# Skew cyclic codes over $\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ with derivation:
structural properties and computational results

Djoko Suprijanto 111Combinatorial Mathematics Research Group, Faculty of
Mathematics and Natural Sciences, Institut Teknologi Bandung, Jl. Ganesha 10,
Bandung, 40132, INDONESIA<EMAIL_ADDRESS>and Hopein Christofen
Tang 222Combinatorial Mathematics Research Group, Faculty of Mathematics and
Natural Sciences, Institut Teknologi Bandung, Jl. Ganesha 10, Bandung, 40132,
INDONESIA<EMAIL_ADDRESS>
###### Abstract

In this work, we study a class of skew cyclic codes over the ring
$R:=\mathbb{Z}_{4}+v\mathbb{Z}_{4},$ where $v^{2}=v,$ with an automorphism
$\theta$ and a derivation $\Delta_{\theta},$ namely codes as modules over a
skew polynomial ring $R[x;\theta,\Delta_{\theta}],$ whose multiplication is
defined using an automorphism $\theta$ and a derivation $\Delta_{\theta}.$ We
investigate the structures of a skew polynomial ring
$R[x;\theta,\Delta_{\theta}].$ We define $\Delta_{\theta}$-cyclic codes as a
generalization of the notion of cyclic codes. The properties of
$\Delta_{\theta}$-cyclic codes as well as dual $\Delta_{\theta}$-cyclic codes
are derived. As an application, some new linear codes over $\mathbb{Z}_{4}$
with good parameters are obtained by Plotkin sum construction, also via a Gray
map as well as residue and torsion codes of these codes.

Keywords: Cyclic codes, quasi-cyclic codes, skew polynomial ring, skew cyclic
codes, derivation.

2020 MSC: 94B05, 94B15, 11T71

##  1 Introduction

Cyclic codes are an important class of codes from both theoretical and
practical viewpoints. Theoretically, cyclic codes have a rich mathematical
theory, in particular, they have additional algebraic structures to make,
practically, the process of encoding and decoding cyclic codes is more
efficient.

Cyclic codes over finite fields were first studied by Prange [22] in 1957.
Since then, many coding theorists have made significant progress in studying
cyclic codes for both the so-called random-error correction and burst-error
correction (See, for example, [14] for the detailed description of random-
error and burst-error correction).

In 2007, Boucher, Geiselmann, and Ulmer [6] (see also [7],[8]) extended the
notion of cyclic codes over finite fields by using generator polynomials in
non-commutative skew polynomial rings. The new notion of codes is called skew
cyclic codes over finite fields. In general, a skew polynomial ring is not a
unique factorization ring. In this case there are typically more factors of
$x^{n}-1$ in this ring than in the commutative case. Hence there are more skew
cyclic codes than cyclic codes (over finite fields). This new class of codes
increases the number of possibilities for finding codes with good parameters.
Boucher, Geiselmann, and Ulmer [6] also give many examples of codes that
improve the previously best-known linear codes over finite fields.

Later, the notion of skew cyclic codes over finite fields was generalized to
the skew cyclic codes over several kinds of finite rings. Abualrub, Aydin, and
Seneviratne [1] considered skew cyclic codes over
${\mathbb{F}}_{2}+v{\mathbb{F}}_{2},$ where $v^{2}=v.$ They [1] also
constructed optimal self-dual codes over this ring. Gursoy, Siap, and Yildiz
[12] investigated structural properties of skew cyclic codes over
${\mathbb{F}}_{q}+v{\mathbb{F}}_{q},$ with $v^{2}=v.$ They [12] showed that
skew cyclic codes over this ring are principally generated. Later, the first
author together with his coauthors considered structural aspects of skew
cyclic codes over the ring $A_{k}$ [15] and $B_{k}$ [16] (c.f. [17]),
respectively. Very recently, Benbelkacem, Ezerman, Abualrub, Aydin, and Batoul
[4] considered the skew cyclic codes over the mixed alphabet which are also a
finite ring, denoted by ${\mathbb{F}}_{4}R,$ where
$R={\mathbb{F}}_{4}+v{\mathbb{F}}_{4}$ with $v^{2}=v.$ They [4] showed a
natural connection between the skew cyclic codes over this ring to DNA codes.

In the next development, Boucher and Ulmer [9] generalized the notion of skew
cyclic codes over finite fields to the skew cyclic codes over finite fields
with derivation. They [9] also constructed MDS as well as MRD codes from
certain families of the skew cyclic codes (see Section 4.3 in [9]). Sharma and
Bhaintwal [23] extended the study of these skew cyclic codes over a finite
ring, namely over the ring $\mathbb{Z}_{4}+u\mathbb{Z}_{4},$ with $u^{2}=1.$
Via residue codes, Plotkin sum, or the Gray map they [23] defined, they
obtained several linear codes over $\mathbb{Z}_{4}$ with good parameters. Very
recently, Patel and Prakash [21] have investigated the same object over the
ring
${\mathbb{F}}_{q}+u{\mathbb{F}}_{q}+v{\mathbb{F}}_{q}+uv{\mathbb{F}}_{q},$
with $u^{2}=u$ and $v^{2}=v,$ where $q$ is a prime power. By the decomposition
method, they [21] obtained several optimal linear codes over
${\mathbb{F}}_{q}.$

Continuing the study of [23] and [21], in this paper, we investigate a class
of skew cyclic codes with derivation over the ring
$\mathbb{Z}_{4}+v\mathbb{Z}_{4},$ with $v^{2}=v.$ We derive several structural
properties of skew cyclic codes with derivation over the ring
$\mathbb{Z}_{4}+v\mathbb{Z}_{4}.$ As a by-product, we construct many new
linear codes over $\mathbb{Z}_{4}$ with good parameters.

The organization of the paper is as follows. In Section 2, we provide some
definitions and basic facts related to the ring
$\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ and also the linear codes over the ring
$\mathbb{Z}_{4}+v\mathbb{Z}_{4}.$ We also define a Gray map from
$\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ to $\mathbb{Z}_{4}^{2},$ which can be
extended naturally to define the Gray map from
$(\mathbb{Z}_{4}+v\mathbb{Z}_{4})^{n}$ to $\mathbb{Z}_{4}^{2n}.$ Several
properties of the skew-polynomial ring
$(\mathbb{Z}_{4}+v\mathbb{Z}_{4})[x;\theta,\Delta_{\theta}]$ are derived. The
notion of $\Delta_{\theta}$-cyclic codes, as well as dual of
$\Delta_{\theta}$-cyclic codes as a generalization of cyclic codes together
with their properties are investigated in Section 3 and Section 4,
respectively. Several examples of linear codes over $\mathbb{Z}_{4}$ with good
parameters obtained by using the Plotkin sum construction, a Gray map, or
residue and torsion codes of these classes of codes are provided in Section 5.
The paper is ended with concluding remarks. We follow [14] for undefined terms
in coding theory.

##  2 Preliminaries

In this section, we present some definitions together with some basic facts
regarding the ring $\mathbb{Z}_{4}+v\mathbb{Z}_{4},$ linear codes over the
ring, and the skew-polynomial ring
$(\mathbb{Z}_{4}+v\mathbb{Z}_{4})[x;\theta,\Delta_{\theta}]$ that is required
in the next sections.

###  2.1 The ring $\mathbb{Z}_{4}+v\mathbb{Z}_{4}$

Let $R:=\mathbb{Z}_{4}+v\mathbb{Z}_{4}=\\{a+bv:~{}a,b\in\mathbb{Z}_{4}\\},$
with $v^{2}=v.$ This ring is isomorphic to a polynomial ring, namely
$R\cong\displaystyle\frac{\mathbb{Z}_{4}[v]}{\langle v^{2}-v\rangle}.$ An
element $a+bv\in R$ is a unit if and only if $a$ and $a+b$ both are units in
$\mathbb{Z}_{4}$. Since the units of $\mathbb{Z}_{4}$ are $1$ and $3,$ the
units of $R$ are $1,$ $3,$ $1+2v,$ and $3+2v.$ Hence, the non-units of $R$ are

$\\{0,~{}2,~{}v,~{}2v,~{}3v,~{}1+v,~{}1+3v,~{}2+v,~{}2+2v,~{}2+3v,~{}3+v,~{}3+3v\\}.$

$R$ is a principal ideal ring with $7$ non-trivial ideals, namely

$\displaystyle\langle 2v\rangle$ $\displaystyle=\\{0,2v\\},$
$\displaystyle\langle 2+2v\rangle$ $\displaystyle=\\{0,2+2v\\},$
$\displaystyle\langle 2\rangle$ $\displaystyle=\\{0,2,2v,2+2v\\},$
$\displaystyle\langle v\rangle$ $\displaystyle=\\{0,v,2v,3v\\}=\langle
3v\rangle,$ $\displaystyle\langle 3+v\rangle$
$\displaystyle=\\{0,1+3v,2+2v,3+v\\}=\langle 1+3v\rangle,$
$\displaystyle\langle 1+v\rangle$
$\displaystyle=\\{0,2,2v,1+v,1+3v,2+2v,3+v,3+3v\\}=\langle 3+3v\rangle,$
$\displaystyle\langle 2+v\rangle$
$\displaystyle=\\{0,2,v,2v,3v,2+v,2+2v,2+3v\\}=\langle 2+3v\rangle.$

The maximal ideals of $R$ are $\langle 1+v\rangle$ and $\langle 2+v\rangle$.
Hence, $R$ is a semi-local ring.

For more information on the structures of the ring
$R=\mathbb{Z}_{4}+v\mathbb{Z}_{4},$ the reader can refer to [3], [11], and
[18].

###  2.2 Linear codes over $R$

Lee weight is an important weight to consider on $\mathbb{Z}_{4}$. For
$x\in\mathbb{Z}_{4},$ the Lee weight of $x,$ denoted by $w_{L}(x),$ is defined
as $w_{L}(0)=0,$ $w_{L}(1)=1=w_{L}(3),$ $w_{L}(2)=2.$ The Lee weight for any
vector $(r_{0},r_{1},\ldots,r_{n-1})\in\mathbb{Z}_{4}^{n}$ is defined as the
rational sum of Lee weights of its coordinates, namely
$w_{L}((r_{0},r_{1},\ldots,r_{n-1}))=w_{L}(r_{0})+w_{L}(r_{1})+\cdots+w_{L}(r_{n-1}).$

Define a Gray map $\phi:R\longrightarrow\mathbb{Z}_{4}^{2}$ as

$\phi(a+bv)=(a,a+b).$

The Gray weight $w_{G}(a+bv)$ for any $a+bv\in R$ is defined as
$w_{G}(a+bv)=w_{L}(\phi(a+bv)).$ The Gray weights of the elements of $R$ are
given as follows.

