Reversible Cyclic Codes over ${\mathbb{F}}_2+{u\mathbb{F}}_2+{u^2\mathbb{F}}_2$

Abstract

In this article, we investigate the formation of reversible cyclic codes (i.e., its codewords forms a symmetry) over the ring $\mathcal{S}={\mathbb{F}}_2+u{\mathbb{F}}_2+u^2{\mathbb{F}}_2$, where $u^3=0$. We find a unique set of generators for cyclic codes over $\mathcal{S}$ and classify reversible cyclic codes to their generators. The dual reversible cyclic codes are studied as well. Moreover, we provide some examples of reversible cyclic codes.

1. Introduction

In the early 1970s, linear codes over finite rings gained much attention in the field of algebraic coding theory after the introduction of Gray maps. In 1994, Hammons et al. [1] calculated non-linear binary codes over ${\mathbb{Z}}_4$ under the Gray map. The research of linear codes over finite rings has since gained more interest than the binary field, and numerous families of codes have been examined in [2,3,4,5], such as over ${\mathbb{Z}}_4$, ${\mathbb{F}}_2+v{\mathbb{F}}_2,v^2=v$; ${\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2$, $u^2=v^2=0$; ${\mathbb{Z}}_{p^r}+u{\mathbb{Z}}_{p^r}+\cdots +u^{k-1}{\mathbb{Z}}_{p^r},u^k=0$ where p is a prime. Observe that cyclic codes are linear block codes in which each codeword’s cyclic shift is again a codeword. Because of their rich algebraic structure, these error-correcting codes are a significant family of linear codes. Numerous findings and new codes were discovered when cyclic codes were investigated across different finite rings in [4,6,7].

A reversible code is a linear code in which its components form a symmetry, i.e., it remains unchanged when the digits of codewords are reversed. By virtue of this symmetry, it is one of the many well-known kinds of cyclic codes. These codes are employed in data storage, retrieval systems, and DNA computing. It is important to note that reversible cyclic codes of length n over a finite field ${\mathbb{F}}_2$ are closely related to Linear complementary dual codes shortened as LCD codes and can be used for application in cryptography as well when $gcd(n,2)=1$ (see [8,9]). In 1964, James L. Massey [6] first defined the characteristic properties of reversible cyclic codes. After that, the results of the formation of reversible cyclic codes over ${\mathbb{Z}}_4$ were presented by Siap and Abualrub in [3] in 2007. In 2015, Srinivasulu and Bhaintwal [10] investigated reversible cyclic codes over ${\mathbb{F}}_4+u{\mathbb{F}}_4,u^2=0$ and their implications for DNA codes. In 2021, Prakash et al. [11] examined the reversible cyclic codes and their applications over the ring ${\mathbb{F}}_q+u{\mathbb{F}}_q$ with $u^2=0.$ Motivated by this work, we examine reversible cyclic codes of any length n over ${\mathbb{F}}_2+u{\mathbb{F}}_2+u^2{\mathbb{F}}_2,$$u^3=0$.

The paper is structured as follows: we give some preliminaries in Section 2, while in Section 3, we provide the structure of cyclic codes of arbitrary length n over the ring $\mathcal{S}$. Section 4 has some crucial results on reversible cyclic codes over ${\mathbb{F}}_2+u{\mathbb{F}}_2+u^2{\mathbb{F}}_2$. Section 5 discusses the dual of reversible cyclic code. The distance of the code is discussed in Section 6. Finally, Section 7 gives some examples to support our results.

2. Preliminary Results

A code $\mathcal{A}$ of length n over $\mathcal{S}$ is a non-empty subset of ${\mathcal{S}}^n$, and its element is called a codeword. A code $\mathcal{A}$ is linear over $\mathcal{S}$ if it is an $\mathcal{S}$-submodule of ${\mathcal{S}}^n$. A linear code $\mathcal{A}$ over $\mathcal{S}$ is called a cyclic code if for any $e=(e_0,e_1,\dots ,e_{n-1})\in \mathcal{A}$, $(e_{n-1},e_0,e_1,\dots ,e_{n-2})\in \mathcal{A}.$ Additionally, a cyclic code over $\mathcal{S}$ can be viewed as an ideal in ${\mathcal{S}}_n=\mathcal{S}\left[\vartheta \right]/({\vartheta }^n-1)$ identifying $(e_0,e_1,e_2,\dots ,e_{n-2},e_{n-1})$ by $e_0+e_1\vartheta +e_2{\vartheta }^2+\dots +e_{n-2}{\vartheta }^{n-2}+e_{n-1}{\vartheta }^{n-1}$.

The number of the nonzero component in $\mathbf{e}$ is referred to as the Hamming weight. The Hamming distance between any two codewords $\mathbf{e},\mathbf{c}$ is defined such that $\mathbf{e}\ne \mathbf{c}$ is the number of components in which these codewords differ. We define the Hamming distance of the code $\mathcal{A}$ as $d_H\left(\mathcal{A}\right)=min\left\{d_H(\mathbf{e},\mathbf{c})\right|\mathbf{e}\ne \mathbf{c};\forall \mathbf{e},\mathbf{c}\in \mathcal{A}\}$.

Every polynomial $f\left(\vartheta \right)=f_0+f_1\vartheta +\dots +f_{n-1}{\vartheta }^{n-1}\in \mathcal{S}\left[\vartheta \right]/({\vartheta }^n-1)$ with $f_{n-1}\ne 0$, the reciprocal of $f\left(\vartheta \right)$ is defined to be the poynomial $f^{*}\left(\vartheta \right)={\vartheta }^{n-1}f(1/\vartheta )=f_{n-1}+f_{n-2}\vartheta +\dots +f_0{\vartheta }^{n-1}$. Notice that $deg{f}^{*}\left(\vartheta \right)\le degf\left(\vartheta \right)$, and if $f_0\ne 0$, then $deg{f}^{*}\left(\vartheta \right)=degf\left(\vartheta \right)$. We say that $f\left(\vartheta \right)$ is self reciprocal if and only if $f^{*}\left(\vartheta \right)=f\left(\vartheta \right)$. Additionally, for any two polynomials $f\left(\vartheta \right)$, $g\left(\vartheta \right)$ satisfying $f\left(\vartheta \right)$∣$g\left(\vartheta \right)$ implies $f^{*}\left(\vartheta \right)$∣$g^{*}\left(\vartheta \right)$.

3. Construction of Cyclic Codes over $\mathcal{S}$

Throughout this paper, we assume that $\mathcal{S}={\mathbb{F}}_2+u{\mathbb{F}}_2+u^2{\mathbb{F}}_2$ where $u^3=0$. The structure of the generators of the cyclic codes over $\mathcal{S}$ depends on the ring $T={\mathbb{F}}_2+u{\mathbb{F}}_2=\{0,1,u,1+u\}$ where $u^2=0$.

Now, let ${\mathcal{A}}_1$ be a cyclic code in $\mathcal{S}$. Define ${\psi }_1:\mathcal{S}\to T$ by ${\psi }_1\left(a\right)=a,$${\psi }_1$ is a ring homomorphism that can be extended to a homomorphism ${\eta }_1:{\mathcal{A}}_1\to T$ defined by

$${\eta }_1(e_0+e_1\vartheta +e_2{\vartheta }^2+\cdots +e_{n-2}{\vartheta }^{n-2}+e_{n-1}{\vartheta }^{n-1})={\psi }_1\left(e_0\right)+{\psi }_1\left(e_1\right)\vartheta +\dots +{\psi }_1\left(e_{n-1}\right){\vartheta }^{n-1}.$$

where

$$ker{\eta }_1=\{u^{2}r\left(\vartheta \right):r\left(\vartheta \right)\in {\mathbb{F}}_2\left[\vartheta \right]\}.$$

Let $J=\{r\left(\vartheta \right):u^{2}r\left(\vartheta \right)\in ker{\eta }_1\}$. Notice that J is a cyclic code in ${\mathbb{F}}_2\left[\vartheta \right]/\langle {\vartheta }^n-1\rangle$ since it is an ideal in ${\mathbb{F}}_2\left[\vartheta \right]/\langle {\vartheta }^n-1\rangle$. So, $J=\langle {\alpha }_2\left(\vartheta \right)\rangle$ and $ker{\eta }_1=\langle u^2{\alpha }_2\left(\vartheta \right)\rangle$ with ${\alpha }_2\left(\vartheta \right)\mid ({\vartheta }^n-1)$, i.e., ${\alpha }_2\left(\vartheta \right)$ divides $({\vartheta }^n-1)$.

