# Cyclic codes over a non-chain ring $R_{e,q}$ and their application to LCD
codes

Habibul Islam  Department of Mathematics, Indian Institute of Technology
Patna, India<EMAIL_ADDRESS>Edgar Martínez-Moro  Institute of
Mathematics, University of Valladolid, Spain<EMAIL_ADDRESS>Om Prakash
Department of Mathematics, Indian Institute of Technology Patna, India
<EMAIL_ADDRESS>
###### Abstract

Let $\mathbb{F}_{q}$ be a finite field of order $q$, a prime power integer
such that $q=et+1$ where $t\geq 1,e\geq 2$ are integers. In this paper, we
study cyclic codes of length $n$ over a non-chain ring
$R_{e,q}=\mathbb{F}_{q}[u]/\langle u^{e}-1\rangle$. We define a Gray map
$\varphi$ and obtain many maximum-distance-separable (MDS) and optimal
$\mathbb{F}_{q}$-linear codes from the Gray images of cyclic codes. Under
certain conditions we determine linear complementary dual (LCD) codes of
length $n$ when $\gcd(n,q)\neq 1$ and $\gcd(n,q)=1$, respectively. It is
proved that a cyclic code $\mathcal{C}$ of length $n$ is an LCD code if and
only if its Gray image $\varphi(\mathcal{C})$ is an LCD code of length $4n$
over $\mathbb{F}_{q}$. Among others, we present the conditions for existence
of free and non-free LCD codes. Moreover, we obtain many optimal LCD codes as
the Gray images of non-free LCD codes over $R_{e,q}$.

###### keywords:

Cyclic code , LCD code , Optimal code , MDS code , self-dual code , Gray map.

###### MSC:

[2010] 94B15, 94B05, 94B60.

††journal: Discrete Math.

##  1 Introduction

Cyclic codes are one of the most important classes of linear codes and they
possess rich algebraic properties. A cyclic code of length $n$ over a ring (or
field) $R$ is defined as an $R$-submodule (resp. subspace) of $R^{n}$ closed
under the cyclic shift. Over a finite field $\mathbb{F}_{q}$ they can be
easily classified via ideals of the ring $\mathbb{F}_{q}[x]/\langle
x^{n}-1\rangle$ but usually their structure is non-trivial for the case of
finite rings. The algebraic structure such as generator polynomials, generator
matrices are useful to obtain their parameters like rank or minimum distances.
In the field case the standard notation $[n,k,d]$ is used to represent a
linear code of length $n$, dimension $k$ and distance $d$. There are many
well-known theoretical relations among these parameters, and some of them are
Singleton bound, Plotkin bound, Gilbert-Varshamov bound, Hamming bound and
Griesmer bound, etc. A code attaining any of these bounds is called maximum-
distance-separable (MDS) for that bound. In 1997, Kanwar and Lopez-Permouth
[23] determined the structure of cyclic codes over $\mathbb{Z}_{p^{m}}$ while
Pless et al. [31] obtained self-dual cyclic codes over $\mathbb{Z}_{4}$.
Later, in 1999, Bonnecaze and Udaya [6] studied cyclic codes over the ring
$\mathbb{F}_{2}+u\mathbb{F}_{2},u^{2}=0$. Also, Blackford [5] classified all
cyclic codes of length $2n$ ($n$ is odd integer) over $\mathbb{Z}_{4}$. After
that, Abualrub et al. [2] discussed these codes of length $2^{e}$ over
$\mathbb{Z}_{4}$. Also, in a more general setup, cyclic and negacyclic codes
over finite chain rings are extensively studied in [10]. Later, many papers
contributed significant tools and results to obtain the structural properties
of these codes over finite rings, see [1, 3, 15, 20, 21, 22, 37].

Linear complementary dual (shortly LCD) codes were introduced by Massey [29],
in 1992. An LCD code is defined as a linear code having a trivial intersection
with its dual code. These codes have gained serious attention due to their
recent successful application in cryptography [8]. In 2016, Carlet and Guilley
[8] shown that LCD codes with possibly large minimum distances prevent the
resistance against side-channel attacks in a cryptosystem. In addition, these
codes have a much simpler nearest-codeword decoding algorithm than linear
codes, and are widely used in communications systems, data storage, and
consumer electronics. In 1994, Yang and Massey [35] completely classified
cyclic LCD codes over finite fields. Later, Sendrier [32] shown that LCD codes
attain the Gilbert-Varshamov bound. In 2009 Esmaeili and Yari [13] studied
quasi-cyclic LCD codes. Later, Li et al. [24] obtained many optimal and good
LCD codes over finite fields. Also, Dougherty et al. [11] discussed LCD codes
along with the linear programming bounds. Further, Galvez et al. [14]
determined many bounds on LCD codes over $\mathbb{F}_{2}$.

Recently, LCD codes over finite chain rings are extensively studied in [12,
25, 26]. They have determined necessary and sufficient conditions of a linear
code to be an LCD code and show the existence of asymptotically good LCD codes
over finite chain rings. Therefore, it is still open to investigate the
structure of LCD codes over finite commutative non-chain rings. Very recently,
these codes have been explored over a non-chain ring
$\mathbb{F}_{q}+u\mathbb{F}_{q}+v\mathbb{F}_{q}$ by Yadav et al. [34]. Here,
we consider a class of finite commutative non-chain rings with unity
$R_{e,q}=\mathbb{F}_{q}[u]/\langle u^{e}-1\rangle$ where $q=et+1$ and $e\geq
2,t\geq 1$ are integers. Then $R_{e,q}$ is a semi-local Frobenius ring of
order $q^{e}$. The main concern of this paper is to obtain the structure of
cyclic and LCD codes of length $n$ over $R_{e,q}$ and to construct optimal
linear (or LCD) codes from them. With the help of a new Gray map $\varphi$, we
derive many optimal and MDS codes from cyclic codes over $R_{e,q}$. Further,
we obtain necessary and sufficient conditions of cyclic codes to LCD codes.
Finally, many optimal LCD codes are presented to validate our obtained
results.

##  2 Preliminary

Let $p$ be an odd prime and $q=p^{m}$ such that $q=et+1$ where $t,m\geq
1,e\geq 2$ are integers. Let $\mathbb{F}_{q}$ be a field of order $q$ and
denote $R_{e,q}=\mathbb{F}_{q}[u]/\langle u^{e}-1\rangle$. Then it is easy to
check that $R_{e,q}$ is a finite commutative semi-local, Frobenius and non-
chain ring with unity and characteristic $p$. Also $R_{e,q}$ can be written as
$R_{e,q}=\mathbb{F}_{q}+u\mathbb{F}_{q}+u^{2}\mathbb{F}_{p}+\dots+u^{e-1}\mathbb{F}_{q}$,
where $u^{e}=1$, thus an arbitrary element $r\in R_{e,q}$ can be written as
$r=a_{0}+ua_{1}+u^{2}a_{2}+\dots+u^{e-1}a_{e-1}$, where
$a_{i}\in\mathbb{F}_{q}$ for $0\leq i\leq e-1$. Recall that a nonempty subset
$\mathcal{C}\subseteq R^{n}_{e,q}$ is said to be a linear code of length $n$
if it is an $R_{e,q}$-submodule of $R^{n}_{e,q}$ and elements in $\mathcal{C}$
are called codewords. Also, the rank of a code $\mathcal{C}$ is the minimum
number of generators for $\mathcal{C}$ and the free rank is the rank of
$\mathcal{C}$ when it is a free module over $R_{e,q}$. A typical linear code
of length $n$ and rank $k$ is denoted by $[n,k]$. The inner product between
two vectors $x=(x_{0},x_{1},\cdots,x_{n-1}),y=(y_{0},y_{1},\cdots,y_{n-1})\in
R_{e,q}^{n}$ is defined as $x\cdot y=\sum_{i=0}^{n-1}x_{i}y_{i}$. The
orthogonal of a linear code $\mathcal{C}$ is denoted by $\mathcal{C}^{\perp}$
and defined by $\mathcal{C}^{\perp}=\\{x\in R_{e,q}^{n}\mid x\cdot
y=0,~{}\forall~{}y\in\mathcal{C}\\}$. Note that the orthogonal code
$\mathcal{C}^{\perp}$ is also a linear code of length $n$ over $R_{e,q}$. The
linear code $\mathcal{C}$ is said to be self-orthogonal if
$\mathcal{C}\subseteq\mathcal{C}^{\perp}$ and self-dual if
$\mathcal{C}=\mathcal{C}^{\perp}$.
Since $e\mid(q-1)$, the polynomial $u^{e}-1$ uniquely splits into linear
factors over the ring $\mathbb{F}_{q}[u]$. Suppose

$\displaystyle u^{e}-1=\prod_{i=1}^{e}(u-\alpha_{i})\in\mathbb{F}_{q}[u].$

Let $G_{i}=u-\alpha_{i}$ and $\widehat{G}_{i}=\frac{(u^{e}-1)}{G_{i}}$, for
$1\leq i\leq e$. Since $\gcd(G_{i},\widehat{G}_{i})=1$, there exist
$z_{i},h_{i}\in\mathbb{F}_{q}[u]$ such that
$z_{i}G_{i}+h_{i}\widehat{G}_{i}=1$, for $1\leq i\leq e$. Let
$\mu_{i}=h_{i}\widehat{G}_{i}$, for $1\leq i\leq e$.

