Additive conjucyclic codes over a class of Galois rings

Abstract

As a tool towards quantum error correction, additive conjucyclic codes have gained great attention. But, their algebraic structure is completely unknown over finite fields (except ${\mathbb {F}}_{q^2}$) as well as rings. In this article, we investigate the structure of additive conjucyclic codes over Galois rings $GR(2^r,2)$, where $r\ge 2$ is an integer. We develop a one-to-one correspondence between the family of additive conjucyclic codes of length n over $GR(2^r,2)$ and the family of linear cyclic codes of length 2n over ${\mathbb {Z}}_{2^r}$. This correspondence helps to obtain additive conjucyclic codes over $GR(2^r,2)$ via known linear cyclic codes over ${\mathbb {Z}}_{2^r}$. We prove that the trace dual ${\mathscr {C}}^{Tr}$ of an additive conjucyclic code ${\mathscr {C}}$ is also an additive conjucyclic code. Moreover, we derive a necessary and sufficient condition of additive conjucyclic codes to be self-dual. We further propose a technique for constructing linear cyclic codes over ${\mathbb {Z}}_{2^r}$ contained in additive conjucyclic codes over $GR(2^r,2)$. Last but not least, we explicitly derive the generator matrices for these codes.

1 Introduction

Let ${\mathbb {F}}_q$ be the finite field of order q. A code ${\mathscr {C}}$ of length n over ${\mathbb {F}}_q$ is a nonempty subset of ${\mathbb {F}}^n_q$, and it is an additive code if ${\mathscr {C}}$ is a subgroup of the additive group ${\mathbb {F}}_q^n$. A $k\times n$ matrix G is said to be an additive generator matrix of an additive code ${\mathscr {C}}$ if G contains a minimum number of rows such that all codewords can be found by the addition operation of all rows of G. In that case, ${\mathscr {C}}$ contains $2^k$ codewords, and is denoted by $(n,2^k)$. Additive quaternary codes, that is, additive codes over ${\mathbb {F}}_4$ have been classified in [4] for a small length, up to 12. In 2007, Huffman studied the structure of additive cyclic codes of even length over ${\mathbb {F}}_4$ in [16], and again in 2012, these codes were revisited by [5]. Later, additive codes have gained great attention in several contexts of coding theory, for example, in complementary dual codes [7, 25, 26], due to their efficient application in preventing cryptographic attacks [8].

Additive quaternary codes invariant under conjucyclic operators (see the definition in Sect. 2), popularly known by additive conjucyclic codes over ${\mathbb {F}}_4$ have been introduced in [9] as a tool towards quantum error-correction. After a long gap, in 2020, Abualrub and Dougherty studied additive conjucyclic codes over ${\mathbb {F}}_4$ in [2]. Fundamentally, this class of codes does not have a polynomial representation like linear cyclic, skew cyclic codes, etc. Consequently, the standard algebraic (ring theoretical) approach fails to classify their properties. This is one of the major challenges in analyzing their algebraic behaviour. However, in [1, 2], the authors used a linear algebraic approach to determine their algebraic structure. In fact, they established a one-to-one correspondence between the set of all binary cyclic codes of length 2n and the set of all additive conjucyclic codes of length n over ${\mathbb {F}}_4$. Further, they obtained several algebraic properties including generator and parity check matrices for such codes. Later on, following a similar approach, the work given in [1, 2] was generalized in [19], by studying additive conjucyclic codes over ${\mathbb {F}}_{q^2}$. As per our survey, [1, 2, 14, 15, 19] are the only five publicly known articles that are involved in the investigation of such codes over finite fields. In other words, the algebraic properties of additive conjucyclic codes over finite fields (except ${\mathbb {F}}_{q^2}$), as well as rings, are left yet to be determined.

This is the first attempt to study these codes over finite rings. It is well-known that towards the development of finite commutative chain rings, the class of Galois rings is the fundamental choice. To avoid the computational difficulties, here we restrict our study to a special class of Galois rings $GR(2^r,2)$, where $r\ge 2$ is an integer. In fact, in this article, we obtain the complete algebraic structure of additive conjucyclic codes over $GR(2^r,2)$. Similar to [1], first, we establish a one-to-one correspondence between the family of linear cyclic codes of length 2n over ${\mathbb {Z}}_{2^r}$ and the family of additive conjucyclic codes of length n over $GR(2^r,2)$. This correspondence allows us to classify additive conjucyclic codes via known linear cyclic codes over ${\mathbb {Z}}_{2^r}$. Also, by considering the trace inner product, we show that the trace dual ${\mathscr {C}}^{Tr}$ of an additive conjucyclic code over $GR(2^r,2)$ is also an additive conjucyclic code. Moreover, a necessary and sufficient condition for such codes to be self-dual has been obtained. Eventually, by using the proposed correspondence, we explicitly discuss the form of generator matrices.