$\begin{array}[]{c|cccccccc}\hline\cr\hline\cr x&0&1&2&3&v&2v&3v&1+v\\\
w_{G}(x)&0&2&4&2&1&2&1&3\\\ \hline\cr
x&1+2v&1+3v&2+v&2+2v&2+3v&3+v&3+2v&3+3v\\\ w_{G}(x)&2&1&3&2&3&1&2&3\\\
\hline\cr\hline\cr\end{array}$

The Gray map $\phi$ is extended naturally to
$\Phi:R^{n}\longrightarrow\mathbb{Z}_{4}^{2n}$ as

$\Phi((a_{0}+b_{0}v,a_{1}+b_{1}v,\ldots,a_{n-1}+b_{n-1}v))=(a_{0},a_{0}+b_{0},a_{1},a_{1}+b_{1},\ldots,a_{n-1},a_{n-1}+b_{n-1}),$

and the Gray weight of any vector $\mathbf{x}\in R^{n}$ is defined as the
rational sum of Gray weights of its coordinates.

A code $C$ of length $n$ over $R$ is a non-empty subset of $R^{n}.$ A code $C$
is called linear over $R$ if it is an $R$-submodule of $R^{n}.$ A linear code
over $R$ is called free if it is free as an $R$-submodule. The Gray distance
of any vectors $\mathbf{x},\mathbf{y}\in R^{n}$ is defined as
$d_{G}(\mathbf{x},\mathbf{y})=w_{G}(\mathbf{x}-\mathbf{y})$ and the Lee
distance of any vectors $\mathbf{x},\mathbf{y}\in\mathbb{Z}_{4}^{n}$ is
defined as $d_{L}(\mathbf{x},\mathbf{y})=w_{L}(\mathbf{x}-\mathbf{y})$. The
minimum Gray distance $d_{G}(C)$ and the minimum Lee distance $d_{L}(C)$ of
$C$ is defined as
$d_{G}(C):=\text{min}\\{d_{G}(\mathbf{x},\mathbf{y}):~{}\mathbf{x},\mathbf{y}\in
C,~{}\mathbf{x}\neq\mathbf{y}\\}$ and
$d_{L}(C):=\text{min}\\{d_{L}(\mathbf{x},\mathbf{y}):~{}\mathbf{x},\mathbf{y}\in
C,~{}\mathbf{x}\neq\mathbf{y}\\},$ respectively. It is easy to verify that the
Gray map $\Phi$ is a distance-preserving map (isometry) from $(R^{n},d_{G})$
to $(\mathbb{Z}_{4}^{2n},d_{L}).$

A (linear) code over the ring $\mathbb{Z}_{4}$ is defined similarly. We write
the parameters of a linear code $C$ over $\mathbb{Z}_{4}$ as
$[n,4^{k_{1}}2^{k_{2}},d_{L}],$ where $n$ is the length of $C,$
$|C|=4^{k_{1}}2^{k_{2}},$ and $d_{L}=d_{L}(C).$ Moreover, following Hammons,
Kumar, Calderbank, Sloane, and Solé [13] (cf. [25]), we say that the code $C$
is of type $4^{k_{1}}2^{k_{2}}.$

For a linear code $C\subseteq R^{n}$ over $R,$ we define the residue code
$Res(C)$ and the torsion code $Tor(C)$ of $C,$ respectively, as

$Res(C):=\\{\mathbf{a}:~{}\mathbf{a}+\mathbf{b}v\in C,\text{ for some
}\mathbf{b}\in\mathbb{Z}_{4}^{n}\\},$

and

$Tor(C):=\\{\mathbf{b}:~{}\mathbf{b}v\in C\\}.$

We note that $Res(C)$ and $Tor(C)$ are linear codes of length $n$ over
$\mathbb{Z}_{4}.$

Regarding the residue and torsion codes, we have the following property.

######  Lemma 2.1.

Let $C$ be a free linear code of length $n$ over $R$ and
$\\{\mathbf{c}_{1},\mathbf{c}_{2},\ldots,\mathbf{c}_{k}\\},$ with
$\mathbf{c}_{i}=\mathbf{a}_{i}+\mathbf{b}_{i}v,$ be a basis of $C.$ Then the
following statements hold:

  * (1)

$Res(C)$ is a free linear code over $\mathbb{Z}_{4}$ with
$\\{\mathbf{a}_{1},\mathbf{a}_{2},\ldots,\mathbf{a}_{k}\\}$ as a basis.

  * (2)

$Tor(C)$ is a free linear code over $\mathbb{Z}_{4}$ with
$\\{\mathbf{a}_{1}+\mathbf{b}_{1},\mathbf{a}_{2}+\mathbf{b}_{2},\ldots,\mathbf{a}_{k}+\mathbf{b}_{k}\\}$
as a basis.

###### Proof.

We prove only part (1). Let $\mathbf{a}\in Res(C).$ Then there exists
$\mathbf{b}\in\mathbb{Z}_{4}^{n}$ such that $\mathbf{a}+\mathbf{b}v\in C.$
Since $\\{\mathbf{c}_{1},\mathbf{c}_{2},\ldots,\mathbf{c}_{k}\\}$ is a basis
of $C,$ there exist $r_{1},r_{2},\ldots,r_{k}\in R,$ with
$r_{i}=s_{i}+t_{i}v,$ for $i\in[1,k]_{\mathbb{Z}},$ such that

$\mathbf{a}+\mathbf{b}v=\sum_{i=1}^{k}s_{i}\mathbf{a}_{i}+\left(\sum_{i=1}^{k}(s_{i}\mathbf{b}_{i}+t_{i}\mathbf{a}_{i}+t_{i}\mathbf{b}_{i})\right)v,$

which means $\displaystyle\mathbf{a}=\sum_{i=1}^{k}s_{i}\mathbf{a}_{i},$ and
$\\{\mathbf{a}_{1},\mathbf{a}_{2},\ldots,\mathbf{a}_{k}\\}$ generates
$Res(C).$

Next, suppose on the contrary that
$\\{\mathbf{a}_{1},\mathbf{a}_{2},\ldots,\mathbf{a}_{k}\\}$ is linearly
dependent. It is easy to prove that
$\\{\mathbf{c}_{1},\mathbf{c}_{2},\ldots,\mathbf{c}_{k}\\}$ is also linearly
dependent, a contradiction. Therefore,
$\\{\mathbf{a}_{1},\mathbf{a}_{2},\ldots,\mathbf{a}_{k}\\}$ must be linear-ly
independent and we conclude that
$\\{\mathbf{a}_{1},\mathbf{a}_{2},\ldots,\mathbf{a}_{k}\\}$ is a basis of
$Res(C).$

∎

Similarly, we obtain the following theorem.

######  Theorem 2.2.

If $C$ be a free linear code of length $n$ over $R$ having a basis
$\\{\mathbf{c}_{1},\mathbf{c}_{2},\ldots,\mathbf{c}_{k}\\},$ then
$\\{\Phi(\mathbf{c}_{1}),\Phi(\mathbf{c}_{1}v),$
$\Phi(\mathbf{c}_{2}),\Phi(\mathbf{c}_{2}v),\ldots,\Phi(\mathbf{c}_{k}),\Phi(\mathbf{c}_{k}v)\\}$
is a basis of a free linear code $\Phi(C).$

###  2.3 Skew-polynomial ring $R[x;\theta,\Delta_{\theta}]$

We first recall the definition of a derivation on a finite ring $\mathbf{R}$
following [9].

######  Definition 1.

Let $\mathbf{R}$ be a finite ring and
$\Theta:\mathbf{R}\longrightarrow\mathbf{R}$ be an automorphism on
$\mathbf{R}.$ Then a map $\Delta_{\Theta}:\mathbf{R}\longrightarrow\mathbf{R}$
is called a derivation on $\mathbf{R}$ if the following two conditions are
satisfied:

  * (i)

$\Delta_{\Theta}(x+y)=\Delta_{\Theta}(x)+\Delta_{\Theta}(y),$ and

  * (ii)

$\Delta_{\Theta}(xy)=\Delta_{\Theta}(x)y+\Theta(x)\Delta_{\Theta}(y).$

Let $\mathbf{R}$ be a finite ring with an automorphism $\Theta$ and a
derivation $\Delta_{\Theta}.$ The skew-polynomial ring
$\mathbf{R}[x;\Theta,\Delta_{\Theta}]$ is the set of all polynomials over
$\mathbf{R}$ with ordinary addition of polynomials and multiplication defined
by

$xa:=\Theta(a)x+\Delta_{\Theta}(a),$

for any $a\in\mathbf{R}.$ This multiplication is extended to all polynomials
in $\mathbf{R}[x;\Theta,\Delta_{\Theta}]$ in the usual manner. This kind of
ring was introduced by Ore [20] in 1933, where $\mathbf{R}$ is the finite
field ${\mathbb{F}}_{q}.$ See also McDonald [19].

Consider a map $\theta:R\longrightarrow R$ defined by $\theta(a+bv)=a+b-bv.$
It is easy to see that $\theta$ defines an automorphism of $R.$ Moreover,
since for all $a+bv\in R$ we have $\theta^{2}(a+bv)=a+bv,$ we conclude that
the order of $\theta$ is $2.$

######  Lemma 2.3.

For all $x\in R,$ a map $\Delta_{\theta}:R\longrightarrow R$ such that

$\Delta_{\theta}(x)=(1+2v)(\theta(x)-x)$

defines a derivation on $R.$

###### Proof.

Straightforward by definition. ∎

We prove several properties related to the derivation $\Delta_{\theta}$ on
$R.$ We begin with the following that can be derived by a routine computation.

######  Lemma 2.4.

Let $\Delta_{\theta}(x)=(1+2v)(\theta(x)-x)$ be a derivation on $R.$ Then the
following statements hold:

  * (1)

$\Delta_{\theta}\theta+\theta\Delta_{\theta}\equiv 0.$

  * (2)

$\Delta_{\theta}\Delta_{\theta}\equiv 0.$

  * (3)

for all $x\in R,$ $\Delta_{\theta}(x)=0$ $\iff$ $\theta(x)=x.$

Next, we have

######  Lemma 2.5.

For all $a\in R,$ we have $x^{2}a=ax^{2}.$

###### Proof.

Since $xa=\theta(a)x+\Delta_{\theta}(a),$

$\displaystyle x^{2}a$ $\displaystyle=x\theta(a)x+x\Delta_{\theta}(a)$
$\displaystyle=(\theta^{2}(a)x+\Delta_{\theta}(\theta(a)))x+\theta(\Delta_{\theta}(a))x+\Delta_{\theta}^{2}(a)$
$\displaystyle=\theta^{2}(a)x^{2}+((\Delta_{\theta}\theta+\theta\Delta_{\theta})(a))x+\Delta_{\theta}^{2}(a)$
$\displaystyle=ax^{2}\hskip 28.45274pt\text{(by Lemma \ref{Property D1})}.$

∎

We can generalize Lemma 2.5 using mathematical induction.