Further, let ${\mathcal{A}}_2$ be a cyclic code in $T_n$. Define ${\psi }_2:T\to {\mathbb{F}}_2$ by ${\psi }_2\left(a\right)=a^2$, ${\psi }_2$ is a ring homomorphism. Extend ${\psi }_2$ to a homomorphism ${\eta }_2:{\mathcal{A}}_2\to {\mathbb{F}}_2\left[\vartheta \right]/\langle {\vartheta }^n-1\rangle$ defined by

$${\eta }_2(e_0+e_1\vartheta +e_2{\vartheta }^2+\dots +e_{n-2}{\vartheta }^{n-2}+e_{n-1}{\vartheta }^{n-1})={\psi }_2\left(e_0\right)+{\psi }_2\left(e_1\right)\vartheta +\dots +{\psi }_2\left(e_{n-1}\right){\vartheta }^{n-1}$$

where

$$ker{\eta }_2=\{ur\left(\vartheta \right):r\left(\vartheta \right)\in {\mathbb{F}}_2\left[\vartheta \right]\}=\langle u{\alpha }_1\left(\vartheta \right)\rangle$$

with ${\alpha }_1\left(\vartheta \right)\mid ({\vartheta }^n-1)$, i.e., ${\alpha }_1\left(\vartheta \right)$ divides $({\vartheta }^n-1)$. The image of ${\eta }_2$ is also an ideal and hence a binary cyclic code generated by $\mathrm{g}\left(\vartheta \right)$ with $g\left(\vartheta \right)\mid ({\vartheta }^n-1)$. So, $\mathcal{A}=\langle g\left(\vartheta \right)+up\left(\vartheta \right),u{\alpha }_1\left(\vartheta \right)\rangle$ for some binary polynomial $p\left(\vartheta \right)$.

Note that ${\alpha }_1\mid$$\left(p{\displaystyle \frac{{\vartheta }^n-1}{g}}\right)$ because ${\eta }_2\left({\displaystyle \frac{{\vartheta }^n-1}{g}}\left[g+up\right]\right)={\eta }_2\left(up{\displaystyle \frac{{\vartheta }^n-1}{g}}\right)=0$ implies $\left(up{\displaystyle \frac{{\vartheta }^n-1}{g}}\right)\in ker{\eta }_2$= $\langle u{\alpha }_1\left(\vartheta \right)\rangle$. Since $ug\in ker{\eta }_2$⇒ ${\alpha }_1\left(\vartheta \right)\mid g\left(\vartheta \right)$.

In the rest of the article, we use $g\left(\vartheta \right)$, $p\left(\vartheta \right)$, ${\alpha }_1\left(\vartheta \right)$, and ${\alpha }_2\left(\vartheta \right)$ as mentioned above.

Abualrub and Siap [3], in their paper, classify all of the cyclic codes in ${\mathcal{S}}_n$ as follows.

Theorem 1 

([3], Theorem 2). Let $\mathcal{A}$ be a cyclic code in ${\mathcal{S}}_n=\mathcal{S}\left[\vartheta \right]/\langle {\vartheta }^n-1\rangle$, $\mathcal{S}={\mathbb{F}}_2+u{\mathbb{F}}_2+u^2{\mathbb{F}}_2=\{0,1,u,1+u,u^2,1+u^2,1+u+u^2,u+u^2\}$ where $u^3=0$.

(1)

If n is odd, then ${\mathcal{S}}_n$ is a principal ideal ring and $\mathcal{A}=\langle g,u{\alpha }_1,u^2{\alpha }_2\rangle =\langle g+u{\alpha }_1+u^2{\alpha }_2\rangle$ where $g\left(\vartheta \right),{\alpha }_1\left(\vartheta \right),{\alpha }_2\left(\vartheta \right)$ are binary polynomials with ${\alpha }_2\left(\vartheta \right)\mid {\alpha }_1\left(\vartheta \right)\mid g\left(\vartheta \right)\mid ({\vartheta }^n-1)$, i.e., ${\alpha }_2\left(\vartheta \right)\mid {\alpha }_1\left(\vartheta \right)$, ${\alpha }_1\left(\vartheta \right)\mid g\left(\vartheta \right)$ and $g\left(\vartheta \right)\mid ({\vartheta }^n-1)$.

(2)

If n is not odd, then

(a) 

$\mathcal{A}=\langle g+u{p}_1+u^2{p}_2\rangle$ where ${\alpha }_2\left(\vartheta \right)\mid {\alpha }_1\left(\vartheta \right)\mid g\left(\vartheta \right)\mid ({\vartheta }^n-1)$, $(g+up)\mid ({\vartheta }^n-1)$ in T and $(g+u{p}_1+u^2{p}_2)\mid ({\vartheta }^n-1)$ in $\mathcal{S}$ and $deg{p}_2<deg{p}_1$. Or,

(b) 

$\mathcal{A}=\langle g+u{p}_1+u^2{p}_2,u^2{\alpha }_2\rangle$ where ${\alpha }_2\left(\vartheta \right)\mid g\left(\vartheta \right)\mid ({\vartheta }^n-1)$, $(g+up)\mid ({\vartheta }^n-1)$ in T, $g\left(\vartheta \right)\mid p_1\left(\vartheta \right)\left({\displaystyle \frac{{\vartheta }^n-1}{g\left(\vartheta \right)}}\right)$, ${\alpha }_2$ divides $p_1\left(\vartheta \right)\left({\displaystyle \frac{{\vartheta }^n-1}{g\left(\vartheta \right)}}\right)$ and $p_2\left(\vartheta \right)\left({\displaystyle \frac{{\vartheta }^n-1}{g\left(\vartheta \right)}}\right)$$\left({\displaystyle \frac{{\vartheta }^n-1}{{\alpha }_1\left(\vartheta \right)}}\right)$ and $deg{p}_2<deg{\alpha }_2$. Or,

(c) 

$\mathcal{A}=\langle g+u{p}_1+u^2{p}_2,u{\alpha }_1+u^2{q}_1,u^2{\alpha }_2\rangle$ with ${\alpha }_2\mid {\alpha }_1\mid g\mid ({\vartheta }^n-1)$, ${\alpha }_1\left(\vartheta \right)\mid p_1\left(\vartheta \right)\left({\displaystyle \frac{{\vartheta }^n-1}{g\left(\vartheta \right)}}\right)$ and ${\alpha }_2\left(\vartheta \right)\mid$$q_1\left(\vartheta \right)\left({\displaystyle \frac{{\vartheta }^n-1}{{\alpha }_1\left(\vartheta \right)}}\right)$ and $p_2\left(\vartheta \right)\left({\displaystyle \frac{{\vartheta }^n-1}{g\left(\vartheta \right)}}\right)\left({\displaystyle \frac{{\vartheta }^n-1}{{\alpha }_1\left(\vartheta \right)}}\right)$. Moreover, $deg{p}_2<$$deg{\alpha }_2$, $deg{q}_1<$$deg{\alpha }_2$ and $deg{p}_1<$$deg{\alpha }_1$.

4. Reversible Cyclic Code over $\mathcal{S}$

In this part, we focus on reversible codes for arbitrary lengths and identify the necessary and sufficient criteria for the reversibility of cyclic codes $\mathcal{A}$ over $\mathcal{S}$. Some cases for even length have been discussed in [12]. For any codeword $e=(e_0,e_1,\dots ,e_{n-2},e_{n-1})\in \mathcal{A}$, the reverse of the codeword (symmetry of the codeword) is represented by $e^r$, where $e^r=(e_{n-1},e_{n-2},\dots ,e_0).$

Definition 1. 

A linear code $\mathcal{A}$ of length n over a ring $\mathcal{S}$ is said to be reversible if the symmetry of each codeword is in $\mathcal{A}$, i.e., $e^r\in \mathcal{A}$, for all $e\in \mathcal{A}$.

In [6], Massey-characterized cyclic codes tend to be reversible over the finite field, as follows:

Theorem 2 

([6] Theorem 1). The cyclic code over ${\mathbb{F}}_q$ generated by the monic polynomial $g\left(\vartheta \right)$ is reversible if and only if $g\left(\vartheta \right)$ is self-reciprocal.

We mention some of the results of Mostafanasab, and of Yousefian Darani [12], which are necessary to prove our results.

Lemma 1 

([12] Propsition 2.5). Let $\mathcal{A}$ be a cyclic code of length n over a commutative ring $\mathcal{S}$ and $f\left(\vartheta \right)\in \mathcal{S}\left[\vartheta \right]/({\vartheta }^n-1)$. Then, $f{\left(\vartheta \right)}^r\in \mathcal{A}$ if and only if $f^{*}\left(\vartheta \right)\in \mathcal{A}$.

