then
$wt_{L}(c_{a})=|L|$ and let $f_{52}$ be the frequency in this case.

      * *

If $\chi(supp(p)|D)=0$, $\chi(supp(q)|F)=1$, $\chi(supp(r)|E)=1$, then
$wt_{L}(c_{a})=|L|$ and let $f_{53}$ be the frequency in this case. Now,
$\sum\limits_{i=50}^{53}f_{i}=(2^{m}-2^{m-|D|})(2^{m}-1)$.

      * *

If $\chi(supp(p)|D)=1$, $\chi(supp(q)|F)=0$, $\chi(supp(r)|E)=0$, then
$wt_{L}(c_{a})=|L|$ and $f_{54}=(2^{m-|D|}-1)[2^{m}-(2^{m-|E\cup
F|})\\{(2^{|E|-|E\cap F|})+(2^{|F|-|E\cap F|})-1\\}]$.

      * *

If $\chi(supp(p)|D)=1$, $\chi(supp(q)|F)=0$, $\chi(supp(r)|E)=1$, then
$wt_{L}(c_{a})=|L|$ and $f_{55}=(2^{m-|D|}-1)[(2^{m-|E\cup F|})(2^{|F|-|E\cap
F|}-1)]$.

      * *

If $\chi(supp(p)|D)=1$, $\chi(supp(q)|F)=1$, $\chi(supp(r)|E)=0$, then
$wt_{L}(c_{a})=|L|-\frac{1}{2}[(2^{|D|})(2^{m}-2^{|E|})(2^{|F|})]$ and
$f_{56}=(2^{m-|D|}-1)[(2^{m-|E\cup F|})(2^{|E|-|E\cap F|}-1)]$.

      * *

If $\chi(supp(p)|D)=1$, $\chi(supp(q)|F)=1$, $\chi(supp(r)|E)=1$, then
$wt_{L}(c_{a})=|L|+\frac{1}{2}[(2^{|D|})(2^{|E|})(2^{|F|})]-\frac{1}{2}[(2^{|D|})(2^{m}-2^{|E|})(2^{|F|})]$
and $f_{57}=(2^{m-|D|}-1)[2^{m-|E\cup F|}-1]$.

We must note that $|Ker(f)|=|\\{c_{a}~{}|~{}wt_{L}(c_{a})=0\\}|$. So, to
compute the size of $C_{L}$, first we observe from Table 1 that
$wt_{L}(c_{a})=0$, i.e., $((a\cdot l))_{l\in L}=0$ in the following three
cases depending on the choices of $|D|,~{}|E|,~{}|F|$.

  * •

$a=0$ for all $|D|,~{}|E|,~{}|F|$. In this case $f_{0}=1$.

  * •

$a=(0,q+ur)$ satisfying $\chi(supp(q)|F)=1$, $\chi(supp(r)|E)=1$,
$\chi(supp(q+r)|E)=1$ for $|E|=|F|=m-1$. In this case $f_{15}=0$.

  * •

$a=(p,ur)$ satisfying $\chi(supp(p)|D)=1$, $\chi(supp(r)|E)=1$ for
$|D|=|E|=m-1$. In this case $f_{25}=1$.

