# Parabolic Singular Integrals with Nonhomogeneous Kernels

S. Bortz  Department of Mathematics
University of Alabama
Tuscaloosa, AL, 35487, USA<EMAIL_ADDRESS>,  J. Hoffman  Department of
Mathematics
University of Missouri
Columbia, MO 65211, USA<EMAIL_ADDRESS>,  S. Hofmann  Department of
Mathematics
University of Missouri
Columbia, MO 65211, USA<EMAIL_ADDRESS>,  J-L. Luna Garcia  Department
of Mathematics
University of Missouri
Columbia, MO 65211, USA<EMAIL_ADDRESS>and  K. Nyström  Department
of Mathematics, Uppsala University, S-751 06 Uppsala, Sweden
<EMAIL_ADDRESS>
###### Abstract.

We establish $L^{2}$ boundedness of all “nice” parabolic singular integrals on
“Good Parabolic Graphs”, aka regular Lip(1,1/2) graphs. The novelty here is
that we include non-homogeneous kernels, which are relevant to the theory of
parabolic uniform rectifiability. Previously, the third named author had
treated the case of homogeneous kernels. The present proof combines the
methods of that work (which in turn was based on methods described in Christ’s
CBMS lecture notes), with the techniques of Coifman-David-Meyer.

This is a very preliminary draft. Eventually, these results will be part of a
more extensive work on parabolic uniform rectifiability and singular
integrals.

The authors J. H., S. H., and J-L. L. G. were partially supported by NSF
grants DMS-1664047 and DMS-2000048. K.N. was partially supported by grant
2017-03805 from the Swedish research council (VR)

##  1\. Notation, definitions, and statement of results

Notation: Our ambient space is $(n+1)$-dimensional space-time:

$\mathbb{R}^{n+1}=\big{\\{}{\bf
X}:=(X,t)=(x_{0},x,t)\in\mathbb{R}\times\mathbb{R}^{n-1}\times\mathbb{R}\big{\\}}\,,$

and we shall also at times work with $n$-dimensional space-time

$\mathbb{R}^{n}=\big{\\{}{\bf
x}:=(x,t)\in\mathbb{R}^{n-1}\times\mathbb{R}\big{\\}}\,.$

We write $(\xi,\tau)\in\mathbb{R}^{n-1}\times\mathbb{R}$ to denote points on
the Fourier transform side of space-time. We also write $\mathbb{R}^{n}_{sp}$
to denote purely spatial $n$-dimensional Euclidean space. Note in particular
that we shall often distinguish one spatial variable in $\mathbb{R}^{n}_{sp}$,
and write, e.g., $X=(x_{0},x)\in\mathbb{R}\times\mathbb{R}^{n-1}$, and that we
use lower case letters $x,y,z$ to denote spatial points in $\mathbb{R}^{n-1}$,
and capital letters $X=(x_{0},x),Y=(y_{0},y),Z=(z_{0},z)$ to denote points in
$\mathbb{R}^{n}_{sp}=\mathbb{R}\times\mathbb{R}^{n-1}$. In accordance with the
notation introduced above, we let $\|{\bf X}\|=\|(X,t)\|=\|(x_{0},x,t)\|$ and
$\|{\bf x}\|=\|(x,t)\|$ denote the parabolic length of the vectors ${\bf
X}=(X,t)\in\mathbb{R}^{n+1}$ and ${\bf x}=(x,t)\in\mathbb{R}^{n}$,
respectively.

We let $d=n+1$ denote the parabolic homogeneous dimension of space-time
$\mathbb{R}^{n}$.

We define a fractional integral operator $I_{p}$ of parabolic order 1 on
$\mathbb{R}^{n}$ by means of the Fourier transform:

$\widehat{I_{p}f}(\xi,\tau):=\|(\xi,\tau)\|^{-1}\widehat{f}(\xi,\tau)\,.$

######  Definition 1 (Parabolic Calderón-Zygmund kernels).

For an integer $N\geq 1$, we shall say that a kernel $K=K(X,t)$ satisfies a
($d$-dimensional) parabolic C-Z(N) condition, and we write $K\in
C\text{-}Z(N)$, if it satisfies the following properties:

  1. (i)

Smoothness with C-Z estimates: $K\in C^{N}(\mathbb{R}^{n+1}\setminus\\{0\\})$,
with the estimate

$|\nabla_{X}^{j}\,\partial_{t}^{k}K(X,t)|\,\leq\,C_{j,k}\,\|(X,t)\|^{-d-j-2k}\,,\quad\forall\,0\leq
j+k\leq N\,.$

  2. (ii)

Oddness in spatial variables: $K(X,t)=-K(-X,t)$, for each
$(X,t)\in\mathbb{R}^{n}_{sp}\times\mathbb{R}$.

We shall simply say that $K$ is a C-Z kernel (and we write $K\in C\text{-}Z$),
if $K\in C\text{-}Z(N)$ for some positive integer $N$. We shall also consider
analogous kernels $H(x,t)$ defined on $\mathbb{R}^{n}$, not necessarily odd in
the space variables, but still satisfying the $n$-dimensional version of
property (i) above, i.e.,

(2)
$|\nabla_{x}^{j}\,\partial_{t}^{k}H(x,t)|\,\leq\,C_{j,k}\,\|(x,t)\|^{-d-j-2k}\,,\quad\forall\,0\leq
j+k\leq N\,.$

In this case we shall say that $H$ satisfies the $C\text{-}Z(N)$(i) condition,
or $H\in C\text{-}Z(N)$(i).

Note that we do not assume homogeneity of $K$ and $H$ in the preceding
definition.

######  Definition 3 (Regular Lip(1,1/2) functions and Good Parabolic Graphs).

We say that a function $A:\mathbb{R}^{n}\to\mathbb{R}$ is a regular Lip(1,1/2)
function if it is Lip(1,1/2), and if in addition $\mathbb{D}_{n}A\in BMO$,
where $\mathbb{D}_{n}:=I_{p}\circ\partial_{t}$ is the half-order time
derivative. Following [H1], we endow the regular Lip(1,1/2) functions with the
norm

$\|A\|_{comm}:=\|\nabla_{x}A\|_{L^{\infty}(\mathbb{R}^{n})}+\|\mathbb{D}_{n}A\|_{BMO(\mathbb{R}^{n})}\,.$

Of course, BMO is parabolic BMO, defined with respect to parabolic cubes (or
balls).

We say that a graph $\Gamma:=\left\\{\big{(}A({\bf x}),{\bf
x}\big{)}\right\\}$ is a Good Parabolic Graph, and we write $\Gamma\in GPG$,
if $A$ is a regular Lip(1,1/2) function.

“Surface measure” on $\Gamma$ is defined to be
$d\sigma(x,t):=\sqrt{1+|\nabla_{x}A(x,t)|^{2}}dxdt$.

Given a Good Parabolic Graph $\Gamma$, and a C-Z kernel $K$, we define
associated truncated singular integral operators as follows, for each
$\varepsilon>0$:

$T^{\Gamma}_{\varepsilon}f({\bf X}):=\int_{\Gamma\cap\|{\bf X}-{\bf
Y}\|>\varepsilon}K\big{(}{\bf X}-{\bf Y}\big{)}\,f({\bf Y})\,d\sigma({\bf
Y})\,,\quad{\bf X}\in\Gamma\,,$

and using graph co-ordinates $\Gamma:=\left\\{\big{(}A({\bf x}),{\bf
x}\big{)}\right\\}$, we write the corresponding Euclidean version

(4) $T_{\varepsilon}f({\bf x}):=\int_{\|{\bf x}-{\bf
y}\|>\varepsilon}K\big{(}A({\bf x})-A({\bf y}),{\bf x}-{\bf y}\big{)}\,f({\bf
y})\,d{\bf y}\,,\quad{\bf x}\in\mathbb{R}^{n}\,.$

We have the following.

######  Theorem 5.

Let $K\in C\text{-}Z(2N)$, and suppose that $\Gamma:=\left\\{\big{(}A({\bf
x}),{\bf x}\big{)}\right\\}\in GPG$. If $N$ is large enough, then for some
universal constant $N_{1}$, we have the uniform $L^{2}$ bound

$\sup_{\varepsilon>0}\|T^{\Gamma}_{\varepsilon}f\|_{L^{2}(\Gamma)}\,\leq\,C(1+\|A\|_{comm})^{N_{1}}\|f\|_{L^{2}(\Gamma)}\,,$

where $C$ depends on $K$ an $n$.

The case that $K$ is homogeneous was previously treated in [H2]; some
particular homogeneous kernels arising in the theory of parabolic layer
potentials were originally treated by Lewis and Murray [LM].

Combining Theorem 5 with the results of [BHHLN], and using the “big pieces/
good-$\lambda$” method of G. David (see [D, Proposition III.3.2]), we obtain
as an immediate corollary the following (see [BHHLN, Section 4] for more
details):

######  Corollary 6.

The conclusion of Theorem 5 continues to hold when the graph $\Gamma$ is
replaced by a parabolic uniformly rectifiable set.

In order to prove Theorem 5, we will need the following pair of results.

######  Theorem 7.

Let $H$ be a kernel defined on $\mathbb{R}^{n}\setminus\\{0\\}$, odd in the
space variable, and satisfying (2) with $N=1$, i.e., $H\in C\text{-}Z(1)$(i).
Define truncated convolution singular integral operators

$S_{\varepsilon}f({\bf x}):=\int_{\|{\bf x}-{\bf y}\|>\varepsilon}H({\bf
x}-{\bf y})\,f({\bf y})\,d{\bf y}\,,\quad{\bf x}\in\mathbb{R}^{n}\,.$

Then $S_{\varepsilon}:L^{2}(\mathbb{R}^{n})\to L^{2}(\mathbb{R}^{n})$
uniformly in $\varepsilon$, i.e.,

$\sup_{\varepsilon>0}\|S_{\varepsilon}f\|_{L^{2}(\mathbb{R}^{n})}\leq\,C\|f\|_{L^{2}(\mathbb{R}^{n})}$

The theorem is well known, and we omit the proof. One can either estimate the
Fourier transform of the truncated kernel, and use Plancherel’s theorem, or
else use the parabolic $T1$ theorem.

######  Theorem 8.

Let $H$ be a kernel defined on $\mathbb{R}^{n}\setminus\\{0\\}$, even in the
space variable, and satisfying (2) with $N=1$, i.e., $H\in C\text{-}Z(1)$(i).
Let $A$ be a regular Lip(1,1/2) function. Define the truncated first
commutator

$\mathcal{C}_{\varepsilon}f({\bf x}):=\int_{\|{\bf x}-{\bf
y}\|>\varepsilon}\frac{A({\bf x})-A({\bf y})}{\|{\bf x}-{\bf y}\|}\,H({\bf
x}-{\bf y})\,f({\bf y})\,d{\bf y}\,,\quad{\bf x}\in\mathbb{R}^{n}\,.$

Then $\mathcal{C}_{\varepsilon}:L^{2}(\mathbb{R}^{n})\to
L^{2}(\mathbb{R}^{n})$ uniformly in $\varepsilon$, i.e.,

$\sup_{\varepsilon>0}\|\mathcal{C}_{\varepsilon}f\|_{L^{2}(\mathbb{R}^{n})}\leq\,C\|A\|_{comm}\,\|f\|_{L^{2}(\mathbb{R}^{n})}$

In the case that $H$ is homogeneous, the result is proved in [H1]. We shall
prove the general case here (which is not much harder). We defer the proof to
the end of this note, and take Theorem 8 for granted for the moment.

##  2\. Proofs of the Theorems

We take Theorem 8 for granted until the end of this section.

###### Proof of Theorem 5.

Note that it is equivalent to prove

(9)
$\sup_{\varepsilon>0}\|T_{\varepsilon}f\|_{L^{2}(\mathbb{R}^{n})}\,\leq\,C(1+\|A\|_{comm})^{N_{0}}\|f\|_{L^{2}(\mathbb{R}^{n})}\,,$

for some universal constant $N_{0}$, where $T_{\varepsilon}$ is defined as in
(4). To this end, we first use the technique of [CDM] to reduce matters to the
case of “singular integrals of Calderón type”. The method is nowadays well-
known, but for the reader’s convenience, we provide the details. Observe that

$K(x_{0},{\bf x})=K\left(\|{\bf x}\|\left(\frac{x_{0}}{\|{\bf
x}\|}\,,\frac{{\bf x}}{\|{\bf x}\|}\right)\right)\,,$

so therefore

$K(x_{0},{\bf x})=\int_{\mathbb{R}}e^{2\pi i\frac{x_{0}}{\|{\bf
x}\|}\zeta}\,\,\widehat{K}\left(\|{\bf x}\|\left(\cdot\,,\frac{{\bf x}}{\|{\bf
x}\|}\right)\right)(\zeta)\,d\zeta\,=:\,\int_{\mathbb{R}}e^{2\pi
i\frac{x_{0}}{\|{\bf x}\|}\zeta}\,H_{\zeta}({\bf x})\,d\zeta\,,$

where $\widehat{K}$ denotes the 1-dimensional Fourier transform of $K$, taken
in the $x_{0}$ variable, i.e.,

$H_{\zeta}({\bf x}):=\int_{\mathbb{R}}e^{-2\pi i\zeta\kappa}\,K\big{(}\|{\bf
x}\|\kappa,{\bf x}\big{)}\,d\kappa\,.$

Using the $C\text{-}Z(2N)$ condition, we find that for $0\leq j+k\leq N$, and
$0\leq m\leq N$,

$\|(X,t)\|^{d+j+2k}\,\left|\zeta^{m}\nabla_{X}^{j}\,\partial_{t}^{k}\,H_{\zeta}({\bf
x})\right|\\\\[4.0pt]
=\,C_{m}\|(X,t)\|^{d+j+2k}\left|\int_{\mathbb{R}}e^{-2\pi
i\zeta\kappa}\,\partial_{\kappa}^{m}\nabla_{X}^{j}\,\partial_{t}^{k}K\big{(}\|{\bf
x}\|\kappa,{\bf x}\big{)}\,d\kappa\right|\\\\[4.0pt] \leq
C_{m,j}\int_{\mathbb{R}}(1+\kappa)^{-d}\,d\kappa\leq C_{m,j}\,,$

i.e., $(1+|\zeta|)^{N}H_{\zeta}({\bf x})$ satisfies (2), for each
$\zeta\in\mathbb{R}$, with constants that are uniform in $\zeta$.

Next, we split $H_{\zeta}(\cdot,t)$ into its (spatial) even and odd parts,
i.e., $H_{\zeta}=H_{\zeta}^{even}+H_{\zeta}^{odd}$, where

$H_{\zeta}^{even}(x,t)=\frac{1}{2}\big{(}H_{\zeta}(x,t)+H_{\zeta}(-x,t)\big{)}\,,\quad
H_{\zeta}^{odd}(x,t)=\frac{1}{2}\big{(}H_{\zeta}(x,t)-H_{\zeta}(-x,t)\big{)}\,.$

We now claim that $H_{\zeta}^{even}$ and $H_{\zeta}^{odd}$ have opposite
parity, when viewed as functions of $\zeta$, i.e.,

(10) $H_{\zeta}^{even}(x,t)=-H_{-\zeta}^{even}(x,t)\,,\quad
H_{\zeta}^{odd}(x,t)=H_{-\zeta}^{odd}(x,t)\,.$

To see this, observe first that

$H_{-\zeta}({\bf x}):=\int_{\mathbb{R}}e^{-2\pi
i(-\zeta)\kappa}\,K\big{(}\|{\bf x}\|\kappa,{\bf
x}\big{)}\,d\kappa\,=\,\int_{\mathbb{R}}e^{-2\pi
i\zeta\kappa}\,K\big{(}-\|{\bf x}\|\kappa,{\bf x}\big{)}\,d\kappa\\\\[4.0pt]
=\,-\int_{\mathbb{R}}e^{-2\pi i\zeta\kappa}\,K\big{(}\|{\bf
x}\|\kappa,-x,t\big{)}\,d\kappa\,,$

where we have made the change of variable $\kappa\to-\kappa$, and then used
oddness of $K$ in the space variables. Taking, for example, the even part in
$x$, we have

$H_{-\zeta}^{even}(x,t)=-\frac{1}{2}\int_{\mathbb{R}}e^{-2\pi
i\zeta\kappa}\,\left(K\big{(}\|{\bf x}\|\kappa,-x,t\big{)}+K\big{(}\|{\bf
x}\|\kappa,x,t\big{)}\right)\,d\kappa=-H_{\zeta}^{even}(x,t)\,,$

as claimed. The odd part $H_{\zeta}^{odd}$ may be treated by a similar
argument. We omit the details. Thus (10) holds.