Lemma 2 

([12] Corollary 2.6). Let $\mathcal{A}$ be a linear code. Then, $\mathcal{A}$ is reversible if and only if $f^{*}\left(\vartheta \right)\in \mathcal{A}$ for all $f\left(\vartheta \right)\in \mathcal{A}$.

Lemma 3 

([12] Lemma 2.7). Let $f_i\left(\vartheta \right),g_i\left(\vartheta \right)\in {\mathbb{F}}_2\left[\vartheta \right]$ for $i=1,2,3$. If

$$f_1\left(\vartheta \right)+u{f}_2\left(\vartheta \right)+u^2{f}_3\left(\vartheta \right)=g_1\left(\vartheta \right)+u{g}_2\left(\vartheta \right)+u^2{g}_3\left(\vartheta \right),$$

then $f_i\left(\vartheta \right)=g_i\left(\vartheta \right)$ for $i=1,2,3$.

Lemma 4 

([12] Lemma 2.11). Let $f_i\left(\vartheta \right)\in {\mathbb{F}}_2\left[\vartheta \right]$ for $i=1,2,3$. Suppose that $deg\left(f_1\left(\vartheta \right)\right)=r$, $deg\left(f_2\left(\vartheta \right)\right)=s$ and $deg\left(f_3\left(\vartheta \right)\right)=t$ where $r>max\{s,t\}$. Then,

$${\left(f_1\left(\vartheta \right)+u{f}_2\left(\vartheta \right)+u^2{f}_3\left(\vartheta \right)\right)}^{*}=f_1^{*}\left(\vartheta \right)+u{\vartheta }^{r-s}{f}_2^{*}\left(\vartheta \right)+u^2{\vartheta }^{r-t}{f}_3^{*}\left(\vartheta \right).$$

Lemma 5. 

A reversible cyclic code $\mathcal{A}$ of length n over the ring $\mathcal{S}$ under the homomorphism $\eta :\mathcal{A}\to {\mathbb{F}}_2\left[\vartheta \right]/\langle {\vartheta }^n-1\rangle$ (as defined in the Section 3) is also reversible.

Proof. 

Let $\eta \left(c\right)\in \eta \left(\mathcal{A}\right)$, where $e=(e_0,e_1,\cdots ,e_{n-1})\in \mathcal{A}$. Hence, $\eta \left(e\right)=(e_0^2,e_1^2,\cdots ,e_{n-1}^2)\in \eta \left(\mathcal{A}\right)$. Now as $\mathcal{A}$ is a reversible cyclic code, we have $e^r=(e_{n-1},e_{n-2},\cdots ,e_0)\in \mathcal{A}$. Now, consider

$$\begin{array}{cc}\hfill \eta {\left(e\right)}^r& ={(e_0^2,e_1^2,\cdots ,e_{n-1}^2)}^r\hfill \\ \hfill & =(e_{n-1}^2,e_{n-2}^2,\cdots ,e_0^2)\hfill \\ \hfill & =\eta (e_{n-1},e_{n-2},\cdots ,e_0)\in \eta \left(\mathcal{A}\right).\hfill \end{array}$$

so $\eta \left(\mathcal{A}\right)$ is reversible. □

Lemma 6. 

For any reversible cyclic code $\mathcal{A}$ over $\mathcal{S}$, the generated polymials $\langle g\left(\vartheta \right)\rangle ,\langle {\alpha }_1\left(\vartheta \right)\rangle$, and $\langle {\alpha }_2\left(\vartheta \right)\rangle$ are also reversible cyclic codes over ${\mathbb{F}}_2$.

Proof. 

We know that from Section 3, $\eta \left(\mathcal{A}\right)=\langle g\left(\vartheta \right)\rangle$ and using Lemma 5, $\eta \left(\mathcal{A}\right)$ is reversible code over ${\mathbb{F}}_2$. Hence, $\langle g\left(\vartheta \right)\rangle$ is reversible cyclic code over ${\mathbb{F}}_2$.

As $ker{\eta }_1=\{u^{2}r\left(\vartheta \right)\mid r\left(\vartheta \right)\in \mathcal{A}\}$ and $J_1=\{r\left(\vartheta \right)\mid u^{2}r\left(\vartheta \right)\in$$ker{\eta }_1\}=\langle {\alpha }_2\left(\vartheta \right)\rangle$, it is sufficient to show that $J_1$ is reversible.

Let $r\left(\vartheta \right)=r_0+r_1\vartheta +\cdots +r_{n-1}{\vartheta }^{n-1}\in J_1$. Then, $r\left(\vartheta \right)\in {\mathbb{F}}_2\left[\vartheta \right]$ is a polynomial in $\mathcal{A}$, which gives $r^{*}\left(\vartheta \right)\in \mathcal{A}$$\left(\mathrm{by}\mathrm{the}\mathrm{reversibility}\mathrm{of}\mathrm{cyclic}\mathrm{code}\mathcal{A}\right)$. Moreover, $u^2{r}^{*}\left(\vartheta \right)\in$$ker{\eta }_1$ i.e., $r^{*}\left(\vartheta \right)\in J_1$. Hence, $\langle {\alpha }_2\left(\vartheta \right)\rangle$ is a reversible cyclic code.

In a similar way, $ker{\eta }_2=\{u^{2}t\left(\vartheta \right)\mid t\left(\vartheta \right)$ is a polynomial in $\mathcal{A}$ with coefficient in ${\mathbb{F}}_2$} and $J_2=\{t\left(\vartheta \right)\mid u^{2}t\left(\vartheta \right)\in$$ker{\eta }_2\}=\langle {\alpha }_1\left(\vartheta \right)\rangle$; it is sufficient to show that $J_2$ is reversible.

Let $t\left(\vartheta \right)=t_0+t_1\vartheta +\cdots +t_{n-1}{\vartheta }^{n-1}\in J_2$ be arbitrary. Then, $t\left(\vartheta \right)\in {\mathbb{F}}_2\left[\vartheta \right]$ is a polynomial in $\mathcal{A}$. Since $\mathcal{A}$ is reversible cyclic code in $\mathcal{S}$, $t^{*}\left(\vartheta \right)$ is also in $\mathcal{A}$. Thus, $u^2{t}^{*}\left(\vartheta \right)\in$$ker{\eta }_2$, i.e., $t^{*}\left(\vartheta \right)\in J_2$. As a consequence, we obtain the desired outcome. □

Theorem 3. 

Let $\mathcal{A}=\langle g\left(\vartheta \right),u{\alpha }_1\left(\vartheta \right),u^2{\alpha }_2\left(\vartheta \right)\rangle =\langle g\left(\vartheta \right)+u{\alpha }_1\left(\vartheta \right)+u^2{\alpha }_2\left(\vartheta \right)\rangle$ be a linear cyclic code of odd length n over $\mathcal{S}$, where $g\left(\vartheta \right),{\alpha }_1\left(\vartheta \right)$, and ${\alpha }_2\left(\vartheta \right)$ are binary polynomials with ${\alpha }_2\left(\vartheta \right)\mid {\alpha }_1\left(\vartheta \right)\mid g\left(\vartheta \right)\mid ({\vartheta }^n-1)$. Then, $\mathcal{A}$ is reversible if and only if $g\left(\vartheta \right)$, ${\alpha }_1\left(\vartheta \right)$, and ${\alpha }_2\left(\vartheta \right)$ are self reciprocal.

Proof. 

Assume that $\mathcal{A}$ is a reversible cyclic code over $\mathcal{S}$. By using Lemma 6, we have that $g\left(\vartheta \right)$, ${\alpha }_1\left(\vartheta \right)$, and ${\alpha }_2\left(\vartheta \right)$ are self reciprocal.