So, we have

$|Ker(f)|=\begin{cases}2&if~{}|D|=|E|=m-1,\\\ 1&otherwise.\end{cases}$

This gives the size of $C_{L}$. ∎

Table 1. Lee weight distribution of $C_{L}$ Lee weight | frequency
---|---
$0$ | $f_{1}=1$
$|L|$ | $\sum\limits_{i=2}^{i=57}f_{i},~{}i\neq 3,6,7,13,14,15,18,19,$
| $21,25,32,33,47,48,49,56,57$
$|L|+(2^{m}-2^{|D|})(2^{|E|})(2^{m}-2^{|F|})$ | $f_{3}$
$|L|+\frac{1}{2}[(2^{m}-2^{|D|})(2^{m}-2^{|E|})(2^{|F|})]$ | $f_{6}+f_{18}$
$|L|+\frac{1}{2}[(2^{m}-2^{|D|})(2^{m}-2^{|E|})(2^{|F|})-(2^{m}-2^{|D|})(2^{|E|})(2^{|F|})]$ | $f_{7}+f_{19}$
$|L|-\frac{1}{2}[(2^{m}-2^{|D|})(2^{|E|})(2^{|F|})]$ | $f_{13}+f_{14}$
$|L|-(2^{m}-2^{|D|})(2^{|E|})(2^{|F|})$ | $f_{15}$
$|L|+(2^{|D|})(2^{m}-2^{|E|})(2^{m}-2^{|F|})$ | $f_{21}$
$|L|-(2^{|D|})(2^{|E|})(2^{m}-2^{|F|})$ | $f_{25}$
$|L|-\frac{1}{2}[(2^{|D|})(2^{m}-2^{|E|})(2^{|F|})]$ | $f_{32}+f_{56}$
$|L|-\frac{1}{2}[(2^{|D|})(2^{m}-2^{|E|})(2^{|F|})-(2^{|D|})(2^{|E|})(2^{|F|})]$ | $f_{33}+f_{57}$
$|L|+\frac{1}{2}[(2^{|D|})(2^{|E|})(2^{|F|})]$ | $f_{47}+f_{48}$
$|L|+(2^{|D|})(2^{|E|})(2^{|F|})$ | $f_{49}$
Table 2. Lee weight distribution of $C_{L}$ if $|D|=|E|=|F|=m-1$ Lee weight | frequency
---|---
$0$ | $\frac{1}{2}(f_{1}+f_{15}+f_{25})=1$
$|L|=2^{3m-3}$ | $\frac{1}{2}\sum\limits_{i=2}^{i=57}f_{i},~{}(i\neq 3,6,13,14,15,18,$
| $21,25,32,47,48,49,56)=2^{3m-1}-6$
$|L|+(2^{m}-2^{|D|})(2^{|E|})(2^{m}-2^{|F|})=2^{3m-2}$ | $\frac{1}{2}(f_{3}+f_{21}+f_{49})=1$
$|L|+\frac{1}{2}[(2^{m}-2^{|D|})(2^{m}-2^{|E|})(2^{|F|})]=2^{3m-3}+2^{3m-4}$ | $\frac{1}{2}(f_{6}+f_{18}+f_{47}+f_{48})=2$
$|L|-\frac{1}{2}[(2^{m}-2^{|D|})(2^{|E|})(2^{|F|})]=2^{3m-4}$ | $\frac{1}{2}(f_{13}+f_{14}+f_{32}+f_{56})=2$

######  Remark 3.3.

Let $|D|=|E|=m-1$, i.e., $|Ker(f)|=2$. Then for each $a\in\mathcal{R}^{m}$,
there exists $b\in\mathcal{R}^{m}$ such that $((a\cdot l))_{l\in L}=((b\cdot
l))_{l\in L}$ as this happens if and only if $(((a-b)\cdot l))_{l\in L}=0$ (as
$C_{L}$ is linear), i.e., $(a-b)\in Ker(f)$ and $|Ker(f)|=2$. So, in this case
the frequencies in Table 1 corresponding to a particular Lee weight will be
half of that. In Table 2, the Lee weight distribution of $C_{L}$ is given for
$|D|=|E|=|F|=m-1$.

######  Example 3.4.

Let $m=3$ and $D=\\{1,2\\}$, $E=\\{1,3\\}$, $F=\\{2,3\\}$. Then the Lee weight
enumerator of the code $C_{L}$ (as in Theorem 3.2 and Table 2), denoted by
$lwe_{C_{L}}(X,Y)$, is given by
$lwe_{C_{L}}(X,Y)=X^{128}+2X^{96}Y^{32}+250X^{64}Y^{64}+2X^{32}Y^{96}+Y^{128}$.

######  Example 3.5.

Let $m=3$ and $D=\\{1\\}$, $E=\\{2\\}$, $F=\\{3\\}$. Then the Lee weight
enumerator of the code $C_{L}$ (as in Theorem 3.2 and Table 1), denoted by
$lwe_{C_{L}}(X,Y)$, is given by
$lwe_{C_{L}}(X,Y)=X^{432}+11X^{240}Y^{192}+24X^{228}Y^{204}+6X^{224}Y^{208}+416X^{216}Y^{216}+36X^{212}Y^{220}+6X^{208}Y^{224}+2X^{192}Y^{240}+4X^{180}Y^{252}+6X^{144}Y^{288}$.