Taking even and odd parts, and using (10), we see that

(11) $K(x_{0},{\bf x})=\int_{\mathbb{R}}\sin\left(2\pi\frac{x_{0}}{\|{\bf
x}\|}\zeta\right)\,H^{even}_{\zeta}({\bf
x})\,d\zeta\,+\,\int_{\mathbb{R}}\cos\left(2\pi\frac{x_{0}}{\|{\bf
x}\|}\zeta\right)\,H^{odd}_{\zeta}({\bf x})\,d\zeta\,.$

To conclude the proof of Theorem 5, it is therefore enough to prove the
following.

######  Theorem 12.

Let $H$ be a singular kernel satisfying (2), with $N=1$. Define truncated
singular integrals of “Calderón-type” by

(13) $\mathcal{T}_{\varepsilon}f({\bf x}):=\int_{\|{\bf x}-{\bf
y}\|>\varepsilon}E\left(\frac{A({\bf x})-A({\bf y})}{\|{\bf x}-{\bf
y}\|}\right)\,H({\bf x}-{\bf y})\,f({\bf y})\,d{\bf y}\,,\quad{\bf
x}\in\mathbb{R}^{n}\,,$

where either $H(x,t)$ is odd in $x$, and $E=\cos$, or $H(x,t)$ is even in $x$,
and

Then there is a universal constant $N_{0}$ such that

$\sup_{\varepsilon>0}\|\mathcal{T}_{\varepsilon}f\|_{L^{2}(\mathbb{R}^{n})}\leq\,C(1+\|A\|_{comm})^{N_{0}}\|f\|_{L^{2}(\mathbb{R}^{n})}\,,$

where $C$ depends on $n$ and the constants in (2).

Let us take Theorem 12 for granted momentarily. Setting $x_{0}=A({\bf
x})-A({\bf y})$ in (11), we see that the truncated singular integral
$T_{\varepsilon}$ defined in (4) can be represented as a sum of two terms of
the form

$\int_{\mathbb{R}}\mathcal{T}_{\varepsilon}^{\zeta}f\,d\zeta\,,$

where $\mathcal{T}_{\varepsilon}^{\zeta}$ is defined as in (13), with $A$
replaced by $\zeta A$, and either with $E=\cos$ and $H=H^{odd}_{\zeta}$, or
with $E=\sin$ and $H=H^{even}_{\zeta}$. Using our previous observation that
$(1+|\zeta|)^{N}H_{\zeta}$ satisfies the bounds in (2), uniformly in $\zeta$,
and invoking Theorem 12, we find that

$\sup_{\varepsilon>0}\|T_{\varepsilon}f\|_{L^{2}(\mathbb{R}^{n})}\,\lesssim\int_{\mathbb{R}}(1+|\zeta|)^{-N}(1+|\zeta|\,\|A\|_{comm})^{N_{0}}\,d\zeta\,\|f\|_{L^{2}(\mathbb{R}^{n})}\\\\[4.0pt]
\lesssim(1+\|A\|_{comm})^{N_{0}}\,\|f\|_{L^{2}(\mathbb{R}^{n})}\,,$

provided that $N>N_{0}+1$. Thus (9) holds, and the conclusion of Theorem 5
follows.

To complete the proof of Theorem 5, it therefore remains to prove Theorem 12.

###### Proof of Theorem 12.

The theorem was already proved in [H2], in the case that $H$ is homogeneous.
The present proof is a modified version of the argument in [H2], which in turn
follows that of [Ch]. For specificity, we treat the case that $E=\cos$, and
$H(x,t)$ is odd in $x$, for each fixed $t$; the proof in the other case is
essentially the same. To simplify notation, we set

$M:=\|A\|_{comm}\,.$

It is enough to verify the localized $T1$ estimate

(14) $\fint_{B}|\mathcal{T}_{\varepsilon}\eta_{B}|\leq C(1+M)^{N_{0}}$

(and the same for the transpose of $\mathcal{T}_{\varepsilon}$, but the latter
is of the same form as $\mathcal{T}_{\varepsilon}$), uniformly in
$\varepsilon>0$ and in every parabolic ball

$B=B_{r}({\bf x}_{B}):=\\{{\bf x}\in\mathbb{R}^{n}:\,\|{\bf x}-{\bf
x}_{B}\|<r\\}\,,$

where $\eta_{B}\in C_{0}^{\infty}(5B)$, with $\eta_{B}\equiv 1$ on $4B$, and
$0\leq\eta_{B}\leq 1$.

By scale-invariance111Even though our individual kernels are not homogeneous,
they do form a scale-invariant class, with scale-invariant control of the
various relevant constants., we may suppose that the ball $B$ in (14) has
radius $r=1$.

Following [Ch] , we choose $\varphi\in C_{0}^{\infty}(1/4,1)$ such that

(15) $\int\limits_{0}^{\infty}\varphi(\rho)\,\frac{d\rho}{\rho}=1,$

We may then replace $\mathcal{T}_{\varepsilon}\eta_{B}({\bf x})$ by

(16) $\int_{\varepsilon}^{1}\\!\int_{\mathbb{R}^{n}}\varphi\left(\frac{\|{\bf
x}-{\bf y}\|}{\delta}\right)\cos\left(\frac{A({\bf x})-A({\bf y})}{\|{\bf
x}-{\bf y}\|}\right)\,H({\bf x}-{\bf y})\,d{\bf y}\,\frac{d\delta}{\delta}\,,$

since for ${\bf x}\in B$, the error is bounded by a uniform constant depending
only on $n$ and $H$. For $\delta>0$, we define a nice parabolic approximate
identity $P_{\delta}f:=\Phi_{\delta}*f$, where $\Phi\in
C_{0}^{\infty}(B_{1}(0))$, $0\leq\Phi$, $\int\Phi=1$, and

$\Phi_{\delta}(x,t):=\delta^{-d}\Phi\left(\delta^{-1}x,\delta^{-2}t\right)\,.$

We then write

$\cos\left(\frac{A({\bf x})-A({\bf y})}{\parallel{\bf x}-{\bf
y}\parallel}\right)\,=\,\cos\left(\frac{(x-y)\cdot P_{\delta}\nabla_{x}A({\bf
x})}{\parallel{\bf x}-{\bf y}\parallel}\right)\\\\[4.0pt] +\,\,\frac{A({\bf
x})-A({\bf y})-(x-y)\cdot P_{\delta}\nabla_{x}A({\bf x})}{\parallel{\bf
x}-{\bf y}\parallel}\,\sin\left(\frac{(x-y)\cdot P_{\delta}\nabla_{x}A({\bf
x})}{\parallel{\bf x}-{\bf y}\parallel}\right)\\\\[4.0pt]
+\,\,O\left(\frac{\mid A({\bf x})-A({\bf y})-(x-y)\cdot
P_{\delta}\nabla_{x}A({\bf x})\mid^{2}}{\parallel{\bf x}-{\bf
y}\parallel^{2}}\right)\\\\[4.0pt] =:\,I({\bf x},{\bf y})+II({\bf x},{\bf
y})+III({\bf x},{\bf y})$

Since $H(x,t)$ is odd in $x$, term $I({\bf x},{\bf y})$ contributes zero to
the integral in (16), as does the part of term $II({\bf x},{\bf y})$ involving
$(x-y)\cdot P_{\delta}\nabla_{x}A({\bf x})$ in the numerator of the first
factor; thus, the contribution of term $II({\bf x},{\bf y})$ equals

(17) $\int_{\varepsilon}^{1}\\!\int_{\mathbb{R}^{n}}\varphi\left(\frac{\|{\bf
x}-{\bf y}\|}{\delta}\right)\frac{A({\bf x})-A({\bf y})}{\|{\bf x}-{\bf
y}\|}\sin\left(\frac{(x-y)\cdot P_{\delta}\nabla_{x}A({\bf x})}{\parallel{\bf
x}-{\bf y}\parallel}\right)H({\bf x}-{\bf y})\,d{\bf
y}\,\frac{d\delta}{\delta}\\\\[4.0pt]
=\sum_{j,k}a_{k,j}\big{(}P_{\delta}\nabla_{x}A({\bf
x})\big{)}\int_{\varepsilon}^{1}\\!\int_{\mathbb{R}^{n}}\varphi\left(\frac{\|{\bf
x}-{\bf y}\|}{\delta}\right)\frac{A({\bf x})-A({\bf y})}{\|{\bf x}-{\bf
y}\|}\widetilde{H}_{k,j}({\bf x}-{\bf y})\,d{\bf
y}\,\frac{d\delta}{\delta}\,,$

where $\widetilde{H}_{k,j}=Y_{k,j}H$, and with
$\vec{b}=P_{\delta}\nabla_{x}A({\bf x})$,

$\sum_{j,k}a_{k,j}(\vec{b})\,Y_{k,j}\left(\frac{{\bf y}}{\|{\bf
y}\|}\right)=\sin\left(\frac{y\cdot\vec{b}}{\parallel{\bf
y}\parallel}\right)=:f(\vec{b},{\bf y})$

is the orthonormal spherical harmonic expansion (with $k$ being the degree of
$Y_{k,j}$) of the function $f(\vec{b},{\bf y}):=\sin(\|{\bf
y}\|^{-1}y\cdot\vec{b})$, which is homogeneous of degree zero (with respect to
parabolic dilations), and thus is determined by its values on the sphere. Note
that by oddness of the Sine function, the $Y_{k,j}$ may be chosen to be odd in
the spatial variable (since the even and odd parts of a spherical harmonic are
also spherical harmonics), and therefore, since $H$ has the same property, it
follows that $\widetilde{H}_{k,j}(x,t)=Y_{k,j}(x,t)H(x,t)$ is even in $x$.
Furthermore, for every non-negative integer $m$,

$\sup_{|\vec{b}|\leq M}\,\sup_{{\bf y}\in
S^{n-1}}|\nabla_{y,s}^{m}f(\vec{b},{\bf y})|\leq C_{m}M^{m}$

so by a well known property of the spherical harmonics (see, e.g., [S, pp.
70-71]) we have

(18) $\parallel a_{k,j}\parallel_{L^{\infty}(\mid\vec{b}\mid\leq
M)}\,\leq\,C_{m}\,M^{m}\,k^{-m}$

with $m$ arbitrarily large. Let us recall also that by standard facts for
spherical harmonics (see, e.g., [CZ]), the dimension $h_{k}$ of the space of
spherical harmonics of degree $k$, satisfies

(19) $h_{k}\leq\,C_{n}\,k^{n-2}\,,$

and also, the normalized spherical harmonics $Y_{k,j}$ satisfy

(20) $\sup_{{\bf y}\in S^{n-1}}|Y_{k,j}({\bf y})|\,+\,\sup_{{\bf y}\in
S^{n-1}}k^{-1}|\nabla_{y,s}Y_{k,j}({\bf y})|\,\leq\,C_{n}\,k^{(n-2)/2}\,.$

Let us further note that for ${\bf x}\in B$, the integral in (17) is unchanged
if we insert the factor $\eta_{B}({\bf y})$, since the latter equals 1 in
$4B$. Combining these observations, we find that (17) is bounded in absolute
value by

$F({\bf x}):=\,C_{m}M^{m}\sum_{k\geq
1}k^{-m+1+(n-2)/2}\,\sum_{j=1}^{h_{k}}\left|\mathcal{C}^{k,j}_{\varepsilon}\,\eta_{B}({\bf
x})\right|\,,$

for ${\bf x}\in B$, where $\mathcal{C}^{k,j}_{\varepsilon}$ is a smoothly
truncated version of a parabolic Calderón-type commutator, defined as in
Theorem 8, with respect to the normalized kernel
$k^{-1-(n-2)/2}\widetilde{H}_{k,j}$, which as noted above is even in $x$. By
this normalization, (20), and Theorem 8, we have

$\sup_{j,k}\sup_{\varepsilon>0}\|\mathcal{C}^{k,j}_{\varepsilon}\|_{L^{2}(\mathbb{R}^{n})\to
L^{2}(\mathbb{R}^{n})}\leq\,CM\,.$

Consequently, by (19), we have

$\fint_{B}|F({\bf x})|d{\bf x}\leq\left(\fint_{B}|F({\bf x})|^{2}d{\bf
x}\right)^{1/2}\lesssim M^{m+1}\,,$

provided that we fix $m>1+3(n-2)/2$. For the contribution of term $II({\bf
x},{\bf y})$, we therefore obtain the desired bound in (14), with $N_{0}=m+1$.

Finally , we consider term $III({\bf x},{\bf y})$. Define

$\gamma_{A}({\bf x},\delta):=\left(\delta^{-d-2}\int_{\delta/4\leq\|{\bf
x}-{\bf y}\|\leq\delta}\mid A({\bf x})-A({\bf y})-(x-y)\cdot
P_{\delta}\nabla_{x}A({\bf x})\mid^{2}\,d{\bf y}\right)^{1/2}\,.$

After plugging $III({\bf x},{\bf y})$ into the integral in (16), and then
averaging over the (unit) ball $B$, we obtain the bound

$|B|^{-1}\int_{B}\int_{0}^{1}\gamma^{2}_{A}({\bf
x},\delta)\,\frac{d\delta}{\delta}d{\bf x}\leq\,CM^{2}\,,$

by the parabolic Dorronsoro-type result proved in [H2, p.249, estimate (35)].
∎

With Theorem 12 in hand, this concludes the proof of Theorem 5. ∎

###### Proof of Theorem 8.

As above, it is enough to verify the localized $T1$ estimate (14), but now
with $\mathcal{C}_{\varepsilon}$ in place of $\mathcal{T}_{\varepsilon}$. Once
again by dilation invariance (within the class of kernels under
consideration), we may suppose that $B$ has radius $r=1$. By the localization
lemma in [H2, Appendix, Lemma 2], we may assume that $A$ is supported in
$10B$, and that $\nabla_{x}A$ and $\mathbb{D}_{n}A$ belong to
$L^{2}(\mathbb{R}^{n})$, with

(21) $\left(|B|^{-1}\int_{\mathbb{R}^{n}}\left(|\nabla_{x}A({\bf
x})|^{2}+|\mathbb{D}_{n}A({\bf x})|^{2}\right)\,d{\bf
x}\right)^{1/2}\,\lesssim\,\|A\|_{comm}=:M\,.$

For notational convenience, with ${\bf x}=(x,t)$, we shall denote the
parabolic dilations as follows:

$\rho^{(1,2)}{\bf x}:=(\rho x,\rho^{2}t)\,,\quad\rho>0\,.$

We also denote points on the unit sphere as
$\omega=(\omega^{\prime},\omega_{n})$, where $\omega_{n}$ is the component of
$\omega$ in the time direction. Using parabolic polar coordinates (see, e.g.,
[FR1]), and integrating by parts, we may write

$C_{\varepsilon}\eta_{B}({\bf x})\\\\[4.0pt]
=\,\int_{S^{n-1}}\int_{\varepsilon}^{\infty}\frac{A({\bf x})-A\left({\bf
x}-\rho^{(1,2)}\omega\right)}{\rho}H\left(\rho^{(1,2)}\omega\right)\,\rho^{d-1}\,d\rho\,(1+\omega_{n}^{2})\,d\sigma(\omega)\\\\[4.0pt]
=\,\int_{S^{n-1}}\int_{\varepsilon}^{\infty}\omega^{\prime}\cdot\nabla_{x}A\big{(}{\bf
x}-\rho^{(1,2)}\omega\big{)}\,\widetilde{H}(\rho^{(1,2)}\omega)\,\rho^{d-1}\,d\rho\,(1+\omega_{n}^{2})\,d\sigma(\omega)\\\\[4.0pt]
+\,\,\int_{S^{n-1}}\int_{\varepsilon}^{\infty}2\rho\omega_{n}\,\partial_{t}A\big{(}{\bf
x}-\rho^{(1,2)}\omega\big{)}\,\widetilde{H}(\rho^{(1,2)}\omega)\,\rho^{d-1}\,d\rho\,(1+\omega_{n}^{2})\,d\sigma(\omega)\\\\[4.0pt]
+\,\,\text{boundary terms}\,=:\,I+II+\text{boundary terms},$

where the boundary term at $\rho=\varepsilon$ is uniformly bounded by $CM$,
and where

$\widetilde{H}(\rho^{(1,2)}\omega):=\rho^{1-d}\int_{\rho}^{\infty}r^{d}H\left(r^{(1,2)}\omega\right)\,\frac{dr}{r^{2}}\,=\,\int_{1}^{\infty}r^{d}H\left((r\rho)^{(1,2)}\omega\right)\,\frac{dr}{r^{2}}\,,$

i.e., with $\rho^{(1,2)}\omega={\bf x}-{\bf y}$,

$\widetilde{H}({\bf x}-{\bf y})=\int_{1}^{\infty}r^{d}H\left(r^{(1,2)}({\bf
x}-{\bf y})\right)\,\frac{dr}{r^{2}}$

Note that $\widetilde{H}$ satisfies (2) with $N=1$, and $\widetilde{H}$ is
even in the space variable (since these properties holds for $H$, by
assumption). Returning to rectangular coordinates, we see that

$I=\int_{\|{\bf x}-{\bf y}\|>\varepsilon}\frac{x-y}{\|{\bf x}-{\bf
y}\|}\cdot\nabla_{y}A({\bf y})\,\widetilde{H}({\bf x}-{\bf y})\,d{\bf
y}=:S_{\varepsilon}(\nabla A)({\bf x})\,,$

where $S_{\varepsilon}$ is the truncated convolution singular integral
operator with the vector-valued C-Z kernel $H_{0}({\bf x}):=x\|{\bf
x}\|^{-1}\widetilde{H}({\bf x})$, which is odd in $x$. Thus,

$\fint_{B}|I|\,d{\bf x}\leq\left(\fint_{B}|S_{\varepsilon}(\nabla A)({\bf
x})|^{2}\,d{\bf x}\right)\,\leq\,CM\,,$

by Theorem 7 and (21).