For the remaining part, we assume that $g\left(\vartheta \right)$, ${\alpha }_1\left(\vartheta \right)$, and ${\alpha }_2\left(\vartheta \right)$ are binary self-reciprocal polynomials. Let $e\left(\vartheta \right)\in \mathcal{A}$, i.e., $e\left(\vartheta \right)=g\left(\vartheta \right)n_1\left(\vartheta \right)+u{\alpha }_1\left(\vartheta \right)n_2\left(\vartheta \right)+u^2{\alpha }_2\left(\vartheta \right)n_3\left(\vartheta \right)$ for some polynomials $n_1\left(\vartheta \right),n_2\left(\vartheta \right)$, and $n_3\left(\vartheta \right)$ over $\mathcal{S}$. $\mathcal{A}$ is reversible if and only if $c^{*}\left(\vartheta \right)\in \mathcal{A}.$

For this consider

$$\begin{array}{cc}\hfill e^{*}\left(\vartheta \right)& ={(g\left(\vartheta \right)n_1\left(\vartheta \right)+u{\alpha }_1\left(\vartheta \right)n_2\left(\vartheta \right)+u^2{\alpha }_2\left(\vartheta \right)n_3\left(\vartheta \right))}^{*}\hfill \\ \hfill & =(g^{*}\left(\vartheta \right)n_1^{*}\left(\vartheta \right)+u{\vartheta }^i{\alpha }_1^{*}\left(\vartheta \right)n_2^{*}\left(\vartheta \right)+u^2{\vartheta }^j{\alpha }_2^{*}\left(\vartheta \right)n_3^{*}\left(\vartheta \right))\hfill \\ \hfill & =(g\left(\vartheta \right)n_1^{*}\left(\vartheta \right)+u{\alpha }_1\left(\vartheta \right){\vartheta }^i{n}_2^{*}\left(\vartheta \right)+u^2{\alpha }_2\left(\vartheta \right){\vartheta }^j{n}_3^{*}\left(\vartheta \right))\hfill \end{array}$$

where $n_1^{*}\left(\vartheta \right),n_2^{*}\left(\vartheta \right)$, and $n_3^{*}\left(\vartheta \right)$ are polynomials over $\mathcal{S}$. Therefore,

$$e^{*}\left(\vartheta \right)\in \langle g\left(\vartheta \right),u{\alpha }_1\left(\vartheta \right),u^2{\alpha }_2\left(\vartheta \right)\rangle .$$

Thus, $\mathcal{A}$ is a reversible cyclic code over $\mathcal{S}$. □

Theorem 4. 

For $\mathcal{A}=\langle g\left(\vartheta \right)+u{p}_1\left(\vartheta \right)+u^2{p}_2\left(\vartheta \right),u{\alpha }_1\left(\vartheta \right)+u^{2}q\left(\vartheta \right),u^2{\alpha }_2\left(\vartheta \right)\rangle ,$ where $\mathcal{A}$ is a cyclic code of length n over $\mathcal{S}$ such that n is even with ${\alpha }_2\left(\vartheta \right)\mid {\alpha }_1\left(\vartheta \right)\mid g\left(\vartheta \right)\mid ({\vartheta }^n-1)$ with $deg\left(p_1\left(\vartheta \right)\right)<$$deg{\alpha }_1\left(\vartheta \right)$, $deg\left(p_2\left(\vartheta \right)\right)<$$deg\left({\alpha }_2\left(\vartheta \right)\right)$, and $degq\left(\vartheta \right)<$$deg{\alpha }_2\left(\vartheta \right).$ Then, $\mathcal{A}$ is reversible if and only if

(1)

$g\left(\vartheta \right)$ and ${\alpha }_2\left(\vartheta \right)$ are self reciprocal.

(2)

 

(a)

${\vartheta }^i{p}_1^{*}\left(\vartheta \right)=p_1\left(\vartheta \right)+t_0\left(\vartheta \right){\alpha }_1\left(\vartheta \right)$ and ${\alpha }_2\left(\vartheta \right)\mid {\vartheta }^j{p}_2^{*}\left(\vartheta \right)+p_2\left(\vartheta \right)+t_0\left(\vartheta \right)q\left(\vartheta \right)$. or

(b)

${\vartheta }^i{p}_1^{*}\left(\vartheta \right)=p_1\left(\vartheta \right)+g\left(\vartheta \right)+t_0\left(\vartheta \right){\alpha }_1\left(\vartheta \right)$ and

${\alpha }_2\left(\vartheta \right)\mid {\vartheta }^j{p}_2^{*}\left(\vartheta \right)+p_2\left(\vartheta \right)+p_1\left(\vartheta \right)+t_0\left(\vartheta \right)q\left(\vartheta \right)$.

Proof. 

We know that $\mathcal{A}$ is an $\mathcal{S}\left[\vartheta \right]$-module. First, we consider $\mathcal{A}$ reversible. Then, using Lemma 4 gives

$$\begin{array}{cc}\hfill {(g\left(\vartheta \right)+u{p}_1\left(\vartheta \right)+u^2{p}_2\left(\vartheta \right))}^{*}& =g^{*}\left(\vartheta \right)+u{\vartheta }^i{p}_1^{*}\left(\vartheta \right)+u^2{\vartheta }^j{p}_2^{*}\left(\vartheta \right)\hfill \\ \hfill & =s\left(\vartheta \right)(g\left(\vartheta \right)+u{p}_1\left(\vartheta \right)+u^2{p}_2\left(\vartheta \right))\hfill \\ \hfill & +t\left(\vartheta \right)((u{\alpha }_1\left(\vartheta \right)+u^{2}q\left(\vartheta \right))+w\left(\vartheta \right)\left(u^2{\alpha }_2\left(\vartheta \right)\right)\in \mathcal{A}.\hfill \end{array}$$

For $s\left(\vartheta \right),t\left(\vartheta \right),w\left(\vartheta \right)\in \mathcal{S}\left[\vartheta \right].$ Let $s\left(\vartheta \right)=s_0\left(\vartheta \right)+u{s}_1\left(\vartheta \right)+u^2{s}_2\left(\vartheta \right)$ and $t\left(\vartheta \right)=t_0+u{t}_1\left(\vartheta \right)+u^2{t}_2\left(\vartheta \right)$ where $s_i\left(\vartheta \right),$$t_i\left(\vartheta \right)\in {\mathbb{F}}_2\left[\vartheta \right]$. Then,

$$\begin{array}{cc}\hfill (g^{*}\left(\vartheta \right)+u{\vartheta }^i{p}_1^{*}\left(\vartheta \right)+u^2{\vartheta }^j{p}_2^{*}\left(\vartheta \right)& =(s_0\left(\vartheta \right)+u{s}_1\left(\vartheta \right)+u^2{s}_2\left(\vartheta \right))(g\left(\vartheta \right)+u{p}_1\left(\vartheta \right)+u^2{p}_2\left(\vartheta \right))\hfill \\ \hfill & +(t_0\left(\vartheta \right)+u{t}_1\left(\vartheta \right)+u^2{t}_2\left(\vartheta \right))(u{\alpha }_1\left(\vartheta \right)+u^{2}q\left(\vartheta \right))\hfill \\ \hfill & +w\left(\vartheta \right)\left(u^2{\alpha }_2\left(\vartheta \right)\right)\hfill \\ \hfill & =s_0\left(\vartheta \right)g\left(\vartheta \right)+s_0\left(\vartheta \right)u{p}_1\left(\vartheta \right)+s_0\left(\vartheta \right)u^2{p}_2\left(\vartheta \right)+u{s}_1\left(\vartheta \right)g\left(\vartheta \right)\hfill \\ \hfill & +u^2{s}_1\left(\vartheta \right)p_1\left(\vartheta \right)+u^2{s}_2\left(\vartheta \right)g\left(\vartheta \right)+t_0\left(\vartheta \right)u{\alpha }_1\left(\vartheta \right)+t_0\left(\vartheta \right)u^{2}q\left(\vartheta \right)\hfill \\ \hfill & +u^2{t}_1\left(\vartheta \right){\alpha }_1\left(\vartheta \right)+w\left(\vartheta \right)u^2{\alpha }_2\left(\vartheta \right)\hfill \\ \hfill & =s_0\left(\vartheta \right)g\left(\vartheta \right)+u(s_0\left(\vartheta \right)p_1\left(\vartheta \right)+s_1\left(\vartheta \right)g\left(\vartheta \right)+t_0\left(\vartheta \right){\alpha }_1\left(\vartheta \right))\hfill \\ \hfill & +u^2(s_0\left(\vartheta \right)p_2\left(\vartheta \right)+s_1\left(\vartheta \right)p_1\left(\vartheta \right)+s_2\left(\vartheta \right)g\left(\vartheta \right)+t_0\left(\vartheta \right)q\left(\vartheta \right)\hfill \\ \hfill & +t_1\left(\vartheta \right){\alpha }_1\left(\vartheta \right)+w\left(\vartheta \right)a_2\left(\vartheta \right)).\hfill \end{array}$$