######  Lemma 1.

Assuming the same notations discussed above and mod $(u^{e}-1)$, we have
$\sum_{i=1}^{e}\mu_{i}=1,$ and

$\mu_{i}\mu_{j}=\begin{cases}\mu_{i},&\text{if $i=j$}\\\ 0,&\text{if $i\neq
j$.}\end{cases}$

Therefore, $\\{\mu_{1},\mu_{2}\dots,\mu_{e}\\}$ is a set of orthogonal
idempotent elements in $R_{e,q}$.

###### Proof.

For $i\neq j$, we have $(u-\alpha_{i})\mid h_{j}\widehat{G}_{j}$ and
$(u-\alpha_{j})\mid h_{i}\widehat{G}_{i}$. Thus,
$\mu_{i}\mu_{j}=h_{i}\widehat{G}_{i}h_{j}\widehat{G}_{j}=0\pmod{u^{e}-1}$. To
prove $\mu_{i}^{2}=\mu_{i}$, it is enough to show
$(u-\alpha_{s})\mid(\mu_{i}^{2}-\mu_{i})$ for all $s=1,2,\dots,e$. For $i\neq
s$, it follows from the construction. If $i=s$, then

$\displaystyle\mu_{i}^{2}-\mu_{i}$ $\displaystyle=\mu_{i}(\mu_{i}-1)$
$\displaystyle=h_{i}\widehat{G}_{i}(h_{i}\widehat{G}_{i}-1)$
$\displaystyle=z_{i}G_{i}(z_{i}G_{i}-1),~{}(\text{since}~{}z_{i}G_{i}+h_{i}\widehat{G}_{i}=1).$

Therefore, $(u-\alpha_{i})\mid(\mu_{i}^{2}-\mu_{i})$, and hence
$\mu_{i}^{2}=\mu_{i}$. Again,

$\displaystyle 1-\sum_{i=1}^{e}\mu_{i}$
$\displaystyle=\prod_{i=1}^{e}(1-\mu_{i})~{}(\text{since}~{}\mu_{i}^{2}=\mu_{i},\mu_{i}\mu_{j}=0,i\neq
j)$ $\displaystyle=\prod_{i=1}^{e}(1-h_{i}\widehat{G}_{i})$
$\displaystyle=\prod_{i=1}^{e}z_{i}G_{i}$ $\displaystyle=0~{}\pmod{u^{e}-1}.$

Hence, $\sum_{i=1}^{e}\mu_{i}=1\pmod{u^{e}-1}$. ∎

Taking into account Lemma 1 and the Chinese Remainder Theorem, we have the
decomposition

$\displaystyle
R_{e,q}=\bigoplus_{i=1}^{e}\mu_{i}R_{e,q}\cong\bigoplus_{i=1}^{e}\mu_{i}\mathbb{F}_{q}.$

Therefore, any element $r\in R_{e,q}$ can be uniquely expressed as
$r=\sum_{i=1}^{e}s_{i}\mu_{i}$, for $s_{i}\in\mathbb{F}_{q},1\leq i\leq e$.
Then the units of $R_{e,q}$ can be classified by the result that $r$ is a unit
in $R_{e,q}$ if and only if $s_{i}$ is a unit in $\mathbb{F}_{q}$, for $1\leq
i\leq e$.

Let $GL_{e}(\mathbb{F}_{q})$ be the set of all $e\times e$ invertible matrices
over $\mathbb{F}_{q}$. Let $M\in GL_{e}(\mathbb{F}_{q})$ such that
$MM^{T}=\gamma I_{e}$, where $M^{T}$ is the transpose matrix of $M$, $I_{e}$
is the identity matrix of order $e$ and
$\gamma\in\mathbb{F}_{q}^{*}=\mathbb{F}_{q}\setminus\\{0\\}$. Since $q$ is
odd, so for the existence of such matrix $M$, we restrict
$\gamma\in\mathbb{F}_{q}^{*}$ such that $\gamma^{e}$ is a square in
$\mathbb{F}_{q}^{*}$ (see [[17], Definition 7.6.4 and Proposition 7.6.6]).
Now, we define a map $\varphi:R_{e,q}\longrightarrow\mathbb{F}_{q}^{e}$ by

$\displaystyle\varphi(r)=(s_{1},s_{2},\dots,s_{e})M=\boldsymbol{r}M,$ (1)

where $r=\sum_{i=1}^{e}s_{i}\mu_{i}\in R_{e,q}$, for
$s_{i}\in\mathbb{F}_{q},1\leq i\leq e$. Here, we use $\boldsymbol{r}$ for the
vector $(s_{1},s_{2},\dots,s_{e})$. The map $\varphi$ can be extended to
$R_{e,q}^{n}$ as follows:

$\displaystyle\varphi:R_{e,q}^{n}\longrightarrow\mathbb{F}_{q}^{en}$

by

$\displaystyle\varphi(r_{0},r_{1},\dots,r_{n-1})=$
$\displaystyle[(s_{0,1},s_{0,2},\dots,s_{0,e})M,(s_{1,1},s_{1,2},\dots,s_{1,e})M,\dots,(s_{n-1,1},s_{n-1,2},\dots,s_{n-1,e})M]$
$\displaystyle=$
$\displaystyle(\boldsymbol{r_{0}}M,\boldsymbol{r_{1}}M,\dots,\boldsymbol{r_{n-1}}M),$
(2)

where $r_{i}=\sum_{j=1}^{e}s_{i,j}\mu_{j}$, for $0\leq i\leq n-1$.

The Hamming weight $w_{H}(c)$ is the number of non-zero components of the
codeword $c=(c_{0},c_{1},\dots,c_{n-1})\in\mathcal{C}$ and the distance
between two codewords is given by
$d_{H}(c^{\prime},c^{\prime\prime})=w_{H}(c^{\prime}-c^{\prime\prime})$. The
Hamming distance of $\mathcal{C}$ is defined as
$d_{H}(\mathcal{C})=\min\\{d_{H}(c^{\prime},c^{\prime\prime})\mid
c^{\prime}\neq c^{\prime\prime},$ for all
$c^{\prime},c^{\prime\prime}\in\mathcal{C}\\}$. The Gray weight of any element
$r\in R_{e,q}$ is $w_{G}(r)=w_{H}(\varphi(r))$ and the Gray weight for
$\overline{r}=(r_{0},r_{1},\dots,r_{n-1})\in R_{e,q}^{n}$ is
$w_{G}(\overline{r})=\sum_{i=0}^{n-1}w_{G}(r_{i})$. The Gray distance between
any two codewords $c^{\prime},c^{\prime\prime}$ is defined as
$d_{G}(c^{\prime},c^{\prime\prime})=w_{G}(c^{\prime}-c^{\prime\prime})$ and
the Gray distance of $\mathcal{C}$ is
$d_{G}(\mathcal{C})=\min\\{d_{G}(c^{\prime},c^{\prime\prime})\mid
c^{\prime}\neq c^{\prime\prime},c^{\prime},c^{\prime\prime}\in\mathcal{C}\\}$.
Now, we present an example for $e=3,q=7$ to show the procedure of finding
orthogonal idempotent elements and the corresponding Gray map.

######  Example 1.

Let $e=3,t=2,q=3\cdot 2+1=7$ and $R_{3,7}=\mathbb{F}_{7}[u]/\langle
u^{3}-1\rangle$. Then in $\mathbb{F}_{7}[u]$, we have

$\displaystyle u^{3}-1=(u+3)(u+5)(u+6).$

Let $G_{1}=u+3,G_{2}=u+5,G_{3}=u+6$. Then

$\displaystyle\widehat{G}_{1}=$
$\displaystyle\frac{u^{3}-1}{G_{1}}=(u+5)(u+6),$
$\displaystyle\widehat{G}_{2}=$
$\displaystyle\frac{u^{3}-1}{G_{2}}=(u+3)(u+6),$
$\displaystyle\widehat{G}_{3}=$
$\displaystyle\frac{u^{3}-1}{G_{3}}=(u+3)(u+5).$

Therefore,

$\displaystyle h_{1}=6,z_{1}=(u+3),\mu_{1}=6(u+5)(u+6),$ $\displaystyle
h_{2}=3,z_{2}=4(u+4),\mu_{2}=3(u+3)(u+6),$ $\displaystyle
h_{3}=5,z_{3}=2(u+2),\mu_{3}=5(u+3)(u+5).$

Then $\\{\mu_{1},\mu_{2},\mu_{3}\\}$ is the set of orthogonal idempotent
elements in $R_{3,7}=\mathbb{F}_{7}[u]/\langle u^{3}-1\rangle$, and hence by
the Chinese Remainder Theorem, we have
$R_{3,7}=\mu_{1}R_{3,7}\oplus\mu_{2}R_{3,7}\oplus\mu_{3}R_{3,7}\cong\mu_{1}\mathbb{F}_{7}\oplus\mu_{2}\mathbb{F}_{7}\oplus\mu_{3}\mathbb{F}_{7}$.
Again, any element $r\in R_{3,7}$ can be written as

$\displaystyle r=a_{0}+ua_{1}+u^{2}a_{2}=$
$\displaystyle(a_{0}+4a_{1}+2a_{2})\mu_{1}+(a_{0}+2a_{1}+4a_{2})\mu_{2}+(a_{0}+a_{1}+a_{2})\mu_{3}.$

Thus, the map $\varphi:R_{3,7}\longrightarrow\mathbb{F}^{3}_{7}$ is given by

$\displaystyle\varphi(r=a_{0}+ua_{1}+u^{2}a_{2})=(a_{0}+4a_{1}+2a_{2},a_{0}+2a_{1}+4a_{2},a_{0}+a_{1}+a_{2})M,$

where $a_{i}\in\mathbb{F}_{7}$, for $0\leq i\leq 2$ and $M\in
GL_{3}(\mathbb{F}_{7})$ with $MM^{T}=\gamma I_{3}$,
$\gamma^{3}\in\mathbb{F}_{7}^{*}$ is a square.