Finally, writing term $II$ in rectangular coordinates yields

$II=2\int_{\|{\bf x}-{\bf y}\|>\varepsilon}\frac{t-s}{\|{\bf x}-{\bf
y}\|}\,\partial_{s}A({\bf y})\,\widetilde{H}({\bf x}-{\bf y})\,d{\bf
y}=:J_{\varepsilon}*\partial_{s}A({\bf x})\,,$

where

$J_{\varepsilon}({\bf x}):=\frac{t-s}{\|{\bf x}-{\bf y}\|}\,\widetilde{H}({\bf
x}-{\bf y})\,1_{\|{\bf x}\|>\varepsilon}\,.$

A routine computation shows that

$|\widehat{J_{\varepsilon}}(\xi,\tau)|\lesssim\|(\xi,\tau)\|^{-1}\,,$

uniformly in $\varepsilon$, and hence that

$\|J_{\varepsilon}*\partial_{s}A\|_{L^{2}(\mathbb{R}^{n})}\,\lesssim\,\|\mathbb{D}_{n}A\|_{L^{2}(\mathbb{R}^{n})}\,\lesssim\,M\,,$

by (21) and the fact that $B$ has radius 1. Consequently,

$\fint_{B}|II|\,d{\bf x}\,\lesssim\,M\,.$

∎

## References

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  * [H2] S. Hofmann, Parabolic singular integrals of Calderon-type, rough operators, and caloric layer potentials, Duke Math. J., Vol.  90 (1997), pp 209-259.
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∎

22institutetext: 1Department of Mathematics, Indian Institute of Technology
Patna, Patna 801 106, India
2School of Computer Science, University of St. Gallen, Switzerland
22email: om@iitp.ac.in(*corresponding author<EMAIL_ADDRESS><EMAIL_ADDRESS>
#  $\mathbb{F}_{q}\mathcal{R}$-skew cyclic codes and their application to
quantum codes

Om Prakash∗1 Shikha Patel1 Habibul Islam2

(Received: date / Accepted: date)

###### Abstract

Let $p$ be a prime and $\mathbb{F}_{q}$ be the finite field of order
$q=p^{m}$. In this paper, we study $\mathbb{F}_{q}\mathcal{R}$-skew cyclic
codes where $\mathcal{R}=\mathbb{F}_{q}+u\mathbb{F}_{q}$ with $u^{2}=u$. To
characterize $\mathbb{F}_{q}\mathcal{R}$-skew cyclic codes, we first establish
their algebraic structure and then discuss the dual-containing properties by
considering a non-degenerate inner product. Further, we define a Gray map over
$\mathbb{F}_{q}\mathcal{R}$ and obtain their $\mathbb{F}_{q}$-Gray images. As
an application, we apply the CSS (Calderbank-Shor-Steane) construction on Gray
images of dual containing $\mathbb{F}_{q}\mathcal{R}$-skew cyclic codes and
obtain many quantum codes with better parameters than the best-known codes
available in the literature.

###### Keywords:

Cyclic codes Skew cyclic codes Self-orthogonal codes Gray map CSS construction
Quantum codes.

###### MSC:

11T71 11T06 94B05 94B15.

††journal: Quantum Inf. Process.

##  1 Introduction

The family of skew cyclic codes is an emerging and interesting class of linear
codes introduced by Boucher et al. Boucher1  in $2007$. In $2009$, they have
obtained some new linear codes with record-breaking parameters from skew
cyclic codes Boucher2 . Remarkably, these works have considered the length $n$
of the code as an integral multiple of the order of underlying automorphism.
In $2011$, Siap et al. Siap  relaxed this restriction from $n$ to investigate
skew cyclic codes of any length over a finite field. In addition to quasi-
cyclic (QC) and quasi-twisted (QT) codes over commutative rings Ak ; Pandey22
, skew linear codes such as skew cyclic, skew QC and skew QT codes produce
some excellent parameters Boucher1 ; Boucher2 . In fact, in the last two
decades, skew cyclic codes produced some new parameters. Therefore, exploring
skew cyclic codes remains one of the active research areas. Currently, many
researchers have been working on these codes over different alphabets,
particularly to produce new codes over finite rings, we refer Gursoy ; P1 ;
Siap . Also, the number of skew cyclic codes is directly proportional to the
number of factors of $x^{n}-1$. Since the skew polynomial ring is a non-UFD,
it generally possesses more factorization of a polynomial in comparison to a
commutative ring. Hence, it is one of the reasons to study such noncommutative
skew codes.

On the other hand, many researchers have also worked on codes over mixed
alphabets for the last two decades. Linear codes over mixed alphabets were
introduced by Brouwer et al. Bro  in $1998$. In $2010$, Borges et al. Bor1
studied $\mathbb{Z}_{2}\mathbb{Z}_{4}$-linear codes and discussed their
generator matrices and parity-check matrices. In $2014$, Abualrub et al. A
investigated $\mathbb{Z}_{2}\mathbb{Z}_{4}$-cyclic codes and proved that the
dual of a $\mathbb{Z}_{2}\mathbb{Z}_{4}$-cyclic code is again a
$\mathbb{Z}_{2}\mathbb{Z}_{4}$-cyclic code. Further, in $2015$ Aydogdu et al.
Aydo4  generalized these additive codes to
$\mathbb{Z}_{p^{r}}\mathbb{Z}_{p^{s}}$. Again, in $2018$ Aydogdu et al. Aydo3
generalized $\mathbb{Z}_{2}\mathbb{Z}_{4}$-cyclic codes to
$\mathbb{Z}_{2}\mathbb{Z}_{4}\mathbb{Z}_{8}$-cyclic codes. For more related
works on mixed alphabets, we refer Aydo1 ; Aydo2 ; Aydo5 ; Bor2 ; Islam.amca ;
Prakash.amc ; Wu . Recently, in $2020$, Benbelkacem et al. MN  studied mixed
alphabet skew cyclic codes. Here, we discuss structural properties of skew
cyclic codes of length $(\alpha,\beta)$ over the mixed alphabet
$\mathbb{F}_{q}\mathcal{R}$ where $\mathcal{R}=\mathbb{F}_{q}+u\mathbb{F}_{q}$
with $u^{2}=u$. In particular, if $\alpha=0$, we get skew cyclic codes over
$\mathcal{R}$ discussed by Gursoy et al. Gursoy , and if $\beta=0$, we get
skew cyclic codes over $\mathbb{F}_{q}$. Therefore, we generalize both codes
over $\mathbb{F}_{q}$ and $\mathcal{R}$, which is one of our motivations for
this study.

On the other hand, quantum information theory is a fascinating area of
research. It introduces concepts from computer science, classical information
theory, and quantum mechanics all at once. Results and techniques of various
branches of mathematics, mathematical physics, and quantum statistical physics
find applications in this fast-growing field. Classical information theory
deals with information-processing functions like storage and transmission of
information, whereas quantum information theory studies how these tools can be
accomplished using quantum mechanical systems. In classical information
theory, error correction occurs in bits; however, in quantum information
theory, error correction occurs in quantum states (qubits). Both are concerned
with one of the fundamental problems of communication namely reliable
transmission of information over a noisy channel. The implementation of
quantum error-correcting codes is widely acknowledged as necessary for the
development of large-scale, viable quantum computers and the usage of quantum
communication. Quantum error-correcting codes (QECCs) have been used to
protect quantum information from errors caused by decoherence and other types
of quantum noise. Classical computers use a large number of electrons, so when
one goes wrong, it is not too severe. However, in quantum computers single
qubit will probably be just one or a small number of particles. This single
qubit already creates a need for some sort of error correction. Therefore, to
build reliable quantum computers, quantum error correction will be necessarily
needed, while classical computers do not use error correction Gottesman . The
construction of quantum error-correcting codes from classical error-correcting
codes was independently invented by Shor Shor  and Steane Steane . In 1998,
Calderbank et al. Calderbank  proposed a systematic method to obtain quantum
codes from classical codes, known as CSS (Calderbank-Shor-Steane)
construction. Under this construction, several investigations have been
carried out, for instance, quantum codes from cyclic codes over
$\mathbb{F}_{4}+u\mathbb{F}_{4},u^{2}=0$ are found in Kai11 . In the
literature Dertli15 ; Dertli15a ; Islam18b ; Islam.ccds ; Qian ; Sari , binary
and non-binary quantum codes from cyclic codes over finite non-chain rings are
well studied. For a detailed study of quantum codes from linear codes, we
refer Alahmadi ; Edel ; Grassl04 ; Islam19 ; Islam20b ; Islam20a ; Shi20 . The
number of (dual containing) skew codes depends on the factorization of
$x^{n}-1$, and hence the major reason behind the investigation of skew cyclic
codes in the construction of quantum code lies in the factorization of
$x^{n}-1$, which sometimes has more choices for $h(x)$ in a skew polynomial
ring. We note that quantum codes from skew codes have appeared in very few
papers bag ; Prakash2 ; Prakash ; Verma21 .

Further, we notice that in the case of mixed alphabets, researchers mainly
focused on exploring the structural properties such as generator matrices,
parity check matrices, generating polynomials, minimal generating sets,
generating polynomials for dual codes, etc. Presently, only a few works are
available in the literature on the applications of mixed alphabet codes
Islam20b ; Li20 . Recently, Li et al. Li20  studied $\mathbb{F}_{q}R$-linear
skew constacyclic codes and constructed quantum codes by the Hermitian
construction. In continuation with that framework, here we find quantum codes
in the case of noncommutative mixed alphabets codes. Two significant
contributions of the paper are given below.

  1. 1.

We discuss the algebraic structure of $\mathbb{F}_{q}\mathcal{R}$-skew cyclic
codes and their dual codes. Then a necessary and sufficient condition of these
codes to contain their dual is provided.

  2. 2.

As an application, we obtain quantum codes with better parameters (compared to
the recent literature) from dual containing codes.

The paper is structured as follows: In Section $2$, we present some basic
results and definitions followed by the algebraic structure of the ring
$\mathbb{F}_{q}\mathcal{R}$. Section $3$ presents the algebraic properties of
$\mathbb{F}_{q}\mathcal{R}$-skew cyclic codes. We have shown that the dual of
an $\mathbb{F}_{q}\mathcal{R}$-skew cyclic code is again an
$\mathbb{F}_{q}\mathcal{R}$-skew cyclic code. Section $4$ includes the Gray
map through which we discuss images of $\mathbb{F}_{q}\mathcal{R}$-skew cyclic
codes. Moreover, we have provided some examples of
$\mathbb{F}_{q}\mathcal{R}$-skew cyclic codes and their Gray images over
$\mathbb{F}_{q}$. Section $5$ gives a method to construct quantum codes.
Section $6$ concludes the paper.

##  2 Preliminary

Let $\mathbb{F}_{q}$ be a finite field and $q=p^{m}$ for some prime $p$ and a
positive integer $m$. A $k$-dimensional subspace $\mathcal{C}$ of
$\mathbb{F}_{q}^{n}$ is said to be a linear code of length $n$ over
$\mathbb{F}_{q}$ and every element
$c=(c_{0},c_{1},\dots,c_{n-1})\in\mathcal{C}$ is called a codeword. Let
$c\in\mathbb{F}_{q}^{n}$. The Hamming weight of $c$, denoted by $w_{H}(c)$, is
defined as the number of nonzero coordinates in $c$. For any two vectors $c$
and $c^{\prime}$ in $\mathbb{F}_{q}^{n}$, the Hamming distance between $c$ and
$c^{\prime}$, denoted by $d_{H}(c,c^{\prime})$, is defined as the number of
places in which $c$ and $c^{\prime}$ differ. For a linear code $\mathcal{C}$,
the Hamming distance is given by

$d_{H}(\mathcal{C})=\min\\{d_{H}(c,c^{\prime})~{}|~{}c\neq
c^{\prime},\text{for all}~{}c,c^{\prime}\in\mathcal{C}\\}.$

Let $c=(c_{0},c_{1},\dots,c_{n-1})$ and
$c^{\prime}=(c^{\prime}_{0},c^{\prime}_{1},\dots,c^{\prime}_{n-1})$ be two
vectors. The Euclidean inner product of $c$ and $c^{\prime}$ in
$\mathbb{F}_{q}^{n}$ is defined by $c\cdot
c^{\prime}=\sum_{i=0}^{n-1}c_{i}c^{\prime}_{i}$. The dual code of
$\mathcal{C}$ is defined by
$\mathcal{C}^{\perp}=\\{c\in\mathbb{F}_{q}^{n}~{}|~{}c\cdot
c^{\prime}=0,~{}\text{for all}~{}c^{\prime}\in\mathcal{C}\\}$. A linear code
$\mathcal{C}$ is self-orthogonal if $\mathcal{C}\subseteq\mathcal{C}^{\perp}$
and self-dual if $\mathcal{C}=\mathcal{C}^{\perp}$. The parameters of a code
with minimum distance $d$ and dimension $k$ are written as $[n,k,d]$. If
$\mathcal{C}$ is an $[n,k,d]$ code, then from the Singleton bound, its minimum
distance is bounded above by

$d\leq n-k+1.$

A code meeting the above bound with equality is called maximum-distance-
separable (MDS). We call a code is almost MDS if its minimum distance is one
unit less than the MDS bound. A code is called optimal if it has the highest
possible minimum distance for a given length and dimension.
Throughout this paper, $\mathcal{R}$ represents the quotient ring
$\mathbb{F}_{q}[u]/\langle u^{2}-u\rangle$. It can be easily seen that
$\mathcal{R}$ is a finite commutative non-chain ring containing $q^{2}$
elements with characteristic $p$. It is a semi-local ring with maximal ideals
$\langle u\rangle$ and $\langle 1-u\rangle$. Let $\xi_{1}=1-u,\xi_{2}=u$. Then
$\xi_{i}^{2}=\xi_{i}$, $\xi_{i}\xi_{j}=0$ and $\sum_{k=1}^{2}\xi_{k}=1$ where
$i,j=1,2$ and $i\neq j$.Thus, we have

$\mathcal{R}=\xi_{1}\mathcal{R}\oplus\xi_{2}\mathcal{R}.$

Further, $\xi_{i}\mathcal{R}\cong\xi_{i}\mathbb{F}_{q}$, $i=1,2$. Therefore,
any $r\in\mathcal{R}$ can be uniquely written as
$r=\sum_{i=1}^{2}\xi_{i}a_{i}$ where $a_{i}\in\mathbb{F}_{q}$ for $i=1,2$. A
linear code $\mathcal{C}$ of length $n$ over $\mathcal{R}$ is an
$\mathcal{R}$-submodule of $\mathcal{R}^{n}$. Let $B_{i}$ be a linear code
over $\mathcal{R}$, for $i=1,2$. Then the operations $\oplus$ and $\otimes$
are defined as

$B_{1}\oplus B_{2}=\\{b_{1}+b_{2}~{}|~{}b_{i}\in B_{i}~{}\text{for
all}~{}i\\}$

and

$B_{1}\otimes B_{2}=\\{(b_{1},b_{2})~{}|~{}b_{i}\in B_{i}~{}\text{for
all}~{}i\\},$

respectively. We know that any element of the ring $\mathcal{R}$ can be
uniquely written as

$a+ub=(a+b)u+a(1-u)$

where $a,b\in\mathbb{F}_{q}$. Therefore, $a+ub\in\mathcal{R}$ is a unit if and
only if $a,a+b\in\mathbb{F}_{q}^{*}$. Now, following Gursoy , let
$\mathcal{C}$ be a linear code of length $n$ over $\mathcal{R}$ and

$\displaystyle\mathcal{C}_{1}$
$\displaystyle=\\{\mathbb{a}\in\mathbb{F}_{q}^{n}~{}|~{}\mathbb{a}+u\mathbb{b}\in\mathcal{C},~{}\text{for
some}~{}\mathbb{b}\in\mathbb{F}_{q}^{n}\\};$ $\displaystyle\mathcal{C}_{2}$
$\displaystyle=\\{\mathbb{a}+\mathbb{b}\in\mathbb{F}_{q}^{n}~{}|~{}\mathbb{a}+u\mathbb{b}\in\mathcal{C}\\}$

Then $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ are linear codes of length $n$
over $\mathbb{F}_{q}$ and $\mathcal{C}$ can be uniquely written as
$\mathcal{C}=\xi_{1}\mathcal{C}_{1}\oplus\xi_{2}\mathcal{C}_{2}$. Let
$s=a+ub\in\mathcal{R}$ and $c=(c_{0},c_{1},\dots,c_{n-1})\in\mathcal{C}$ where
$c_{i}=a_{i}+ub_{i}$ with $a_{i},b_{i}\in\mathbb{F}_{q}$ for $0\leq i\leq
n-1$. The $i^{th}$-entry of $sc$ is given as

$\displaystyle(a+ub)(a_{i}+ub_{i})$
$\displaystyle=((a+b)u-a(u-1))(a_{i}+ub_{i})$
$\displaystyle=aa_{i}+u(ba_{i}+(b+a)b_{i})$ $\displaystyle=(1-u)r+u(r+t)$

where $r=aa_{i}$ and $t=ba_{i}+bb_{i}+ab_{i}$. Hence, $sc$ can be written in
terms of $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ with
$r=a(a_{0},a_{1},\dots,a_{n-1})$ and
$t=(a+b)(b_{0},b_{1},\dots,b_{n-1})+b(a_{0},a_{1},\dots,a_{n-1})$.
Now, we introduce two classes of noncommutative rings
$\mathbb{F}_{q}[x;\Theta]$ and $\mathcal{R}[x;\theta]$.