The application of Lemma 3 implies that $g^{*}\left(\vartheta \right)=$$s_0\left(\vartheta \right)g\left(\vartheta \right)$, ${\vartheta }^i{p}_1^{*}\left(\vartheta \right)=s_0\left(\vartheta \right)p_1\left(\vartheta \right)+s_1\left(\vartheta \right)g\left(\vartheta \right)+t_0\left(\vartheta \right){\alpha }_1\left(\vartheta \right)$ and ${\vartheta }^j{p}_2^{*}\left(\vartheta \right)=s_0\left(\vartheta \right)p_2\left(\vartheta \right)+s_1\left(\vartheta \right)p_1\left(\vartheta \right)+s_2\left(\vartheta \right)g\left(\vartheta \right)+t_0\left(\vartheta \right)q\left(\vartheta \right)+t_1\left(\vartheta \right){\alpha }_1\left(\vartheta \right)+w\left(\vartheta \right){\alpha }_2\left(\vartheta \right)$. Now since $g^{*}\left(\vartheta \right)=s_0\left(\vartheta \right)g\left(\vartheta \right)$ and $d\left(g^{*}\left(\vartheta \right)\right)\le$$d\left(g\right(\vartheta \left)\right)$, where $d\left(g\right(\vartheta \left)\right)$ denotes degree of $g\left(\vartheta \right)$, which implies $s_0\left(\vartheta \right)=1$; hence, $g\left(\vartheta \right)$ is self-reciprocal. Now, ${\vartheta }^i{p}_1^{*}\left(\vartheta \right)=p_1\left(\vartheta \right)+s_1\left(\vartheta \right)g\left(\vartheta \right)+t_0\left(\vartheta \right){\alpha }_1\left(\vartheta \right)$; equating the degrees on both sides gives that $s_1\left(\vartheta \right)$ is either 0 or 1. In a similar manner, ${\vartheta }^j{p}_2^{*}\left(\vartheta \right)=p_2\left(\vartheta \right)+s_1\left(\vartheta \right)p_1\left(\vartheta \right)+s_2\left(\vartheta \right)g\left(\vartheta \right)+t_0\left(\vartheta \right)q\left(\vartheta \right)+t_1\left(\vartheta \right){\alpha }_1\left(\vartheta \right)+w\left(\vartheta \right){\alpha }_2\left(\vartheta \right)$. Similarly, equating the degrees on both sides gives that $s_2\left(\vartheta \right)$ is either 0 or 1; hence, the following four cases arise:

Case 1. 

If $s_1\left(\vartheta \right)=s_2\left(\vartheta \right)=0$, then ${\vartheta }^i{p}_1^{*}\left(\vartheta \right)=p_1\left(\vartheta \right)+t_0\left(\vartheta \right){\alpha }_1\left(\vartheta \right)$ and ${\vartheta }^j{p}_2^{*}\left(\vartheta \right)=p_2\left(\vartheta \right)+t_0\left(\vartheta \right)q\left(\vartheta \right)+t_1\left(\vartheta \right){\alpha }_1\left(\vartheta \right)+w\left(\vartheta \right){\alpha }_2\left(\vartheta \right)$.

Case 2. 

If $s_1\left(\vartheta \right)=0$ and $s_2\left(\vartheta \right)=1$, then ${\vartheta }^i{p}_1^{*}\left(\vartheta \right)=p_1\left(\vartheta \right)+t_0\left(\vartheta \right){\alpha }_1\left(\vartheta \right)$ and ${\vartheta }^j{p}_2^{*}\left(\vartheta \right)=p_2\left(\vartheta \right)+g\left(\vartheta \right)+t_0\left(\vartheta \right)q\left(\vartheta \right)+t_1\left(\vartheta \right){\alpha }_1\left(\vartheta \right)+w\left(\vartheta \right){\alpha }_2\left(\vartheta \right)$.

Case 3. 

If $s_1\left(\vartheta \right)=1$ and $s_2\left(\vartheta \right)=0$, then ${\vartheta }^i{p}_1^{*}\left(\vartheta \right)=p_1\left(\vartheta \right)+g\left(\vartheta \right)+t_0\left(\vartheta \right){\alpha }_1\left(\vartheta \right)$ and ${\vartheta }^j{p}_2^{*}\left(\vartheta \right)=p_2\left(\vartheta \right)+p_1\left(\vartheta \right)+t_0\left(\vartheta \right)q\left(\vartheta \right)+t_1\left(\vartheta \right){\alpha }_1\left(\vartheta \right)+w\left(\vartheta \right){\alpha }_2\left(\vartheta \right)$.

Case 4. 

If $s_1\left(\vartheta \right)=1$ and $s_2\left(\vartheta \right)=1$, then ${\vartheta }^i{p}_1^{*}\left(\vartheta \right)=p_1\left(\vartheta \right)+g\left(\vartheta \right)+t_0\left(\vartheta \right){\alpha }_1\left(\vartheta \right)$ and ${\vartheta }^j{p}_2^{*}\left(\vartheta \right)=p_2\left(\vartheta \right)+p_1\left(\vartheta \right)+g\left(\vartheta \right)+t_0\left(\vartheta \right)q\left(\vartheta \right)+t_1\left(\vartheta \right){\alpha }_1\left(\vartheta \right)+w\left(\vartheta \right){\alpha }_2\left(\vartheta \right)$.

By assumption, we have ${\alpha }_2\left(\vartheta \right)\mid {\alpha }_1\left(\vartheta \right)\mid g\left(\vartheta \right)$. so ${\vartheta }^i{p}_1^{*}\left(\vartheta \right)=p_1\left(\vartheta \right)+t_0\left(\vartheta \right){\alpha }_1\left(\vartheta \right)$ and ${\alpha }_2\left(\vartheta \right)\mid {\vartheta }^j{p}_2^{*}\left(\vartheta \right)+p_2\left(\vartheta \right)+t_0\left(\vartheta \right)q\left(\vartheta \right)$. Similarly, ${\vartheta }^i{p}_1^{*}\left(\vartheta \right)=p_1\left(\vartheta \right)+g\left(\vartheta \right)+t_0\left(\vartheta \right){\alpha }_1\left(\vartheta \right)$ and ${\alpha }_2\left(\vartheta \right)\mid {\vartheta }^j{p}_2^{*}\left(\vartheta \right)+p_2\left(\vartheta \right)+p_1\left(\vartheta \right)+t_0\left(\vartheta \right)q\left(\vartheta \right)$.

Moreover, $u^2{\alpha }_2^{*}\left(\vartheta \right)\in \mathcal{A}$. Then, the polynomials ${\lambda }_0,{\lambda }_1,{\lambda }_2,{\mu }_1,{\mu }_2\in \mathcal{S}\left[\vartheta \right]$, and $\delta \in {\mathbb{F}}_2\left[\vartheta \right]$ exist such that

$$\begin{array}{cc}\hfill u^2{\alpha }_2^{*}\left(\vartheta \right)& ={\lambda }_0\left(\vartheta \right)g\left(\vartheta \right)+u({\lambda }_0\left(\vartheta \right)p_1\left(\vartheta \right)+{\lambda }_1\left(\vartheta \right)g\left(\vartheta \right)+{\mu }_0\left(\vartheta \right){\alpha }_1\left(\vartheta \right))\hfill \\ \hfill & +u^2({\lambda }_0\left(\vartheta \right)p_2\left(\vartheta \right)+{\lambda }_1\left(\vartheta \right)p_1\left(\vartheta \right)+{\lambda }_2\left(\vartheta \right)g\left(\vartheta \right)+{\mu }_0\left(\vartheta \right)q\left(\vartheta \right)\hfill \\ \hfill & +{\mu }_1\left(\vartheta \right){\alpha }_1\left(\vartheta \right)+\delta \left(\vartheta \right){\alpha }_2\left(\vartheta \right)).\hfill \end{array}$$

Therefore, ${\lambda }_0\left(\vartheta \right)={\lambda }_1\left(\vartheta \right)={\mu }_0\left(\vartheta \right)=0$ and ${\alpha }_2^{*}\left(\vartheta \right)={\lambda }_2\left(\vartheta \right)g\left(\vartheta \right)+{\mu }_1\left(\vartheta \right){\alpha }_1\left(\vartheta \right)+\delta \left(\vartheta \right){\alpha }_2\left(\vartheta \right)$. Since ${\alpha }_2\left(\vartheta \right)\mid {\alpha }_1\left(\vartheta \right)\mid g\left(\vartheta \right)$, we have ${\alpha }_2\left(\vartheta \right)\mid {\alpha }_2^{*}\left(\vartheta \right)$ and so ${\alpha }_2^{*}\left(\vartheta \right)={\alpha }_2\left(\vartheta \right)$, i.e., ${\alpha }_2\left(\vartheta \right)$ is self reciprocal.