In Table 1, we give some examples of orthogonal idempotent set for different
values of $e$ and $q$. Also, we calculate the canonical components of $r\in
R_{e,q}$ such that $r=\sum_{i=1}^{e}\mu_{i}s_{i}$, which are useful in the
subsequent examples.

Now, we remind some well-known techniques from [15] as follows: Let $A_{i}$ be
a linear code for $1\leq i\leq e$, we denote
$\bigoplus_{i=1}^{e}A_{i}=\\{a_{1}+a_{2}+\dots+a_{e}\mid a_{i}\in
A_{i}~{}\forall~{}i\\}$. Let $\mathcal{C}$ be a linear code of length $n$ over
$R_{e,q}$. For $1\leq i\leq e$, define
$\mathcal{C}_{i}:=\\{s_{i}\in\mathbb{F}_{q}^{n}\mid\exists~{}s_{1},s_{2},\dots,s_{i-1},s_{i+1},\dots,s_{e}$
such that $\sum_{i=1}^{e}s_{i}\mu_{i}\in\mathcal{C}\\}$. Then
$\mathcal{C}_{i}$ is a linear code of length $n$ over $\mathbb{F}_{q}$ for
$1\leq i\leq e$. Also, the code $\mathcal{C}$ can be uniquely expressed as
$\mathcal{C}=\bigoplus_{i=1}^{e}\mu_{i}\mathcal{C}_{i}$ and
$\mid\mathcal{C}\mid=\prod_{i=1}^{e}\mid\mathcal{C}_{i}\mid$. Moreover, a
generator matrix for the code $\varphi(\mathcal{C})$ can be obtained as
$M=\begin{pmatrix}\varphi(\mu_{1}M_{1})\\\ \varphi(\mu_{2}M_{2})\\\ \vdots\\\
\varphi(\mu_{e}M_{e})\end{pmatrix}$, where $M_{i}$ is a generator matrix of
$\mathcal{C}_{i}$, for $1\leq i\leq e$.

######  Lemma 2.

The map $\varphi$ defined in equation (2) is linear and isometric from
($R_{e,q}^{n}$, Gray distance) to ($\mathbb{F}_{q}^{en}$, Hamming distance).

###### Proof.

It follows from the definition of the Gray weight. ∎

######  Theorem 1.

Let $\mathcal{C}$ be a linear code of length $n$ over $R_{e,q}$,
$\mid\mathcal{C}\mid=q^{ek}$ and Gray distance $d_{G}$.

  1. 1.

Then $\varphi(\mathcal{C})$ is an $[en,k,d_{H}]$ linear code over
$\mathbb{F}_{q}$ where $d_{H}=d_{G}$.

  2. 2.

$\varphi(\mathcal{C}^{\perp})=(\varphi(\mathcal{C}))^{\perp}$.

  3. 3.

$\varphi(\mathcal{C})$ is a self-orthogonal (Euclidean) linear code of length
$en$ over $\mathbb{F}_{q}$ if $\mathcal{C}$ is a self-orthogonal (Euclidean)
linear code over $R_{e,q}$.

  4. 4.

$\varphi(\mathcal{C})$ is a self-dual code if and only if $\mathcal{C}$ is a
self-dual code.

###### Proof.

  1. 1.

Since $\varphi$ is a linear map, $\varphi(\mathcal{C})$ is a linear code of
length $en$ over $\mathbb{F}_{q}$. Also, $\varphi$ being isometric,
$\varphi(\mathcal{C})$ is an $[en,k,d_{H}]$ linear code over $\mathbb{F}_{q}$
with $d_{H}=d_{G}$.

  2. 2.

Let $c=(c_{0},c_{1},\dots,c_{n-1})\in\mathcal{C}$ and
$d=(d_{0},d_{1},\dots,d_{n-1})\in\mathcal{C}^{\perp}$, where
$c_{j}=\sum_{i=1}^{e}\mu_{i}t_{j}^{i},~{}d_{j}=\sum_{i=1}^{e}\mu_{i}m_{j}^{i}$,
$t^{i}_{j},m^{i}_{j}\in\mathbb{F}_{q}$ for all $i,j$. Now, $c\cdot
d=\sum_{j=0}^{n-1}c_{j}d_{j}=0$ gives,
$\sum_{j=0}^{n-1}(t^{1}_{j}m^{1}_{j}+t^{2}_{j}m^{2}_{j}+\dots+t^{e}_{j}m^{e}_{j})=0$.
Again,

$\displaystyle\varphi(c)=[(t_{0}^{1},t_{0}^{2},\dots,t_{0}^{e})M,\dots,(t_{n-1}^{1},t_{n-1}^{2},\dots,t_{n-1}^{e})M]=(\alpha_{0}M,\dots,\alpha_{n-1}M)$
and
$\displaystyle\varphi(d)=[(m_{0}^{1},m_{0}^{2},\dots,m_{0}^{e})M,\dots,(m_{n-1}^{1},m_{n-1}^{2},\dots,m_{n-1}^{e})M]=(\beta_{0}M,\dots,\beta_{n-1}M),$

where $\alpha_{j}=(t_{j}^{1},t_{j}^{2},\dots,t_{j}^{e}),$
$\beta_{j}=(m_{j}^{1},m_{j}^{2},\dots,m_{j}^{e})$ for $0\leq j\leq n-1$ and
$MM^{T}=\gamma I_{e}$. Now,

$\displaystyle\varphi(c)\cdot\varphi(d)=\varphi(c)\varphi(d)^{T}=\sum_{j=0}^{n-1}\alpha_{j}MM^{T}\beta_{j}^{T}=\gamma\sum_{j=0}^{n-1}\alpha_{j}\beta_{j}^{T}=\gamma\sum_{j=0}^{n-1}(t^{1}_{j}m^{1}_{j}+t^{2}_{j}m^{2}_{j}+\dots+t^{e}_{j}m^{e}_{j})=0.$

Since $c\in\mathcal{C}$ and $d\in\mathcal{C}^{\perp}$ are arbitrary, so
$\varphi(\mathcal{C}^{\perp})\subseteq(\varphi(\mathcal{C}))^{\perp}$. On the
other side, as $\varphi$ is a bijective linear map, so
$\mid\varphi(\mathcal{C}^{\perp})\mid=\mid(\varphi(\mathcal{C}))^{\perp}\mid$.
Therefore, $\varphi(\mathcal{C}^{\perp})=(\varphi(\mathcal{C}))^{\perp}$

  3. 3.

Let $\mathcal{C}$ be self-orthogonal, that is,
$\mathcal{C}\subseteq\mathcal{C}^{\perp}$. Now,
$\varphi(\mathcal{C})\subseteq\varphi(\mathcal{C}^{\perp})=(\varphi(\mathcal{C}))^{\perp}$.
Hence $\varphi(\mathcal{C})$ is a self-orthogonal linear code of length $en$
over $\mathbb{F}_{q}$.

  4. 4.

Let $\mathcal{C}$ be self-dual, that is $\mathcal{C}=\mathcal{C}^{\perp}$.
Then
$\varphi(\mathcal{C})=\varphi(\mathcal{C}^{\perp})=(\varphi(\mathcal{C}))^{\perp}$.
Hence $\varphi(\mathcal{C})$ is a self-dual linear code over $\mathbb{F}_{q}$.
Conversely, let $\varphi(\mathcal{C})$ be a self-dual code. Then
$\varphi(\mathcal{C})=(\varphi(\mathcal{C}))^{\perp}=\varphi(\mathcal{C}^{\perp})$.
Since $\varphi$ is bijection, $\mathcal{C}=\mathcal{C}^{\perp}$. Hence
$\mathcal{C}$ is a self-dual code of length $n$ over $R_{e,q}$.

∎

Now, to obtain the necessary and sufficient condition for the existence of
self-dual codes we calculate dual codes in the next result, analogous to
Theorem 5 of [36].