######  Corollary 2.6.

For all $a\in R,~{}n\in\mathbb{Z}^{+},$ we have

$x^{n}a=\begin{cases}(\theta(a)x+\Delta_{\theta}(a))x^{n-1},&\text{if }n\text{
is odd},\\\ ax^{n},&\text{if }n\text{ is even}.\end{cases}$

###### Proof.

We know that $xa=\theta(a)x+\Delta_{\theta}(a)$ and $x^{2}a=ax^{2}.$ Suppose
the above statement holds for all $n\leq k,$ with $k\geq 2.$ Consider two
cases. If $k+1$ is even, then $k-1$ is also even. From the induction
hypothesis,

$x^{k+1}a=x^{2}(x^{k-1}a)=x^{2}ax^{k-1}=(ax^{2})x^{k-1}=ax^{k+1}.$

If $k+1$ is odd, then $k$ is even. Again, from the induction hypothesis,

$x^{k+1}a=x(x^{k}a)=x(ax^{k})=(xa)x^{k}=(\theta(a)x+\Delta_{\theta}(a))x^{k}.$

Then the result follows. ∎

Let $R^{\theta}$ be a subset of $R,$ fixed element-wise by $\theta,$ namely
$R^{\theta}:=\\{a\in R:~{}\theta(a)=a\\}.$ It is easy to verify that
$R^{\theta}$ is a subring of $R.$ In our case,
$R^{\theta}=\\{0,1,2,3\\}=\mathbb{Z}_{4}.$ Also, for all $a\in R^{\theta},$ we
have $\Delta_{\theta}(a)=0.$ It implies, by Corollary 2.6, that for all $a\in
R^{\theta}$ and $n\in\mathbb{Z}^{+}$, we have $x^{n}a=ax^{n}.$

######  Definition 2.

A polynomial $f(x)\in R[x;\theta,\Delta_{\theta}]$ is called a central element
if it satisfies

$f(x)a(x)=a(x)f(x),$

for all $a(x)\in R[x;\theta,\Delta_{\theta}].$ The center of
$R[x;\theta,\Delta_{\theta}],$ denoted by $Z(R[x;\theta,\Delta_{\theta}]),$ is
defined as

$Z(R[x;\theta,\Delta_{\theta}]):=\\{f(x)\in
R[x;\theta,\Delta_{\theta}]:~{}f(x)a(x)=a(x)f(x),~{}\text{for all }a(x)\in
R[x;\theta,\Delta_{\theta}]\\}.$

The central element satisfies the following property.

######  Theorem 2.7.

$f(x)\in Z(R[x;\theta,\Delta_{\theta}])$ if and only if $f(x)\in
R^{\theta}[x]$ and for all odd integers $i,$ the coefficient of $x^{i}$ is
equal to $0.$

###### Proof.

($\Longrightarrow$) Let $f(x)=f_{0}+f_{1}x+f_{2}x^{2}+\cdots+f_{k}x^{k}.$
Observe that

$xf(x)=\sum_{i=0}^{k}(xf_{i})x^{i}=\sum_{i=0}^{k}(\theta(f_{i})x+\Delta_{\theta}(f_{i}))x^{i}=\Delta_{\theta}(f_{0})+\sum_{i=1}^{k}\left(\theta(f_{i-1})+\Delta_{\theta}(f_{i})\right)x^{i}+\theta(f_{k})x^{k+1}$

and

$f(x)x=f_{0}x+f_{1}x^{2}+\cdots+f_{k}x^{k+1}.$

Since $f(x)$ is a central element, $xf(x)=f(x)x,$ which implies

$\Delta_{\theta}(f_{0})=0,$ (1)
$\theta(f_{i-1})+\Delta_{\theta}(f_{i})=f_{i-1},\text{ for }1\leq i\leq k,$
(2) $\theta(f_{k})=f_{k}.$ (3)

The Equation (3) implies $f_{k}\in R^{\theta}.$ By Lemma 2.4 we have
$\Delta_{\theta}(f_{k})=0.$ By substituting the Equation (2) repeatedly we
obtain $f_{1},f_{2},\ldots,f_{k-1}\in R^{\theta}.$ Moreover, by Equation (1)
and Lemma 2.4 obtain $f_{0}\in R^{\theta}.$ Thus, $f(x)\in R^{\theta}[x].$

Now, take $a=v\in R.$ Observe that

$af(x)=af_{0}+af_{1}x+\cdots+af_{k}x^{k}\text{ and
}f(x)a=f_{0}a+\sum_{i=1}^{k}f_{i}(x^{i}a).$

In this case, for $0\leq 2j<k,$ the coefficient of $x^{2j}$ in $af(x)$ and
$f(x)a$ is equal to $af_{2j}$ and $f_{2j}a+f_{2j+1}\Delta_{\theta}(a),$
respectively. Since $f(x)$ is a central element and $\Delta_{\theta}(a)=1$, we
conclude that $f_{2j+1}=0.$

$(\Longleftarrow)$ Let
$f(x)=f_{0}+f_{2}x^{2}+f_{4}x^{4}+\cdots+f_{2m}x^{2m}\in R^{\theta}[x].$ Let
$a(x)=a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{k}x^{k}\in
R[x;\theta,\Delta_{\theta}].$ Observe that

$(f_{2i}x^{2i})(a_{j}x^{j})=f_{2i}(x^{2i}a_{j})x^{j}=f_{2i}a_{j}x^{2i+j},$

and

$\displaystyle(a_{j}x^{j})(f_{2i}x^{2i})$
$\displaystyle=a_{j}(x^{j}f_{2i})x^{2i}$
$\displaystyle=\begin{cases}a_{j}\theta(f_{2i})x^{2i+j}+a_{j}\Delta_{\theta}(f_{2i})x^{2i+j-1},&\text{if
}j\text{ is odd},\\\ a_{j}f_{2i}x^{2i+j},&\text{if }j\text{ is
even,}\end{cases}$ $\displaystyle=a_{j}f_{2i}x^{2i+j}\text{ (by Lemma
\ref{Property D1}).}$

Then, $(f_{2i}x^{2i})(a_{j}x^{j})=(a_{j}x^{j})(f_{2i}x^{2i}),$ for all $i,j.$
Hence, $f(x)a(x)=a(x)f(x),$ for all $a(x)\in R[x;\theta,\Delta_{\theta}].$ ∎

We end this section by establishing the right-division algorithm below.

######  Lemma 2.8 (Right-Division Algorithm).

Let $f(x),g(x)\in R[x;\theta,\Delta_{\theta}]$ such that the leading
coefficient of $g(x)$ is a unit. Then there exist $q(x),r(x)\in
R[x;\theta,\Delta_{\theta}]$ such that

$f(x)=q(x)g(x)+r(x),$

with $r(x)=0$ or $\deg r(x)<\deg g(x).$

###### Proof.

Similar to the proof of Theorem 2.8 in [23] and Theorem 1 in [21]. ∎

##  3 $\Delta_{\theta}$-cyclic codes over $R$

For $f(x)$ a polynomial of degree $n$ in $R[x;\theta,\Delta_{\theta}],$ let

$\langle f(x)\rangle=\\{a(x)f(x):~{}a(x)\in R[x;\theta,\Delta_{\theta}]\\}.$

It is easy to see that $R[x;\theta,\Delta_{\theta}]/\langle f(x)\rangle$ is a
left module over $R[x;\theta,\Delta_{\theta}],$ where the scalar
multiplication is defined by

$r(x)\left(a(x)+\langle f(x)\rangle\right):=r(x)a(x)+\langle f(x)\rangle.$

######  Definition 3.

A code $C\subseteq R^{n}$ is called a $\Delta_{\theta}$-linear code of length
$n$ over $R$ if $C$ is a left $R[x;\theta,\Delta_{\theta}]$-submodule of
$R[x;\theta,\Delta_{\theta}]/\langle f(x)\rangle$ for $f(x)\in
R[x;\theta,\Delta_{\theta}]$ a polynomial of degree $n.$ If $f(x)$ is a
central element then $C$ is called a central $\Delta_{\theta}$-linear code.

######  Definition 4.

A code $C\subseteq R^{n}$ is called a $\Delta_{\theta}$-cyclic code of length
$n$ over $R$ if $C$ is a $\Delta_{\theta}$-linear code and for all
$\mathbf{c}=(c_{0},c_{1},\ldots,c_{n-1})\in C$ we have

$T_{\Delta_{\theta}}(\mathbf{c}):=(\theta(c_{n-1})+\Delta_{\theta}(c_{0}),\theta(c_{0})+\Delta_{\theta}(c_{1}),\ldots,\theta(c_{n-2})+\Delta_{\theta}(c_{n-1}))\in
C.$

Here, $T_{\Delta_{\theta}}$ is called a $\Delta_{\theta}$-cyclic shift
operator.