For the converse part, we are using only $\left(1\right)$ and condition $\left(b\right)$ of $\left(2\right)$ holds.

$$\begin{array}{cc}\hfill {(g\left(\vartheta \right)+u{p}_1\left(\vartheta \right)+u^2{p}_2\left(\vartheta \right))}^{*}& =g^{*}\left(\vartheta \right)+u{\vartheta }^i{p}_1^{*}\left(\vartheta \right)+u^2{\vartheta }^j{p}_2^{*}\left(\vartheta \right)\hfill \\ \hfill & =g\left(\vartheta \right)+u(p_1\left(\vartheta \right)+g\left(\vartheta \right)+t_0\left(\vartheta \right){\alpha }_1\left(\vartheta \right))+u^2(p_1\left(\vartheta \right)+p_2\left(\vartheta \right)\hfill \\ \hfill & +t_0\left(\vartheta \right)q\left(\vartheta \right)+\lambda \left(\vartheta \right){\alpha }_2\left(\vartheta \right))\mathrm{for}\mathrm{some}\lambda \left(\vartheta \right)\in {\mathbb{F}}_2\left[\vartheta \right].\hfill \\ \hfill & =g\left(\vartheta \right)+u{p}_1\left(\vartheta \right)+ug\left(\vartheta \right)+u{t}_0\left(\vartheta \right){\alpha }_1\left(\vartheta \right)+u^2{p}_1\left(\vartheta \right)+u^2{p}_2\left(\vartheta \right)\hfill \\ \hfill & +u^2{t}_0\left(\vartheta \right)q\left(\vartheta \right)+u^2\lambda \left(\vartheta \right){\alpha }_2\left(\vartheta \right)\hfill \\ \hfill & =(g\left(\vartheta \right)+u{p}_1\left(\vartheta \right)+u^2{p}_2\left(\vartheta \right))+u(g\left(\vartheta \right)+u{p}_1\left(\vartheta \right)+u^2{p}_2\left(\vartheta \right))\hfill \\ \hfill & +u^2{t}_0\left(\vartheta \right)q\left(\vartheta \right)+u{t}_0\left(\vartheta \right){\alpha }_1\left(\vartheta \right)+u^2\lambda \left(\vartheta \right){\alpha }_2\left(\vartheta \right)\hfill \\ \hfill & =(g\left(\vartheta \right)+u{p}_1\left(\vartheta \right)+u^2{p}_2\left(\vartheta \right))+u(g\left(\vartheta \right)+u{p}_1\left(\vartheta \right)+u^2{p}_2\left(\vartheta \right))\hfill \\ \hfill & +t_0\left(\vartheta \right)(u{\alpha }_1\left(\vartheta \right)+u^{2}q\left(\vartheta \right))+\lambda \left(\vartheta \right)\left(u^2{\alpha }_2\left(\vartheta \right)\right)\in \mathcal{A}.\hfill \end{array}$$

where $t_0\left(\vartheta \right)$, $\lambda \left(\vartheta \right)\in {\mathbb{F}}_2\left[\vartheta \right]$. Hence, $\mathcal{A}$ is reversible. □

5. Dual of Reversible Cyclic Code over $\mathcal{S}$

Suppose that $l\left(\vartheta \right)=l_0+l_1\vartheta +\cdots +l_k{\vartheta }^k$ is a parity check polynomial over a cyclic $[n,k]$-code in $\mathcal{A}$ and $\overline{l}\left(\vartheta \right)=l^{*}\left(\vartheta \right)$. Then, for the characterization of ${\mathcal{A}}^{\perp }$ of cyclic code $\mathcal{A}$, we have the following result.

Theorem 5. 

Let $\mathcal{A}$ be a cyclic code over ${\mathbb{F}}_q$, and ${\mathcal{A}}^{\perp }$ be its dual. Then, ${\mathcal{A}}^{\perp }=<\overline{l}\left(\vartheta \right)>$ is reversible iff $l\left(\vartheta \right)\in {\mathcal{A}}^{\perp }$.

Proof. 

Let $\overline{l}\left(\vartheta \right)=(l_k,l_{k-1},\cdots ,l_0)\in {\mathcal{A}}^{\perp }$. Then, ${\left(\overline{l}\left(\vartheta \right)\right)}^r=(l_0,l_1,\cdots ,l_{k-1},l_k)=l\left(\vartheta \right)\in {\mathcal{A}}^{\perp }$. Conversely, suppose that $l\left(\vartheta \right)\in {\mathcal{A}}^{\perp }$, then $l\left(\vartheta \right)={\left(\overline{l}\left(\vartheta \right)\right)}^r$, which means $\overline{l}{\left(\vartheta \right)}^r=l\left(\vartheta \right)\in {\mathcal{A}}^{\perp }$. Hence, ${\mathcal{A}}^{\perp }=<\overline{l}\left(\vartheta \right)>$ is reversible cyclic code. □

Definition 2. 

For any ideal J in ${\mathcal{S}}_n$, the annihilator $\mathbb{Q}\left(J\right)$ of J in ${\mathcal{S}}_n$ is defined as

$$\mathbb{Q}\left(J\right)=\left\{d\right(\vartheta )\mid f(\vartheta \left)d\right(\vartheta )=0\forall f(\vartheta )\in J\}.$$

Another associated ideal of ${\mathcal{A}}^{\perp }$ of cyclic code $\mathcal{A}$ is

$$\mathbb{Q}{\left(J\right)}^{*}=\{t^{*}\left(\vartheta \right)\mid t\left(\vartheta \right)\in J\}.$$

Proposition 1. 

Let $\mathcal{A}$ be a cyclic code of odd length n over $\mathcal{S}$. Then,

$$\mathbb{Q}\left(\mathcal{A}\right)=〈{\displaystyle \frac{{\vartheta }^n-1}{{\alpha }_2\left(\vartheta \right)}},u{\displaystyle \frac{{\vartheta }^n-1}{{\alpha }_1\left(\vartheta \right)}},u^2{\displaystyle \frac{{\vartheta }^n-1}{g\left(\vartheta \right)}}〉,$$

where $\mathbb{Q}\left(\mathcal{A}\right)$ is the annihilator of $\mathcal{A}$, where ${\alpha }_2\left(\vartheta \right)\mid {\alpha }_1\left(\vartheta \right)\mid g\left(\vartheta \right)\mid ({\vartheta }^n-1)$.

Proof. 

It is given that $\mathcal{A}$ is a cyclic code of odd length over ${\mathbb{F}}_2+u{\mathbb{F}}_2+u^2{\mathbb{F}}_2$; then,

$$\mathcal{A}=\langle g\left(\vartheta \right),u{\alpha }_1\left(\vartheta \right),u^2{\alpha }_2\left(\vartheta \right)\rangle =\langle g\left(\vartheta \right)+u{\alpha }_1\left(\vartheta \right)+u^2{\alpha }_2\left(\vartheta \right)\rangle ,$$

where ${\alpha }_2\left(\vartheta \right)\mid {\alpha }_1\left(\vartheta \right)\mid g\left(\vartheta \right)\mid ({\vartheta }^n-1)$. Additionally, $n_1\left(\vartheta \right)$, $n_2\left(\vartheta \right)$ exists and $n_3\left(\vartheta \right)$ such that $g\left(\vartheta \right)={\alpha }_1\left(\vartheta \right)n_1\left(\vartheta \right)$, $g\left(\vartheta \right)={\alpha }_2\left(\vartheta \right)n_2\left(\vartheta \right)$, and ${\alpha }_1\left(\vartheta \right)={\alpha }_2\left(\vartheta \right)n_3\left(\vartheta \right)$. Notice that

$$\begin{array}{cc}\hfill \left({\displaystyle \frac{{\vartheta }^n-1}{{\alpha }_2\left(\vartheta \right)}}\right)(g\left(\vartheta \right)+u{\alpha }_1\left(\vartheta \right)+u^2{\alpha }_2\left(\vartheta \right))& =\left({\displaystyle \frac{{\vartheta }^n-1}{{\alpha }_2\left(\vartheta \right)}}\right)g\left(\vartheta \right)+u\left({\displaystyle \frac{{\vartheta }^n-1}{{\alpha }_2\left(\vartheta \right)}}\right){\alpha }_1\left(\vartheta \right)\hfill \\ & +u^2\left({\displaystyle \frac{{\vartheta }^n-1}{{\alpha }_2\left(\vartheta \right)}}\right){\alpha }_2\left(\vartheta \right)\hfill \\ \hfill & =\left({\displaystyle \frac{{\vartheta }^n-1}{{\alpha }_2\left(\vartheta \right)}}\right){\alpha }_2\left(\vartheta \right)n_2\left(\vartheta \right)+u\left({\displaystyle \frac{{\vartheta }^n-1}{{\alpha }_2\left(\vartheta \right)}}\right){\alpha }_2\left(\vartheta \right)n_3\left(\vartheta \right)\hfill \\ \hfill & =0,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill u\left({\displaystyle \frac{{\vartheta }^n-1}{{\alpha }_1\left(\vartheta \right)}}\right)(g\left(\vartheta \right)+u{\alpha }_1\left(\vartheta \right)+u^2{\alpha }_2\left(\vartheta \right))& =u\left({\displaystyle \frac{{\vartheta }^n-1}{{\alpha }_1\left(\vartheta \right)}}\right)g\left(\vartheta \right)+u^2\left({\displaystyle \frac{{\vartheta }^n-1}{{\alpha }_1\left(\vartheta \right)}}\right){\alpha }_1\left(\vartheta \right)\hfill \\ \hfill & =u\left({\displaystyle \frac{{\vartheta }^n-1}{{\alpha }_1\left(\vartheta \right)}}\right){\alpha }_1\left(\vartheta \right)n_1\left(\vartheta \right)\hfill \\ \hfill & =0.\hfill \end{array}$$