######  Theorem 2.

Let $\mathcal{C}=\bigoplus_{i=1}^{e}\mu_{i}\mathcal{C}_{i}$ be a linear code
of length $n$ over $R_{e,q}$. Then

  1. 1.

$\mathcal{C}^{\perp}=\bigoplus_{i=1}^{e}\mu_{i}\mathcal{C}_{i}^{\perp}$, and

  2. 2.

$\mathcal{C}$ is self-dual (Euclidean) if and only if $\mathcal{C}_{i}$ is
self-dual (Euclidean), for $1\leq i\leq e$ .

###### Proof.

  1. 1.

Let $\mathcal{C}$ be a linear code of length $n$ over $R_{e,q}$. Also, let

$\displaystyle\mathcal{S}_{i}:=\\{{s_{i}}\in\mathbb{F}_{q}^{n}\mid\exists~{}s_{1},s_{2},\cdots,s_{i-1},s_{i+1},\cdots,s_{e}~{}\text{such
that}~{}\sum_{i=1}^{e}s_{i}\mu_{i}\in\mathcal{C}^{\perp}\\},~{}\text{for}~{}1\leq
i\leq e.$

Thus, $\mathcal{C}^{\perp}$ can be uniquely written as
$\mathcal{C}^{\perp}=\bigoplus_{i=1}^{e}\mu_{i}\mathcal{S}_{i}$. Clearly,
$\mathcal{S}_{1}\subseteq\mathcal{C}_{1}^{\perp}$. On the other side, let
$r\in\mathcal{C}_{1}^{\perp}$. Then $r\cdot a_{1}=0$ for all
$a_{1}\in\mathcal{C}_{1}$. Let $z=\sum_{i=1}^{e}\mu_{i}a_{i}\in\mathcal{C}$.
Now, $\mu_{1}r\cdot z=\mu_{1}a_{1}\cdot r=0$, which implies
$\mu_{1}r\in\mathcal{C}^{\perp}$. From the unique expression of
$\mathcal{C}^{\perp}$, we have $r\in\mathcal{S}_{1}$. Therefore,
$\mathcal{C}_{1}^{\perp}\subseteq\mathcal{S}_{1}$. Hence,
$\mathcal{S}_{1}=\mathcal{C}_{1}^{\perp}$. Similarly,
$\mathcal{C}_{i}^{\perp}=\mathcal{S}_{i}$ for $2\leq i\leq e$. Consequently,
$\mathcal{C}^{\perp}=\bigoplus_{i=1}^{e}\mu_{i}\mathcal{C}_{i}^{\perp}$.

  2. 2.

Further, $\mathcal{C}$ is self-dual if and only if
$\mathcal{C}^{\perp}=\mathcal{C}$ if and only if
$\bigoplus_{i=1}^{e}\mu_{i}\mathcal{C}_{i}^{\perp}=\bigoplus_{i=1}^{e}\mu_{i}\mathcal{C}_{i}$.
Therefore, $\mathcal{C}_{i}^{\perp}=\mathcal{C}_{i}$ for $1\leq i\leq e$.

∎

##  3 Cyclic codes over $R_{e,q}$

In this section, we determine the structure of cyclic codes of length $n$ over
$R_{e,q}$ and use these results to obtain LCD codes in the subsequent section.
Note that in [27] the authors considered this type of codes as codes over an
affine algebras with a finite commutative chain coefficient ring where the
affine algebra is $R_{e,q}$. However, we will construct them from scratch
based on the results before so we get an explicit description of the duality
and LCD condition based on Theorem 2 above.

######  Definition 3.1.

A linear code $\mathcal{C}$ of length $n$ over $R_{e,q}$ is said to be a
cyclic code if for any $c=(c_{0},c_{1},\dots,c_{n-1})\in\mathcal{C}$, we have
$\sigma(c):=(c_{n-1},c_{0},\dots,c_{n-2})\in\mathcal{C}$. Here, $\sigma$ is
called the cyclic shift operator.

We identify each codeword $c=(c_{0},c_{1},\dots,c_{n-1})\in\mathcal{C}$ by a
polynomial $c(x)\in R_{e,q}[x]/\langle x^{n}-1\rangle$ under the
correspondence $c=(c_{0},c_{1},\dots,c_{n-1})\longmapsto
c(x)=c_{0}+c_{1}x+\dots+c_{n-1}x^{n-1}\pmod{x^{n}-1}$. Under this
identification, we have the following result.

######  Theorem 3.

Let $\mathcal{C}$ be a linear code of length $n$ over $R_{e,q}$. Then
$\mathcal{C}$ is a cyclic code if and only if $\mathcal{C}$ is an ideal of the
ring $R_{e,q}[x]/\langle x^{n}-1\rangle$.

###### Proof.

Let $\mathcal{C}$ be a cyclic code of length $n$ over $R_{e,q}$. Let $a(x)\in
R_{e,q}[x]$ and $c(x)=c_{0}+c_{1}x+\dots+c_{n-1}x^{n-1}\in\mathcal{C}$. Then
$xc(x)=c_{0}x+c_{1}x^{2}+\dots+c_{n-2}x^{n-1}+c_{n-1}x^{n}=c_{n-1}+c_{0}x+c_{1}x^{2}+\dots+c_{n-2}x^{n-1}{\pmod{x^{n}-1}}=\sigma(c)\in\mathcal{C}$.
In similar manner, for any positive integer $i\geq 2$, we have
$x^{i}c(x)\in\mathcal{C}$. Since $\mathcal{C}$ is an $R_{e,q}$-submodule,
$a(x)c(x)\in\mathcal{C}$. Hence, $\mathcal{C}$ is an ideal of
$R_{e,q}[x]/\langle x^{n}-1\rangle$.

Conversely, let $\mathcal{C}$ be an ideal of $R_{e,q}[x]/\langle
x^{n}-1\rangle$. Let $c=(c_{0},c_{1},\dots,c_{n-1})\in\mathcal{C}$. Now,
$\sigma(c)=c_{n-1}+c_{0}x+c_{1}x^{2}+\dots+c_{n-2}x^{n-1}=xc(x)\in\mathcal{C}$.
Hence $\mathcal{C}$ is a cyclic code of length $n$ over $R_{e,q}$. ∎

######  Theorem 4.

[19] Let $\mathcal{C}$ be a cyclic code of length $n$ over $\mathbb{F}_{q}$.
Then there exists a unique polynomial $g(x)\in\mathbb{F}_{q}[x]$ such that
$\mathcal{C}=\langle g(x)\rangle$ and $\mathcal{C}^{\perp}=\langle
h^{*}(x)\rangle$ where $x^{n}-1=g(x)h(x)$ and $h^{*}(x)$ is the reciprocal
polynomial of $h(x)$.

######  Theorem 5.

A linear code $\mathcal{C}=\bigoplus_{i=1}^{e}\mu_{i}\mathcal{C}_{i}$ of
length $n$ over $R_{e,q}$ is cyclic if and only if $\mathcal{C}_{i}$ is a
cyclic code of length $n$ over $\mathbb{F}_{q}$, for $1\leq i\leq e$.

###### Proof.

Let $\mathcal{C}=\bigoplus_{i=1}^{e}\mu_{i}\mathcal{C}_{i}$ be a cyclic code
of length $n$ over $R_{e,q}$. Let
$a^{i}=(a_{0}^{i},a_{1}^{i},\dots,a_{n-1}^{i})\in\mathcal{C}_{i}$ for $1\leq
i\leq e$. Take $r_{j}=\sum_{i=1}^{e}\mu_{i}a^{i}_{j}$ for $0\leq j\leq n-1$.
Then $r=(r_{0},r_{1},\dots,r_{n-1})\in\mathcal{C}$ and hence
$\sigma(r)=(r_{n-1},r_{0},\dots,r_{n-2})\in\mathcal{C}$. Again
$\sigma(r)=\sum_{i=1}^{e}\mu_{i}\sigma(a^{i})\in\mathcal{C}=\bigoplus_{i=1}^{e}\mu_{i}\mathcal{C}_{i}$.
Therefore, $\sigma(a^{i})\in\mathcal{C}_{i}$ for $1\leq i\leq e$. Hence
$\mathcal{C}_{i}$ is a cyclic code over $\mathbb{F}_{q}$ for $1\leq i\leq e$.

Conversely, let $\mathcal{C}_{i}$ be a cyclic code over $\mathbb{F}_{q}$ for
$1\leq i\leq e$. Let $r=(r_{0},r_{1},\dots,r_{n-1})\in\mathcal{C}$ where
$r_{j}=\sum_{i=1}^{e}\mu_{i}a^{i}_{j}$ for $0\leq j\leq n-1$. Then
$a^{i}=(a_{0}^{i},a_{1}^{i},\dots,a_{n-1}^{i})\in\mathcal{C}_{i}$ for $1\leq
i\leq e$. Now,
$\sigma(r)=\sum_{i=1}^{e}\mu_{i}\sigma(a^{i})\in\bigoplus_{i=1}^{e}\mu_{i}\mathcal{C}_{i}=\mathcal{C}$.
Hence $\mathcal{C}$ is a cyclic code of length $n$ over $R_{e,q}$. ∎

By the help of Theorem 4 and Theorem 5 we prepare Theorem 6 to obtain the
generator polynomial for cyclic codes.