###### Definition 1

Let $\Theta$ be an automorphism of $\mathbb{F}_{q}$ defined by
$\Theta(a)=a^{p^{i}}$ and $q=p^{m}$. We consider the set

$\mathbb{F}_{q}[x;\Theta]=\\{b_{e}x^{e}+\cdots+b_{1}x+b_{0}~{}|~{}b_{i}\in\mathbb{F}_{q}~{}\text{and}~{}e\in\mathbb{N}\\}.$

Then $\mathbb{F}_{q}[x;\Theta]$ is a noncommutative ring unless $\Theta$ is
the identity under the usual addition of polynomials and multiplication
defined by $(ax^{i})(bx^{j})=a\Theta^{i}(b)x^{i+j}$ for $a,b\in\mathbb{F}_{q}$
and associative and distributive rules over $\mathbb{F}_{q}[x;\Theta]$.

Now, consider

$\mathcal{R}[x;\theta]=\\{a_{l}x^{l}+\cdots+a_{1}x+a_{0}~{}|~{}a_{i}\in\mathcal{R}~{}\text{and}~{}l\in\mathbb{N}\\}$

where $\theta$ is the automorphism of $\mathcal{R}$ defined by
$\theta(a+ub)=a^{p^{i}}+ub^{p^{i}}$. Hence the order of the automorphism is
$|\langle\theta\rangle|=\frac{m}{\gcd{(m,i)}}$ and in particular,
$|\langle\theta\rangle|=\frac{m}{i}$ when $i|m$. Also, the subring of
$\mathbb{F}_{p^{i}}+u\mathbb{F}_{p^{i}}$ is invariant under $\theta$. Then
$\mathcal{R}[x;\theta]$ is a noncommutative ring unless $\theta$ is the
identity under the usual addition of polynomials and multiplication defined by
$(sx^{i})(s^{\prime}x^{j})=s\theta^{i}(s^{\prime})x^{i+j}$ for
$s,s^{\prime}\in\mathcal{R}$ and associative and distributive rules over
$\mathcal{R}[x;\theta]$. This ring is known as a skew polynomial ring.

###### Definition 2

A linear code $\mathscr{C}$ of length $\beta$ over $\mathcal{R}$ is said to be
a skew cyclic code with respect to the automorphism $\theta$ if for any
codeword $c=(c_{0},c_{1},...,c_{\beta-1})\in\mathscr{C}$ we have
$\delta(c)=(\theta(c_{\beta-1}),\theta({c_{0}}),...,\theta(c_{\beta-2}))\in\mathscr{C}$
where $\delta$ is a skew cyclic shift of $\mathscr{C}$.

Here, $\frac{\mathcal{R}[x;\theta]}{\langle x^{\beta}-1\rangle}$ forms a ring
when $|\theta|$ divide $\beta$. But for $|\theta|\nmid\beta$,
$\frac{\mathcal{R}[x;\theta]}{\langle x^{\beta}-1\rangle}$ fails to be a ring.
However, in that case it is a left $\mathcal{R}[x;\theta]$-module under the
left multiplication defined by $f(x)\big{(}g(x)+\langle
x^{\beta}-1\rangle\big{)}=f(x)g(x)+\langle x^{\beta}-1\rangle$ where
$f(x),g(x)\in\mathcal{R}[x;\theta]$. As usual, it is convenient to identify
each codeword
$c=(c_{0},c_{1},...,c_{\beta-1})\in\mathscr{C}\subseteq\mathcal{R}^{\beta}$
with the polynomial
$c(x)=c_{0}+c_{1}x+\cdots+c_{\beta-1}x^{\beta-1}\in\frac{\mathcal{R}[x;\theta]}{\langle
x^{\beta}-1\rangle}$ under the correspondence
$c=(c_{0},c_{1},...,c_{\beta-1})\longrightarrow
c(x)=c_{0}+c_{1}x+\cdots+c_{\beta-1}x^{\beta-1}$. As a consequence, every code
of length $\beta$ over $\mathcal{R}$ can be viewed as a subset of
$\frac{\mathcal{R}[x;\theta]}{\langle x^{\beta}-1\rangle}$. The following
theorems are used to derive the results in further sections.

###### Theorem 2.1

(Gursoy, , Theorem 3) Let
$\mathscr{C}=\xi_{1}\mathcal{C}_{1}\oplus\xi_{2}\mathcal{C}_{2}$ be a linear
code of length $\beta$ over $\mathcal{R}$. Then $\mathscr{C}$ is a skew cyclic
code of length $\beta$ over $\mathcal{R}$ if and only if $\mathcal{C}_{1}$ and
$\mathcal{C}_{2}$ are skew cyclic codes of length $\beta$ over
$\mathbb{F}_{q}$.

###### Theorem 2.2

(Gursoy, , Theorem 5) Let
$\mathscr{C}=\xi_{1}\mathcal{C}_{1}\oplus\xi_{2}\mathcal{C}_{2}$ be a skew
cyclic code of length $\beta$ over $\mathcal{R}$. Let $g_{1}(x)$ and
$g_{2}(x)$ be generator polynomials of $\mathcal{C}_{1}$ and
$\mathcal{C}_{2}$, respectively. Then
$\mathscr{C}=\langle\xi_{1}g_{1}(x)+\xi_{2}g_{2}(x)\rangle$.

Let $s=a+ub$ be an element of $\mathcal{R}$. Then we define a map
$\eta:\mathcal{R}\rightarrow\mathbb{F}_{q}$ as follows

$\eta(s)=a.$

It is clear that $\eta$ is a ring homomorphism. Let
$\mathbb{F}_{q}\mathcal{R}=\\{(x,y)~{}|~{}x\in\mathbb{F}_{q}~{}\text{and}~{}y\in\mathcal{R}\\}$.
Now for any $s\in\mathcal{R}$ and $(x,y)\in\mathbb{F}_{q}\mathcal{R}$, we
define an $\mathcal{R}$-scalar multiplication on $\mathbb{F}_{q}\mathcal{R}$
as

$*:\mathcal{R}\times\mathbb{F}_{q}\mathcal{R}\longrightarrow\mathbb{F}_{q}\mathcal{R}$

such that

$s*(x,y)=(\eta(s)x,sy).$

It is easy to verify that $\mathbb{F}_{q}\mathcal{R}$ is an
$\mathcal{R}$-module under this multiplication. Further, it can be extended
componentwise to $\mathbb{F}_{q}^{\alpha}\times\mathcal{R}^{\beta}$ as
follows:

$*:\mathcal{R}\times(\mathbb{F}_{q}^{\alpha}\times\mathcal{R}^{\beta})\longrightarrow\mathbb{F}_{q}^{\alpha}\times\mathcal{R}^{\beta}$

where

$s*l=(\eta(s)x_{0},\eta(s)x_{1},\dots,\eta(s)x_{\alpha-1},sy_{0},sy_{1},\dots,sy_{\beta-1});$

with $s\in\mathcal{R}$ and
$l=(x_{0},x_{1},\dots,x_{\alpha-1},y_{0},y_{1},\dots,y_{\beta-1})\in\mathbb{F}_{q}^{\alpha}\times\mathcal{R}^{\beta}$.
Under this multiplication, $\mathbb{F}_{q}^{\alpha}\times\mathcal{R}^{\beta}$
forms an $\mathcal{R}$-module.

###### Definition 3

A nonempty subset $\mathcal{C}$ of
$\mathbb{F}_{q}^{\alpha}\times\mathcal{R}^{\beta}$ is said to be an
$\mathbb{F}_{q}\mathcal{R}$-linear code of length $(\alpha,\beta)$ if
$\mathcal{C}$ is an $\mathcal{R}$-submodule of
$\mathbb{F}_{q}^{\alpha}\times\mathcal{R}^{\beta}$.

Now, we define an inner product between
$l=(x_{0},x_{1},\dots,x_{\alpha-1},y_{0},y_{1},\dots,y_{\beta-1})$ and
$l^{\prime}=(x_{0}^{\prime},x_{1}^{\prime},\dots,x_{\alpha-1}^{\prime},y_{0}^{\prime},y_{1}^{\prime},\dots,y_{\beta-1}^{\prime})$
as

$l\cdot
l^{\prime}=u\sum_{i=0}^{\alpha-1}x_{i}x_{i}^{\prime}+\sum_{j=0}^{\beta-1}y_{j}y_{j}^{\prime}.$

Let $\mathcal{C}$ be an $\mathbb{F}_{q}\mathcal{R}$-linear code of length
$(\alpha,\beta)$. Then the dual code of $\mathcal{C}$ is defined as

$\mathcal{C}^{\perp}=\\{l^{\prime}\in\mathbb{F}_{q}^{\alpha}\times\mathcal{R}^{\beta}~{}|~{}l\cdot
l^{\prime}=0~{}\forall~{}l\in\mathcal{C}\\}.$

Denote $R_{\alpha,\beta}=\frac{\mathbb{F}_{q}[x;\Theta]}{\langle
x^{\alpha}-1\rangle}\times\frac{\mathcal{R}[x;\theta]}{\langle
x^{\beta}-1\rangle}$,
$k(x)=k_{0}+k_{1}x+\cdots+k_{\alpha-1}x^{\alpha-1}\in\frac{\mathbb{F}_{q}[x;\Theta]}{\langle
x^{\alpha}-1\rangle}$ and
$t(x)=t_{0}+t_{1}x+\cdots+t_{\beta-1}x^{\beta-1}\in\frac{\mathcal{R}[x;\theta]}{\langle
x^{\beta}-1\rangle}$. Then any vector
$c=(k_{0},k_{1},\dots,k_{\alpha-1},t_{0},t_{1},\dots,t_{\beta-1})\in\mathbb{F}_{q}^{\alpha}\times\mathcal{R}^{\beta}$
can be identified with a polynomial of the form

$c(x)=(k(x),t(x))\in R_{\alpha,\beta}$

which gives a one-to-one correspondence between
$\mathbb{F}_{q}^{\alpha}\times\mathcal{R}^{\beta}$ and $R_{\alpha,\beta}$. Let
$r(x)=r_{0}+r_{1}x+\cdots+r_{\gamma-1}x^{\gamma-1}\in\mathcal{R}[x;\theta]$
and $(k(x),t(x))\in R_{\alpha,\beta}$. Then the multiplication $*$ in
$R_{\alpha,\beta}$ is defined as follows:

$r(x)*(k(x),t(x))=(\eta(r(x))k(x),r(x)t(x))$

where
$\eta(r(x))=\eta(r_{0})+\eta(r_{1})x+\cdots+\eta(r_{\gamma-1})x^{\gamma-1}$.
Then under the above defined multiplication $*$, $R_{\alpha,\beta}$ forms a
left $\mathcal{R}[x;\theta]$-module.

##  3 Structural Properties of $\mathbb{F}_{q}\mathcal{R}$-Skew Cyclic Codes

In this section, first, we define $\mathbb{F}_{q}\mathcal{R}$-skew cyclic
codes. Then algebraic properties of these codes are discussed in detail. Also,
the form of a generator polynomial of an $\mathbb{F}_{q}\mathcal{R}$-skew
cyclic code in $R_{\alpha,\beta}$ is established.

###### Definition 4

An $\mathcal{R}$-submodule $\mathcal{C}$ of
$\mathbb{F}_{q}^{\alpha}\times\mathcal{R}^{\beta}$ is said to be an
$\mathbb{F}_{q}\mathcal{R}$-skew cyclic code of length $n=(\alpha,\beta)$ if
for any
$c=(x_{0},x_{1},\dots,x_{\alpha-1},y_{0},y_{1},\dots,y_{\beta-1})\in\mathcal{C}$
we have
$\sigma(c):=(\Theta(x_{\alpha-1}),\Theta(x_{0}),\Theta(x_{1}),\dots,\Theta(x_{\alpha-2}),\theta(y_{\beta-1}),\theta(y_{0}),\theta(y_{1}),\dots,\theta(y_{\beta-2}))\in\mathcal{C}$
where $\sigma$ is the skew cyclic shift of $c$.

###### Theorem 3.1

Let $\mathcal{C}$ be an $\mathbb{F}_{q}\mathcal{R}$-skew cyclic code of length
$n=(\alpha,\beta)$ and $|\langle\theta\rangle|$ divide $\beta$. Then
$\mathcal{C}^{\perp}$ is also an $\mathbb{F}_{q}\mathcal{R}$-skew cyclic code
of length $n=(\alpha,\beta)$.

###### Proof

Let $\mathcal{C}$ be an $\mathbb{F}_{q}\mathcal{R}$-skew cyclic code of length
$n=(\alpha,\beta)$ and
$l^{\prime}=(x_{0}^{\prime},x_{1}^{\prime},\dots,x_{\alpha-1}^{\prime},y_{0}^{\prime},y_{1}^{\prime},\\\
\dots,y_{\beta-1}^{\prime})\in\mathcal{C}^{\perp}$. Assume that
$l=(x_{0},x_{1},\dots,x_{\alpha-1},y_{0},y_{1},\dots,y_{\beta-1})\in\mathcal{C}$
and $lcm(\alpha,\beta)=\mathcal{L}$. Now, we have to show that
$\sigma(l^{\prime})\in\mathcal{C}^{\perp}$. Consider the inner product of $l$
and $\sigma(l^{\prime})$. We have

$\displaystyle l\cdot\sigma(l^{\prime})=$
$\displaystyle(x_{0},x_{1},\dots,x_{\alpha-1},y_{0},y_{1},\dots,y_{\beta-1})\cdot$
$\displaystyle(\Theta(x_{\alpha-1}^{\prime}),\Theta(x_{0}^{\prime}),\Theta(x_{1}^{\prime}),\dots,\Theta(x_{\alpha-2}^{\prime}),\theta(y_{\beta-1}^{\prime}),\theta(y_{0}^{\prime}),\theta(y_{1}^{\prime}),\dots,\theta(y_{\beta-2}^{\prime}))$
$\displaystyle=$ $\displaystyle
u(x_{0}\Theta(x_{\alpha-1}^{\prime})+x_{1}\Theta(x_{0}^{\prime})+\cdots+x_{\alpha-1}\Theta(x_{\alpha-2}^{\prime}))$
$\displaystyle+(y_{0}\theta(y_{\beta-1}^{\prime})+y_{1}\theta(y_{0}^{\prime})+\cdots+y_{\beta-1}\theta(y_{\beta-2}^{\prime})).$