######  Example 3.6.

Let $m=4$ and $D=\\{1,2,3\\}$, $E=\\{1,2,4\\}$, $F=\\{1,3,4\\}$. Then the Lee
weight enumerator of the code $C_{L}$ (as in Theorem 3.2 and Table 2), denoted
by $lwe_{C_{L}}(X,Y)$, is given by
$lwe_{C_{L}}(X,Y)=X^{1024}+2X^{768}Y^{256}+2042X^{512}Y^{512}+2X^{256}Y^{768}+Y^{1024}$.

######  Example 3.7.

Let $m=4$ and $D=\\{1,2\\}$, $E=\\{2,3\\}$, $F=\\{3,4\\}$. Then the Lee weight
enumerator of the code $C_{L}$ (as in Theorem 3.2 and Table 1), denoted by
$lwe_{C_{L}}(X,Y)$, is given by
$lwe_{C_{L}}(X,Y)=X^{3456}+11X^{1920}Y^{1536}+24X^{1824}Y^{1632}+6X^{1792}Y^{1664}+4000X^{1728}Y^{1728}+36X^{1696}Y^{1760}+6X^{1664}Y^{1792}+2X^{1536}Y^{1920}+4X^{1440}Y^{2016}+6X^{1152}Y^{2304}$.

##  4\. Gray images and Minimal codes

First of all we observe that: Since $\Phi$ is a linear map and $C_{L}$ is a
linear code over $\mathbb{Z}_{2}[u]$, $\Phi(C_{L})$ is linear over
$\mathbb{Z}_{2}$. Next we will use some characterizations of binary linear
codes regarding orthogonality and minimality.

######  Proposition 4.1.

Let $C_{L}$ be the linear code over $\mathbb{Z}_{2}[u]$ as in Theorem 3.2.
Then the binary linear code $\phi(C_{L})$ is self-orthogonal with respect to
the Euclidean inner product if $|D|,|E|,|F|\neq 0$.

###### Proof.

Let $0<|D|,|E|,|F|\leq m-1$. We know that if all the codewords of a binary
linear codes is of weight $4k$ for some $k\in\mathbb{N}$, then the linear code
is self-orthogonal (see (ii) of [10, Theorem 1.4.8]). Now,
$(2^{m}-2^{|D|})\geq 2^{|D|}\geq 2$ (as $m>1$ and $0<|D|\leq m-1$). So, from
Table 1, it is easy to observe that 4 divides all the Lee weights. Hence
$\phi(C_{L})$ is self-orthogonal. ∎

######  Example 4.2.

Let $|D|=|E|=|F|=m-1$. Then $\phi(C_{L})$ is self-orthogonal with parameters
$[2^{3m-2},3m-1,2^{3m-4}]$.

######  Example 4.3.

Let $|D|=|E|=|F|=n<m-1$. Then $\phi(C_{L})$ has parameters
$[2(2^{m}-2^{n})^{3},3m,(2^{m}-2^{n})(2^{m})(2^{m}-2^{n+1})]$. Moreover, if
$n\neq 0$, $\phi(C_{L})$ is self orthogonal.

We call a linear code $C$ over $\mathbb{F}_{p}$ is minimal if all of its
nonzero codewords are minimal. Further a codeword $c\in C$ is said to be
minimal if $supp(c^{\prime})\subseteq supp(c)$ if and only if $c^{\prime}=ac$,
where $a(\neq 0)\in\mathbb{F}_{p}$. In general, it is hard to determine
whether a linear code is minimal or not but the famous Ashikhmin-Barg
condition ([2]) is a sufficient condition for a linear code to be minimal and
it is relatively easy to check.

######  Lemma 4.4.

(Ashikhmin-Barg condition for p=2) [2] Let $C$ be a linear code over
$\mathbb{Z}_{2}$. Denote $w_{0}$ as the minimum(nonzero) and $w_{\infty}$ as
the maximum Hamming weights. If $\frac{w_{0}}{w_{\infty}}>\frac{1}{2}$, then
$C$ is minimal.

######  Theorem 4.5.

Let $C_{L}$ be the linear code over $\mathbb{Z}_{2}[u]$ as in Theorem 3.2 with
$|D|=|E|=|F|=n(<m)$. Then the code $\Phi(C_{L})$ is minimal if $n\leq m-2$.