######  Theorem 6.

Let $\mathcal{C}=\bigoplus_{i=1}^{e}\mu_{i}\mathcal{C}_{i}$ be a cyclic code
of length $n$ over $R_{e,q}$. Then there exists a unique polynomial $g(x)\in
R_{e,q}[x]$ such that $\mathcal{C}=\langle g(x)\rangle$ and $x^{n}-1=g(x)h(x)$
where $g(x)=\sum_{i=1}^{e}\mu_{i}g_{i}(x)$ and $x^{n}-1=g_{i}(x)h_{i}(x)$, for
$1\leq i\leq e$.

###### Proof.

Let $\mathcal{C}=\bigoplus_{i=1}^{e}\mu_{i}\mathcal{C}_{i}$ be a cyclic code
of length $n$ over $R_{e,q}$. Then by Theorem 5, $\mathcal{C}_{i}$ is a cyclic
code over $\mathbb{F}_{q}$ for $1\leq i\leq e$. By Theorem 4, we have
$\mathcal{C}_{i}=\langle g_{i}(x)\rangle$ where $x^{n}-1=g_{i}(x)h_{i}(x)$ for
$1\leq i\leq e$. Let $g(x)=\sum_{i=1}^{e}\mu_{i}g_{i}(x)$. Then $\langle
g(x)\rangle\subseteq\mathcal{C}$. On the other hand,
$\mu_{i}g_{i}(x)=\mu_{i}g(x)\in\langle g(x)\rangle$ for $1\leq i\leq e$. Thus
$\mathcal{C}\subseteq\langle g(x)\rangle$. Therefore, $\mathcal{C}=\langle
g(x)\rangle$. Since $g_{i}(x)$ is unique for $1\leq i\leq e$, then $g(x)$ is
unique.
Further, we have
$g(x)\sum_{i=1}^{e}\mu_{i}h_{i}(x)=\sum_{i=1}^{e}\mu_{i}g_{i}(x)h_{i}(x)=x^{n}-1\sum_{i=1}^{e}\mu_{i}=x^{n}-1$.
Hence $x^{n}-1=g(x)h(x)$ where $h(x)=\sum_{i=1}^{e}\mu_{i}h_{i}(x)$. ∎

######  Corollary 3.1.

Every cyclic code of length $n$ over $R_{e,q}$ is principally generated, that
is, $R_{e,q}[x]/\langle x^{n}-1\rangle$ is a principal ideal ring.

######  Corollary 3.2.

Let $\mathcal{C}=\bigoplus_{i=1}^{e}\mu_{i}\mathcal{C}_{i}$ be a cyclic code
of length $n$ over $R_{e,q}$ and $\mathcal{C}_{i}=\langle g_{i}(x)\rangle$
such that $x^{n}-1=g_{i}(x)h_{i}(x)$ for $1\leq i\leq e$. Then

  1. 1.

$\mathcal{C}^{\perp}=\bigoplus_{i=1}^{e}\mu_{i}\mathcal{C}_{i}^{\perp}$ is a
cyclic code over $R_{e,q}$,

  2. 2.

$\mathcal{C}^{\perp}=\langle\sum_{i=1}^{e}\mu_{i}h_{i}^{*}(x)\rangle$, where
$h_{i}^{*}(x)$ is the reciprocal polynomial of $h_{i}(x)$, i.e.,
$h_{i}^{*}(x)=x^{\deg(h_{i}(x))}h_{i}(1/x)$ for $1\leq i\leq e$,

  3. 3.

$\mid\mathcal{C}^{\perp}\mid=q^{\sum_{i=1}^{e}\deg(g_{i}(x))}$.

###### Proof.

  1. 1.

Let $\mathcal{C}=\bigoplus_{i=1}^{e}\mu_{i}\mathcal{C}_{i}$ be a cyclic code
of length $n$ over $R_{e,q}$. Then by Theorem 5, $\mathcal{C}_{i}$ is a cyclic
code over $\mathbb{F}_{q}$ for $1\leq i\leq e$. Then $\mathcal{C}_{i}^{\perp}$
is a cyclic code over $\mathbb{F}_{q}$ for $1\leq i\leq e$. By Theorem 5,
$\mathcal{C}^{\perp}=\bigoplus_{i=1}^{e}\mu_{i}\mathcal{C}_{i}^{\perp}$ is a
cyclic code over $R_{e,q}$.

  2. 2.

Let $\mathcal{C}=\bigoplus_{i=1}^{e}\mu_{i}\mathcal{C}_{i}$ be a cyclic code
of length $n$ over $R_{e,q}$. Then by Theorem 4,
$\mathcal{C}_{i}^{\perp}=\langle h^{*}_{i}(x)\rangle$ where
$x^{n}-1=g_{i}(x)h_{i}(x)$ for $1\leq i\leq e$. Therefore, similar to the
proof of Theorem 6, we have
$\mathcal{C}^{\perp}=\langle\sum_{i=1}^{e}\mu_{i}h^{*}_{i}(x)\rangle$, where
$h_{i}^{*}(x)$ is the reciprocal polynomial of $h_{i}(x)$.

  3. 3.

Since $\mid\mathcal{C}_{i}^{\perp}\mid=q^{\deg(g_{i}(x))}$, so
$\mid\mathcal{C}^{\perp}\mid=\prod_{i=1}^{e}\mid\mathcal{C}_{i}^{\perp}\mid=q^{\sum_{i=1}^{e}\deg(g_{i}(x))}$.

∎

Now, we present some examples of cyclic codes over $R_{e,q}$ and their
$\mathbb{F}_{q}$-images which include MDS and optimal linear codes. The
calculation involved in these examples is carried out by the Magma computation
system [7].

######  Example 2.

Let $e=2,q=5,n=10$ and $R_{2,5}=\mathbb{F}_{5}[u]/\langle u^{2}-1\rangle$.
Then $\mu_{1}=\frac{1+u}{2}$ and $\mu_{2}=\frac{1-u}{2}$. Let

$M=\left[{\begin{array}[]{cccc}3&2\\\ 2&2\end{array}}\right]\in
GL_{2}(\mathbb{F}_{5}).$

Then $M$ satisfies $MM^{T}=3I_{2}$ and the Gray map
$\varphi:R_{2,5}\longrightarrow\mathbb{F}_{5}^{2}$ is defined by

$\displaystyle\varphi(a_{0}+ua_{1})=(a_{0}+a_{1},a_{0}-a_{1})\left[{\begin{array}[]{cccc}3&2\\\
2&2\end{array}}\right].$

Again

$\displaystyle x^{10}-1=(x+1)^{5}(x+4)^{5}\in\mathbb{F}_{5}[x].$

Let $\mathcal{C}=\langle\mu_{1}g_{1}(x)+\mu_{2}g_{2}(x)\rangle$ be a cyclic
code of length $10$ over $R_{2,5}$ where $g_{1}(x)=x+4$ and
$g_{2}(x)=(x+1)(x+4)^{3}=x^{4}+3x^{3}+2x+4$. Therefore, $\varphi(\mathcal{C})$
is a $[20,15,4]$ linear code over $\mathbb{F}_{5}$ and as per the database
[16], it is an optimal linear code.

######  Example 3.

Let $e=2,q=11,n=5$ and $R_{2,11}=\mathbb{F}_{11}[u]/\langle u^{2}-1\rangle$.
Now, in $\mathbb{F}_{11}[x]$, we have

$\displaystyle x^{5}-1=(x+2)(x+6)(x+7)(x+8)(x+10).$

Let $g_{1}(x)=(x+8)(x+10)=x^{2}+7x+3$ and $g_{2}(x)=(x+6)(x+7)=x^{2}+2x+9$.
Then $\mathcal{C}=\langle\mu_{1}g_{1}(x)+\mu_{2}g_{2}(x)\rangle$ is a cyclic
code of length $5$ over $R_{2,11}$. Again, let

$M=\left[{\begin{array}[]{cccc}9&2\\\ 2&2\end{array}}\right]\in
GL_{2}(\mathbb{F}_{11}),$

satisfying $MM^{T}=8I_{2}$. Therefore, the Gray image $\varphi(\mathcal{C})$
has parameters $[10,6,5]$, which is an MDS code satisfying $n-k+1-d=0$.

######  Example 4.

Let $\mathcal{C}=\langle\mu_{1}g_{1}(x)+\mu_{2}g_{2}(x)\rangle$ be a cyclic
code of length $n$ over $R_{2,q}=\mathbb{F}_{q}[u]/\langle u^{2}-1\rangle$,
where $g_{i}(x)\mid x^{n}-1$ for $i=1,2$. Let

$M=\left[{\begin{array}[]{cccc}-2&2\\\ 2&2\end{array}}\right]\in
GL_{2}(\mathbb{F}_{q}),$

satisfying $MM^{T}=8I_{2}$. Now, in Table 2 and Table 3, the generator
polynomials $g_{i}(x)~{}(i=1,2)$ are presented in the third and fourth column,
respectively and the Gray images $\varphi(\mathcal{C})$ are included in the
fifth column. As per the database [16], the codes obtained in the fifth column
of Table 2 are optimal and the best-known linear codes (BKLC). On the other
hand, the codes obtained in the fifth column of Table 3 attaining the
singleton bound $n-k+1-d=0$ are MDS.