2 Background

Let ${\mathbb {Z}}_{2^r}$ be the ring of integers modulo $2^r$, where $r\ge 2$ is an integer. Recall that a monic polynomial f(x) in ${\mathbb {Z}}_{2^r}[x]$ is said to be basic irreducible if its reduction modulo 2, i.e., $f(x) \pmod {2}$ has the same degree, $\deg f(x)$ and is irreducible in ${\mathbb {Z}}_{2}[x]$. Clearly, the monic polynomial $f(x)=x^2+x+1$ is a basic irreducible polynomial in ${\mathbb {Z}}_{2^r}[x]$. Denote

$$\begin{aligned} GR(2^r,2):={\mathbb {Z}}_{2^r}[x]/\langle x^2+x+1\rangle . \end{aligned}$$

Note that $GR(2^r,2)$ is a finite commutative local chain ring with unique maximal ideal $\langle 2\rangle$ and the set of all ideals forms a chain under set inclusion as

$$\begin{aligned} \{0\}={\langle 2^r\rangle } \subsetneq \langle 2^{r-1}\rangle \subsetneq \cdots \subsetneq \langle 2\rangle \subsetneq \langle 1\rangle =GR(2^r,2). \end{aligned}$$

Conventionally, $GR(2^r,2)$ is known as a Galois ring of characteristic $2^r$ and degree 2. If $r=1$, then $GR(2^r,2)={\mathbb {F}}_4$. Since additive conjucyclic codes over ${\mathbb {F}}_4$ are already known in [1], we always consider $r\ge 2$. Let $\omega$ be a primitive 3-th root of unity in $GR(2^r,2)$, which is also a root of $x^2+x+1$, that is, $\omega ^3=1$. Then, the set ${\mathcal {T}}=\{0,1,\omega ,\omega ^2\}$ is known as the Teichm$\ddot{\text {u}}$ller set and any element $\alpha \in GR(2^r,2)$ can be uniquely written as

$$\begin{aligned} \alpha =\beta _0+2\beta _1+\dots +2^{r-1}\beta _{r-1}~\text {where}~ \beta _i\in {\mathcal {T}},0\le i\le r-1. \end{aligned}$$

(1)

We consider the Frobenius map $\theta :GR(2^r,2)\longrightarrow GR(2^r,2)$ defined by

$$\begin{aligned} \theta (\alpha )=\beta _0^2+2\beta _1^2+\dots +2^{r-1}\beta _{r-1}^2, \end{aligned}$$

where $\alpha \in GR(2^r,2)$ has the expression given by Eq. (1). For any $\alpha \in {\mathbb {Z}}_{2^r}$, we know the 2-adic expression of $\alpha$ is $\alpha =\beta _0+2\beta _1+\dots +2^{r-1}\beta _{r-1}$, where $\beta _i\in \{0,1\}$ for all $0\le i\le r-1$. As $\beta _i^2=\beta _i$ for $0\le i\le r-1$, we have that $\theta (\alpha )=\alpha$. Let $\Theta :GR(2^r,2)^n\longrightarrow GR(2^r,2)^n$ be the extension of $\theta$ given by $\Theta (\varvec{\alpha })=(\theta (\alpha _1), \dots , \theta (\alpha _n))$, for $\varvec{\alpha }=(\alpha _1,\dots ,\alpha _n)\in GR(2^r,2)^n$. We define the conjugate of $\alpha$ by ${\overline{\alpha }}:=\theta (\alpha )$. So, the conjugate operator is identity in ${\mathbb {Z}}_{2^r}$. From [24], we have that ${\mathcal {T}}\equiv {\mathbb {F}}_4$ (mod 2) and $tr: {\mathbb {F}}_4 \longrightarrow {\mathbb {F}}_2$ is the standard trace function defined by $tr(a) =a+{\overline{a}}= a + a^2$, where $a\in {\mathbb {F}}_4$. Using tr, we define trace function $Tr: GR(2^r,2) \longrightarrow {\mathbb {Z}}_{2^r}$ by

$$\begin{aligned} Tr\Big (\sum _{i=0}^{r-1}2^i\beta _i\Big ) = \sum _{i=0}^{r-1}2^itr(\beta _i), \end{aligned}$$

where $\beta _i\in {\mathcal {T}}$. For more details about Galois rings, we refer to [28]. Now, we recall some basic definitions of algebraic codes over $GR(2^r,2)$ which are useful in the subsequent discussion. We know that $GR(2^r,2)^n:=\{(\gamma _0, \gamma _1,\dots , \gamma _{n-1}): \gamma _i\in GR(2^r,2)$ for all $0\le i\le n-1\}$ is a $GR(2^r,2)$-module. Let ${\mathscr {C}}\subseteq GR(2^r,2)^n$. Then, ${\mathscr {C}}$ is said to be

• an additive code of length n if ${\mathscr {C}}$ is a subgroup of the additive group $GR(2^r,2)^n$.