Similarly,

$$u^2\left({\displaystyle \frac{{\vartheta }^n-1}{g\left(\vartheta \right)}}\right)(g\left(\vartheta \right)+u{\alpha }_1\left(\vartheta \right)+u^2{\alpha }_2\left(\vartheta \right))=0.$$

So,

$$N=〈{\displaystyle \frac{{\vartheta }^n-1}{{\alpha }_2\left(\vartheta \right)}},u{\displaystyle \frac{{\vartheta }^n-1}{{\alpha }_1\left(\vartheta \right)}},u^2{\displaystyle \frac{{\vartheta }^n-1}{g\left(\vartheta \right)}}〉\subseteq \mathbb{Q}\left(\mathcal{A}\right).$$

Now, we prove that $\mathbb{Q}\left(\mathcal{A}\right)\subseteq N$. Suppose that $\mathbb{Q}\left(\mathcal{A}\right)=〈l\left(\vartheta \right),uj\left(\vartheta \right),u^{2}k\left(\vartheta \right)〉$. Then,

$$u^{2}k\left(\vartheta \right)(g\left(\vartheta \right)+u{\alpha }_1\left(\vartheta \right)+u^2{\alpha }_2\left(\vartheta \right))=0.$$

From here, we conclude that a polynomial $b_1\left(\vartheta \right)\in {\mathbb{F}}_2$ exists such that

$$k\left(\vartheta \right)=\left({\displaystyle \frac{{\vartheta }^n-1}{g\left(\vartheta \right)}}\right)b_1\left(\vartheta \right)\in N.$$

Similarly, one can get

$$\begin{array}{cc}\hfill uj\left(\vartheta \right)(g\left(\vartheta \right)+u{\alpha }_1\left(\vartheta \right)+u^2{\alpha }_2\left(\vartheta \right))& =0\hfill \\ \hfill uj\left(\vartheta \right)g\left(\vartheta \right)+u^{2}j\left(\vartheta \right){\alpha }_1\left(\vartheta \right)& =0\hfill \end{array}$$

Since $j\left(\vartheta \right)g\left(\vartheta \right)=0$, $u^{2}j\left(\vartheta \right){\alpha }_1\left(\vartheta \right)=0$ and so, there exists polynomial $b_2\left(\vartheta \right)$ in ${\mathbb{F}}_2$ such that

$$j\left(\vartheta \right)=\left({\displaystyle \frac{{\vartheta }^n-1}{{\alpha }_1\left(\vartheta \right)}}\right)b_2\left(\vartheta \right).$$

Additionally,

$$\begin{array}{ccc}l\left(\vartheta \right)(g\left(\vartheta \right)+u{\alpha }_1\left(\vartheta \right)+u^2{\alpha }_2\left(\vartheta \right))& =& 0\\ l\left(\vartheta \right)g\left(\vartheta \right)+ul\left(\vartheta \right){\alpha }_1\left(\vartheta \right)+u^{2}l\left(\vartheta \right){\alpha }_2\left(\vartheta \right)& =& 0\end{array}$$

which implies $l\left(\vartheta \right)=\left({\displaystyle \frac{{\vartheta }^n-1}{{\alpha }_2\left(\vartheta \right)}}\right)b_3\left(\vartheta \right)$ for some $b_3\left(\vartheta \right)\in {\mathbb{F}}_2$.

Hence,

$$\mathbb{Q}\left(\mathcal{A}\right)=〈l\left(\vartheta \right),uj\left(\vartheta \right),u^{2}k\left(\vartheta \right)〉\subseteq 〈{\displaystyle \frac{{\vartheta }^n-1}{{\alpha }_2\left(\vartheta \right)}},u{\displaystyle \frac{{\vartheta }^n-1}{{\alpha }_1\left(\vartheta \right)}},u^2{\displaystyle \frac{{\vartheta }^n-1}{g\left(\vartheta \right)}}〉\in N.$$

Therefore, $\mathbb{Q}\left(\mathcal{A}\right)=〈{\displaystyle \frac{{\vartheta }^n-1}{{\alpha }_2\left(\vartheta \right)}},u{\displaystyle \frac{{\vartheta }^n-1}{{\alpha }_1\left(\vartheta \right)}},u^2{\displaystyle \frac{{\vartheta }^n-1}{g\left(\vartheta \right)}}〉.$ □

The implications of Proposition 1 are as follows.

Theorem 6. 

Let $\mathcal{A}$ be a cyclic code of odd length n over $\mathcal{S}$. Then,

$${\mathcal{A}}^{\perp }=〈{\left({\displaystyle \frac{{\vartheta }^n-1}{{\alpha }_2\left(\vartheta \right)}}\right)}^{*},u{\left({\displaystyle \frac{{\vartheta }^n-1}{{\alpha }_1\left(\vartheta \right)}}\right)}^{*},u^2{\left({\displaystyle \frac{{\vartheta }^n-1}{g\left(\vartheta \right)}}\right)}^{*}〉.$$

Theorem 7. 

Let $\mathcal{A}$ be a reversible cyclic code of odd length n over $\mathcal{S}$ with ${\alpha }_2\left(\vartheta \right)\mid {\alpha }_1\left(\vartheta \right)\mid g\left(\vartheta \right)|({\vartheta }^n-1)$ and ${\mathcal{A}}^{\perp }=〈{\left({\displaystyle \frac{{\vartheta }^n-1}{{\alpha }_2\left(\vartheta \right)}}\right)}^{*},u{\left({\displaystyle \frac{{\vartheta }^n-1}{{\alpha }_1\left(\vartheta \right)}}\right)}^{*},u^2{\left({\displaystyle \frac{{\vartheta }^n-1}{g\left(\vartheta \right)}}\right)}^{*}〉.$ Then, ${\mathcal{A}}^{\perp }$ is a reversible cyclic code over $\mathcal{S}$.

Proof. 

Assume that $\mathcal{A}$ is a reversible cyclic code of odd length n over $\mathcal{S}$. Hence, by Theorem 3, $g\left(\vartheta \right)$, ${\alpha }_1\left(\vartheta \right)$, and ${\alpha }_2\left(\vartheta \right)$ are self reciprocal. Suppose that $\left({\displaystyle \frac{{\vartheta }^n-1}{{\alpha }_2\left(\vartheta \right)}}\right)=v_1\left(\vartheta \right)$, $\left({\displaystyle \frac{{\vartheta }^n-1}{{\alpha }_1\left(\vartheta \right)}}\right)=v_2\left(\vartheta \right)$, and $\left({\displaystyle \frac{{\vartheta }^n-1}{g\left(\vartheta \right)}}\right)=v_3\left(\vartheta \right).$

Therefore,

$$\begin{array}{ccc}{({\vartheta }^n-1)}^{*}& =& {\alpha }_2^{*}\left(\vartheta \right)v_1^{*}\left(\vartheta \right),\\ {({\vartheta }^n-1)}^{*}& =& {\alpha }_1^{*}\left(\vartheta \right)v_2^{*}\left(\vartheta \right),\end{array}$$

and

$${({\vartheta }^n-1)}^{*}=g^{*}\left(\vartheta \right)v_3^{*}\left(\vartheta \right).$$

This implies

$$v_1^{*}\left(\vartheta \right)={\displaystyle \frac{{({\vartheta }^n-1)}^{*}}{{\alpha }_2^{*}\left(\vartheta \right)}}={\displaystyle \frac{-({\vartheta }^n-1)}{{\alpha }_2\left(\vartheta \right)}}=-v_1\left(\vartheta \right),$$

$$v_2^{*}\left(\vartheta \right)={\displaystyle \frac{{({\vartheta }^n-1)}^{*}}{{\alpha }_1^{*}\left(\vartheta \right)}}={\displaystyle \frac{-({\vartheta }^n-1)}{{\alpha }_1\left(\vartheta \right)}}=-v_2\left(\vartheta \right),$$

and

$$v_3^{*}\left(\vartheta \right)={\displaystyle \frac{{({\vartheta }^n-1)}^{*}}{g^{*}\left(\vartheta \right)}}={\displaystyle \frac{-({\vartheta }^n-1)}{g\left(\vartheta \right)}}=-v_3\left(\vartheta \right).$$