##  4 LCD codes over $R_{e,q}$

In the previous section we characterized cyclic codes of length $n$ over
$R_{e,q}$. Here, using their structure we construct LCD codes of length $n$
(Corollary 4.1 and 4.2). Moreover, we discuss about the existence of free and
non-free LCD codes. Recall that a linear code $\mathcal{C}$ of length $n$ is
said to be a linear complementary dual (shortly, LCD) code if
$\mathcal{C}\cap\mathcal{C}^{\perp}=\\{0\\}$.

######  Definition 4.1.

A cyclic code $\mathcal{C}$ is said to be a reversible cyclic code if for any
$(c_{0},c_{1},\dots,c_{n-1})\in\mathcal{C}$, its reverse translated codeword
$(c_{n-1},c_{n-2},\dots,c_{1},c_{0})$ also is in $\mathcal{C}$.

Recall that a cyclic code of length $n$ over a field $\mathbb{F}_{q}$ is a
reversible cyclic code if it is generated by a self-reciprocal factor of
$x^{n}-1$. Also, LCD codes over $\mathbb{F}_{q}$ are characterized by Yang and
Massey [35]. Using these results here we determine the structure of LCD codes
over $R_{e,q}$. Now, we recall the following results.

######  Lemma 3.

[35] Let $\mathcal{C}=\langle g(x)\rangle$ be a cyclic code of length $n$ over
$\mathbb{F}_{q}$.

  1. 1.

Let $\gcd(n,q)\neq 1$. Then $\mathcal{C}$ is an LCD code if and only if $g(x)$
is self-reciprocal and all monic irreducible factors of $g(x)$ have the same
multiplicity in $g(x)$ and $x^{n}-1$.

  2. 2.

Let $\gcd(n,q)=1$. Then $\mathcal{C}$ is an LCD code if and only if it is a
reversible code.

######  Theorem 7.

Let $\mathcal{C}=\bigoplus_{i=1}^{e}\mu_{i}\mathcal{C}_{i}$ be a cyclic code
of length $n$ over $R_{e,q}$. Then $\mathcal{C}$ is an LCD code if and only if
$\mathcal{C}_{i}$ is an LCD cyclic code of length $n$ over $\mathbb{F}_{q}$
for $1\leq i\leq e$.

###### Proof.

Note that $\mathcal{C}\cap\mathcal{C}^{\perp}=\\{0\\}$ if and only if
$\mathcal{C}_{i}\cap\mathcal{C}_{i}^{\perp}=\\{0\\}$. Hence the result
follows. ∎

######  Corollary 4.1.

Let $\gcd(n,q)\neq 1$. Also, with the notations of Theorem 6, let
$\mathcal{C}=\langle\sum_{i=1}^{e}\mu_{i}g_{i}(x)\rangle$ be a cyclic code of
length $n$ over $R_{e,q}$, where $g_{i}(x)\in\mathbb{F}_{q}[x]$ and
$g_{i}(x)\mid(x^{n}-1)$ for $1\leq i\leq e$. Then $\mathcal{C}$ is an LCD code
if and only if $g_{i}(x)$ is self-reciprocal and each monic irreducible factor
of $g_{i}(x)$ has the same multiplicity in $g_{i}(x)$ and in $(x^{n}-1)$, for
$1\leq i\leq e$.

###### Proof.

It follows from part 1 of Lemma 3 and Theorem 7. ∎

######  Theorem 8.

Let $\gcd(n,q)=1$ and $\mathcal{C}=\bigoplus_{i=1}^{e}\mu_{i}\mathcal{C}_{i}$
be a cyclic code of length $n$ over $R_{e,q}$. Then $\mathcal{C}$ is an LCD
code if and only if $\mathcal{C}_{i}$ is a reversible code of length $n$ over
$\mathbb{F}_{q}$ for $1\leq i\leq e$.

###### Proof.

It follows from the combination of part 2 of Lemma 3 and Theorem 7. ∎

######  Corollary 4.2.

Let $\gcd(n,q)=1$ and $\mathcal{C}=\bigoplus_{i=1}^{e}\mu_{i}\mathcal{C}_{i}$
be a cyclic code of length $n$ over $R_{e,q}$ where $\mathcal{C}_{i}=\langle
g_{i}(x)\rangle$ is a cyclic code of length $n$ over $\mathbb{F}_{q}$ for
$1\leq i\leq e$. Then $\mathcal{C}$ is an LCD code if and only if $g_{i}(x)$
is self-reciprocal polynomial in $\mathbb{F}_{q}$, for $1\leq i\leq e$.

######  Lemma 4.

Let $\mathcal{C}$ be a linear code of length $n$ over $R_{e,q}$. Then
$\varphi(\mathcal{C}\cap\mathcal{C}^{\perp})=\varphi(\mathcal{C})\cap\varphi(\mathcal{C})^{\perp}$,
where $\varphi$ is the Gray map defined in equation (2).

###### Proof.

Let $a\in\varphi(\mathcal{C})\cap\varphi(\mathcal{C}^{\perp})$. Then there
exist $c\in\mathcal{C}$ and $c^{\prime}\in\mathcal{C}^{\perp}$ such that
$\varphi(c)=\varphi(c^{\prime})=a$. Now, $\varphi$ being injective, we have
$c=c^{\prime}\in\mathcal{C}\cap\mathcal{C}^{\perp}$. Therefore,
$a=\varphi(c)\in\varphi(\mathcal{C}\cap\mathcal{C}^{\perp})$. Hence,
$\varphi(\mathcal{C})\cap\varphi(\mathcal{C}^{\perp})\subseteq\varphi(\mathcal{C}\cap\mathcal{C}^{\perp})$.
On the other side, let $a\in\varphi(\mathcal{C}\cap\mathcal{C}^{\perp})$. Then
there exists $c\in\mathcal{C}\cap\mathcal{C}^{\perp}$ such that
$\varphi(c)=a$. Again, $c\in\mathcal{C}\cap\mathcal{C}^{\perp}$ implies that
$c\in\mathcal{C}$ and $c\in\mathcal{C}^{\perp}$. Therefore,
$\varphi(c)\in\varphi(\mathcal{C})$ and
$\varphi(c)\in\varphi(\mathcal{C}^{\perp})$. Hence,
$a=\varphi(c)\in\varphi(\mathcal{C})\cap\varphi(\mathcal{C}^{\perp})$. This
shows that
$\varphi(\mathcal{C}\cap\mathcal{C}^{\perp})\subseteq\varphi(\mathcal{C})\cap\varphi(\mathcal{C}^{\perp})$.
Hence, combining both sides we have
$\varphi(\mathcal{C})\cap\varphi(\mathcal{C}^{\perp})=\varphi(\mathcal{C}\cap\mathcal{C}^{\perp})$.
Also, by Theorem 1,
$\varphi(\mathcal{C}^{\perp})=\varphi(\mathcal{C})^{\perp}$. Thus,
$\varphi(\mathcal{C})\cap\varphi(\mathcal{C})^{\perp}=\varphi(\mathcal{C}\cap\mathcal{C}^{\perp})$.
∎

######  Theorem 9.

Let $\mathcal{C}$ be a linear code of length $n$ over $R_{e,q}$. Then
$\mathcal{C}$ is an LCD code if and only if its Gray image
$\varphi(\mathcal{C})$ is an LCD code of length $en$ over $\mathbb{F}_{q}$.

###### Proof.

Let $\mathcal{C}$ be an LCD code of length $n$ over $R_{e,q}$. Then
$\mathcal{C}\cap\mathcal{C}^{\perp}=\\{0\\}$. Now, by Lemma 4,
$\varphi(\mathcal{C})\cap\varphi(\mathcal{C})^{\perp}=\varphi(\mathcal{C}\cap\mathcal{C}^{\perp})=\varphi(\\{0\\})=\\{0\\}$.
This implies that $\varphi(\mathcal{C})$ is an LCD code of length $en$ over
$\mathbb{F}_{q}$.

Conversely, let $\varphi(\mathcal{C})$ be an LCD code of length $en$ over
$\mathbb{F}_{q}$. Then
$\varphi(\mathcal{C})\cap\varphi(\mathcal{C})^{\perp}=\\{0\\}$. Also, by Lemma
4, we have
$\varphi(\mathcal{C}\cap\mathcal{C}^{\perp})=\varphi(\mathcal{C})\cap\varphi(\mathcal{C})^{\perp}=\\{0\\}$.
Since $\varphi$ is injective, we have
$\mathcal{C}\cap\mathcal{C}^{\perp}=\\{0\\}$. Hence, $\mathcal{C}$ is an LCD
code of length $n$ over $\mathcal{R}$. ∎

###  4.1 Existence of non-free LCD codes over $R_{e,q}$

In [4], the authors proved that there does not exist any non-free LCD codes
over finite commutative local Frobenius ring, or in other words, every LCD
code over commutative local Frobenius ring is free. Also, the authors gave the
conditions in case of a non-local ring for a code to be LCD. Here we show that
both free and non-free LCD cyclic codes exist over the commutative semi-local
Frobenius ring $R_{e,q}$ in terms of the degree of the polynomials defining
them. To do so we use the following result.