######  Remark 1.

If $\theta$ is an identity automorphism and $\Delta_{\theta}\equiv 0,$ then we
obtain a (usual) cyclic code. Hence, the $\Delta_{\theta}$-cyclic code is a
generalization of a cyclic code. Moreover, recall that if $\theta$ is an
identity automorphism and $\Delta_{\theta}\equiv 0,$ then a linear code
$C\subseteq R^{n}$ is called quasi-cyclic of index $k$ ($k$ is a divisor of
$n$) if for all $\mathbf{c}\in C,$ we have
$T^{k}_{\Delta_{\theta}}(\mathbf{c})\in C.$

For our purpose, to convert the algebraic structures of
$\Delta_{\theta}$-cyclic codes into combinatorial structures and vice versa,
we consider the following correspondence:

$\begin{array}[]{ccc}R[x;\theta,\Delta_{\theta}]/\langle
f(x)\rangle&\longrightarrow&R^{n},\\\
c_{0}+c_{1}x+c_{2}x^{2}+\cdots+c_{n-1}x^{n-1}&\longmapsto&(c_{0},c_{1},\ldots,c_{n-1}).\end{array}$

From now on, let $R_{n,\Delta_{\theta}}:=R[x;\theta,\Delta_{\theta}]/\langle
x^{n}-1\rangle.$

######  Lemma 3.1.

If $c(x)=c_{0}+c_{1}x+c_{2}x^{2}+\cdots+c_{n-1}x^{n-1}\in
R_{n,\Delta_{\theta}}$ is identified by a codeword
$\mathbf{c}=(c_{0},c_{1},\ldots,c_{n-1})\in R^{n},$ then $xc(x)\in
R_{n,\Delta_{\theta}}$ is identified by $T_{\Delta_{\theta}}(\mathbf{c})\in
R^{n}.$

###### Proof.

$\displaystyle xc(x)$ $\displaystyle=x\left(\sum_{i=0}^{n-1}c_{i}x^{i}\right)$
$\displaystyle=\sum_{i=1}^{n-1}\left(\theta(c_{i-1})+\Delta_{\theta}(c_{i})\right)x^{i}+(\theta(c_{-1})+\Delta_{\theta}(c_{0}))~{}(\text{denoting
that }c_{-1}=c_{n-1})$
$\displaystyle=\sum_{i=0}^{n-1}(\theta(c_{i-1})+\Delta_{\theta}(c_{i}))x^{i}.$

It means that $xc(x)$ is identified by $T_{\Delta_{\theta}}(\mathbf{c})\in
R^{n}.$ ∎

######  Lemma 3.2.

A code $C\subseteq R^{n}$ is $\Delta_{\theta}$-cyclic code if and only if $C$
is a left $R[x;\theta,\Delta_{\theta}]$-submodule of $R_{n,\Delta_{\theta}}.$

###### Proof.

$(\Longrightarrow)$ Since $C$ is a $\Delta_{\theta}$-linear code, $(C,+)$ is a
subgroup of $(R_{n,\Delta_{\theta}},+),$ and for any $\mathbf{c}\in C,$
identified by $c(x)\in R_{n,\Delta_{\theta}},$ we have
$T_{\Delta_{\theta}}(\mathbf{c})\in C.$ By Lemma 3.1,
$T_{\Delta_{\theta}}(\mathbf{c})\in C$ can be identified by $xc(x)\in
R_{n,\Delta_{\theta}}.$ Inductively, we obtain $x^{i}c(x)\in C,$ for all
$i\in\mathbb{Z}^{+}.$ By the linearity of the scalar multiplication, we have
$a(x)c(x)\in C$ for all $a(x)\in R[x;\theta,\Delta_{\theta}].$ Hence, $C$ is a
left submodule of $R_{n,\Delta_{\theta}}.$

$(\Longleftarrow)$ If $C$ is a left submodule of $R_{n,\Delta_{\theta}},$ then
for all $\mathbf{c}\in C,$ identified by $c(x)\in R_{n,\Delta_{\theta}},$ we
have $xc(x)\in C.$ By Lemma 3.1, $xc(x)$ can be identified by
$T_{\Delta_{\theta}}(\mathbf{c})\in R^{n}.$ Hence,
$T_{\Delta_{\theta}}(\mathbf{c})\in C,$ which implies that $C$ is a
$\Delta_{\theta}$-cyclic code. ∎

######  Corollary 3.3.

If $C\subseteq R^{n}$ is a $\Delta_{\theta}$-cyclic code of even length $n,$
then $C$ is an ideal of $R_{n,\Delta_{\theta}}.$

###### Proof.

Since $n$ is even, by Theorem 2.7, $x^{n}-1$ is a central element. Then for
all $a(x)\in R[x;\theta,\Delta_{\theta}],$ we have
$a(x)(x^{n}-1)=(x^{n}-1)a(x).$ Then, $\langle x^{n}-1\rangle$ is a two-sided
ideal of $R[x;\theta,\Delta_{\theta}].$ Since $C$ is a left submodule of
$R_{n,\Delta_{\theta}},$ $C$ is an ideal of $R_{n,\Delta_{\theta}}.$ ∎

######  Lemma 3.4.

Let $C\subseteq R^{n}$ be a $\Delta_{\theta}$-cyclic code. Then the following
two statements hold.

  * (1)

$C$ is a cyclic code of length $n$ over $R$ if $n$ is odd;

  * (2)

$C$ is a quasi-cyclic code of length $n$ and index $2$ over $R$ if $n$ is
even.

###### Proof.

  * (1)

If $n$ is odd, then there exists $b\in\mathbb{Z}$ such that $2b=n+1.$ Let
$\mathbf{c}=(c_{0},c_{1},\ldots,c_{n-1})\in C$ be a codeword identified by
$c(x)=c_{0}+c_{1}x+c_{2}x^{2}+\cdots+c_{n-1}x^{n-1}\in R_{n,\Delta_{\theta}}.$
It is clear that $x^{2b}c(x)\in C.$ Observe that

$x^{2b}c(x)=x^{2b}\sum_{i=0}^{n-1}c_{i}x^{i}=\sum_{i=0}^{n-1}(x^{2b}c_{i})x^{i}=\sum_{i=0}^{n-1}c_{i}x^{n+1+i}=\sum_{i=0}^{n-1}c_{i-1}x^{i}~{}(\text{denoting
that }c_{-1}=c_{n-1}).$

Note that $x^{2b}c_{i}=c_{i}x^{2b}$ is derived from Corollary 2.6. The
equation above says that the codeword $x^{2b}c(x)\in C$ can be identified by
the vector $(c_{n-1},c_{0},c_{1},\ldots,c_{n-2})\in C.$ Thus, $C$ is a cyclic
code.

  * (2)

Observe that for every vector $\mathbf{c}=(c_{0},c_{1},\ldots,c_{n-1})\in C$
identified by the polynomial
$c(x)=c_{0}+c_{1}x+c_{2}x^{2}+\cdots+c_{n-1}x^{n-1}\in R_{n,\Delta_{\theta}}$
we have $x^{2}c(x)\in C.$ By the similar way as in part (1), we can show that
$x^{2}c(x)$ can be identified by $(c_{n-2},c_{n-1},c_{0},\ldots,c_{n-3})\in
C.$ Then, $C$ is a quasi-cyclic code of index $2.$

∎

######  Lemma 3.5.

If $C\subseteq R^{n}$ is a $\Delta_{\theta}$-cyclic code and $g(x)$ is a
nonzero polynomial in $C$ of smallest degree with leading coefficient is a
unit, then the following three statements hold.

  * (1)

$C=\langle g(x)\rangle.$

  * (2)

$g(x)$ is a right divisor of $x^{n}-1.$

  * (3)

$\\{g(x),xg(x),\ldots,x^{n-k-1}g(x)\\}$ is a basis of $C,$ with
$k=\deg(g(x)).$

###### Proof.

  * (1)

It is clear that $\langle g(x)\rangle\subseteq C.$ Let $c(x)\in C.$ By the
right-division algorithm, there exist $q(x),r(x)\in
R[x;\theta,\Delta_{\theta}]$ such that

$c(x)=q(x)g(x)+r(x),$

with $r(x)=0$ or $\deg(r(x))<\deg(g(x)).$ Since $C$ is a left submodule of
$R_{n,\Delta_{\theta}}$ over $R[x;\theta,\Delta_{\theta}],$
$r(x)=c(x)-q(x)g(x)\in C.$ Moreover, since $g(x)$ is a nonzero polynomial of
smallest degree, $r(x)=0.$ Thus $c(x)=q(x)g(x)\in\langle g(x)\rangle,$ and
hence $C\subseteq\langle g(x)\rangle.$

  * (2)

Again, by right-division algorithm, there exist $q(x),r(x)\in
R[x;\theta,\Delta_{\theta}]$ such that

$x^{n}-1=q(x)g(x)+r(x),$

with $r(x)=0$ or $\deg(r(x))<\deg(g(x)).$ By the same argument with above, we
have $x^{n}-1=q(x)g(x).$ It means that $g(x)$ is is a right divisor of
$x^{n}-1.$

  * (3)

From the above result, let $x^{n}-1=h(x)g(x),$ for some $h(x)\in
R[x;\theta,\Delta_{\theta}].$ Let $c(x)\in C=\langle g(x)\rangle$ and let
$c(x)=\ell(x)g(x)$ for some $\ell(x)\in R[x;\theta,\Delta_{\theta}].$ If
$\deg(\ell(x))\leq n-k-1,$ then $c(x)\in\langle
g(x),xg(x),\ldots,x^{n-k-1}g(x)\rangle.$ If $\deg(\ell(x))>n-k-1,$ by using
the right-division algorithm we come up with the conclusion that
$c(x)=\ell(x)g(x)=r(x)g(x)$ (in $R_{n,\Delta_{\theta}}$) for some polynomial
$r(x)\in R[x;\theta,\Delta_{\theta}]$ with $r(x)=0$ or
$\deg(r(x))<\deg(h(x))=n-k$, so $\\{g(x),xg(x),\ldots,x^{n-k-1}g(x)\\}$
generates $C.$ Moreover, it is easy to see that the set is also linearly
independent. Hence, we conclude that $\\{g(x),xg(x),\ldots,x^{n-k-1}g(x)\\}$
is a basis for $C.$

∎

Since $\langle g(x)\rangle$ is a submodule of $R_{n,\Delta_{\theta}}$ over
$R[x;\theta,\Delta_{\theta}],$ by Lemma 3.2, $C$ is a $\Delta_{\theta}$-cyclic
code. The Lemma 3.5 above says that if $C=\langle g(x)\rangle$ is a
$\Delta_{\theta}$-cyclic code, where $g(x)$ is a nonzero polynomial in $C$ of
smallest degree with a unit leading coefficient and $g(x)$ is also a right
divisor of $x^{n}-1,$ then $C$ is free. Hence, we obtained a construction
method for $\Delta_{\theta}$-cyclic codes as described by the following.

######  Corollary 3.6.

If $g(x)$ is a right divisor of $x^{n}-1,$ with leading coefficient a unit,
then $C=\langle g(x)\rangle$ is a free $\Delta_{\theta}$-cyclic code.