We only need to show that
$x_{0}\Theta(x_{\alpha-1}^{\prime})+x_{1}\Theta(x_{0}^{\prime})+\cdots+x_{\alpha-1}\Theta(x_{\alpha-2}^{\prime})=0$
and
$y_{0}\theta(y_{\beta-1}^{\prime})+y_{1}\theta(y_{0}^{\prime})+\cdots+y_{\beta-1}\theta(y_{\beta-2}^{\prime})=0$.
As $\mathcal{C}$ is an $\mathbb{F}_{q}\mathcal{R}$-skew cyclic code,
$\sigma^{\mathcal{L}-1}(l)\in\mathcal{C}$ where
$\sigma^{\mathcal{L}-1}(l)=(\Theta^{\mathcal{L}-1}(x_{1}),\Theta^{\mathcal{L}-1}(x_{2}),\dots,\Theta^{\mathcal{L}-1}(x_{\alpha-1}),\Theta^{\mathcal{L}-1}(x_{0}),\theta^{\mathcal{L}-1}(y_{1}),\theta^{\mathcal{L}-1}(y_{2}),\dots,\\\
\theta^{\mathcal{L}-1}(y_{\beta-1}),\theta^{\mathcal{L}-1}(y_{0}))$. Now, we
get that $\sigma^{\mathcal{L}-1}(l)\cdot l^{\prime}=0$, where

$\displaystyle\sigma^{\mathcal{L}-1}(l)\cdot l^{\prime}$
$\displaystyle=u\sum_{i=0}^{\alpha-1}\Theta^{\mathcal{L}-1}(x_{i+1})x_{i}^{\prime}+\sum_{j=0}^{\beta-1}\theta^{\mathcal{L}-1}(y_{j+1})y_{j}^{\prime}.$

Comparing the coefficients on both sides, we get

$\Theta^{\mathcal{L}-1}(x_{0})x_{\alpha-1}^{\prime}+\Theta^{\mathcal{L}-1}(x_{1})x_{0}^{\prime}+\cdots+\Theta^{\mathcal{L}-1}(x_{\alpha-1})x_{\alpha-2}^{\prime}=0$

and

$\theta^{\mathcal{L}-1}(y_{0})y_{\beta-1}^{\prime}+\theta^{\mathcal{L}-1}(y_{1})y_{0}^{\prime}+\cdots+\theta^{\mathcal{L}-1}(y_{\beta-1})y_{\beta-2}^{\prime}=0.$

Since the order of $\theta$ divides $\beta$, $|\langle\theta\rangle|$ divide
$lcm(\alpha,\beta)=\mathcal{L}$ and hence
$\theta^{\mathcal{L}}(a)=\Theta^{\mathcal{L}}(a)=a$ for any
$a\in\mathbb{F}_{q}$. Then
$\theta(\theta^{\mathcal{L}-1}(y_{0})y_{\beta-1}^{\prime}+\theta^{\mathcal{L}-1}(y_{1})y_{0}^{\prime}+\cdots+\theta^{\mathcal{L}-1}(y_{\beta-1})y_{\beta-2}^{\prime})=\theta(0)=0$
and
$\Theta(\Theta^{\mathcal{L}-1}(x_{0})x_{\alpha-1}^{\prime}+\Theta^{\mathcal{L}-1}(x_{1})x_{0}^{\prime}+\cdots+\Theta^{\mathcal{L}-1}(x_{\alpha-1})x_{\alpha-2}^{\prime})=\Theta(0)=0$.
This implies
$y_{0}\theta(y_{\beta-1}^{\prime})+y_{1}\theta(y_{0}^{\prime})+\cdots+y_{\beta-1}\theta(y_{\beta-2}^{\prime})=0$
and
$x_{0}\Theta(x_{\alpha-1}^{\prime})+x_{1}\Theta(x_{0}^{\prime})+\cdots+x_{\alpha-1}\Theta(x_{\alpha-2}^{\prime})=0$.
Therefore, $l\cdot\sigma(l^{\prime})=0$ and hence
$\sigma(l^{\prime})\in\mathcal{C}^{\perp}$. Thus, $\mathcal{C}^{\perp}$ is an
$\mathbb{F}_{q}\mathcal{R}$-skew cyclic code of length $(\alpha,\beta)$.∎

###### Theorem 3.2

A code $\mathcal{C}$ is an $\mathbb{F}_{q}\mathcal{R}$-skew cyclic code of
length $(\alpha,\beta)$ if and only if $\mathcal{C}$ is a left
$\mathcal{R}[x;\theta]$-submodule of $R_{\alpha,\beta}$.

###### Proof

Let $\mathcal{C}$ be an $\mathbb{F}_{q}\mathcal{R}$-skew cyclic code of length
$(\alpha,\beta)$ and $c=(k_{0},k_{1},\dots,k_{\alpha-1},t_{0},t_{1},\dots,\\\
t_{\beta-1})\in\mathcal{C}$; and the corresponding element of $c$ in
$R_{\alpha,\beta}$ be $c(x)=(k(x),t(x))$. Now, consider

$\displaystyle x*c(x)$
$\displaystyle=(\Theta(k_{\alpha-1})+\Theta(k_{0})x+\cdots+\Theta(k_{\alpha-2})x^{\alpha-1},\theta(t_{\beta-1})+\theta(t_{0})x+\cdots+\theta(t_{\beta-2})x^{\beta-1})$

which corresponds to the skew cyclic shift
$(\Theta(k_{\alpha-1}),\Theta(k_{1}),\dots,\Theta(k_{\alpha-2}),\theta(t_{\beta-1}),\theta(t_{1}),\dots,\\\
\theta(t_{\beta-2}))$ of
$(k_{0},k_{1},\dots,k_{\alpha-1},t_{0},t_{1},\dots,t_{\beta-1})$. Therefore,
$x*c(x)\in\mathcal{C}$. Then by linearity of $\mathcal{C}$,
$s(x)*c(x)\in\mathcal{C}$ for any $s(x)\in\mathcal{R}[x;\theta]$ and hence
$\mathcal{C}$ is a left $\mathcal{R}[x;\theta]$-submodule of
$R_{\alpha,\beta}$.
The converse part follows from the definition.∎

Let $\mathcal{C}$ be an $\mathbb{F}_{q}\mathcal{R}$-skew cyclic code of length
$(\alpha,\beta)$. Let $c(x)=(k(x),t(x))\in\mathcal{C}$ and
$m(x)\in\frac{\mathbb{F}_{q}[x;\Theta]}{\langle x^{\alpha}-1\rangle}$. Now, we
consider

$M=\\{t(x)\in\mathcal{R}[x;\theta]/\langle
x^{\beta}-1\rangle~{}|~{}(m(x),t(x))\in\mathcal{C}\\}$

and

$N=\\{k(x)\in\mathbb{F}_{q}[x;\Theta]/\langle
x^{\alpha}-1\rangle~{}|~{}(k(x),0)\in\mathcal{C}\\}.$

Next, we present the algebraic properties of these two sets.

###### Lemma 1

The above defined set $N$ is a left $\mathbb{F}_{q}[x;\Theta]$-submodule of
$\mathbb{F}_{q}[x;\Theta]/\langle x^{\alpha}-1\rangle$ generated by a right
divisor of $x^{\alpha}-1$.

###### Proof

Let $k_{1}(x)$ and $k_{2}(x)$ be two elements of $N$. Then from definition,
$(k_{1}(x),0)\in\mathcal{C}$ and $(k_{2}(x),0)\in\mathcal{C}.$ Further,
$(k_{1}(x),0)+(k_{2}(x),0)=(k_{1}(x)+k_{2}(x),0)\in\mathcal{C}$. This implies
that $k_{1}(x)+k_{2}(x)\in N$. Again, let
$t^{\prime}(x)\in\mathbb{F}_{q}[x;\Theta]/\langle x^{\alpha}-1\rangle$ and
$k(x)\in N$. Then $(k(x),0)\in\mathcal{C}$. As $\mathcal{C}$ is a left
$\mathcal{R}[x;\theta]$-submodule, we have

$t^{\prime}(x)*(k(x),0)=(t^{\prime}(x)k(x),0)\in\mathcal{C}.$

This implies $t^{\prime}(x)k(x)\in N$ where $t^{\prime}(x)k(x)$ is taken
modulo $x^{\alpha}-1$. Hence, $N$ is a left
$\mathbb{F}_{q}[x;\Theta]$-submodule of $\mathbb{F}_{q}[x;\Theta]/\langle
x^{\alpha}-1\rangle$ generated by a right divisor $f(x)$ of $x^{\alpha}-1$.∎

###### Lemma 2

The set $M$ is a left $\mathcal{R}[x;\theta]$-submodule of
$\mathcal{R}[x;\theta]/\langle x^{\beta}-1\rangle$ which is generated by a
single element.

###### Proof

Let $t_{1}(x)$ and $t_{2}(x)$ be two elements of $M$. Then from definition
there exist $m_{1}(x)$ and $m_{2}(x)$ in $\mathbb{F}_{q}[x;\Theta]/\langle
x^{\alpha}-1\rangle$ such that $(m_{1}(x),t_{1}(x))\in\mathcal{C}$,
$(m_{2}(x),t_{2}(x))\in\mathcal{C}$. Thus,

$(m_{1}(x),t_{1}(x))+(m_{2}(x),t_{2}(x))=(m_{1}(x)+m_{2}(x),t_{1}(x)+t_{2}(x))\in\mathcal{C}.$

This implies that $t_{1}(x)+t_{2}(x)\in M$. Let
$s(x)\in\mathcal{R}[x;\theta]/\langle x^{\beta}-1\rangle$ and
$(m(x),t(x))\in\mathcal{C}$. Since $\mathcal{C}$ is a left
$\mathcal{R}[x;\theta]$-submodule of $R_{\alpha,\beta}$, we have

$s(x)*(m(x),t(x))=(\eta(s(x))m(x),s(x)t(x))\in\mathcal{C}.$

This implies $s(x)t(x)\in M$, where $s(x)t(x)$ is taken modulo $x^{\beta}-1$
and $\eta(s(x))m(x)$ is taken modulo $x^{\alpha}-1$. Therefore, $M$ is a left
$\mathcal{R}[x;\theta]$-submodule of $\mathcal{R}[x;\theta]/\langle
x^{\beta}-1\rangle$. Further, from Theorem 2.2, $M=\langle g(x)\rangle$ where
$g(x):=\xi_{1}g_{1}(x)+\xi_{2}g_{2}(x)$.∎

###### Theorem 3.3

Let $\mathcal{C}$ be an $\mathbb{F}_{q}\mathcal{R}$-skew cyclic code of length
$(\alpha,\beta)$. Then $\mathcal{C}$ is generated as a left
$\mathcal{R}[x;\theta]$-submodule of $R_{\alpha,\beta}$ by $(f(x),0)$ and
$(m(x),g(x))$ where $m(x)\in\mathbb{F}_{q}[x;\Theta]/\langle
x^{\alpha}-1\rangle$, $g(x)=\xi_{1}g_{1}(x)+\xi_{2}g_{2}(x)$ and $f(x)$ is a
right divisor of $x^{\alpha}-1$.

###### Proof

Let $c=(c_{1},c_{2})\in\mathcal{C}$ with $c_{1}\in\mathbb{F}_{q}^{\alpha}$ and
$c_{2}\in\mathcal{R}^{\beta}$. It has the polynomial representation as
$c(x)=(c_{1}(x),c_{2}(x))$ in $R_{\alpha,\beta}$. Then $c_{2}(x)\in M$ and
from Lemma 2, $c_{2}(x)=h(x)g(x)$ for some
$h(x)\in\mathcal{R}[x;\theta]/\langle x^{\beta}-1\rangle$. As $g(x)\in M$,
there exist $m(x)\in\mathbb{F}_{q}[x;\Theta]/\langle x^{\alpha}-1\rangle$ such
that $(m(x),g(x))\in\mathcal{C}$. Now,

$\displaystyle c(x)$ $\displaystyle=(c_{1}(x),c_{2}(x))$
$\displaystyle=(c_{1}(x),0)+(0,h(x)g(x))$
$\displaystyle=(c_{1}(x),0)+p(\eta(h(x)m(x)),0)+(0,h(x)g(x))$
$\displaystyle=(c_{1}(x),0)+(p-1)(\eta(h(x))m(x),0)+(\eta(h(x)m(x)),0)+(0,h(x)g(x))$
$\displaystyle=(c_{1}(x),0)+(p-1)(\eta(h(x))m(x),0)+(\eta(h(x))m(x),h(x)g(x))$
$\displaystyle=(c_{1}(x)+(p-1)\eta(h(x))m(x),0)+(\eta(h(x))m(x),h(x)g(x)).$

where $p$ is the characteristic of $\mathbb{F}_{q}$. This gives
$(c_{1}(x)+(p-1)\eta(h(x))m(x),0)\in\mathcal{C}$ and hence
$(c_{1}(x)+(p-1)\eta(h(x))m(x),0)\in N$. Moreover, by Lemma 1 there exists
$d(x)\in N$ such that $c_{1}(x)+(p-1)\eta(h(x))m(x)=d(x)f(x)$. Therefore,
$c(x)=(\eta(h(x))m(x),h(x)g(x))+(d(x)f(x),0)=h(x)*(m(x),g(x))+d(x)*(f(x),0)$.∎

###### Theorem 3.4

Let $\mathcal{C}=\langle(f(x),0),(m(x),g(x))\rangle$ be an
$\mathbb{F}_{q}\mathcal{R}$-skew cyclic code of length $(\alpha,\beta)$ with
$m(x)=0$. Then $\mathcal{C}=\mathcal{C}^{\prime}\otimes\mathscr{C}$ where
$\mathcal{C}^{\prime}$ is a skew cyclic code of length $\alpha$ over
$\mathbb{F}_{q}$ and $\mathscr{C}$ is a skew cyclic code of length $\beta$
over $\mathcal{R}$.

###### Proof

Let $c=(c_{1},c_{2})\in\mathcal{C}$ with $c_{1}\in\mathbb{F}_{q}^{\alpha}$ and
$c_{2}\in\mathcal{R}^{\beta}$. Then, from Lemma 1 and Lemma 2,
$c_{1}\in\mathcal{C}^{\prime}=\langle f(x)\rangle$ and
$c_{2}\in\mathscr{C}=\langle g(x)\rangle$. This implies
$\mathcal{C}=\mathcal{C}^{\prime}\otimes\mathscr{C}$ where
$\mathcal{C}^{\prime}=\langle f(x)\rangle$ and $\mathscr{C}=\langle
g(x)\rangle$.∎

###### Corollary 1

Let $\gcd(\alpha,|\langle\Theta\rangle|)=1$ and
$\mathcal{C}=\langle(f(x),0),(0,g(x))\rangle$ be an
$\mathbb{F}_{q}\mathcal{R}$-skew cyclic code of length $(\alpha,\beta)$. Then
$\mathcal{C}=\mathcal{C}^{\prime}\otimes\mathscr{C}$ where
$\mathcal{C}^{\prime}$ is a cyclic code of length $\alpha$ over
$\mathbb{F}_{q}$ and $\mathscr{C}$ is a skew cyclic code of length $\beta$
over $\mathcal{R}$.

###### Proof

Straightforward.∎

###### Definition 5

Let $\mathcal{C}$ be an $\mathbb{F}_{q}\mathcal{R}$-linear code of length
$(\alpha,\beta)$ and $\mathcal{C}_{\alpha}$ (respectively,
$\mathcal{C}_{\beta}$) be the canonical projection of $\mathcal{C}_{\alpha}$
on the first $\alpha$ (respectively, on the last $\beta$) coordinates. The
code $\mathcal{C}$ is called separable if $\mathcal{C}$ is the direct product
of $\mathcal{C}_{\alpha}$ and $\mathcal{C}_{\beta}$, i.e.,
$\mathcal{C}=\mathcal{C}_{\alpha}\otimes\mathcal{C}_{\beta}$.

Let $\mathcal{C}^{\prime}$ be a skew cyclic code over $\mathbb{F}_{q}$, and
$\mathscr{C}$ be a skew cyclic code over $\mathcal{R}$. If $\mathcal{C}$ is
separable, then $\mathcal{C}=\mathcal{C}^{\prime}\otimes\mathscr{C}$, i.e.
$\mathcal{C}=\langle(f(x),0),(0,g(x))\rangle$ where
$\mathcal{C}^{\prime}=\langle f(x)\rangle$ with $f(x)$ is a right divisor of
$x^{\alpha}-1$ and $\mathscr{C}=\langle g(x)\rangle$ with $g(x)$ is a right
divisor of $x^{\beta}-1$.

###### Theorem 3.5

Let $\mathcal{C}=\mathcal{C}^{\prime}\otimes\mathscr{C}$ be an
$\mathbb{F}_{q}\mathcal{R}$-skew cyclic code of length $(\alpha,\beta)$ where
$\mathcal{C}^{\prime}$ and $\mathscr{C}$ are skew cyclic codes over
$\mathbb{F}_{q}$ and $\mathcal{R}$, respectively and
$\mathcal{C}=\langle(f(x),0),(0,g(x))\rangle$. Then
$\mathcal{C}^{\prime\perp}\subseteq\mathcal{C}^{\prime}$ and
$\mathscr{C}^{\perp}\subseteq\mathscr{C}$ if and only if
$\mathcal{C}^{\perp}\subseteq\mathcal{C}$.