• a linear code of length n if ${\mathscr {C}}$ is a $GR(2^r,2)$-submodule of $GR(2^r,2)^n$.

• a cyclic code if $(\gamma _{n-1},\gamma _0,\cdots ,\gamma _{n-2})\in {\mathscr {C}}$ whenever $(\gamma _0,\gamma _1,\dots ,\gamma _{n-1})\in {\mathscr {C}}$.

• a conjucyclic code if for each $\varvec{\gamma }=(\gamma _0,\gamma _1,\ldots ,\gamma _{n-1}) \in {\mathscr {C}}$, we have that $T(\varvec{\gamma }):=(\overline{\gamma _{n-1}},\gamma _0,\ldots ,\gamma _{n-2}) \in {\mathscr {C}}$. The operator $T: GR(2^r,2)^n \rightarrow GR(2^r,2)^n$ is known as conjucyclic operator and $\overline{\gamma _{n-1}}$ represents the conjugate of $\gamma _{n-1}$. Note that, if ${\mathscr {C}}$ is conjucyclic, then we have that $T^n(\varvec{\gamma })=(\overline{\gamma _0}, \overline{\gamma _1}, \dots , \overline{\gamma _{n-1}}) \in {\mathscr {C}}$ for $\varvec{\gamma } \in {\mathscr {C}}$.

• an additive (resp. linear) conjucyclic code if it is additive (resp. linear) and conjucyclic over $GR(2^r,2)$.

We recall that for any two vectors $\varvec{\gamma }=(\gamma _0,\gamma _1,\dots ,\gamma _{n-1}),~ \varvec{\gamma '}=(\gamma '_0,\gamma '_1,\dots , \gamma '_{n-1})\in GR(2^r,2)^n$, the Euclidean inner product is defined by $[\varvec{\gamma },\varvec{\gamma '}]_e=\sum \limits _{i=0}^{n-1}\gamma _i\gamma '_i$. For a linear code ${\mathscr {C}}$ of length n over $GR(2^r,2)$, the Euclidean dual defined by

$$\begin{aligned} {\mathscr {C}}^{\perp }=\{\varvec{\gamma }\in GR(2^r,2)^n~:~[\varvec{\gamma },\varvec{\gamma '}]_e=0, \forall ~\varvec{\gamma '}\in {\mathscr {C}}\} \end{aligned}$$

is also a linear code of length n over $GR(2^r,2)$. Again, for any two vectors $\varvec{\gamma },\varvec{\gamma '}\in GR(2^r,2)^n$, the trace inner product based on trace function Tr is defined by $[\varvec{\gamma },\varvec{\gamma '}]_{Tr} = Tr\big ([\varvec{\gamma },\varvec{\gamma '}]_e\big )$. In this way, the dual code is given by

$$\begin{aligned} {\mathscr {C}}^{Tr} = \{\varvec{\gamma }\in GR(2^r,2)^n : [\varvec{\gamma },\varvec{\gamma '}]_{Tr}=0,~\forall ~\varvec{\gamma '}\in {\mathscr {C}}\}. \end{aligned}$$

Note that, with respect to trace inner product, ${\mathscr {C}}$ is said to be self-orthogonal if and only if ${\mathscr {C}}\subseteq {\mathscr {C}}^{Tr}$, and it is said to be self-dual if and only if ${\mathscr {C}}={\mathscr {C}}^{Tr}$.

3 Structure of additive conjucyclic codes over $GR(2^r,2)$

The main contribution starts with this section. We begin by defining ${\mathbb {Z}}_{2^r}$-linear isomorphism which helps us to establish a correspondence between additive conjucyclic codes over $GR(2^r,2)$ and linear cyclic codes over ${\mathbb {Z}}_{2^r}$.

As we recalled in the last section, $a\in {\mathbb {Z}}_{2^r}$ has 2-adic expression as $a=c_{0}+2c_{1}+\dots +2^{r-1}c_{r-1 }$, where $c_{i}\in \{0,1\}$ for all $0\le i\le r-1$. Now, we define the map $\Phi : {\mathbb {Z}}_{2^r}^{2n} \longrightarrow GR(2^r,2)^n$ by

$$\begin{aligned}&\Phi (a_0,a_1,\dots ,a_{n-1}, a_n,a_{n+1},\dots , a_{2n-1})\nonumber \\&\quad = \Bigg (\sum _{s=0}^{r-1}2^s (c_{s0}\omega +c_{sn}\omega ^2), \sum _{s=0}^{r-1}2^s (c_{s1}\omega +c_{s(n+1)}\omega ^2), \dots ,\nonumber \\&\qquad \quad \,\,\sum _{s=0}^{r-1}2^s (c_{s(n-1)}\omega +c_{s(2n-1)}\omega ^2)\Bigg )\nonumber \\&\quad =\Bigg (\sum _{s=0}^{r-1}2^s u_{s 0}, \sum _{s=0}^{r-1}2^s u_{s1}, \dots ,\sum _{s=0}^{r-1}2^s u_{s(n-1)}\Bigg ) \end{aligned}$$

(2)

where $a_i=c_{0i}+2c_{1i}+\dots +2^{r-1}c_{(r-1) i}$ and $c_{j i}\in \{0,1\}$ for $0\le j\le r-1$ and $0\le i\le 2n-1$, and $u_{s t}=c_{s t}\omega +c_{s (n+t)} \omega ^2$ for $0 \le t\le n-1$ and $0\le s\le r-1$.