Let $C=\langle g(x)\rangle$ be a free $\Delta_{\theta}$-cyclic code of length
$n$ generated by a right divisor $g(x)$ of $x^{n}-1,$ whose leading
coefficient is a unit. Then the generator matrix $G$ of $C$ of dimension
$(n-k)\times n$ is given by

$G=\begin{pmatrix}g(x)\\\ xg(x)\\\ x^{2}g(x)\\\ \vdots\\\
x^{n-k-1}g(x)\end{pmatrix},$

where $g(x)=g_{0}+g_{1}x+g_{2}x^{2}+\cdots+g_{k}x^{k}.$ To be more precise, if
$n-k$ is odd, then

$G=\left(\begin{smallmatrix}g_{0}&g_{1}&g_{2}&\cdots&g_{k}&0&\cdots&0\\\
\Delta_{\theta}(g_{0})&\theta(g_{0})+\Delta_{\theta}(g_{1})&\theta(g_{1})+\Delta_{\theta}(g_{2})&\cdots&\theta(g_{k-1})+\Delta_{\theta}(g_{k})&\theta(g_{k})&\cdots&0\\\
0&0&g_{0}&\cdots&g_{k-3}&g_{k-2}&\cdots&0\\\
\vdots&\vdots&\vdots&\cdots&\vdots&&&\vdots\\\
0&0&\cdots&g_{0}&g_{k-2}&\cdots&g_{k-1}&g_{k}\end{smallmatrix}\right),$

and if $n-k$ is even, then

$G=\left(\begin{smallmatrix}g_{0}&g_{1}&g_{2}&\cdots&g_{k}&0&\cdots&0&0\\\
\Delta_{\theta}(g_{0})&\theta(g_{0})+\Delta_{\theta}(g_{1})&\theta(g_{1})+\Delta_{\theta}(g_{2})&\cdots&\theta(g_{k-1})+\Delta_{\theta}(g_{k})&\theta(g_{k})&\cdots&0&0\\\
0&0&g_{0}&\cdots&g_{k-3}&g_{k-2}&\cdots&0&0\\\
\vdots&\vdots&\vdots&\cdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&0&\cdots&\Delta_{\theta}(g_{0})&\theta(g_{0})+\Delta_{\theta}(g_{1})&\theta(g_{1})+\Delta_{\theta}(g_{2})&\cdots&\theta(g_{k-1})+\Delta_{\theta}(g_{k})&\theta(g_{k})\\\
\end{smallmatrix}\right).$

##  4 Dual of $\Delta_{\theta}$-cyclic codes over $R$

In this section, we investigate the structure of the dual of a free
$\Delta_{\theta}$-cyclic code over $R$ with even length.

######  Definition 5.

Let $C\subseteq R^{n}$ is a $\Delta_{\theta}$-cyclic code. Dual of $C,$
denoted by $C^{\perp},$ is defined by

$C^{\perp}=\\{\mathbf{x}\in R^{n}:~{}\mathbf{x}\cdot\mathbf{y}=0,\text{ for
all }\mathbf{y}\in C\\}.$

Here, $\mathbf{x}\cdot\mathbf{y}=x_{0}y_{0}+x_{1}y_{1}+\cdots+x_{n-1}y_{n-1}$
denotes the usual inner product in $R^{n}.$

It is easy to check that $C^{\perp}$ is a linear code over $R.$ Moreover, for
any even $n,$ if $C$ is a free $\Delta_{\theta}$-cyclic code principally
generated by a polynomial whose leading coefficient is a unit, then
$C^{\perp}$ is free, as we show below.

######  Lemma 4.1.

Let $n$ be an even integer. Let $g(x),h(x)\in R[x;\theta,\Delta_{\theta}],$
where the leading coefficient of $g(x)$ is a unit, and $h(x)g(x)=x^{n}-1.$
Then we have

$h(x)g(x)=g(x)h(x).$

###### Proof.

If $n$ is even, then $x^{n}-1=h(x)g(x)$ is a central element (by Theorem 2.7).
Then we have

$h(x)h(x)g(x)=h(x)g(x)h(x).$

Since $h(x)$ is not a zero divisor, $x^{n}-1=h(x)g(x)=g(x)h(x).$ ∎

######  Lemma 4.2.

Let $C$ be a $\Delta_{\theta}$-cyclic code, where $C=\langle g(x)\rangle,$ for
some right divisor $g(x)$ of $x^{n}-1,$ whose leading coefficient is a unit
and $n$ is even. Let $x^{n}-1=h(x)g(x).$ Then $c(x)\in R_{n,\Delta_{\theta}}$
is contained in $C$ if and only if $c(x)h(x)=0$ (in $R_{n,\Delta_{\theta}}$).

###### Proof.

$(\Longrightarrow)$ Since $c(x)\in C,$ there exists $a(x)\in
R[x;\theta,\Delta_{\theta}]$ such that $c(x)=a(x)g(x).$ Hence,

$c(x)h(x)=a(x)g(x)h(x)=a(x)h(x)g(x)=a(x)(x^{n}-1)=0\text{ (in
}R_{n,\Delta_{\theta}}\text{).}$

$(\Longleftarrow)$ Since $c(x)h(x)=0$ (in $R_{n,\Delta_{\theta}}$) for some
$c(x)\in R_{n,\Delta_{\theta}},$ we have $c(x)h(x)=q(x)(x^{n}-1)$ for some
$q(x)\in R[x;\theta,\Delta_{\theta}].$ In this case

$c(x)h(x)=q(x)(x^{n}-1)=q(x)h(x)g(x)=q(x)g(x)h(x).$

Since $h(x)$ is not a zero divisor, we have $c(x)=q(x)g(x)\in\langle
g(x)\rangle=C.$

∎

We also have known that all unit in $R$ are $\\{1,3,1+2v,3+2v\\}.$ It is easy
to check that for all units $a$ in $R,$ we have $\theta(a)^{\prime}$s are also
units in $R.$ Hence, we have the following.

######  Lemma 4.3.

If $a\in R$ is a unit then $\theta(a)\in R$ is also a unit.

Now, consider a $\Delta_{\theta}$-cyclic code $C$ of even length $n.$ Let
$C=\langle g(x)\rangle,$ where $g(x)$ is a right divisor of $x^{n}-1$ and its
leading coefficient is a unit. Then there exists
$h(x)=h_{0}+h_{1}x+\cdots+h_{k}x^{k}\in R_{n,\Delta_{\theta}}$ such that
$x^{n}-1=h(x)g(x).$ For $c(x)=c_{0}+c_{1}x+\cdots+c_{n-1}x^{n-1}\in C,$ by
Lemma 4.2, we have $c(x)h(x)=0$ (in $R_{n,\Delta_{\theta}}$). By considering
the coefficients of $x^{k},x^{k+1},\ldots,x^{n-1}$ in the last equation, we
obtain two systems of equations each consisting of $n-k$ linear equations in
$c_{0},c_{1},\ldots,c_{n-1}.$ To be more precise, we have the following two
systems of equations, for $k$ is odd and even, respectively.

$\begin{cases}c_{i}h_{k}+c_{i+1}(\theta(h_{k-1})+\Delta_{\theta}(h_{k}))+c_{i+2}h_{k-2}+\cdots+c_{k+i}(\theta(h_{0})+\Delta_{\theta}(h_{1}))=0,&\text{if
}i\text{ is even,}\\\
c_{i}\theta(h_{k})+c_{i+1}h_{k-1}+c_{i+2}(\theta(h_{k-2})+\Delta_{\theta}(h_{k-1}))+\cdots+c_{k+i}h_{0}+c_{k+i+1}\Delta_{\theta}(h_{0})=0,&\text{if
}i\text{ is odd.}\end{cases}$
$\begin{cases}c_{i}h_{k}+c_{i+1}(\theta(h_{k-1})+\Delta_{\theta}(h_{k}))+c_{i+2}h_{k-2}+\cdots+c_{k+i}h_{0}+c_{k+i+1}\Delta_{\theta}(h_{0})=0,&\text{if
}i\text{ is even,}\\\
c_{i}\theta(h_{k})+c_{i+1}h_{k-1}+c_{i+2}(\theta(h_{k-2})+\Delta_{\theta}(h_{k-1}))+\cdots+c_{k+i}(\theta(h_{0})+\Delta_{\theta}(h_{1}))=0,&\text{if
}i\text{ is odd,}\end{cases}$

If we write the system of equations in a matrix form, we obtain
$H\mathbf{c}^{T}=0,$ for the matrix $H$ of dimension $(n-k)\times n.$ It
implies that $GH^{T}=0,$ for a generator matrix $G$ of $C.$ Moreover, it is
easy to check that $H$ is of the row echelon form with diagonal elements
$h_{k}$ or $\theta(h_{k}),$ and having a submatrix of dimension
$(n-k)\times(n-k).$ Since $h_{k}$ is a unit, by Lemma 4.3, we have
$\theta(h_{k})$ is also a unit. Then, the rows of $H$ are linearly
independent. Hence,

$|\text{Row space of }H|=|R|^{n-k}=|C^{\perp}|.$

Thus, $H$ is a parity check matrix of $C$ and we have proven the following
theorem.

######  Theorem 4.4.

Let $C$ be a $\Delta_{\theta}$-cyclic code of even length $n$, where
$C=\langle g(x)\rangle,$ for some right divisor $g(x)$ of $x^{n}-1,$ whose
leading coefficient is a unit and $x^{n}-1=h(x)g(x),$ for some
$h(x)=h_{0}+h_{1}x+h_{2}x^{2}+\cdots+h_{k}x^{k}\in R_{n,\Delta_{\theta}}.$
Then the $(n-k)\times n$ parity-check matrix $H$ for $C$ is given by

$\left(\setcounter{MaxMatrixCols}{14}\begin{smallmatrix}h_{k}&\theta(h_{k-1})+\Delta_{\theta}(h_{k})&h_{k-2}&\cdots&\cdots&\cdots&\theta(h_{0})+\Delta_{\theta}(h_{1})&\cdots&\cdots&\cdots&\cdots&\cdots&0&0\\\
0&\theta(h_{k})&h_{k-1}&\theta(h_{k-2})&\cdots&\cdots&\cdots&\Delta_{\theta}(h_{0})&\cdots&\cdots&\cdots&\cdots&0&0\\\
0&0&h_{k}&\theta(h_{k-1})+\Delta_{\theta}(h_{k})&h_{k-2}&\cdots&\cdots&\cdots&\theta(h_{0})+\Delta_{\theta}(h_{1})&\cdots&\cdots&\cdots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots&\vdots\\\
0&0&\cdots&\cdots&0&h_{k}&\theta(h_{k-1})+\Delta_{\theta}(h_{k})&h_{k-2}&\cdots&\cdots&\cdots&\cdots&h_{1}&\theta(h_{0})+\Delta_{\theta}(h_{1})\end{smallmatrix}\right)$

for an odd $k,$ and

$\left(\setcounter{MaxMatrixCols}{14}\begin{smallmatrix}h_{k}&\theta(h_{k-1})+\Delta_{\theta}(h_{k})&h_{k-2}&\cdots&\cdots&\cdots&h_{0}&\Delta_{\theta}(h_{0})&0&\cdots&\cdots&\cdots&0&0\\\
0&\theta(h_{k})&h_{k-1}&\theta(h_{k-2})&\cdots&\cdots&h_{1}&\theta(h_{0})+\Delta_{\theta}(h_{1})&0&\cdots&\cdots&\cdots&0&0\\\
0&0&h_{k}&\theta(h_{k-1})+\Delta_{\theta}(h_{k})&h_{k-2}&\cdots&h_{2}&\theta(h_{1})+\Delta_{\theta}(h_{2})&\cdots&\cdots&\cdots&\cdots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots&\vdots&\vdots\\\
0&0&\cdots&\cdots&0&\theta(h_{k})&h_{k-1}&\cdots&\cdots&\cdots&\cdots&\cdots&h_{1}&\theta(h_{0})+\Delta_{\theta}(h_{1})\end{smallmatrix}\right)$

for an even $k.$

##  5 New linear codes over $\mathbb{Z}_{4}$

In this section, we obtained many linear codes over $\mathbb{Z}_{4}$ with new
parameters from the Gray image, residue code, and torsion code of skew-linear
and skew-cyclic codes with derivation over $R$. First, we construct linear
codes over $\mathbb{Z}_{4}$ from principally generated free
$\Delta_{\theta}$-cyclic codes over $R$ in Corollary 3.6. The linear codes
over $\mathbb{Z}_{4},$ some of them are with new parameters, are listed in the
table below.