###### Proof

Let $\mathcal{C}^{\prime\perp}\subseteq\mathcal{C}^{\prime}$ and
$\mathscr{C}^{\perp}\subseteq\mathscr{C}$. Let
$c_{1},c_{1}^{\prime}\in\mathcal{C}^{\prime\perp}$ and
$c_{2},c_{2}^{\prime}\in\mathscr{C}^{\perp}$. Then
$c=(c_{1},c_{2}),c^{\prime}=(c_{1}^{\prime},c_{2}^{\prime})\in\mathcal{C}^{\perp}$.
This implies $c_{1}\cdot c_{1}^{\prime}=0$ in $\mathbb{F}_{q}$ and $c_{2}\cdot
c_{2}^{\prime}=0$ in $\mathcal{R}$. Therefore, $c\cdot
c^{\prime}=uc_{1}c_{1}^{\prime}+c_{2}c_{2}^{\prime}=0$. Hence,
$\mathcal{C}^{\perp}\subseteq\mathcal{C}$.
Converse follows directly from the definition.∎

##  4 Gray Map

Any arbitrary element of $\mathbb{F}_{q}\mathcal{R}$ can be uniquely written
as $(a,r)=(a,\xi_{1}r_{1}+\xi_{2}r_{2})$, where $a\in\mathbb{F}_{q}$ and
$r\in\mathcal{R}$. Let $GL_{2}(\mathbb{F}_{q})$ be the set of all $2\times 2$
invertible matrices over $\mathbb{F}_{q}$. Consider a Gray map (see Li20 ;
Prakash  for similar mapping)
$\phi:\mathcal{R}\longrightarrow\mathbb{F}_{q}^{2}$ defined by

$\phi(r_{1}+ur_{2})=(r_{1},r_{2})M,$

where $M\in GL_{2}(\mathbb{F}_{q})$ such that $MM^{T}=\gamma I_{2}$, $M^{T}$
is the transpose matrix of $M$, $\gamma\in\mathbb{F}_{q}^{*}$ and $I_{2}$ is
the identity matrix of order $2$. Then $\phi$ is a linear bijection and can be
extended to $\mathcal{R}^{n}$ componentwise. Particularly, when $M=I_{2}$, we
have the following Gray map
$\phi_{1}:\mathcal{R}\longrightarrow\mathbb{F}_{q}^{2}$

$\phi_{1}(r_{1}+ur_{2})=(r_{1},r_{2}).$ (1)

Now, we define a Gray map
$\varphi:\mathbb{F}_{q}\mathcal{R}\longrightarrow\mathbb{F}_{q}^{3}$ as
follows:

$\varphi(a,r)=(a,\phi(r))=(a,(r_{1},r_{2})M).$

It is easy to verify that $\varphi$ is an $\mathbb{F}_{q}$-linear map and can
be extended componentwise to $\mathbb{F}_{q}^{\alpha}\mathcal{R}^{\beta}$ in
the following manner:

$\varphi:\mathbb{F}_{q}^{\alpha}\mathcal{R}^{\beta}\longrightarrow\mathbb{F}_{q}^{\alpha+2\beta}$
$\displaystyle\varphi(a_{0},a_{1},\dots,a_{\alpha-1},r_{0},r_{1},\dots,r_{\beta-1})=($
$\displaystyle
a_{0},a_{1},\dots,a_{\alpha-1},(r_{0,1},r_{0,2})M,(r_{1,1},r_{1,2})M,$
$\displaystyle\dots,(r_{\beta-1,1},r_{\beta-1,2})M),$

where $(a_{0},a_{1},\dots,a_{\alpha-1})\in\mathbb{F}_{q}^{\alpha}$ and
$(r_{0},r_{1},\dots,r_{\beta-1})\in\mathcal{R}^{\beta}$ such that
$r_{j}=\xi_{1}r_{j,1}+\xi_{2}r_{j,2}\in\mathcal{R}$ for $j=0,1,\dots,\beta-1$.
The Lee weight of an element $(a,r)\in\mathbb{F}_{q}\mathcal{R}$ as
$w_{L}(a,r)=w_{H}(a)+w_{L}(r)$ for any
$(a,r)\in\mathbb{F}_{q}^{\alpha}\times\mathcal{R}^{\beta}$ where $w_{H}$
denotes the Hamming weight and $w_{L}$ denotes the Lee weight. Furthermore,
the Lee distance between
$c,c^{\prime}\in\mathbb{F}_{q}^{\alpha}\times\mathcal{R}^{\beta}$ is defined
as $d_{L}(c,c^{\prime})=w_{L}(c-c^{\prime})=w_{H}(\varphi(c-c^{\prime}))$.

###### Proposition 1

The Gray map $\varphi$ is an $\mathbb{F}_{q}$-linear and distance preserving
map from $\mathbb{F}_{q}^{\alpha}\mathcal{R}^{\beta}$ (Lee distance) to
$\mathbb{F}_{q}^{\alpha+2\beta}$ (Hamming distance).

###### Proof

Let
$c=(a,r),c^{\prime}=(a^{\prime},r^{\prime})\in\mathbb{F}_{q}^{\alpha}\mathcal{R}^{\beta}$,
where $a=(a_{0},a_{1},\dots,a_{\alpha-1})\in\mathbb{F}_{q}^{\alpha}$,
$r=(r_{0},r_{1},\dots,r_{\beta-1})\in\mathcal{R}^{\beta}$,
$a^{\prime}=(a_{0}^{\prime},a_{1}^{\prime},\dots,a_{\alpha-1}^{\prime})\in\mathbb{F}_{q}^{\alpha}$,
$r^{\prime}=(r_{0}^{\prime},r_{1}^{\prime},\dots,r_{\beta-1}^{\prime})\in\mathcal{R}^{\beta}$
such that $r_{j}=\xi_{1}r_{j,1}+\xi_{2}r_{j,2}$ and
$r_{j}^{\prime}=\xi_{1}r_{j,1}^{\prime}+\xi_{2}r_{j,2}^{\prime}$ for
$j=0,1,\dots,\beta-1$. Then

$\displaystyle\varphi(c+c^{\prime})=$
$\displaystyle\varphi(a+a^{\prime},r+r^{\prime})$ $\displaystyle=$
$\displaystyle\varphi(a_{0}+a_{0}^{\prime},a_{1}+a_{1}^{\prime},\dots,a_{\alpha-1}+a_{\alpha-1}^{\prime},r_{0}+r_{0}^{\prime},r_{1}+r_{1}^{\prime},\dots,r_{\beta-1}+r_{\beta-1}^{\prime})$
$\displaystyle=$
$\displaystyle\varphi(a_{0}+a_{0}^{\prime},a_{1}+a_{1}^{\prime},\dots,a_{\alpha-1}+a_{\alpha-1}^{\prime},\xi_{1}(r_{0,1}+r_{0,1}^{\prime})+\xi_{2}(r_{0,2}+r_{0,2}^{\prime}),$
$\displaystyle\xi_{1}(r_{1,1}+r_{1,1}^{\prime})+\xi_{2}(r_{1,2}+r_{1,2}^{\prime}),\dots,\xi_{1}(r_{\beta-1,1}+r_{\beta-1,1}^{\prime})+\xi_{2}(r_{\beta-1,2}+r_{\beta-1,2}^{\prime})$
$\displaystyle=$
$\displaystyle(a_{0}+a_{0}^{\prime},a_{1}+a_{1}^{\prime},\dots,a_{\alpha-1}+a_{\alpha-1}^{\prime},(r_{0,1}+r_{0,1}^{\prime},r_{0,2}+r_{0,2}^{\prime})M,(r_{1,1}+r_{1,1}^{\prime},r_{1,2}$
$\displaystyle+r_{1,2}^{\prime})M,\dots,(r_{\beta-1,1}+r_{\beta-1,1}^{\prime},r_{\beta-1,2}+r_{\beta-1,2}^{\prime})M)$
$\displaystyle=$
$\displaystyle(a_{0},a_{1},\dots,a_{\alpha-1},(r_{0,1},r_{0,2})M,(r_{1,1},r_{1,2})M,\dots,(r_{\beta-1,1},r_{\beta-1,2})M)+(a_{0}^{\prime},a_{1}^{\prime},\dots,$
$\displaystyle
a_{\alpha-1}^{\prime},(r_{0,1}^{\prime},r_{0,2}^{\prime})M,(r_{1,1}^{\prime},r_{1,2}^{\prime})M,\dots,(r_{\beta-1,1}^{\prime},r_{\beta-1,2}^{\prime})M)$
$\displaystyle=$ $\displaystyle\varphi(a,r)+\varphi(a^{\prime},r^{\prime})$
$\displaystyle=$ $\displaystyle\varphi(c)+\varphi(c^{\prime}).$

Now, for any $\lambda\in\mathbb{F}_{q}$, we have

$\displaystyle\varphi(\lambda c)$ $\displaystyle=\varphi(\lambda a,\lambda r)$
$\displaystyle=(\lambda(a_{0},a_{1},\dots,a_{\alpha-1}),\lambda(r_{0,1},r_{0,2})M,\lambda(r_{1,1},r_{1,2})M,\dots,\lambda(r_{\beta-1,1},r_{\beta-1,2})M)$
$\displaystyle=\lambda(a_{0},a_{1},\dots,a_{\alpha-1},(r_{0,1},r_{0,2})M,(r_{1,1},r_{1,2})M,\dots,(r_{\beta-1,1},r_{\beta-1,2})M)$
$\displaystyle=\lambda\varphi(a,r)$ $\displaystyle=\lambda\varphi(c).$

Thus, $\varphi$ is an $\mathbb{F}_{q}$-linear map. Moreover,
$d_{L}(c,c^{\prime})=w_{L}(c-c^{\prime})=w_{H}(\varphi(c-c^{\prime}))=w_{H}(\varphi(c)-\varphi(c^{\prime}))=d_{H}(\varphi(c),\varphi(c^{\prime}))$.
Therefore, $\varphi$ is a distance preserving map.∎

###### Theorem 4.1

If $\mathcal{C}$ is an $[n,k,d_{L}]$ $\mathbb{F}_{q}\mathcal{R}$-linear code,
then $\varphi(\mathcal{C})$ is a $[\alpha+2\beta,k,d_{H}]$ linear code over
$\mathbb{F}_{q}$.

###### Proof

It follows directly from Proposition 1 and the definition of the Gray map.

Now, the following result shows the Gray map $\varphi$ preserves the
orthogonality.∎

###### Theorem 4.2

Let $\mathcal{C}$ be an $\mathbb{F}_{q}\mathcal{R}$-skew cyclic code of length
$(\alpha,\beta)$. Then
$\varphi(\mathcal{C})^{\perp}=\varphi(\mathcal{C}^{\perp})$. Further,
$\mathcal{C}$ is self-dual if and only if $\varphi(\mathcal{C})$ is self-dual.

###### Proof

Let $\mathcal{C}$ be an $\mathbb{F}_{q}\mathcal{R}$-skew cyclic code of length
$(\alpha,\beta)$. Assume that $l=(x_{0},x_{1},\dots,x_{\alpha-1},\\\
y_{0},y_{1},\dots,y_{\beta-1})\in\mathcal{C}^{\perp}$ where
$y_{j}=\xi_{1}a_{j}+\xi_{2}b_{j}$ for $0\leq j\leq\beta-1$. Then
$\varphi(l)\in\varphi(\mathcal{C^{\perp}})$. To show that
$\varphi(l)\in\varphi(\mathcal{C})^{\perp}$, we consider
$l^{\prime}=(x_{0}^{\prime},x_{1}^{\prime},\dots,x_{\alpha-1}^{\prime},y_{0}^{\prime},y_{1}^{\prime},\dots,y_{\beta-1}^{\prime})\in\mathcal{C}$
where $y_{j}^{\prime}=\xi_{1}a_{j}^{\prime}+\xi_{2}b_{j}^{\prime}$ for $0\leq
j\leq\beta-1$. Now, $l\cdot l^{\prime}=0$ implies that

$\displaystyle l\cdot l^{\prime}$
$\displaystyle=u\sum_{i=0}^{\alpha-1}x_{i}x_{i}^{\prime}+\sum_{j=0}^{\beta-1}y_{j}y_{j}^{\prime}$
$\displaystyle=u\sum_{i=0}^{\alpha-1}x_{i}x_{i}^{\prime}+\sum_{j=0}^{\beta-1}(\xi_{1}a_{j}a_{j}^{\prime}+\xi_{2}b_{j}b_{j}^{\prime})=0.$

Further, it gives $\sum_{i=0}^{\alpha-1}x_{i}x_{i}^{\prime}=0$ and
$\sum_{j=0}^{\beta-1}(a_{j}a_{j}^{\prime}+b_{j}b_{j}^{\prime})=0$. Next, we
have

$\displaystyle\varphi(l)$
$\displaystyle=(x_{0},x_{1},\dots,x_{\alpha-1},(a_{0},b_{0})M,(a_{1},b_{1})M,\cdots,(a_{\beta-1},b_{\beta-1})M)$
$\displaystyle=(x_{0},x_{1},\dots,x_{\alpha-1},c_{0}M,c_{1}M,\cdots,c_{\beta-1}M)$
$\displaystyle\varphi(l^{\prime})$
$\displaystyle=(x_{0}^{\prime},x_{1}^{\prime},\dots,x_{\alpha-1}^{\prime},(a_{0}^{\prime},b_{0}^{\prime})M,(a_{1}^{\prime},b_{1}^{\prime})M,\cdots,(a_{\beta-1}^{\prime},b_{\beta-1})^{\prime}M)$
$\displaystyle=(x_{0}^{\prime},x_{1}^{\prime},\dots,x_{\alpha-1}^{\prime},c_{0}^{\prime}M,c_{1}^{\prime}M,\cdots,c_{\beta-1}^{\prime}M)$

where $c_{j}=(a_{j},b_{j})$ and
$c_{j}^{\prime}=(a_{j}^{\prime},b_{j}^{\prime})$ for $0\leq j\leq\beta-1$.
Also,

$\displaystyle\varphi(l)\cdot\varphi(l^{\prime})$
$\displaystyle=\varphi(l)\varphi(l^{\prime})^{T}=u\sum_{i=0}^{\alpha-1}x_{i}x_{i}^{\prime}+\sum_{j=0}^{\beta-1}y_{j}MM^{T}y_{j}^{\prime
T}$
$\displaystyle=u\sum_{i=0}^{\alpha-1}x_{i}x_{i}^{\prime}+\gamma\sum_{j=0}^{\beta-1}y_{j}y_{j}^{\prime
T}$
$\displaystyle=u\sum_{i=0}^{\alpha-1}x_{i}x_{i}^{\prime}+\gamma\sum_{j=0}^{\beta-1}y_{j}y_{j}^{\prime
T}$
$\displaystyle=u\sum_{i=0}^{\alpha-1}x_{i}x_{i}^{\prime}+\gamma\sum_{j=0}^{\beta-1}(a_{j}a_{j}^{\prime}+b_{j}b_{j}^{\prime}).$

Then, $\varphi(l)\in\varphi(\mathcal{C})^{\perp}$ and hence
$\varphi(\mathcal{C}^{\perp})\subseteq\varphi(\mathcal{C})^{\perp}$. As
$\varphi$ is a bijection,
$|\varphi(\mathcal{C}^{\perp})|=|\varphi(\mathcal{C})^{\perp}|$. Thus,
$\varphi(\mathcal{C}^{\perp})=\varphi(\mathcal{C})^{\perp}$.
Let $\mathcal{C}$ be self-dual, i.e., $\mathcal{C}=\mathcal{C}^{\perp}$. Then
$\varphi(\mathcal{C})=\varphi(\mathcal{C}^{\perp})=\varphi(\mathcal{C})^{\perp}$,
i.e., $\varphi(\mathcal{C})$ is self-dual. Conversely, let
$\varphi(\mathcal{C})$ be self-dual. Then
$\varphi(\mathcal{C})=\varphi(\mathcal{C}^{\perp})=\varphi(\mathcal{C})^{\perp}$.
Since, $\varphi$ is a bijection, $\mathcal{C}=\mathcal{C}^{\perp}$. Hence,
$\mathcal{C}$ is self-dual.∎

###### Theorem 4.3

Let $\mathcal{C}$ be an $\mathbb{F}_{q}\mathcal{R}$-skew cyclic code of length
$(\alpha,\beta)$. Then
$\varphi(\mathcal{C})=\mathcal{C}^{\prime}\otimes\mathcal{C}_{1}\otimes\mathcal{C}_{2}$,
$\mathcal{C}^{\prime}$ is a skew cyclic code of length $\alpha$ in
$\mathbb{F}_{q}[x;\Theta]/\langle x^{\alpha}-1\rangle$ and $\mathcal{C}_{1}$,
$\mathcal{C}_{2}$ are skew cyclic codes of length $\beta$ over
$\mathbb{F}_{q}$. Furthermore,
$|\varphi(\mathcal{C})|=|\mathcal{C}^{\prime}||\mathcal{C}_{1}||\mathcal{C}_{2}|$.