###### Proof.

Suppose $|D|=|E|=|F|=n(<m)$. Then $|L|=(2^{m}-2^{n})^{3}$ and
$(2^{m}-2^{n})\geq 2^{n},\forall m,n$. Now we divide this proof in two cases.
Case I: Let $n<m-1$. Then we observe that $(2^{m}-2^{n})>2^{n}$. Now with
these assumptions and using Table 1, we have
$w_{0}=|L|-[(2^{m}-2^{n})(2^{n})^{2}]=(2^{m}-2^{n})(2^{m})(2^{m}-2^{n+1})$ and
$w_{\infty}=|L|+[(2^{m}-2^{n})^{2}(2^{n})]=(2^{m}-2^{n})^{2}(2^{m})$. So,
$\frac{w_{0}}{w_{\infty}}=\frac{2^{m}-2^{n+1}}{2^{m}-2^{n}}$ (as $m\neq 0$ and
$n\neq m$). Now to satisfy Lemma 4.4, it must be $2^{m+1}-2^{n+2}>2^{m}-2^{n}$
i.e. $2^{m}>2^{n+2}-2^{n}$ and this will be true if and only if $n+2\leq m$,
which is also our assumption in this case.
Case II: Let $n=m-1$. Then we observe that $|L|=(2^{n})^{3}$ and
$(2^{m}-2^{n})=2^{n}$. Now with these assumptions and using Table 2, we have
$w_{0}=|L|-\frac{1}{2}(2^{n})^{3}=\frac{1}{2}|L|$ and
$w_{\infty}=|L|+(2^{n})^{3}=2|L|$. So, $\frac{w_{0}}{w_{\infty}}=\frac{1}{4}$
(as $n\neq 0$). Since Lemma 4.4 is only a sufficient condition, we cannot
conclude anything in this case. Hence the theorem. ∎

######  Example 4.6.

Let $m=3$ in Theorem 4.5. Then

  * (1)

If $n=2$, we can not conclude anything about the minimality of the binary
linear self-orthogonal four-weight code $\Phi(C_{L})$ having parameters
$[128,8,32]$.

  * (2)

If $n=1$, then $\Phi(C_{L})$ is a minimal binary linear self-orthogonal nine-
weight code with parameters $[432,9,192]$.

  * (3)

If $n=0$, then $\Phi(C_{L})$ is a minimal binary linear seven-weight code with
parameters $[686,9,336]$.

######  Example 4.7.

Let $m=4$ in Theorem 4.5. Then

  * (1)

If $n=3$, we can not conclude anything about the minimality of the binary
linear self-orthogonal four-weight code $\Phi(C_{L})$ having parameters
$[1024,11,256]$.

  * (2)

If $n=2$, then $\Phi(C_{L})$ is a minimal binary linear self-orthogonal nine-
weight code with parameters $[3456,12,1536]$.

  * (3)

If $n=1$, then $\Phi(C_{L})$ is a minimal binary linear self-orthogonal nine-
weight code with parameters $[5488,12,2688]$.

  * (4)

If $n=0$, then $\Phi(C_{L})$ is a minimal binary linear seven-weight code with
parameters $[6750,12,3360]$.

##  5\. Conclusion

In [19], the authors constructed few-Lee weight additive codes over the mixed
alphabet $\mathbb{Z}_{p}\mathbb{Z}_{p}[u],u^{2}=0$ with optimal Gray images by
using a suitable defining set. In [20], the authors used the mixed alphabet
ring $\mathbb{Z}_{p}\mathbb{Z}_{p}[u]$ with $u^{2}=u$ and $p>2$ to study some
few-Lee weight additive codes by employing down-sets. In this paper, we use
the mixed alphabet ring $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$ to construct a class
of few-Lee weight linear codes using simplicial complexes generated by a
single element and also have an infinite family of self-orthogonal binary
minimal codes over as Gray images. This shows that like in the case of single
alphabet rings, we may employ all three popular means of constructing few-Lee
weight codes including simplicial complexes when mixed alphabet rings are
involved and hence fully answer the question raised in [20] about the use of
simplicial complexes or down-sets in the case of mixed alphabet rings.

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