Table 1: Free linear codes over $\mathbb{Z}_{4}$ $C$ | $\Phi(C)$ | $Res(C)$ | $Tor(C)$ | $C^{PS}$
---|---|---|---|---
$[n,4^{k_{1}}2^{k_{2}},d_{L}]$ | $[n,4^{k_{1}}2^{k_{2}},d_{L}]$ | $[n,4^{k_{1}}2^{k_{2}},d_{L}]$ | $[n,4^{k_{1}}2^{k_{2}},d_{L}]$
$\langle 3+x\rangle$ | $[8,4^{6}2^{0},2]^{\ast)}$ | $[4,4^{3}2^{0},2]^{\ast\ast}$ | $[4,4^{3}2^{0},2]^{\ast\ast}$ | $[8,4^{6}2^{0},2]^{\ast)}$
$\begin{aligned} \langle(3+2v)+(3+2v)x+2x^{2}\\\ +(1+2v)x^{3}+(3+2v)x^{4}\rangle\end{aligned}$ |  |  | $[6,4^{2}2^{0},6]^{\ast\ast}$ |
$\begin{aligned} \langle 3v+(3+v)x+(3+v)x^{2}+(1+2v)x^{3}\\\ +(2+2v)x^{4}+2x^{5}+vx^{6}+(1+3v)x^{7}\\\ +(1+v)x^{8}+x^{9}\rangle\end{aligned}$ |  | $[12,4^{3}2^{0},10]^{\ast)}$ |  |
$\begin{aligned} \langle(1+3v)+3x+x^{2}+(3+2v)x^{4}\\\ +2x^{5}+2vx^{6}+2x^{7}+(1+3v)x^{8}\\\ +3x^{9}+3x^{10}+x^{12}\rangle\end{aligned}$ |  | $[16,4^{4}2^{0},12]^{\ast\ast}$ |  |

Notes for Table LABEL:Tabel-1, Table LABEL:Tabel-2, Example 5.2, and Example
5.3:

  * •

∗ means that the code has new $k_{1}$ and $k_{2},$ but there is/are other
codes of equal length in the database [2] with the same minimum Lee distance
but with greater or equal cardinalities.

  * •

∗∗ means that the code has minimum Lee distance greater than all existing
linear codes of equal length with the same values of $k_{1}$ and $k_{2}$ in
the database [2].

  * •

∗∗∗ means that the code has new $k_{1}$ and $k_{2},$ with greater
cardinalities compared with all existing linear codes of equal length with the
same value of minimum Lee distance in the database [2].

  * •

∗) means that the code has the same parameters as some existing good linear
codes of equal length in the database [2].

  * •

$C^{PS}$ is the code obtained by Plotkin-sum construction, namely
$C^{PS}:=\\{(\mathbf{x}~{}|~{}\mathbf{x}+\mathbf{y}):\mathbf{x},\mathbf{y}\in
Res(C)\\}$ or
$C^{PS}:=\\{(\mathbf{x}~{}|~{}\mathbf{x}+\mathbf{y}):\mathbf{x},\mathbf{y}\in
Tor(C)\\}.$

In the Table LABEL:Tabel-1 above, notice that all linear codes over
$\mathbb{Z}_{4}$ constructed by this method are free. This is not a
coincidence. In fact, this is a direct consequence of Theorem 2.2, as follows.

######  Corollary 5.1.

Let $C$ be a free $\Delta_{\theta}$-cyclic code of length $n$ over $R$ with
$|C|=16^{k}.$ Then the free linear codes $Res(C)$ and $Tor(C)$ have parameter
$4^{k}2^{0},$ and the free linear code $\Phi(C)$ has parameter $4^{2k}2^{0}.$

######  Remark 2.

The part of Corollary 5.1 related to the parameters of $Res(C)$ and $Tor(C)$
does not hold for linear codes over $\mathbb{Z}_{4}+u\mathbb{Z}_{4},$ with
$u^{2}=1$ considered by Sharma and Bhaintwal [23] (e.g., see Example 7 in
[23]).

###  5.1 Notes on a computational simplification

From Corollary 5.1, we conclude that we need non-free linear codes over $R$ to
obtain non-free Gray images, residue codes, and torsion codes. Here, we use a
slightly modified construction from principally generated
$\Delta_{\theta}$-cyclic codes. Notice that for any $g(x)\in
R_{n,\Delta_{\theta}},$ the set $\\{g(x),xg(x),\ldots,x^{n-1}g(x)\\}$ is a
generating set, which is not necessary minimal, of the (not necessarily free)
$\Delta_{\theta}$-cyclic code $C=\langle g(x)\rangle$ over $R.$ We consider
the subcode $C_{k}=\langle g(x),xg(x),\ldots,x^{k-1}g(x)\rangle$ of $C$ for
some $k\leq n$. From there, we can obtain $Res(C_{k}),Tor(C_{k}),$ and
$\Phi(C_{k}).$ For the case of $\Delta_{\theta}$-cyclic codes over $R,$ we can
simplify our computation as follows.

Let $C$ be a $\Delta_{\theta}$-cyclic code over $R$ of length $n$ generated by
$\\{g(x),xg(x),\ldots,x^{n-1}g(x)\\},$ where
$g(x)=c_{0}+c_{1}x+c_{2}x^{2}+\cdots+c_{n-1}x^{n-1}\in R_{n,\Delta_{\theta}}$
and $c_{i}=a_{i}+b_{i}v,$ for $0\leq i\leq n-1.$ To be more precise, for an
even and an odd $n,$ the generator of $C\subseteq R^{n}$ consists of the
following vectors, respectively:

$\displaystyle g_{1}$
$\displaystyle=(a_{0}+b_{0}v,~{}a_{1}+b_{1}v,~{}\ldots,~{}a_{n-2}+b_{n-2}v,~{}a_{n-1}+b_{n-1}v),$
$\displaystyle g_{2}$
$\displaystyle=(a_{n-1}+b_{n-1}+b_{0}-b_{n-1}v,~{}a_{0}+b_{0}+b_{1}-b_{0}v,~{}\ldots,~{}a_{n-3}+b_{n-3}+b_{n-2}-b_{n-3}v,$
$\displaystyle\quad~{}a_{n-2}+b_{n-2}+b_{n-1}-b_{n-2}v),$ $\displaystyle
g_{3}$
$\displaystyle=(a_{n-2}+b_{n-2}v,~{}a_{n-1}+b_{n-1}v,~{}\ldots,~{}a_{n-4}+b_{n-4}v,~{}a_{n-3}+b_{n-3}v),$
$\displaystyle\quad\vdots$ $\displaystyle g_{n}$
$\displaystyle=(a_{1}+b_{1}+b_{2}-b_{1}v,~{}a_{2}+b_{2}+b_{3}-b_{2}v,~{}\ldots,~{}a_{n-1}+b_{n-1}+b_{0}-b_{n-1}v,~{}a_{0}+b_{0}+b_{1}-b_{0}v),$

and

$\displaystyle g_{1}$
$\displaystyle=(a_{0}+b_{0}v,~{}a_{1}+b_{1}v,~{}\ldots,~{}a_{n-2}+b_{n-2}v,~{}a_{n-1}+b_{n-1}v),$
$\displaystyle g_{2}$
$\displaystyle=(a_{n-1}+b_{n-1}+b_{0}-b_{n-1}v,~{}a_{0}+b_{0}+b_{1}-b_{0}v,~{}\ldots,~{}a_{n-3}+b_{n-3}+b_{n-2}-b_{n-3}v,$
$\displaystyle\quad~{}a_{n-2}+b_{n-2}+b_{n-1}-b_{n-2}v),$ $\displaystyle
g_{3}$
$\displaystyle=(a_{n-2}+b_{n-2}v,~{}a_{n-1}+b_{n-1}v,~{}\ldots,~{}a_{n-4}+b_{n-4}v,~{}a_{n-3}+b_{n-3}v),$
$\displaystyle\quad\vdots$ $\displaystyle g_{n}$
$\displaystyle=(a_{1}+b_{1}v,~{}a_{2}+b_{2}v,~{}\ldots,~{}a_{n-1}+b_{n-1}v,~{}a_{0}+b_{0}v).$

By applying the Lemma 2.1, we obtain the generator of $Res(C)$ for an even and
an odd $n$ that consists of the following vectors, respectively:

$\displaystyle Res(g_{1})$
$\displaystyle=(a_{0},~{}a_{1},~{}a_{2},~{}\ldots,~{}a_{n-2},~{}a_{n-1}),$
$\displaystyle Res(g_{2})$
$\displaystyle=(a_{n-1}+b_{n-1}+b_{0},~{}a_{0}+b_{0}+b_{1},~{}\ldots,~{}a_{n-3}+b_{n-3}+b_{n-2},~{}a_{n-2}+b_{n-2}+b_{n-1}),$
$\displaystyle Res(g_{3})$
$\displaystyle=(a_{n-2},~{}a_{n-1},~{}\ldots,~{}a_{n-4},~{}a_{n-3}),$
$\displaystyle\quad\vdots$ $\displaystyle Res(g_{n})$
$\displaystyle=(a_{1}+b_{1}+b_{2},~{}a_{2}+b_{2}+b_{3},~{}\ldots,~{}a_{n-1}+b_{n-1}+b_{0},~{}a_{0}+b_{0}+b_{1}),$

and

$\displaystyle Res(g_{1})$
$\displaystyle=(a_{0},~{}a_{1},~{}a_{2},~{}\ldots,~{}a_{n-2},~{}a_{n-1}),$
$\displaystyle Res(g_{2})$
$\displaystyle=(a_{n-1}+b_{n-1}+b_{0},~{}a_{0}+b_{0}+b_{1},~{}\ldots,~{}a_{n-3}+b_{n-3}+b_{n-2},~{}a_{n-2}+b_{n-2}+b_{n-1}),$
$\displaystyle Res(g_{3})$
$\displaystyle=(a_{n-2},~{}a_{n-1},~{}\ldots,~{}a_{n-4},~{}a_{n-3}),$
$\displaystyle\quad\vdots$ $\displaystyle Res(g_{n})$
$\displaystyle=(a_{1},~{}a_{2},~{}\ldots,~{}a_{n-1},~{}a_{0}).$

Moreover, we obtain the generator of $Tor(C)$ for an even and an odd $n$ that
consists of the following vectors, respectively:

$\displaystyle Tor(g_{1})$
$\displaystyle=(a_{0}+b_{0},~{}a_{1}+b_{1},~{}a_{2}+b_{2},~{}\ldots,~{}a_{n-2}+b_{n-2},~{}a_{n-1}+b_{n-1}),$
$\displaystyle Tor(g_{2})$
$\displaystyle=(a_{n-1}+b_{0},~{}a_{0}+b_{1},~{}\ldots,~{}a_{n-3}+b_{n-2},~{}a_{n-2}+b_{n-1}),$
$\displaystyle Tor(g_{3})$
$\displaystyle=(a_{n-2}+b_{n-2},~{}a_{n-1}+b_{n-1},~{}\ldots,~{}a_{n-4}+b_{n-4},~{}a_{n-3}+b_{n-3}),$
$\displaystyle\quad\vdots$ $\displaystyle Tor(g_{n})$
$\displaystyle=(a_{1}+b_{2},~{}a_{2}+b_{3},~{}\ldots,~{}a_{n-1}+b_{0},~{}a_{0}+b_{1}),$

and

$\displaystyle Tor(g_{1})$
$\displaystyle=(a_{0}+b_{0},~{}a_{1}+b_{1},~{}a_{2}+b_{2},~{}\ldots,~{}a_{n-2}+b_{n-2},~{}a_{n-1}+b_{n-1}),$
$\displaystyle Tor(g_{2})$
$\displaystyle=(a_{n-1}+b_{0},~{}a_{0}+b_{1},~{}\ldots,~{}a_{n-3}+b_{n-2},~{}a_{n-2}+b_{n-1}),$
$\displaystyle Tor(g_{3})$
$\displaystyle=(a_{n-2}+b_{n-2},~{}a_{n-1}+b_{n-1},~{}\ldots,~{}a_{n-4}+b_{n-4},~{}a_{n-3}+b_{n-3}),$
$\displaystyle\quad\vdots$ $\displaystyle Tor(g_{n})$
$\displaystyle=(a_{1}+b_{1},~{}a_{2}+b_{2},~{}\ldots,~{}a_{n-1}+b_{n-1},~{}a_{0}+b_{0}).$

For any $k\leq n,$ if $C_{k}$ is generated by the set
$\\{g_{1},g_{2},\ldots,g_{k}\\},$ then the $Res(C_{k})$ and $Tor(C_{k})$ is
generated by $\\{Res(g_{1}),Res(g_{2}),\ldots,Res(g_{k})\\}$ and
$\\{Tor(g_{1}),Tor(g_{2}),\ldots,Tor(g_{k})\\},$ respectively. This method can
reduce the computation time significantly. Using this method, we can obtain
some linear codes over $\mathbb{Z}_{4}$ with good parameters, as listed in the
table below.

Table 2: Linear codes over $\mathbb{Z}_{4}$ Generators of $C$ | $Res(C)$ | $Tor(C)$ | $C^{PS}$
---|---|---|---
$[n,4^{k_{1}}2^{k_{2}},d_{L}]$ | $[n,4^{k_{1}}2^{k_{2}},d_{L}]$ | $[n,4^{k_{1}}2^{k_{2}},d_{L}]$
$\\{g_{1}(x),xg_{1}(x)\\}$ | $[4,4^{1}2^{1},4]^{\ast}$ |  | $[8,4^{2}2^{2},4]^{\ast}$
$\\{g_{2}(x),xg_{2}(x),x^{2}g_{2}(x)\\}$ | $[4,4^{1}2^{2},2]^{\ast}$ |  | $[8,4^{2}2^{4},2]^{\ast}$
$\\{g_{3}(x),xg_{3}(x),x^{2}g_{3}(x)\\}$ | $[5,4^{2}2^{1},4]^{\ast\ast\ast}$ |  |
$\\{g_{4}(x),xg_{4}(x),x^{2}g_{4}(x)\\}$ | $[6,4^{2}2^{1},4]^{\ast}$ |  |
$\\{g_{5}(x),xg_{5}(x),x^{2}g_{5}(x),x^{3}g_{5}(x),x^{4}g_{5}(x)\\}$ | $[6,4^{2}2^{3},4]^{\ast)}$ |  | $[12,4^{4}2^{6},4]^{\ast}$
$\\{g_{6}(x),xg_{6}(x),x^{2}g_{6}(x),x^{3}g_{6}(x)\\}$ | $[6,4^{3}2^{1},4]^{\ast}$ |  | $[12,4^{6}2^{2},4]^{\ast}$
$\begin{aligned} \\{g_{7}(x),xg_{7}(x),x^{2}g_{7}(x),x^{3}g_{7}(x),\\\ x^{4}g_{7}(x),x^{5}g_{7}(x),x^{6}g_{7}(x)\\}\end{aligned}$ | $[8,4^{4}2^{3},4]^{\ast\ast\ast}$ |  | $[16,4^{8}2^{6},4]^{\ast}$
$\begin{aligned} \\{g_{8}(x),xg_{8}(x),x^{2}g_{8}(x),x^{3}g_{8}(x),\\\ x^{4}g_{8}(x),x^{5}g_{8}(x)\\}\end{aligned}$ |  | $[8,4^{5}2^{1},4]^{\ast\ast\ast}$ | $[16,4^{10}2^{2},4]^{\ast}$
$\\{g_{9}(x),xg_{9}(x),x^{2}g_{9}(x)\\}$ | $[9,4^{3}2^{0},7]^{\ast\ast}$ |  |
$\\{g_{10}(x),xg_{10}(x),x^{2}g_{10}(x)\\}$ | $[10,4^{3}2^{0},8]^{\ast\ast}$ |  |
$\\{g_{11}(x),xg_{11}(x)\\}$ | $[15,4^{2}2^{0},15]^{\ast\ast}$ |  |
$\\{g_{12}(x),xg_{12}(x),x^{2}g_{12}(x)\\}$ |  | $[18,4^{3}2^{0},14]^{\ast\ast}$ |

In Table LABEL:Tabel-2,

  * •

$g_{1}(x)=(1+3v)+2x+(3+3v)x^{2}.$

  * •

$g_{2}(x)=(1+v)+(2+2v)x+(1+3v)x^{2}.$

  * •

$g_{3}(x)=(1+3v)+2vx+(2+2v)x^{2}+2vx^{3}+(1+3v)x^{4}.$

  * •

$g_{4}(x)=3+(1+3v)x+(3+v)x^{2}+(2+3v)x^{3}.$

  * •

$g_{5}(x)=(3+3v)+(1+3v)x+(3+3v)x^{3}+(3+v)x^{4}+2x^{5}.$

  * •

$g_{6}(x)=(1+v)+x+(2+v)x^{2}+vx^{3}+3vx^{5}.$

  * •

$g_{7}(x)=(1+v)+3x^{1}+(2+3v)x^{2}+(3+v)x^{3}+(2+2v)x^{4}+2x^{6}+x^{7}.$

  * •

$g_{8}(x)=2v+(2+3v)x^{1}+(1+3v)x^{2}+(1+2v)x^{3}+2vx^{4}+(1+v)x^{5}+x^{6}+3vx^{7}.$

  * •

$g_{9}(x)=(1+v)+(1+3v)x^{1}+(3+2v)x^{2}+3x^{3}+(3+3v)x^{4}+(2+2v)x^{5}+(2+3v)x^{6}+(3+3v)x^{7}+(1+3v)x^{8}.$

  * •

$g_{10}(x)=(1+3v)+(2+2v)x^{1}+3x^{2}+vx^{4}+3x^{5}+(3+3v)x^{6}+x^{7}+x^{8}.$

  * •

$g_{11}(x)=(1+v)+(2+v)x^{2}+(3+2v)x^{3}+(3+v)x^{5}+(1+2v)x^{6}+2x^{7}+(3+v)x^{8}+3x^{9}+2vx^{10}+(3+2v)x^{11}+(1+2v)x^{12}+(2+v)x^{13}+(1+3v)x^{14}.$

  * •

$g_{12}(x)=1+2vx^{1}+(3+2v)x^{2}+(2+2v)x^{3}+(1+2v)x^{4}+x^{5}+vx^{6}+x^{7}+(1+2v)x^{8}+(3+2v)x^{9}+(1+3v)x^{10}+(3+3v)x^{11}+(2+v)x^{12}+(3+3v)x^{13}+x^{14}+(2+v)x^{15}+(2+2v)x^{16}.$

Exactly the same method can also be used for $\Phi(C_{k}).$ This observation
brings us to the conclusion that the codes $Res(C_{k}),$ $Tor(C_{k}),$ and
$\Phi(C_{k})$ have at most $4^{k},~{}4^{k}$, and $4^{2k}$ codewords,
respectively. Moreover, the similar observation can also be applied to the
codes over $\mathbb{Z}_{4}+u\mathbb{Z}_{4},$ with $u^{2}=1,$ investigated by
Sharma and Bhaintwal [23] and it is easy to verify that in this case, the
codes $Res(C_{k}),$ $Tor(C_{k}),$ and $\Phi(C_{k})$ have at most $4^{2k}$
codewords.

###  5.2 Construction from another method

By applying the construction in [24], we can obtain even more new linear codes
over $\mathbb{Z}_{4}$ with the highest known minimum Lee distance, previously
unknown to exist in the database [2]. As illustrations, we provide several
examples below. In the following examples, the code $C_{i}$ refers to the code
generated by the polynomial $g_{i}(x)$ in Table LABEL:Tabel-2.