Let $\overline{c}\left(\vartheta \right)\in {\mathcal{A}}^{\perp }$. Then,

$$\begin{array}{cc}\hfill {\left(\overline{c}\left(\vartheta \right)\right)}^{*}& ={\left({\left({\displaystyle \frac{{\vartheta }^n-1}{{\alpha }_2\left(\vartheta \right)}}\right)}^{*}{d}_1\left(\vartheta \right)+{\left({\displaystyle \frac{{\vartheta }^n-1}{{\alpha }_1\left(\vartheta \right)}}\right)}^{*}{d}_2\left(\vartheta \right)+{\left({\displaystyle \frac{{\vartheta }^n-1}{g\left(\vartheta \right)}}\right)}^{*}{d}_3\left(\vartheta \right)\right)}^{*}\hfill \\ \hfill & ={(-v_1\left(\vartheta \right)d_1\left(\vartheta \right)-v_2\left(\vartheta \right)d_2\left(\vartheta \right)-v_3\left(\vartheta \right)d_3\left(\vartheta \right))}^{*}\hfill \\ \hfill & =(-v_1^{*}\left(\vartheta \right)d_1^{*}\left(\vartheta \right)-{\vartheta }^i{v}_2^{*}\left(\vartheta \right)d_2^{*}\left(\vartheta \right)-{\vartheta }^j{v}_3^{*}\left(\vartheta \right)d_3^{*}\left(\vartheta \right))\hfill \\ \hfill & =(v_1^{*}\left(\vartheta \right)i_1\left(\vartheta \right)+v_2^{*}\left(\vartheta \right)i_2\left(\vartheta \right)+v_3^{*}\left(\vartheta \right)i_3\left(\vartheta \right)),\hfill \end{array}$$

where $i_1\left(\vartheta \right)=-d_1^{*}\left(\vartheta \right)$, $i_2\left(\vartheta \right)=-{\vartheta }^i{d}_2^{*}\left(\vartheta \right)$, and $i_3\left(\vartheta \right)=-{\vartheta }^j{d}_3^{*}\left(\vartheta \right)$, over ${\mathbb{F}}_2+u{\mathbb{F}}_2+u^2{\mathbb{F}}_2$. Thus $c\left(\vartheta \right)\in {\mathcal{A}}^{\perp }$. Thus, by Theorem 5, ${\mathcal{A}}^{\perp }$ is a reversible cyclic code over ${\mathbb{F}}_2+u{\mathbb{F}}_2+u^2{\mathbb{F}}_2$. □

6. Minimum Hamming Distance of a Cyclic Code over $\mathcal{S}$

Here, we determine the minimum Hamming distance of a cyclic code of length n over $\mathcal{S}$. Assume that $\mathcal{A}=\langle g\left(\vartheta \right)+u{p}_1\left(\vartheta \right)+u^2{p}_2\left(\vartheta \right),u{\alpha }_1\left(\vartheta \right)+u^2{q}_1\left(\vartheta \right),u^2{\alpha }_2\left(\vartheta \right)\rangle$ is a cyclic code of length n over $\mathcal{S}$. Define ${\mathcal{A}}_{u^2}=\{n\left(\vartheta \right)\mid u^{2}n\left(\vartheta \right)\in \mathcal{A}\}$. Then, ${\mathcal{A}}_{u^2}$ is a cyclic code of length n over ${\mathbb{F}}_2$.

Theorem 8 

([12] Theorem 4.4). If $\mathcal{A}=\langle g\left(\vartheta \right)+u{p}_1\left(\vartheta \right)+u^2{p}_2\left(\vartheta \right),u{\alpha }_1\left(\vartheta \right)+u^2{q}_1\left(\vartheta \right),u^2{\alpha }_2\left(\vartheta \right)\rangle$ is a cyclic code of length n over $\mathcal{S}$. Then, ${\mathcal{A}}_{u^2}=\langle {\alpha }_2\left(\vartheta \right)\rangle$.

Theorem 9 

([12] Theorem 4.4). Let $\mathcal{A}$ be a cyclic code of length n over $\mathcal{S}$. Then, $d_H\left(\mathcal{A}\right)=w_H\left(\mathcal{A}\right)=w_H\left({\mathcal{A}}_{u^2}\right)=d_H\left({\mathcal{A}}_{u^2}\right)$, where $w_H\left(\mathcal{A}\right)$ is the minimum Hamming weight of cyclic code $\mathcal{A}$.

7. Examples

Example 1. 

Let $n=9$, ${\vartheta }^9-1=(\vartheta +1)({\vartheta }^2+\vartheta +1)({\vartheta }^6+{\vartheta }^3+1)$ over ${\mathbb{F}}_2$ with ${\alpha }_2\left(\vartheta \right)=1$, ${\alpha }_1\left(\vartheta \right)={\vartheta }^6+{\vartheta }^3+1$, and $g\left(\vartheta \right)=1+\vartheta +{\vartheta }^2+{\vartheta }^3+{\vartheta }^4+{\vartheta }^5+{\vartheta }^6+{\vartheta }^7+{\vartheta }^8$. Here, ${\alpha }_2\left(\vartheta \right)\mid {\alpha }_1\left(\vartheta \right)\mid g\left(\vartheta \right)\mid ({\vartheta }^9-1)$, then by using Theorem 3, the code generated by $\langle g\left(\vartheta \right)+u{\alpha }_1\left(\vartheta \right)+u^2{\alpha }_2\left(\vartheta \right)\rangle$ is a reversible cyclic code. Additionally, by using Theorem 9 we find that $d_H\left(\mathcal{A}\right)=1$.

Example 2. 

Let ${\vartheta }^{16}-1={(\vartheta +1)}^{16}=g^{16}$ over ${\mathbb{F}}_2$. Let $\mathcal{A}=\langle g\left(\vartheta \right)+u{p}_1\left(\vartheta \right)+u^2{p}_2\left(\vartheta \right)\rangle$, where $g\left(\vartheta \right)=g^{14}$, $p_1\left(\vartheta \right)=\vartheta +{\vartheta }^5+{\vartheta }^9+{\vartheta }^{13}$ and $p_2\left(\vartheta \right)={\vartheta }^2+{\vartheta }^4+{\vartheta }^{10}+{\vartheta }^{12}$. It is easy to check that $g\left(\vartheta \right)={\vartheta }^{14}+{\vartheta }^{12}+{\vartheta }^{10}+{\vartheta }^8+{\vartheta }^6+{\vartheta }^4+{\vartheta }^2+1$ is self-reciprocal. Additionally, ${\vartheta }^i{p}_1^{*}\left(\vartheta \right)=p_1\left(\vartheta \right)$ and ${\vartheta }^j{p}_2^{*}\left(\vartheta \right)=p_2\left(\vartheta \right)$, $i=d\left(g\left(\vartheta \right)\right)-d\left(p_1\left(\vartheta \right)\right)$ & $j=d\left(g\left(\vartheta \right)\right)-d\left(p_2\left(\vartheta \right)\right)$, where $d\left(g\right(\vartheta \left)\right)$ denotes degree of $g\left(\vartheta \right)$. Therefore, $\mathcal{A}$ is a reversible cyclic code with distance $d_H\left(\mathcal{A}\right)=4$.

8. Conclusions and Future Work

The reversible cyclic codes of any length n over the ring $\mathcal{S}={\mathbb{F}}_2+u{\mathbb{F}}_2+u^2{\mathbb{F}}_2$, where $u^3=0$, were examined in this article. As ideals for the ring ${\mathcal{S}}_n=\mathcal{S}\left[\vartheta \right]/({\vartheta }^n-1)$, we have created a unique set of generators for these codes. Additionally, in Section 5, we set certain restrictions on the situations in which a reversible cyclic code’s dual is reversible. Finally, we provided helpful examples to support our results. As an application of our research work, we can use these reversible cyclic codes to obtain linear complementary dual codes, which are used in cryptography, in particular, to secure information from so-called, “side-channel assaults (SCA)” or “fault non-invasive attacks”. Further, for future work, we can also find reversible cyclic codes over more general rings such as ${\mathbb{F}}_q+u{\mathbb{F}}_q+u^2{\mathbb{F}}_q+\dots +u^{k-1}{\mathbb{F}}_q.$