######  Lemma 5.

The element $\sum_{i\in T}\mu_{i}$ is a zero divisor in $R_{e,q}$, for any
non-trivial subset $T$ of $\\{1,2,\dots,e\\}$.

###### Proof.

Since $\mu_{i}(1-\mu_{i})=0$, for $1\leq i\leq e$, then $\sum_{i\in
T}\mu_{i}\prod_{i\in T}(1-\mu_{i})=0$, for any non-trivial subset $T$ of
$\\{1,2,\dots,e\\}$. Also, as $\mu_{i}\mu_{j}=0$, for $i\neq j$, then
$\prod_{i\in T}(1-\mu_{i})=1-(\sum_{i\in T}\mu_{i})\neq 0$. Hence the result.
∎

As we discussed in the previous section a cyclic code of length $n$ is given
by $\mathcal{C}=\langle g(x)\rangle$ where
$g(x)=\sum_{i=1}^{e}\mu_{i}g_{i}(x)$. Since $g_{i}(x)$ is a monic polynomial
in $\mathbb{F}_{q}[x]$, $g(x)$ is monic if the degrees of all polynomials
$g_{i}(x)$, for $1\leq i\leq e$ are same. Also, if the degrees of all
polynomials $g_{i}(x)$ are not same, then by Lemma 5, $g(x)$ is a polynomial
having a zero divisor leading coefficient. Therefore, $g(x)$ is a non-monic
polynomial. Now, for the case when $g(x)$ is monic, by using the division
algorithm on $R_{e,q}[x]$ and a similar argument of Theorem 4 in [33], we
conclude that $\mathcal{C}$ is free. On the other hand, when $g(x)$ is non-
monic, if possible, let us assume that $\mathcal{C}$ is free. Then by similar
argument of Theorem 5 in [33], we have a monic polynomial $g^{\prime}(x)\in
R_{e,q}[x]$ such that $\mathcal{C}=\langle g^{\prime}(x)\rangle$ and
$g^{\prime}(x)\mid x^{n}-1$. Since $g(x)$ is non-monic, this contradicts the
fact that $g(x)$ is the unique polynomial in $R_{e,q}[x]$ such that
$\mathcal{C}=\langle g(x)\rangle$ and $g(x)\mid x^{n}-1$. Therefore, under
above discussion, we have the following results.

######  Theorem 10.

Let $\mathcal{C}=\langle g(x)\rangle$ be a cyclic code of length $n$ over
$R_{e,q}$ where $g(x)=\sum_{i=1}^{e}\mu_{i}g_{i}(x)$.

  1. 1.

$\mathcal{C}$ is free if the degrees of $g_{1}(x),g_{2}(x),\dots,g_{e}(x)$ are
same.

  2. 2.

$\mathcal{C}$ is non-free if the degrees of $g_{1}(x),g_{2}(x),\dots,g_{e}(x)$
are not same.

Unlike commutative local Frobenius ring, both free and non-free LCD codes
exist over $R_{e,q}$ and can be constructed by following Corollary 4.1,
Corollary 4.2 and Theorem 10. Now, we discuss some examples of non-free LCD
codes over $R_{e,q}$ which present many optimal LCD codes over
$\mathbb{F}_{q}$ as the Gray images of these codes.

######  Example 5.

Let $e=2,q=2\cdot 2+1=5,n=6$ and $R_{2,5}=\mathbb{F}_{5}[u]/\langle
u^{2}-1\rangle$. Let $\mathcal{C}$ be a cyclic code of length $6$ over
$R_{2,5}$. Now,

$\displaystyle x^{6}-1=(x+1)(x+4)(x^{2}+x+1)(x^{2}+4x+1)\in\mathbb{F}_{5}[x].$

Let $g_{1}(x)=x+4$ and $g_{2}(x)=(x+1)(x^{2}+x+1)=x^{3}+2x^{2}+2x+1$. Since
$g_{1}(x),g_{2}(x)$ are self-reciprocal polynomials, by Corollary 4.2 and
Theorem 10, we have
$\mathcal{C}=\langle\mu_{1}g_{1}(x)+\mu_{2}g_{2}(x)\rangle$ is a non-free LCD
code of length $6$ over $R_{2,5}$. Let

$M=\left[{\begin{array}[]{ccc}1&4\\\ 1&1\end{array}}\right]\in
GL_{2}(\mathbb{F}_{5}),$

satisfying $MM^{T}=2I_{2}$. Then by Theorem 9, the Gray image
$\varphi(\mathcal{C})$ is an LCD code with parameters $[12,8,4]$ and it is an
optimal code according to [16].

######  Example 6.

Let $e=3,q=3\cdot 2+1=7,n=3$ and $R_{3,7}=\mathbb{F}_{7}[u]/\langle
u^{3}-1\rangle$. Now,

$\displaystyle x^{3}-1=(x+3)(x+5)(x+6)\in\mathbb{F}_{7}[x].$

Let $g_{1}(x)=x+6,g_{2}(x)=1$ and $g_{3}(x)=(x+2)(x+3)=x^{2}+x+1$. Then
$\mathcal{C}=\langle\sum_{i=1}^{3}\mu_{i}g_{i}(x)\rangle$ is a cyclic code of
length $3$ over $R_{3,7}$. Since $\gcd(4,7)=1$ and $g_{i}(x)$ is self-
reciprocal polynomial for $i=1,2$, by Corollary 4.2 and Theorem 10, we say
that $\mathcal{C}$ is a non-free LCD code. Again, let

$M=\left[{\begin{array}[]{cccc}2&1&2\\\ 5&2&1\\\ 1&2&5\\\
\end{array}}\right]\in GL_{3}(\mathbb{F}_{7}),$

satisfying $MM^{T}=2I_{3}$. Then by Theorem 9, the Gray image
$\varphi(\mathcal{C})$ is an LCD code with parameters $[9,6,3]$, which is an
optimal code by [16].

######  Example 7.

Let $\mathcal{C}=\langle\mu_{1}g_{1}(x)+\mu_{2}g_{2}(x)\rangle$ be a non-free
LCD code of length $n$ over $R_{2,q}$ where $\gcd(n,q)=1$ and
$g_{1}(x),g_{2}(x)$ are self-reciprocal factors of $x^{n}-1$. Let

$M=\left[{\begin{array}[]{ccc}1&-1\\\ 1&1\end{array}}\right]\in
GL_{2}(\mathbb{F}_{q}),$

satisfying $MM^{T}=2I_{2}$. Again, by Theorem 9, $\varphi(\mathcal{C})$ is an
LCD code. In Table 4, we present generator polynomials $g_{1}(x),g_{2}(x)$ in
the $3^{rd}$ and $4^{th}$ column, respectively. Also, their
$\mathbb{F}_{q}$-images are included in the $5^{th}$ column which presents
many optimal codes as per [16].

######  Remark 1.

  1. 1.

In order to present Tables 2, 3 and 4 briefly, we write the vector consisting
coefficients of the polynomial $g_{i}(x)$ in decreasing order. For example, we
write the vector $(1,3,0,2,4)$ to represent the polynomial
$x^{4}+3x^{3}+2x+4$.

  2. 2.

As we can see in Tables 2 and 4, the codes of parameters
$[16,11,4],[20,15,4],[24,19,4],[32,27,4],[12,8,3],$
$[12,7,4],[16,12,3],[26,22,3],[50,45,4],46,21,4]$ satisfy $n-k+1-d=2$ while
the remaining codes satisfy $n-k+1-d=1$. Therefore, although these codes are
not MDS (with respect to the singleton bound) they are very close to be MDS,
having good parameters.

##  5 Conclusion

Here, we have obtained many MDS and optimal codes from the Gray images of both
cyclic and LCD codes over $R_{e,q}$. To compute the parameters of codes, we
have used the Magma computation system [7]. Apart from the obtained
parameters, one can further find more optimal codes using the derived results
and the Gray map $\varphi$. The article justifies the fact that LCD codes over
non-chain rings are a useful tool to obtain optimal LCD codes over
$\mathbb{F}_{q}$. We hope our work would motivate researchers to study LCD
codes over other non-chain rings to explore more good LCD codes in the future.

## Acknowledgement

The authors are thankful to the University Grants Commission (UGC), Govt. of
India for financial supports under Ref. No. 20/12/2015(ii)EU-V dated
31/08/2016 and Indian Institute of Technology Patna for providing research
facilities. E. Martínez-Moro is partially funded by Spanish State Research
Agency (AEI) under Grant PGC2018-096446-B-C21.
Also, authors would like to thank the anonymous referee(s) and the Editor for
their valuable comments to improve the presentation of the manuscript.