###### Proof

Consider an element $c=(a,r)\in\mathcal{C}$ such that
$a=(a_{0},a_{1},\dots,a_{\alpha-1})\in\mathbb{F}_{q}^{\alpha}$ and
$r=(r_{0},r_{1},\dots,r_{\beta-1})\in\mathcal{R}^{\beta}$ where
$r_{i}=b_{i}+uc_{i}$ for $i=0,1,\dots,\beta-1$. Now, we define

$\mathcal{C}^{\prime}:=(a_{0},a_{1},\dots,a_{\alpha-1}),$
$\mathcal{C}_{1}:=(b_{0},b_{1},\dots,b_{\beta-1}),$

and

$\mathcal{C}_{2}:=(c_{0},c_{1},\dots,c_{\beta-1}).$

Next, a given codeword
$c^{\prime}=(a_{0},a_{1},\dots,a_{\alpha-1})\in\mathcal{C}^{\prime}$
corresponds to a codeword
$c=(a_{0},a_{1},\dots,a_{\alpha-1},b_{0}+uc_{0},b_{1}+uc_{1},\dots,b_{\beta-1}+uc_{\beta-1})\in\mathcal{C}$.
As $\mathcal{C}$ is an $\mathbb{F}_{q}\mathcal{R}$-skew cyclic code, the
$\sigma$-skew cyclic shift of $c$ is given by
$\sigma(c)=(\Theta(a_{\alpha-1}),\Theta(a_{0}),\dots,\Theta(a_{\alpha-2}),\theta(b_{\beta-1}+uc_{\beta-1}),\theta(b_{0}+uc_{0}),\theta(b_{1}+uc_{1}),\dots,\theta(b_{\beta-2}+uc_{\beta-2}))\in\mathcal{C}$.
Therefore, $(\Theta(a_{\alpha-1}),\Theta(a_{0}),\dots,\\\
\Theta(a_{\alpha-2}))\in\mathcal{C}^{\prime}$. Hence, $\mathcal{C}^{\prime}$
is a skew cyclic code of length $\alpha$ in $\mathbb{F}_{q}[x;\Theta]/\langle
x^{\alpha}-1\rangle.$ Similarly, we get $\mathcal{C}_{1}$, $\mathcal{C}_{2}$
are skew cyclic codes of length $\beta$ over $\mathbb{F}_{q}$. Thus,
$\varphi(\mathcal{C})=\mathcal{C}^{\prime}\otimes\mathcal{C}_{1}\otimes\mathcal{C}_{2}$.
As $\varphi$ is bijective, $|\mathcal{C}|=|\varphi(\mathcal{C})|$ and hence
$|\mathcal{C}|=|\mathcal{C}^{\prime}||\mathcal{C}_{1}||\mathcal{C}_{2}|$.∎

Now, we present an example of an $\mathbb{F}_{q}\mathcal{R}$-skew cyclic code.
To compute Gray image $\varphi(\mathcal{C})$ of an
$\mathbb{F}_{q}\mathcal{R}$-skew cyclic code $\mathcal{C}$, first we find the
corresponding generator polynomials $f(x)$ of skew cyclic code
$\mathcal{C}^{\prime}$ of length $\alpha$ over $\mathbb{F}_{q}$ and $g(x)$ of
skew cyclic code $\mathscr{C}$ of length $\beta$ over $\mathcal{R}$ with
$g(x)=g_{1}(x)\xi_{1}+g_{2}(x)\xi_{2}$, where $g_{1}(x)$ and $g_{2}(x)$ are
the generator polynomials of skew cyclic codes $\mathcal{C}_{1}$ and
$\mathcal{C}_{2}$ over $\mathbb{F}_{q}$ respectively. Then, with the help of
Magma computational algebra system Bosma , we find $\varphi(\mathcal{C})$.

###### Example 1

Let $q=9$, $\alpha=18$, $\beta=36$ and $\mathcal{R}=\mathbb{F}_{9}[u]/\langle
u^{2}-u\rangle$ where $\mathbb{F}_{9}=\mathbb{F}_{3}(w)$ and $w^{2}+2w+2=0$.
In $\mathbb{F}_{9}$, the Frobenius automorphism
$\Theta:\mathbb{F}_{9}\longrightarrow\mathbb{F}_{9}$ is defined by
$\Theta(a)=a^{3}$ for all $a\in\mathbb{F}_{9}$. Therefore,
$\mathbb{F}_{9}[x;\Theta]$ is a skew polynomial ring. In
$\mathbb{F}_{9}[x;\Theta]$, we have

$\displaystyle x^{18}-1$
$\displaystyle=x^{15}+w^{5}x^{14}+x^{13}+2x^{12}+x^{9}+w^{5}x^{8}+x^{7}+2x^{6}+x^{3}+w^{5}x^{2}+x+2)(x^{3}$
$\displaystyle+w^{3}x^{2}+x+1).$ $\displaystyle x^{36}-1$
$\displaystyle=(x^{35}+w^{2}x^{34}+x^{33}+w^{2}x^{32}+x^{31}+w^{2}x^{30}+x^{29}+w^{2}x^{28}+x^{27}+w^{2}x^{26}+x^{25}$
$\displaystyle+w^{2}x^{24}+x^{23}+w^{2}x^{22}+x^{21}+w^{2}x^{20}+x^{19}+w^{2}x^{18}+x^{17}+w^{2}x^{16}+x^{15}+w^{2}x^{14}$
$\displaystyle+x^{13}+w^{2}x^{12}+x^{11}+w^{2}x^{10}+x^{9}+w^{2}x^{8}+x^{7}+w^{2}x^{6}+x^{5}+w^{2}x^{4}+x^{3}+w^{2}x^{2}$
$\displaystyle+x+w^{2})(x+w^{2}).$
$\displaystyle=w^{6}x^{34}+2x^{33}+w^{5}x^{32}+2x^{31}+x^{30}+w^{2}x^{28}+x^{27}+wx^{26}+x^{25}+2x^{24}+w^{6}x^{22}$
$\displaystyle+2x^{21}+w^{5}x^{20}+2x^{19}+x^{18}+w^{2}x^{16}+x^{15}+wx^{14}+x^{13}+2x^{12}+w^{6}x^{10}+2x^{9}$
$\displaystyle+w^{5}x^{8}+2x^{7}+x^{6}+w^{2}x^{4}+x^{3}+wx^{2}+x+2)(w^{2}x^{2}+x+1).$

Now, let $f(x)=x^{3}+w^{3}x^{2}+x+1,$ $g_{1}(x)=x+w^{2}$ and
$g_{2}(x)=w^{2}x^{2}+x+1$. Then $\mathcal{C}$ is an
$\mathbb{F}_{q}\mathcal{R}$-skew cyclic code of length $(18,36)$. Let

$M=\begin{pmatrix}1&1\\\ 1&-1\end{pmatrix}\in GL_{2}(\mathbb{F}_{9})$

such that $MM^{T}=2I_{2}$. Then $\varphi(\mathcal{C})$ is a $[90,84,4]_{9}$
linear code over $\mathbb{F}_{9}$ which is BKLC (best-known linear code) as
per the database Grassl .

##  5 Quantum Codes from $\mathbb{F}_{q}\mathcal{R}$-Skew Cyclic Codes

Recall that the set of $n$-fold tensor product $(\mathbb{C}^{q})^{\otimes
n}=\underbrace{\mathbb{C}^{q}\otimes\mathbb{C}^{q}\otimes\cdots\otimes\mathbb{C}^{q}}_{n\rm\
times}$ is a $q^{n}$-dimensional Hilbert space, where $\mathbb{C}^{q}$ is a
$q$-dimensional complex Hilbert space. A $q$-ary quantum code of length $n$,
dimension $k$ and minimum distance $d$ is denoted by $[[n,k,d]]_{q}$, is a
$q^{k}$-dimensional subspace of $(\mathbb{C}^{q})^{\otimes n}$. A quantum code
$[[n,k,d]]_{q}$ is known as quantum MDS (maximum-distance-separable) if it
attains the singleton bound $k+2d\leq n+2$. Note that a quantum code
$[[n,k,d]]_{q}$ is said to be better than
$[[n^{\prime},k^{\prime},d^{\prime}]]_{q}$ if any one of the following or both
hold:

  1. 1.

$d>d^{\prime}$ when the code rate $\frac{k}{n}=\frac{k^{\prime}}{n^{\prime}}$
(Larger distance with same code rate).

  2. 2.

$\frac{k}{n}>\frac{k^{\prime}}{n^{\prime}}$ when the distance $d=d^{\prime}$
(Larger code rate with same distance).

The relationship between quantum information and classical information is a
matter currently receiving much attention from researchers. One of the
fascinating developments in the study of linear codes is the construction of
quantum codes from classical codes. The first quantum code was independently
studied by Shor Shor  and Steane Steane . However, the construction of quantum
codes from classical codes, their existence proofs and correction methods were
discussed by Calderbank et al. Calderbank . One of the widely used techniques
to obtain quantum codes from classical linear codes is the CSS construction
(Lemma 3) in which dual containing linear codes play an instrumental role.

###### Lemma 3

(Grassl04, , Theorem 3) Let $\mathcal{C}_{1}=[n,k_{1},d_{1}]_{q}$ and
$\mathcal{C}_{2}=[n,k_{2},d_{2}]_{q}$ be two linear codes over $GF(q)$ with
$\mathcal{C}_{2}^{\perp}\subseteq\mathcal{C}_{1}$. Then there exists a quantum
error-correcting code $\mathcal{C}=[[n,k_{1}+k_{2}-n,d]]$ where
$d=\min\\{w(v):v\in(\mathcal{C}_{1}\backslash\mathcal{C}_{2}^{\perp})\cup(\mathcal{C}_{2}\backslash\mathcal{C}_{1}^{\perp})\\}\geq\min\\{d_{1},d_{2}\\}$.
Further, if $\mathcal{C}_{1}^{\perp}\subseteq\mathcal{C}_{1},$ then there
exists a quantum error-correcting code $\mathcal{C}=[[n,2k_{1}-n,d_{1}]]$,
where
$d_{1}=\min\\{w(v):v\in\mathcal{C}_{1}\backslash\mathcal{C}_{1}^{\perp}\\}.$

Now, we recall from Boucher2  that a skew cyclic code of length $n$ over
$\mathbb{F}_{q}$ is a principally generated left
$\mathbb{F}_{q}[x;\Theta]$-submodule of
$\frac{\mathbb{F}_{q}[x;\Theta]}{\langle x^{n}-1\rangle}$ and
$\mathcal{C}=\langle g(x)\rangle$ where $x^{n}-1=h(x)g(x)$, i.e., $g(x)$ is a
right divisor of $x^{n}-1$. Its dual $\mathcal{C}^{\perp}$ is also a skew
cyclic code of length $n$ and $\mathcal{C}^{\perp}=\langle
h^{\dagger}(x)\rangle$ where
$h^{\dagger}(x)=h_{n-r}+\Theta(h_{n-r-1})x+\dots+\Theta^{n-r}(h_{0})x^{n-r}$
for $h(x)=h_{0}+h_{1}x+\dots+h_{n-r}x^{n-r}$. If $\Theta=id$, then
$h^{\dagger}(x)=h^{*}(x)$ where
$h^{*}(x)=h_{n-r}+h_{n-r-1}x+\dots+h_{0}x^{n-r}$. Further, we recall the dual
containing property for skew cyclic codes.

###### Lemma 4

Islam18b  Let $\gcd(\alpha,|\langle\Theta\rangle|)=1$ and $\mathcal{C}=\langle
f(x)\rangle$ be a skew cyclic code of length $\alpha$ over $\mathbb{F}_{q}$.
Then $\mathcal{C}^{\perp}\subseteq\mathcal{C}$ if and only if
$x^{\alpha}-1\equiv 0\pmod{f(x)f^{*}(x)},$ where $f^{*}(x)$ is the reciprocal
polynomial of $f(x)$.

###### Lemma 5

bag ; Li20  Let $|\langle\Theta\rangle|$ divide $\alpha$ and
$\mathcal{C}=\langle g(x)\rangle$ be a skew cyclic code of length $\alpha$
over $\mathbb{F}_{q}$ where $x^{\alpha}-1=h(x)g(x)$. Then
$\mathcal{C}^{\perp}\subseteq\mathcal{C}$ if and only if $h^{\dagger}(x)h(x)$
is divisible by $x^{\alpha}-1$ from the right side.

Now, we derive the necessary and sufficient conditions for
$\mathbb{F}_{q}\mathcal{R}$-skew cyclic codes to contain their duals.

###### Theorem 5.1

Let $|\langle\Theta\rangle|$ divide $\alpha$, $|\langle\theta\rangle|$ divide
$\beta$ and $\mathcal{C}=\mathcal{C}^{\prime}\otimes\mathscr{C}$ be an
$\mathbb{F}_{q}\mathcal{R}$-skew cyclic code of length $(\alpha,\beta)$ where
$\mathcal{C}^{\prime}=\langle f(x)\rangle$ and
$\mathscr{C}=\xi_{1}\mathcal{C}_{1}\oplus\xi_{2}\mathcal{C}_{2}=\langle
g(x)\rangle$ are skew cyclic codes over $\mathbb{F}_{q}$ and $\mathcal{R}$
respectively, with $g(x)=\xi_{1}g_{1}(x)+\xi_{2}g_{2}(x)$ and
$x^{\alpha}-1=h(x)f(x)$, $x^{\beta}-1=h_{i}(x)g_{i}(x)$. Then
$\mathcal{C}^{\perp}\subseteq\mathcal{C}$ if and only if $h^{\dagger}(x)h(x)$
and $h^{\dagger}_{i}(x)h_{i}(x)$ are divisible by $x^{\alpha}-1$ and
$x^{\beta}-1$, respectively for $i=1,2$ from the right side.

###### Proof

Let $\mathcal{C}^{\perp}\subseteq\mathcal{C}$. Then by Theorem 3.5,
$\mathcal{C}^{\prime\perp}\subseteq\mathcal{C}^{\prime}$ and
$\mathscr{C}^{\perp}\subseteq\mathscr{C}$. Also,
$\mathcal{C}_{1}^{\perp}\subseteq\mathcal{C}_{1}$ and
$\mathcal{C}_{2}^{\perp}\subseteq\mathcal{C}_{2}$ where $\mathcal{C}_{1}$ and
$\mathcal{C}_{2}$ are skew cyclic codes of length $\beta$ over
$\mathbb{F}_{q}$. Then from Lemma 5, we have $h^{\dagger}(x)h(x)$ and
$h^{\dagger}_{i}(x)h_{i}(x)$ are divisible by $x^{\alpha}-1$ and
$x^{\beta}-1$, respectively from the right side for $i=1,2$.
Conversely, suppose $h^{\dagger}(x)h(x)$ and $h^{\dagger}_{i}(x)h_{i}(x)$ are
divisible by $x^{\alpha}-1$ and $x^{\beta}-1$, respectively from the right
side for $i=1,2$. Then by Lemma 5, we have
$\mathcal{C}^{\prime\perp}\subseteq\mathcal{C}^{\prime}$,
$\mathcal{C}_{1}^{\perp}\subseteq\mathcal{C}_{1}$ and
$\mathcal{C}_{2}^{\perp}\subseteq\mathcal{C}_{2}$. This implies
$\mathscr{C}^{\perp}\subseteq\mathscr{C}$. Therefore, by Theorem 3.5,
$\mathcal{C}^{\perp}\subseteq\mathcal{C}$. ∎

Finally, we present our main result for the construction of quantum codes from
$\mathbb{F}_{q}\mathcal{R}$-skew cyclic codes with the help of Theorem 5.1.

###### Theorem 5.2

Let $|\langle\Theta\rangle|$ divide $\alpha$, $|\langle\theta\rangle|$ divide
$\beta$ and $\mathcal{C}=\mathcal{C}^{\prime}\otimes\mathscr{C}$ be an
$\mathbb{F}_{q}\mathcal{R}$-skew cyclic code of length $(\alpha,\beta)$ with
$x^{\alpha}-1=h(x)f(x)$, $x^{\beta}-1=h_{i}(x)g_{i}(x)$. If
$h^{\dagger}(x)h(x)$ and $h^{\dagger}_{i}(x)h_{i}(x)$ are divisible by
$x^{\alpha}-1$ and $x^{\beta}-1$, respectively from the right side for
$i=1,2$, then there exists a quantum code
$[[\alpha+2\beta,2k-(\alpha+2\beta),d_{H}]]_{q}$ where $d_{H}$ denotes the
Hamming distance and $k$ denotes the dimension of the code
$\varphi(\mathcal{C})$.