Lemma 1

The map $\Phi$ defined in Eq. (2) is ${\mathbb {Z}}_{2^r}$-linear isomorphism.

Proof

Let ${\varvec{u}}=(a_0,a_1,\dots ,a_{n-1}, a_n,a_{n+1},\dots , a_{2n-1})$, ${\varvec{v}}=(b_0,b_1,\dots ,b_{n-1}, b_n,b_{n+1},\dots , b_{2n-1})\in {\mathbb {Z}}_{2^r}^{2n}$, where $a_i=\sum \nolimits _{j=0}^{r-1}2^jc_{ji}, b_i=\sum \nolimits _{j=0}^{r-1}2^jd_{ji}$, $c_{ji},d_{ji}\in \{0,1\}$ for $0\le j\le r-1, 0\le i\le 2n-1$. Note that $\sum \nolimits _{s=0}^{r-1}2^s (c_{s\ell }\omega +c_{s(n+\ell )}\omega ^2)=a_\ell \omega +a_{n+\ell }\omega ^2$ and $\sum \nolimits _{s=0}^{r-1}2^s (d_{s\ell }\omega +d_{s(n+\ell )}\omega ^2)=b_\ell \omega +b_{n+\ell }\omega ^2$ for all $0\le \ell \le n-1$. We have that

$$\begin{aligned}&\Phi ({\varvec{u}})\\&\quad =\Phi (a_0,a_1,\dots ,a_{n-1}, a_n,a_{n+1},\dots , a_{2n-1})\\&\quad = \Bigg (\sum _{s=0}^{r-1}2^s (c_{s0}\omega +c_{sn}\omega ^2), \sum _{s=0}^{r-1}2^s (c_{s1}\omega +c_{s(n+1)}\omega ^2), \dots ,\\&\qquad \quad \,\,\sum _{s=0}^{r-1}2^s (c_{s(n-1)}\omega +c_{s(2n-1)}\omega ^2)\Bigg )\\&\quad = (a_0\omega +a_n\omega ^2, a_1\omega +a_{n+1}\omega ^2,\dots , a_{n-1}\omega +a_{2n-1}\omega ^2)\\&\text {and } \\&\Phi ({\varvec{v}})\\&\quad =\Phi (b_0,b_1,\dots ,b_{n-1}, b_n,b_{n+1},\dots , b_{2n-1})\\&\quad = \Bigg (\sum _{s=0}^{r-1}2^s (d_{s0}\omega +d_{sn}\omega ^2), \sum _{s=0}^{r-1}2^s (d_{s1}\omega +d_{s(n+1)}\omega ^2), \dots ,\\&\qquad \quad \,\,\sum _{s=0}^{r-1}2^s (d_{s(n-1)}\omega +d_{s(2n-1)}\omega ^2)\Bigg )\\&\quad = (b_0\omega +b_n\omega ^2, b_1\omega +b_{n+1}\omega ^2,\dots , b_{n-1}\omega +b_{2n-1}\omega ^2). \end{aligned}$$

Again,

$$\begin{aligned}&\Phi ({\varvec{u}}+{\varvec{v}})\\&\quad =\Phi (a_0+b_0,a_1+b_1,\dots ,a_n+b_n, a_{n+1}+b_{n+1},\dots ,a_{2n-1}+b_{2n-1}) \\&\quad =\Bigg ((a_0+b_0)\omega +(a_n+b_n)\omega ^2,\dots , (a_{n-1}+b_{n-1})\omega +(a_{2n-1}+b_{2n-1})\Bigg )\\&\quad =(a_0\omega +a_n\omega ^2, a_1\omega +a_{n+1}\omega ^2,\dots , a_{n-1}\omega +a_{2n-1}\omega ^2)\\&\qquad + \, (b_0\omega +b_n\omega ^2, b_1\omega +b_{n+1}\omega ^2,\dots , b_{n-1}\omega +b_{2n-1}\omega ^2)\\&\quad =\Phi ({\varvec{u}})+\Phi ({\varvec{v}}). \end{aligned}$$

Therefore, $\Phi$ is ${\mathbb {Z}}_{2^r}$-additive and hence a ${\mathbb {Z}}_{2^r}$-linear map. Note that $\Phi ({\varvec{u}})=0$ if and only if $a_0=a_1=\dots =a_n=a_{n+1}=\dots =a_{2n-1}=0$, that is, $\Phi ({\varvec{u}})=0$ if and only if ${\varvec{u}}={\varvec{0}}$. Thus, we have that Ker $\Phi =\{{\varvec{0}}\}$ and hence $\Phi$ is an injective ${\mathbb {Z}}_{2^r}$-linear map. Since the cardinality of the domain and co-domain of $\Phi$ is the same, $\Phi$ is surjective. This completes the proof. $\square$

Note that we have used the derived version of map $\Phi$ to prove the linearity. However, we keep both versions in our discussion and use them as per our convenience. Now, we give an example.