Table 1: Some decomposition $\mu_{i}$ and their canonical components $s_{i}$ $e$ | $q$ | $r$ | $\mu_{i}$ | $s_{i}$
---|---|---|---|---
$2$ | $q$ | $a_{0}+ua_{1}$ | $\mu_{1}=\frac{1+u}{2},\mu_{2}=\frac{1-u}{2}$ | $s_{1}=a_{0}+a_{1},s_{2}=a_{0}-a_{1}$
|  |  | $\mu_{1}=6(u+5)(u+6)$ | $s_{1}=a_{0}+4a_{1}+2a_{2}$
3 | 7 | $a_{0}+ua_{1}+u^{2}a_{2}$ | $\mu_{2}=3(u+3)(u+6)$ | $s_{2}=a_{0}+2a_{1}+4a_{2}$
|  |  | $\mu_{3}=5(u+3)(u+5)$ | $s_{3}=a_{0}+a_{1}+a_{2}$
|  |  | $\mu_{1}=3(u+10)(u+12)$ | $s_{1}=a_{0}+9a_{1}+3a_{2}$
3 | 13 | $a_{0}+ua_{1}+u^{2}a_{2}$ | $\mu_{2}=(u+4)(u+12)$ | $s_{2}=a_{0}+3a_{1}+9a_{2}$
|  |  | $\mu_{3}=9(u+4)(u+10)$ | $s_{3}=a_{0}+a_{1}+a_{2}$
|  |  | $\mu_{1}=(u+2)(u+3)(u+4)$ | $s_{1}=a_{0}+4a_{1}+a_{2}+4a_{3}$
4 | 5 | $a_{0}+ua_{1}+u^{2}a_{2}+u^{3}a_{3}$ | $\mu_{2}=2(u+1)(u+3)(u+4)$ | $s_{2}=a_{0}+3a_{1}+4a_{2}+2a_{3}$
|  |  | $\mu_{3}=3(u+1)(u+2)(u+4)$ | $s_{3}=a_{0}+2a_{1}+4a_{2}+3a_{3}$
|  |  | $\mu_{4}=4(u+1)(u+2)(u+3)$ | $s_{4}=a_{0}+a_{1}+a_{2}+a_{3}$
|  |  | $\mu_{1}=3(u+5)(u+8)(u+12)$ | $s_{1}=a_{0}+12a_{1}+a_{2}+12a_{3}$
4 | 13 | $a_{0}+ua_{1}+u^{2}a_{2}+u^{3}a_{3}$ | $\mu_{2}=2(u+1)(u+8)(u+12)$ | $s_{2}=a_{0}+8a_{1}+12a_{2}+5a_{3}$
|  |  | $\mu_{3}=11(u+1)(u+5)(u+12)$ | $s_{3}=a_{0}+5a_{1}+12a_{2}+8a_{3}$
|  |  | $\mu_{4}=10(u+1)(u+5)(u+8)$ | $s_{4}=a_{0}+a_{1}+a_{2}+a_{3}$
|  |  | $\mu_{1}=4(u+4)(u+13)(u+16)$ | $s_{1}=a_{0}+16a_{1}+a_{2}+16a_{3},$
4 | 17 | $a_{0}+ua_{1}+u^{2}a_{2}+u^{3}a_{3}$ | $\mu_{2}=16(u+1)(u+13)(u+16)$ | $s_{2}=a_{0}+13a_{1}+16a_{2}+4a_{2}$
|  |  | $\mu_{3}=(u+1)(u+4)(u+16)$ | $s_{3}=a_{0}+4a_{1}+16a_{2}+13a_{3}$
|  |  | $\mu_{4}=13(u+1)(u+4)(u+13)$ | $s_{4}=a_{0}+a_{1}+a_{2}+a_{3}$
Table 2: Optimal linear as the Gray images of cyclic codes over $R_{2,q}=\mathbb{F}_{q}[u]/\langle u^{2}-1\rangle$ $q$ | $n$ | $g_{1}(x)$ | $g_{2}(x)$ | $[n,k,d]$ | Remark
---|---|---|---|---|---
$5$ | $4$ | $(1,4)$ | $(1,2,2)$ | $[8,5,3]$ | Optimal
$5$ | $4$ | $(1,4,3)$ | $(1,2,2)$ | $[8,4,4]$ | Optimal
$5$ | $4$ | $(1,3,4,2)$ | $(1,4,1,4)$ | $[8,2,6]$ | Optimal
$5$ | $5$ | $(1,3,1)$ | $(1,4)$ | $[10,7,3]$ | Optimal
$5$ | $5$ | $(1,4)$ | $(1,2,3,4)$ | $[10,6,4]$ | Optimal
$5$ | $8$ | $(1,3,2,1)$ | $(1,4,3)$ | $[16,11,4]$ | Optimal
$5$ | $10$ | $(1,4)$ | $(1,3,0,2,4)$ | $[20,15,4]$ | Optimal
$7$ | $6$ | $(1,5,6)$ | $(1,2)$ | $[12,9,3]$ | Optimal
$7$ | $7$ | $(1,5,1)$ | $(1,6)$ | $[14,11,3]$ | Optimal
$7$ | $7$ | $(1,4,3,6)$ | $(1,6)$ | $[14,10,4]$ | Optimal
$7$ | $8$ | $(1,2,5,6)$ | $(1,6)$ | $[16,12,4]$ | Optimal
$7$ | $12$ | $(1,6)$ | $(1,5,0,5,6)$ | $[24,19,4]$ | BKLC
$7$ | $14$ | $(1,5,0,2,6)$ | $(1,6)$ | $[28,24,4]$ | BKLC
$7$ | $14$ | $(1,6)$ | $(1,3,5,4,1)$ | $[32,27,4]$ | Optimal
$3$ | $3$ | $(1,1,1)$ | $(1,2)$ | $[6,3,3]$ | Optimal
$3$ | $6$ | $(1,2,2,1)$ | $(1,2)$ | $[12,8,3]$ | Optimal
$3$ | $6$ | $(1,1,0,2,2)$ | $(1,2)$ | $[12,7,4]$ | Optimal
$3$ | $8$ | $(1,2)$ | $(1,1,0,1,2)$ | $[16,11,4]$ | Optimal
$3$ | $8$ | $(1,2)$ | $(1,2,0,2)$ | $[16,12,3]$ | Optimal
$3$ | $13$ | $(1,0,2,2)$ | $(1,2)$ | $[26,22,3]$ | Optimal
$9$ | $8$ | $(1,w,w^{5},2)$ | $(1,w^{6})$ | $[16,12,4]$ | Optimal
$9$ | $10$ | $(1,w^{2},w^{6},2)$ | $(1,2)$ | $[20,16,4]$ | Optimal
Table 3: linear MDS codes as the Gray images of cyclic codes over $R_{2,q}=\mathbb{F}_{q}[u]/\langle u^{2}-1\rangle$ $q$ | $n$ | $g_{1}(x)$ | $g_{2}(x)$ | $[n,k,d]$ | Remark
---|---|---|---|---|---
$7$ | $3$ | $(1,3)$ | $(1,5)$ | $[6,4,3]$ | MDS
$7$ | $3$ | $(1,3)$ | $(1,4,2)$ | $[6,3,4]$ | MDS
$7$ | $3$ | $(1,1,1)$ | $(1,4,2)$ | $[6,2,5]$ | MDS
$11$ | $5$ | $(1,6)$ | $(1,8)$ | $[10,8,3]$ | MDS
$11$ | $5$ | $(1,7,3)$ | $(1,2,9)$ | $[10,6,5]$ | MDS
$19$ | $9$ | $(1,3)$ | $(1,8)$ | $[18,6,3]$ | MDS
$23$ | $11$ | $(1,5)$ | $(1,7)$ | $[22,20,3]$ | MDS
Table 4: Optimal LCD codes as the Gray images of non-free LCD codes over $R_{2,q}=\mathbb{F}_{q}[u]/\langle u^{2}-1\rangle$ $q$ | $n$ | $g_{1}(x)$ | $g_{2}(x)$ | $[n,k,d]$ | Remark
---|---|---|---|---|---
$5$ | $3$ | $(1,1,1)$ | $(1,4)$ | $[6,3,4]$ | MDS
$5$ | $6$ | $(1,4)$ | $(1,2,2,1)$ | $[12,8,4]$ | Optimal
$5$ | $12$ | $(1,1,2,1,1)$ | $(1,4)$ | $[24,19,4]$ | Optimal
$5$ | $24$ | $(1,2,4,4,2,1)$ | $(1,4)$ | $[48,42,4]$ | Optimal
$7$ | $3$ | $(1,6)$ | $(1,1,1)$ | $[6,3,4]$ | MDS
$7$ | $4$ | $(1,5,2,6)$ | $(1,1)$ | $[12,8,4]$ | Optimal
$7$ | $8$ | $(1,4,4,1)$ | $(1,1)$ | $[16,12,4]$ | Optimal
$7$ | $25$ | $(1,6)$ | $(1,2,4,2,1)$ | $[50,45,4]$ | Optimal
$5$ | $13$ | $(1,1,4,1,1)$ | $(1,4)$ | $[26,21,4]$ | Optimal
$3$ | $4$ | $(1,2,1,2)$ | $(1,2)$ | $[8,4,4]$ | Optimal

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