###### Proof

Let $h^{\dagger}(x)h(x)$ and $h^{\dagger}_{i}(x)h_{i}(x)$ be divisible by
$x^{\alpha}-1$ and $x^{\beta}-1$, respectively from the right side for
$i=1,2$. Then by Theorem 5.1, we have
$\mathcal{C}^{\perp}\subseteq\mathcal{C}$. As
$\varphi(\mathcal{C})=\mathcal{C}^{\prime}\otimes\mathcal{C}_{1}\otimes\mathcal{C}_{2}$,
it is easy to check that
$\varphi(\mathcal{C})^{\perp}=\mathcal{C}^{\prime\perp}\otimes\mathcal{C}_{1}^{\perp}\otimes\mathcal{C}_{2}^{\perp}$.
Also, $\mathcal{C}^{\prime\perp}\subseteq\mathcal{C}^{\prime}$,
$\mathcal{C}_{1}^{\perp}\subseteq\mathcal{C}_{1}$ and
$\mathcal{C}_{2}^{\perp}\subseteq\mathcal{C}_{2}$. Therefore,
$\varphi(\mathcal{C})$ is a dual containing linear code with parameters
$[\alpha+2\beta,k,d_{H}]$. Hence, by Lemma 3, there exists a quantum code with
parameters $[[\alpha+2\beta,2k-(\alpha+2\beta),d_{H}]]_{q}$. ∎

Now, with the help of Corollary 1, Theorem 5.2 can also be stated in the
following form:

###### Corollary 2

Let $|\langle\theta\rangle|$ divide $\beta$,
$\gcd(\alpha,|\langle\Theta\rangle|)=1$ and
$\mathcal{C}=\mathcal{C}^{\prime}\otimes\mathscr{C}$ be an
$\mathbb{F}_{q}\mathcal{R}$-skew cyclic code of length $(\alpha,\beta)$ where
$\mathcal{C}^{\prime}$ is a cyclic code over $\mathbb{F}_{q}$ and
$\mathscr{C}$ is a skew cyclic code over $\mathcal{R}$ with
$x^{\alpha}-1=h(x)f(x)$, $x^{\beta}-1=h_{i}(x)g_{i}(x)$. If $x^{\alpha}-1$ is
divisible by $f(x)f^{*}(x)$ and and $h^{\dagger}_{i}(x)h_{i}(x)$ is divisible
by $x^{\beta}-1$ from the right side for $i=1,2$, then there exists a quantum
code $[[\alpha+2\beta,2k-(\alpha+2\beta),d_{H}]]_{q}$.

###### Remark 1

Notice that the length of the quantum code obtained from the codes over the
ring $\mathcal{R}$ must be an integral multiple of $2$, whereas the code
length in Theorem 5.2 has no such restriction, i.e., we can find code of any
length $\alpha+2\beta$ with some suitable choices of $\alpha$ and $\beta$. For
example, to construct a code of length $60$, there are finitely many choices
for $\alpha$ and $\beta$ such that $\alpha+2\beta=60$. However, in the case of
$\mathcal{R}$, the only choice is $\beta=30$. This is one of the advantages of
studying the $\mathbb{F}_{q}\mathcal{R}$-skew cyclic codes in quantum code
constructions.

Now, with the help of our derived results, we construct several new and better
quantum codes than the existing ones appearing in bag ; Edel ; Verma21 . All
our computations in the examples are carried out using the Magma computation
system Bosma .

###### Example 2

Let $q=9$, $\alpha=49$, $\beta=36$ and $\mathcal{R}=\mathbb{F}_{9}[u]/\langle
u^{2}-u\rangle$ where $\mathbb{F}_{9}=\mathbb{F}_{3}(w)$ and $w^{2}+2w+2=0$.
In $\mathbb{F}_{9}$, the Frobenius automorphism
$\Theta:\mathbb{F}_{9}\longrightarrow\mathbb{F}_{9}$ is defined by
$\Theta(a)=a^{3}$ for all $a\in\mathbb{F}_{9}$. Therefore,
$\mathbb{F}_{9}[x;\Theta]$ is a skew polynomial ring. In
$\mathbb{F}_{9}[x;\Theta]$, we have

$\displaystyle x^{36}-1$
$\displaystyle=w^{2}x^{34}+2x^{33}+w^{7}x^{32}+2x^{31}+x^{30}+w^{6}x^{28}+x^{27}+w^{3}x^{26}+x^{25}+2x^{24}+w^{2}x^{22}$
$\displaystyle+2x^{21}+w^{7}x^{20}+2x^{19}+x^{18}+w^{6}x^{16}+x^{15}+w^{3}x^{14}+x^{13}+2x^{12}+w^{2}x^{10}+2x^{9}$
$\displaystyle+w^{7}x^{8}+2x^{7}+x^{6}+w^{6}x^{4}+x^{3}+w^{3}x^{2}+x+2)(w^{6}x^{2}+x+1),$
$\displaystyle=(w^{6}x^{34}+2x^{33}+w^{5}x^{32}+2x^{31}+x^{30}+w^{2}x^{28}+x^{27}+wx^{26}+x^{25}+2x^{24}+w^{6}x^{22}$
$\displaystyle+2x^{21}+w^{5}x^{20}+2x^{19}+x^{18}+w^{2}x^{16}+x^{15}+wx^{14}+x^{13}+2x^{12}+w^{6}x^{10}+2x^{9}$
$\displaystyle+w^{5}x^{8}+2x^{7}+x^{6}+w^{2}x^{4}+x^{3}+wx^{2}+x+2)(w^{2}x^{2}+x+1).$

In $\mathbb{F}_{9}[x]$, we have

$\displaystyle x^{49}-1$
$\displaystyle=(x+2)(x^{3}+wx^{2}+w^{7}x+2)(x^{3}+w^{3}x^{2}+w^{5}x+2)(x^{21}+wx^{14}+w^{7}x^{7}+2)$
$\displaystyle(x^{21}+w^{3}x^{14}+w^{5}x^{7}+2).$

Now, let $f(x)=x^{3}+w^{3}x^{2}+w^{5}x+2,$ $g_{1}(x)=w^{6}x^{2}+x+1$ and
$g_{2}(x)=w^{2}x^{2}+x+1$. Then $\mathcal{C}$ is an
$\mathbb{F}_{q}\mathcal{R}$-skew cyclic code of length $(49,36)$. Let

$M=\begin{pmatrix}1&1\\\ 1&-1\end{pmatrix}\in GL_{2}(\mathbb{F}_{9})$

such that $MM^{T}=2I_{2}$. Then $\varphi(\mathcal{C})$ is a $[121,114,4]_{9}$
linear code over $\mathbb{F}_{9}$. Now, we have

$\displaystyle h_{1}(x)$
$\displaystyle=w^{2}x^{34}+2x^{33}+w^{7}x^{32}+2x^{31}+x^{30}+w^{6}x^{28}+x^{27}+w^{3}x^{26}+x^{25}+2x^{24}+w^{2}x^{22}$
$\displaystyle+2x^{21}+w^{7}x^{20}+2x^{19}+x^{18}+w^{6}x^{16}+x^{15}+w^{3}x^{14}+x^{13}+2x^{12}+w^{2}x^{10}+2x^{9}$
$\displaystyle+w^{7}x^{8}+2x^{7}+x^{6}+w^{6}x^{4}+x^{3}+w^{3}x^{2}+x+2,$
$\displaystyle h_{1}^{\dagger}(x)$
$\displaystyle=2x^{34}+x^{33}+w^{3}x^{32}+x^{31}+w^{6}x^{30}+x^{28}+2x^{27}+w^{7}x^{26}+2x^{25}+w^{2}x^{24}+2x^{22}$
$\displaystyle+x^{21}+w^{3}x^{20}+x^{19}+w^{6}x^{18}+x^{16}+2x^{15}+w^{7}x^{14}+2x^{13}+w^{2}x^{12}+2x^{10}+x^{9}$
$\displaystyle+w^{3}x^{8}+x^{7}+w^{6}x^{6}+x^{4}+2x^{3}+w^{7}x^{2}+2x+w^{2},$
$\displaystyle h_{1}^{\dagger}(x)h_{1}(x)$
$\displaystyle=(w^{6}x^{32}+w^{5}x^{31}+wx^{30}+w^{6}x^{29}+w^{7}x^{28}+w^{2}x^{27}+2x^{26}+2x^{25}+2x^{24}$
$\displaystyle+w^{6}x^{23}+w^{7}x^{22}+w^{2}x^{21}+wx^{20}+w^{7}x^{19}+w^{6}x^{18}+w^{2}x^{14}+wx^{13}+w^{5}x^{12}$
$\displaystyle+w^{2}x^{11}+w^{3}x^{10}+w^{6}x^{9}+x^{8}+x^{7}+x^{6}+w^{2}x^{5}+w^{3}x^{4}+w^{6}x^{3}+w^{5}x^{2}+w^{3}x$
$\displaystyle+w^{2})(x^{36}-1),$

and

$\displaystyle h_{2}(x)$
$\displaystyle=w^{6}x^{34}+2x^{33}+w^{5}x^{32}+2x^{31}+x^{30}+w^{2}x^{28}+x^{27}+wx^{26}+x^{25}+2x^{24}+w^{6}x^{22}$
$\displaystyle+2x^{21}+w^{5}x^{20}+2x^{19}+x^{18}+w^{2}x^{16}+x^{15}+wx^{14}+x^{13}+2x^{12}+w^{6}x^{10}+2x^{9}$
$\displaystyle+w^{5}x^{8}+2x^{7}+x^{6}+w^{2}x^{4}+x^{3}+wx^{2}+x+2,$
$\displaystyle h_{2}^{\dagger}(x)$
$\displaystyle=2x^{34}+x^{33}+wx^{32}+x^{31}+w^{2}x^{30}+x^{28}+2x^{27}+w^{5}x^{26}+2x^{25}+w^{6}x^{24}+2x^{22}$
$\displaystyle+x^{21}+wx^{20}+x^{19}+w^{2}x^{18}+x^{16}+2x^{15}+w^{5}x^{14}+2x^{13}+w^{6}x^{12}+2x^{10}+x^{9}$
$\displaystyle+wx^{8}+x^{7}+w^{2}x^{6}+x^{4}+2x^{3}+w^{5}x^{2}+2x+w^{6},$
$\displaystyle h_{2}^{\dagger}(x)h_{2}(x)$
$\displaystyle=(w^{2}x^{32}+w^{7}x^{31}+w^{3}x^{30}+w^{2}x^{29}+w^{5}x^{28}+w^{6}x^{27}+2x^{26}+2x^{25}+2x^{24}$
$\displaystyle+w^{2}x^{23}+w^{5}x^{22}+w^{6}x^{21}+w^{3}x^{20}+w^{5}x^{19}+w^{2}x^{18}+w^{6}x^{14}+w^{3}x^{13}+w^{7}x^{12}$
$\displaystyle+w^{6}x^{11}+wx^{10}+w^{2}x^{9}+x^{8}+x^{7}+x^{6}+w^{6}x^{5}+wx^{4}+w^{2}x^{3}+w^{7}x^{2}+wx$
$\displaystyle+w^{6})(x^{36}-1).$

Therefore, $h_{1}^{\dagger}(x)h_{1}(x)$ and $h_{2}^{\dagger}(x)h_{2}(x)$ are
divisible by $x^{36}-1$ from the right sides. Also, $x^{49}-1$ is divisible by
$f(x)f^{*}(x)$. Hence, by Theorem 2, we have a quantum code with parameters
$[[121,107,4]]_{9}$ which has same length and minimum distance but larger code
rate than existing code $[[121,106,4]]_{9}$ given by Edel .

Table 1: New quantum codes from $\mathbb{F}_{q}\mathcal{R}$-skew cyclic codes $q$ | $(\alpha,\beta)$ | $[f(x),g_{1}(x),g_{2}(x)]$ | $\varphi(\mathcal{C})$ | Obtained | Existing
---|---|---|---|---|---
|  |  |  | Codes | Codes
$9$ | $(49,40)$ | $(1ww^{7}2,w^{6}1,1w^{2})$ | $[129,124,3]_{9}$ | $[[129,119,3]]_{9}$ |  $[[129,118,3]]_{9}$ Edel
$9$ | $(49,36)$ | $(1w^{3}w^{5}2,w^{6}11,w^{2}11)$ | $[121,114,4]_{9}$ | $[[121,107,4]]_{9}$ |  $[[121,106,4]]_{9}$ Edel
$9$ | $(75,30)$ | $(1w^{3}1,w^{7}w^{3}0,w^{6}1)$ | $[285,280,3]_{9}$ | $[[285,275,3]]_{9}$ |  $[[286,275,3]]_{9}$ Edel
$9$ | $(729,20)$ | $(12021,w1,w^{5}w^{3}1)$ | $[769,762,3]_{9}$ | $[[769,755,3]]_{9}$ |  $[[769,745,3]]_{9}$ Edel
$25$ | $(21,30)$ | $(1ww^{17}4,w^{15}0w^{3},w^{8}1)$ | $[81,75,3]_{25}$ | $[[81,69,3]]_{25}$ |  $[[80,56,3]]_{25}$ bag
$25$ | $(55,32)$ | $(131,w^{9}1,1w^{21}1)$ | $[119,114,3]_{25}$ | $[[119,109,3]]_{25}$ |  $[[120,106,3]]_{25}$ Verma21
$49$ | $(75,36)$ | $(1ww^{9}6,w^{21}51,w^{39}41)$ | $[147,140,3]_{49}$ | $[[147,133,3]]_{49}$ |  $[[144,126,3]]_{49}$ Verma21

In Table $1$, we present some quantum codes from
$\mathbb{F}_{q}\mathcal{R}$-skew cyclic code for $q=9,25,49$. To compute their
Gray images for $q=9,25,49$, we use

$M=\begin{pmatrix}1&1\\\ 1&-1\end{pmatrix}\in GL_{2}(\mathbb{F}_{q})$

satisfying $MM^{T}=2I_{2}$. Let
$\mathcal{C}=\mathcal{C}^{\prime}\otimes\mathscr{C}$ be an
$\mathbb{F}_{q}\mathcal{R}$-skew cyclic code of length $(\alpha,\beta)$ where
$|\langle\theta\rangle|$ divides $\beta$ and
$\gcd(\alpha,|\langle\Theta\rangle|)=1$. In Table 1, we provide generator
polynomials $f(x)$ for $\mathcal{C}^{\prime}$ and $g_{1}(x),g_{2}(x)$ for
$\mathscr{C}=\xi_{1}\mathcal{C}_{1}\oplus\xi_{2}\mathcal{C}_{2}$ such that
$x^{\alpha}-1=h(x)f(x)$, $x^{\beta}-1=h_{i}(x)g_{i}(x)$ for $i=1,2$. Also, we
construct these codes under the conditions that $x^{\alpha}-1$ is divisible by
$f(x)f^{*}(x)$ and $h^{\dagger}_{i}(x)h_{i}(x)$ is divisible by $x^{\beta}-1$
from the right side for $i=1,2$. Therefore, by Corollary 2, we construct
quantum codes (in the $5^{\text{th}}$ column), which have better parameters
than the existing codes (in the $6^{\text{th}}$ column). To represent
polynomials $f(x),g_{i}(x)$ for $i=1,2$, we write the vectors consisting of
their coefficients in decreasing order of power of $x$. For instance, we write
the vector $1ww^{7}2$ to represent the polynomial $x^{3}+wx+w^{7}x+2$.

##  6 Conclusion

In this paper, we have first discussed the structure of the ring
$\mathbb{F}_{q}\mathcal{R}$ and studied the structural properties of
$\mathbb{F}_{q}\mathcal{R}$-skew cyclic codes. As an application of our
established results, we have constructed many quantum codes with better
parameters using the CSS construction. To show the novelty of our obtained
codes, we have compared them with the best-known codes available in the
literature. We believe that this work still has scope for further study by
taking the direct product of cyclic, constacyclic, skew cyclic and skew
constacyclic codes over finite rings.

## Acknowledgements

The first and second authors thank the Department of Science and Technology
(DST), Govt. of India, for financial support under CRG/2020/005927, vide Diary
No. SERB/F/67
80/ 2020-2021 dated 31 December, 2020, and Ref No. DST/INSPIRE/03/2016/
001445, respectively. H. Islam thanks the University of St. Gallen,
Switzerland, for financial support under International Postdoctoral Fellowship
(IPF). Also, the authors would like to thank the anonymous referee(s) and the
Editor for their valuable comments to improve the presentation of the paper.

## Data Availability

The authors declare that [the/all other] data supporting the findings of this
study are available within the article. Any clarification may be requested
from corresponding author provided it is essential.
Competing interests: The authors declare that there is no conflict of interest
regarding the publication of this manuscript.

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