Example 1

Let $r=3,n=2$ and the map $\Phi : {\mathbb {Z}}_{8}^{4} \longrightarrow GR(2^3,2)^2$ is defined by Eq. (2). Take ${\varvec{u}}=(3,2,0,6), {\varvec{v}}=(7,1,2,5)\in {\mathbb {Z}}_8^{4}$, where

$$\begin{aligned} 3&=1+1\cdot 2^1+0\cdot 2^2, \hspace{.8cm} 7=1+1\cdot 2^1+1\cdot 2^2,\\ 2&=0+1\cdot 2^1+0\cdot 2^2, \hspace{.8cm} 1=1+0\cdot 2^1+0\cdot 2^2,\\ 0&=0+0\cdot 2^1+0\cdot 2^2, \hspace{.8cm} 2=0+1\cdot 2^1+0\cdot 2^2,\\ 6&=0+1\cdot 2^1+1\cdot 2^2, \hspace{.8cm} 5=1+0\cdot 2^1+1\cdot 2^2. \end{aligned}$$

We have that

$$\begin{aligned} \Phi ({\varvec{u}})&=\Phi (3,2,0,6)\\&=\Bigg (1\cdot \omega +0\cdot \omega ^2+2(1\cdot \omega +0\cdot \omega ^2)+2^2(0\cdot \omega +0\cdot \omega ^2), 0\cdot \omega \\&\quad +0\cdot \omega ^2 +\, 2(1\cdot \omega +1\cdot \omega ^2)+2^2(0\cdot \omega +1\cdot \omega ^2)\Bigg )\\&=\big (3\omega , 2\omega +6\omega ^2\big )\\ \Phi ({\varvec{v}})&=\Phi (7,1,2,5)\\&=\Bigg (1\cdot \omega +0\cdot \omega ^2+2(1\cdot \omega +1\cdot \omega ^2)+2^2(1\cdot \omega +0\cdot \omega ^2), 1\cdot \omega \\&\quad +1\cdot \omega ^2 +\, 2(0\cdot \omega +0\cdot \omega ^2)+2^2(0\cdot \omega +1\cdot \omega ^2)\Bigg )\\&=\big (7\omega +2\omega ^2, \omega +5\omega ^2\big )\\ \text {and } \Phi ({\varvec{u}})+\Phi ({\varvec{v}})&=\big (2\omega +2\omega ^2, 3\omega +3\omega ^2\big ). \end{aligned}$$

On the other hand, ${\varvec{u}}+{\varvec{v}}=(3,2,0,6)+(7,1,2,5)=(2,3,2,3)\in {\mathbb {Z}}_8^4$ and

$$\begin{aligned} \Phi (\varvec{u+{\varvec{v}}})&=\Phi (2,3,2,3)\\&=\Bigg (0\cdot \omega +0\cdot \omega ^2+2(1\cdot \omega +1\cdot \omega ^2)+2^2(0\cdot \omega +0\cdot \omega ^2), 1\cdot \omega +1\cdot \omega ^2\\&\quad + 2(1\cdot \omega +1\cdot \omega ^2)+2^2(0\cdot \omega +0\cdot \omega ^2)\Bigg )\\&=\big (2\omega +2\omega ^2, 3\omega +3\omega ^2\big )\\&=\Phi ({\varvec{u}})+\Phi ({\varvec{v}}). \end{aligned}$$

By Lemma 1, the map $\Phi$ is ${\mathbb {Z}}_{2^r}$-linear isomorphism, and hence $\Phi ^{-1}$ is so. In particular, both functions preserve additive subgroups. In the next result, we propose a commutative diagram based on $\Phi$. This diagram allows us to see the conjucyclic codes over $GR(2^r,2)$ via cyclic codes over ${\mathbb {Z}}_{2^r}$.

Lemma 2

The following is a commutative diagram:

In other words, $\Phi \circ \sigma =T\circ \Phi$, where $\sigma$ is the cyclic shift operator.