######  Example 5.2.

In these examples, we use Lemma 4.6 in [24] to construct new linear codes over
$\mathbb{Z}_{4}.$

  * •

From the code with parameters $[6,4^{2}2^{0},6]$ in Table LABEL:Tabel-1, we
obtained linear codes with parameters $[12,4^{2}2^{1},12]^{***},$
$[24,4^{3}2^{0},24]^{**},$ $[24,4^{2}2^{2},24]^{***},$
$[48,4^{3}2^{1},48]^{***},$ and $[48,4^{2}2^{3},48]^{***}.$

  * •

From the code with parameters $[12,4^{3}2^{0},10]$ in Table LABEL:Tabel-1, we
obtained linear codes with parameters $[24,4^{3}2^{1},20]^{***},$
$[48,4^{4}2^{0},40]^{**},$ and $[48,4^{3}2^{2},40]^{***}.$

  * •

From the code $C_{3}$ with parameters $[5,4^{2}2^{1},4]$ we obtained linear
codes with parameters $[10,4^{2}2^{2},8]^{***},$ $[20,4^{3}2^{1},16]^{***},$
$[20,4^{2}2^{3},16]^{***},$ $[40,4^{3}2^{2},32]^{***},$ and
$[40,4^{2}2^{4},32]^{***}.$

  * •

From the code $C_{5}$ with parameters $[6,4^{2}2^{3},4]$ we obtained a linear
code with parameters $[12,4^{2}2^{4},8]^{*}.$

  * •

From the code $C_{6}$ with parameters $[6,4^{3}2^{1},4]$ we obtained a linear
code with parameters $[12,4^{3}2^{2},8]^{*}.$

  * •

From the code $C_{9}$ with parameters $[9,4^{3}2^{0},7]$ we obtained linear
codes with parameters $[18,4^{3}2^{1},14]^{**},$ $[36,4^{4}2^{0},28]^{**},$
and $[36,4^{3}2^{2},28]^{***}.$

  * •

From the code $C_{10}$ with parameters $[10,4^{3}2^{0},8]$ we obtained a
linear code with parameters $[40,4^{4}2^{0},32]^{**}.$

######  Example 5.3.

In these examples, we use Lemma 4.3 in [24] to construct new linear codes over
$\mathbb{Z}_{4}.$

  * •

From the code with parameters $[12,4^{3}2^{0},10]$ in Table LABEL:Tabel-1 and
the code with parameters $[24,4^{3}2^{0},24],$ respectively, we obtained a
linear code with parameters $[36,4^{3}2^{0},34]^{**}.$

  * •

From the code $C_{3}$ with parameters $[5,4^{2}2^{1},4]$ and the code with
parameters $[12,4^{2}2^{1},12]$ we obtained a linear code with parameters
$[17,4^{2}2^{1},16]^{***}.$

  * •

From the code $C_{6}$ with parameters $[6,4^{3}2^{1},4]$ and the code with
parameters $[20,4^{3}2^{1},16]$ we obtained a linear code with parameters
$[26,4^{3}2^{1},20]^{***}.$

##  6 Concluding remarks

We have investigated the algebraic structures of skew-cyclic codes, also known
as $\theta$-cyclic codes, with a derivation $\Delta_{\theta}$ over the ring
$R=\mathbb{Z}_{4}+v\mathbb{Z}_{4},$ with $v^{2}=v,$ extending the observation
of Boucher and Ulmer [9], where they defined and considered the skew-cyclic
codes with derivation over a finite field. To our best knowledge, this is the
third attempt after the paper by Sharma and Bhaintwal [23] and Patel and
Prakash [21]. Our work can also be viewed as a partial answer of the question
of Dougherty [10], page 99:

Question 6.5 Determine the structure of all $\theta$-cyclic codes in
$R[x;\theta]/\langle x^{n}-\lambda\rangle$ where $R$ is a finite commutative
Frobeniüs ring.

As a consequence, we constructed several new codes over $\mathbb{Z}_{4}$
unknown to exist before due to the database [2], with good parameters. All
computations to find the codes were done by Python and Magma computational
algebra system [5].

Regarding the derivation, it is easy to show that the map
$\Delta_{\theta}(x)=(3+2v)(\theta(x)-x)$ also defines a derivation on $R.$ All
properties which hold for the derivation
$\Delta_{\theta}(x)=(1+2v)(\theta(x)-x)$ in this paper also hold for the
derivation $\Delta_{\theta}(x)=(3+2v)(\theta(x)-x).$

The method explained in Section 5 can be modified for the case of derivation
$\Delta_{\theta}(x)=(3+2v)(\theta(x)-x).$ There is some hope to obtain many
more examples of linear codes over $\mathbb{Z}_{4}$ with better parameters. As
an example, $Tor(C)$ of the code $C:=\langle g(x),xg(x),x^{2}g(x)\rangle,$
with $g(x)=3v+(2+v)x+3vx^{2}+vx^{3},$ has parameters $[4,4^{1}2^{2},4],$ which
is better than the one in Table LABEL:Tabel-1.

## Acknowledgments

This research is supported by the Institut Teknologi Bandung (ITB) and the
Ministry of Education, Culture, Research and Technology (_Kementerian
Pendidikan, Kebudayaan, Riset dan Teknologi (Kemdikbudristek)_), Republic of
Indonesia.

## References

  * [1] T. Abualrub, N. Aydin, and P. Seneviratne, "On $\theta$-cyclic codes over ${\mathbb{F}}_{2}+v{\mathbb{F}}_{2},$" _Australas. J. Combin._ 54(2) (2012), 115-126.
  * [2] N. Aydin and T. Asamov, "The database of $\mathbb{Z}_{4}$ codes," available at http://quantumcodes.info/Z4 (accessed at May 2022).
  * [3] R.K. Bandi and M. Bhaintwal, "Codes over $\mathbb{Z}_{4}+v\mathbb{Z}_{4},$" in _Proc. Commun. Inform. ICACCI,_ Sep. 2004, 422-427.
  * [4] N. Benbelkacem, M.F. Ezerman, T. Abualrub, N. Aydin, and A. Batoul, "Skew cyclic codes over ${\mathbb{F}}_{4}R,$" _J. Algebra Appl._ 21(4) 2250065 (2022).
  * [5] W. Bosma, J.J. Cannon, C. Fieker, and A. Steel, _Handbook of magma functions,_ Sydney 2011.
  * [6] D. Boucher, W. Geiselmann, and F. Ulmer, "Skew cyclic codes," _Appl. Algebra Engrg. Comm. Comput._ 18(4) (2007), 379-389.
  * [7] D. Boucher and F. Ulmer, "Coding with skew polynomial rings," _J. Symb. Comput._ 44(3-4) (2009), 1644-1656.
  * [8] D. Boucher and F. Ulmer, "Coding as modules over skew polynomial rings," in Proceedings of the 12th IMA Conference on Cryptography and Coding, _Lecture Notes in Comput. Sci._ 5921 (2009), 38-55.
  * [9] D. Boucher and F. Ulmer, "Linear codes using skew polynomials with automorphisms and derivations," _Des. Codes Cryptogr._ 70(3) (2014), 405-431.
  * [10] S.T. Dougherty, _Algebraic coding theory over finite commutative rings,_ SpringerBriefs in Mathematics, Springer, 2017.
  * [11] J. Gao, F. W. Fu, and Y. Gao, "Some classes of linear codes over $\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ and their applications to construct good and new $\mathbb{Z}_{4}$-linear codes," _Appl. Algebra Engrg. Comm. Comput._ 28(2) (2017), 131-153.
  * [12] F. Gursoy, I. Siap, and B. Yildiz, "Construction of skew cyclic codes over ${\mathbb{F}}_{q}+v{\mathbb{F}}_{q},$" _Adv. Math. Commun._ 8(3) (2014), 313-322.
  * [13] A.R. Hammons, P.V. Kumar, A.R. Calderbank, N.J.A. Sloane, and P. Solé, "The $\mathbb{Z}_{4}$-linearity of Kerdock, Preparata, Goethals and Related Codes," _IEEE Trans. Inform. Theory_ 40(2) (1994), 301-319.
  * [14] W.C. Huffman and V. Pless, _Fundamentals of Error Correcting Codes,_ Cambridge University Press, 2003.
  * [15] Irwansyah, A. Barra, S.T. Dougherty, A. Muchlis, I. Muchtadi-Alamsyah, P. Sole, D. Suprijanto, and O. Yemen, "$\Theta_{S}$-cyclic codes over $A_{k},$" _Int. J. Comput. Math. Comput. Syst. Theory_ 1(1) (2016), 14-31.
  * [16] Irwansyah, A. Barra, A. Muchlis, I. Muchtadi-Alamsyah, and D. Suprijanto, "skew cyclic codes over $B_{k},$" _J. Appl. Math. Comput._ 57(1-2) (2018), 69-84.
  * [17] Irwansyah and D. Suprijanto, "Structure of linear codes over the ring $B_{k},$" _J. Appl. Math. Comput._ 58(1-2) (2018), 755-775.
  * [18] J. Liu and H. Liu, "Construction of cyclic DNA codes over the ring $\mathbb{Z}_{4}+v\mathbb{Z}_{4},$" _IEEE Access_ 8 (2020), 111200-111207.
  * [19] B.R. McDonald, _Finite Rings with Identity_. Marcel Dekker Inc., New York, 1974.
  * [20] O. Ore, "Theory of non-commutative polynomials," _Annals. Math._ 34 (1933), 480-508.
  * [21] S. Patel and O. Prakash, "$(\theta,\delta_{\theta})$-Cyclic codes over ${\mathbb{F}}_{q}[u,v]/\langle u^{2}-u,v^{2}-vuv-vu\rangle$," _Des. Codes Cryptogr._ (to appear) , DOI: https://doi.org/10.1007/s10623-021-00964-7
  * [22] E. Prange, "Cyclic error-correcting codes in two symbols," _Airforce Cambridge Research Center, Cambridge, MA, Tech. Rep. AFCRC-TN_ (1957), 57-103.
  * [23] A. Sharma and M. Bhaintwal, "A class of skew cyclic codes over $\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ with derivation," _Adv. Math. Commun._ 12(4) (2018), 723-739.
  * [24] H.C. Tang and D. Suprijanto, "New optimal linear codes over $\mathbb{Z}_{4},$" _Bull. Aust. Math. Soc._ (to appear), DOI: https://doi.org/10.1017/S0004972722000399.
  * [25] Zhe-Xian Wan, _Quaternary codes,_ World Scientific, 1997.