Proof

Let $(a_0,a_1,\dots ,a_{n-1},a_n,a_{n+1},\dots ,a_{2n-2},a_{2n-1})\in {\mathbb {Z}}_{2^r}^{2n}$, where $a_t=c_{0t}+2c_{1t}+\dots +2^{r-1}c_{(r-1) t}=\sum _{k=0}^{r-1}2^kc_{kt}$, with $c_{jt}\in \{0,1\}$ for $0\le t\le 2n-1$ and $0\le j\le r-1.$ Let $u_{s t}=c_{s t}\omega +c_{s (n+t)} \omega ^2$, where $0 \le t\le n-1$ and $0\le s\le r-1$. Now, we have that

$$\begin{aligned}&\overline{\sum _{s=0}^{r-1}2^s u_{s(n-1)}}\\&\quad =\overline{\sum _{s=0}^{r-1}2^s \big (c_{s(n-1)}\omega +c_{s(2n-1)}\omega ^2\big )}\\&\quad = \sum _{s=0}^{r-1}2^s \big (c_{s(n-1)}\omega ^2+c_{s(2n-1)}\omega ^4\big ) \text { (since } {\overline{\omega }}=\omega ^2, \overline{\omega ^2}=\omega ^4)\\&\quad = \sum _{s=0}^{r-1}2^s \big (c_{s(n-1)}\omega ^2+c_{s(2n-1)}\omega \big ) \text { (since } \omega ^3=1). \end{aligned}$$

Therefore,

$$\begin{aligned}&(\Phi \circ \sigma )(a_0,a_1,\dots ,a_{n-1},a_n,a_{n+1},\dots ,a_{2n-2},a_{2n-1})\\&\quad =\Phi (a_{2n-1},a_0,\dots ,a_{n-1},a_n,a_{n+1},\dots ,a_{2n-2})\\&\quad = \Bigg (\sum _{s=0}^{r-1}2^s \big (c_{s(2n-1)}\omega +c_{s(n-1)}\omega ^2\big ),\\&\qquad \quad \sum _{s=0}^{r-1}2^s \big (c_{s0}\omega +c_{sn}\omega ^2\big ), \dots ,\sum _{s=0}^{r-1}2^s \big (c_{s(n-2)}\omega +c_{s(2n-2)}\omega ^2\big )\Bigg )\\&\quad =\Bigg (\overline{\sum _{s=0}^{r-1}2^s u_{s(n-1)}}, \sum _{s=0}^{r-1}2^s u_{s 0}, \dots ,\sum _{s=0}^{r-1}2^s u_{s(n-2)}\Bigg )\\&\quad =T\Bigg (\sum _{s=0}^{r-1}2^s u_{s 0}, \sum _{s=0}^{r-1}2^s u_{s1}, \dots ,\sum _{s=0}^{r-1}2^s u_{s(n-1)}\Bigg )\\&\quad =(T\circ \Phi )(a_0,a_1,\dots ,a_{n-1},a_n,a_{n+1},\dots ,a_{2n-2},a_{2n-1}). \end{aligned}$$

$\square$

Theorem 1

Let ${\mathscr {C}}$ be an additive code of length n over $GR(2^r,2)$. Then, ${\mathscr {C}}$ is additive conjucyclic if and only if there exists a linear cyclic code ${\mathscr {B}}$ of length 2n over ${\mathbb {Z}}_{2^r}$ such that $\Phi ({\mathscr {B}})={\mathscr {C}}$.

Proof

Let ${\mathscr {C}}$ be conjucyclic. Then, $T({\mathscr {C}})={\mathscr {C}}$. By Lemma 2, we have that $\Phi \circ \sigma =T\circ \Phi$, and hence $\sigma \circ \Phi ^{-1}=\Phi ^{-1}\circ T$. Therefore, $\sigma \big (\Phi ^{-1}({\mathscr {C}})\big )=\Phi ^{-1}\big (T({\mathscr {C}})\big )=\Phi ^{-1}({\mathscr {C}})$. This proves that ${\mathscr {B}}=\Phi ^{-1}({\mathscr {C}})$ is cyclic in ${\mathbb {Z}}_{2^r}^{2n}$. Since $\Phi ^{-1}$ preserves additive subgroups, so ${\mathscr {B}}=\Phi ^{-1}({\mathscr {C}})$ is additive and hence linear in ${\mathbb {Z}}_{2^r}^{2n}$.

Conversely, let ${\mathscr {B}}$ be a linear cyclic code of length 2n over ${\mathbb {Z}}_{2^r}$ such that $\Phi ({\mathscr {B}})={\mathscr {C}}$. By Lemma 2, we have that $T({\mathscr {C}})=T\circ \Phi ({\mathscr {B}})=\Phi \circ \sigma ({\mathscr {B}})=\Phi ({\mathscr {B}})={\mathscr {C}}.$ This shows that ${\mathscr {C}}$ is conjucyclic in $GR(2^r,2)^n$. Further, $\Phi$ being ${\mathbb {Z}}_{2^r}$-linear isomorphism, we have that $\Phi ({\mathscr {B}})={\mathscr {C}}$ is additive in $GR(2^r,2)^n$. $\square$

Remark 1

Theorem 1 ensures that there is a one-to-one correspondence between the set of all linear cyclic codes of length 2n over ${\mathbb {Z}}_{2^r}$ and the set of all additive conjucyclic codes of length n over $GR(2^r,2)$.

Next, we prove that $\Phi$ preserves the inner product on both sides. It helps to derive the dual of conjucyclic codes.

Lemma 3

Let ${\varvec{a}}, {\varvec{b}} \in {\mathbb {Z}}_{2^r}^{2n}$. Then,

$$\begin{aligned}{}[\Phi ({\varvec{a}}), \Phi ({\varvec{b}})]_{Tr} = [{\varvec{a}}, {\varvec{b}}]_e. \end{aligned}$$

Proof

Let ${\varvec{a}}=(a_0,a_1,\dots ,a_{2n-1})$ and ${\varvec{b}}=(b_0,b_1,\dots ,b_{2n-1})$ be two elements in ${\mathbb {Z}}_{2^r}^{2n}$, where $a_t=c_{0t}+2c_{1t}+\dots +2^{r-1}c_{(r-1) t}=\sum _{k=0}^{r-1}2^kc_{kt}$ and $b_t=d_{0t}+2d_{1t}+\dots +2^{r-1}d_{(r-1) t}=\sum _{k=0}^{r-1}2^k d_{kt}$ for $c_{jt},d_{jt}\in \{0,1\}$, $0\le t\le 2n-1$ and $0\le j\le r-1$. Let $u_{st}=\big (c_{st}\omega +c_{s(n+t)}\omega ^2\big )$ and $v_{jt}=\big (d_{jt}\omega +d_{j(n+t)}\omega ^2\big )$, where $0\le s, j\le r-1$ and $0\le t\le n-1$. We have that

$$\begin{aligned}{}[{\varvec{a}},{\varvec{b}}]_e&=\sum _{t=0}^{2n-1}a_tb_t =\sum _{t=0}^{2n-1} \sum _{k=0}^{2r-2} \sum _{s+j=k} 2^k c_{st}d_{jt}. \end{aligned}$$

(3)

Now, from the definition of $\Phi$ given by Eq. (2), we have that

$$\begin{aligned} \Phi (a_0,a_1,\dots ,a_{2n-1})=\Bigg (\sum _{s=0}^{r-1}2^s u_{s 0}, \sum _{s=0}^{r-1}2^s u_{s1}, \dots ,\sum _{s=0}^{r-1}2^s u_{s(n-1)}\Bigg ) \end{aligned}$$

and

$$\begin{aligned} \Phi (b_0,b_1,\dots ,b_{2n-1})=\Bigg (\sum _{s=0}^{r-1}2^s v_{s 0}, \sum _{s=0}^{r-1}2^s v_{s1}, \dots ,\sum _{s=0}^{r-1}2^s v_{s(n-1)}\Bigg ). \end{aligned}$$

Again,

$$\begin{aligned} u_{st}v_{jt}&=\big (c_{st}\omega +c_{s(n+t)}\omega ^2\big )\big (d_{jt}\omega +d_{j(n+t)}\omega ^2\big )\\&=c_{st}d_{jt}\omega ^2+c_{st}d_{j(n+t)}\omega ^3+c_{s(n+t)}d_{jt}\omega ^3+c_{s(n+t)}d_{j(n+t)}\omega ^4\\&=c_{st}d_{jt}\omega ^2+c_{st}d_{j(n+t)}+c_{s(n+t)}d_{jt}+c_{s(n+t)}d_{j(n+t)}\omega . \end{aligned}$$

Since $tr(\omega ^2)=\omega ^2+\omega =1$, we have that

$$\begin{aligned} Tr\big (u_{st}v_{jt}\big )&=Tr(c_{st}d_{jt}\omega ^2+c_{st}d_{j(n+t)}+c_{s(n+t)}d_{jt}+c_{s(n+t)}d_{j(n+t)}\omega )\nonumber \\&=c_{st}d_{jt}+c_{s(n+t)}d_{j(n+t)}. \end{aligned}$$

(4)

Therefore,

$$\begin{aligned}{}[\Phi ({\varvec{a}}),\Phi ({\varvec{b}})]_{Tr}&=Tr\big ([\Phi ({\varvec{a}}),\Phi ({\varvec{b}})]_e\big )\\&=\sum _{t=0}^{n-1} \sum _{k=0}^{2r-2} \sum _{s+j=k} 2^k Tr( u_{st}v_{jt}) \\&=\sum _{t=0}^{n-1} \sum _{k=0}^{2r-2} \sum _{s+j=k} 2^k \big (c_{st}d_{jt}+c_{s(n+t)}d_{j(n+t)}\big ) \, (\text {By Eq.}\,(4))\\&=\sum _{t=0}^{2n-1} \sum _{k=0}^{2r-2} \sum _{s+j=k} 2^k c_{st}d_{jt} \\&=[{\varvec{a}},{\varvec{b}}]_e \,(\text {By Eq.}\,(3)). \end{aligned}$$

$\square$

In the light of the above result, we get the trace dual codes of additive conjucyclic codes shown in the next theorem.

Theorem 2

Let ${\mathscr {C}}$ be an additive conjucyclic code of length n over $GR(2^r,2)$ such that $\Phi ({\mathscr {B}})={\mathscr {C}}$, where ${\mathscr {B}}$ is a linear cyclic code of length 2n over ${\mathbb {Z}}_{2^r}$. Then,

$$\begin{aligned} {\mathscr {C}}^{Tr}=\Phi ({\mathscr {B}}^{\perp }). \end{aligned}$$

Proof

Let ${\varvec{a}}\in \Phi ({\mathscr {B}}^{\perp })$. Then, there exists ${\varvec{c}}\in {\mathscr {B}}^{\perp }$ such that $\Phi ({\varvec{c}})={\varvec{a}}$. Now, for any ${\varvec{y}}=\Phi ({\varvec{z}})\in {\mathscr {C}}$, where ${\varvec{z}}\in {\mathscr {B}}$, we have that

$$\begin{aligned}{}[{\varvec{a}}, {\varvec{y}}]_{Tr}&=[\Phi ({\varvec{c}}), \Phi ({\varvec{z}})]_{Tr}\\&=[{\varvec{c}},{\varvec{z}}]_e \,(\text {by Lemma}~3)\\&=0. \end{aligned}$$

This shows that $\Phi ({\mathscr {B}}^{\perp })\subseteq {\mathscr {C}}^{Tr}$. Conversely, let ${\varvec{a}}\in {\mathscr {C}}^{Tr}$. Then, we have that $[{\varvec{a}},{\varvec{y}}]_{Tr}=[{\varvec{a}},\Phi ({\varvec{z}})]_{Tr}=0$ for all ${\varvec{y}}=\Phi (z)\in {\mathscr {C}}$. Let ${\varvec{b}}\in {\mathbb {Z}}_{2^r}^{2n}$ such that $\Phi ({\varvec{b}})={\varvec{a}}$. By Lemma 3, for any ${\varvec{z}}\in {\mathscr {B}}$, we have that $[{\varvec{b}},{\varvec{z}}]_e=[\Phi ({\varvec{b}}), \Phi ({\varvec{z}})]_{Tr}=[{\varvec{a}}, {\varvec{y}}]_{Tr}=0$. Thus, ${\varvec{b}}\in {\mathscr {B}}^{\perp }$, and hence ${\varvec{a}}=\Phi ({\varvec{b}})\in \Phi ({\mathscr {B}}^{\perp })$. Therefore, ${\mathscr {C}}^{Tr}\subseteq \Phi ({\mathscr {B}}^{\perp })$. Combining both sides, we have that ${\mathscr {C}}^{Tr}=\Phi ({\mathscr {B}}^{\perp })$. $\square$

Corollary 1

Let ${\mathscr {C}}$ be an additive conjucyclic code of length n over $GR(2^r,2)$ such that $\Phi ({\mathscr {B}})={\mathscr {C}}$, where ${\mathscr {B}}$ is a linear cyclic code of length 2n over ${\mathbb {Z}}_{2^r}$. Then, ${\mathscr {C}}$ is (trace) self-dual if and only if ${\mathscr {B}}$ is (Euclidean) self-dual.

Theorem 3

For an additive conjucyclic code ${\mathscr {C}}$ over $GR(2^r,2)$, its dual ${\mathscr {C}}^{Tr}$ is also an additive conjucyclic code.

Proof

Let ${\mathscr {C}}$ be an additive conjucyclic code of length n over $GR(2^r,2)$ such that $\Phi ({\mathscr {B}})={\mathscr {C}}$, where ${\mathscr {B}}$ is a linear cyclic code of length 2n over ${\mathbb {Z}}_{2^r}$. We know that ${\mathscr {B}}^{\perp }$ is also a linear cyclic code of length 2n over ${\mathbb {Z}}_{2^r}$. Again, by Theorem 2, we have that ${\mathscr {C}}^{Tr}=\Phi ({\mathscr {B}}^{\perp })$. Therefore, by Theorem 1, we conclude the result. $\square$

4 Largest cyclic codes over ${\mathbb {Z}}_{2^r}$ contained in ${\mathscr {C}}$

Here we discuss a method for constructing the largest cyclic codes over ${\mathbb {Z}}_{2^r}$ contained in some conjucyclic codes over $GR(2^r,2)$. Let ${\mathscr {C}}$ be an additive conjucyclic code of length n over $GR(2^r,2)$. Let ${\mathcal {L}}({\mathscr {C}})$ denote the largest cyclic code of length n over ${\mathbb {Z}}_{2^r}$ contained in ${\mathscr {C}}$. Clearly, ${\mathcal {L}}({\mathscr {C}})$ is a subgroup of ${\mathscr {C}}$, and $a+{\mathcal {L}}({\mathscr {C}})$ is the coset of ${\mathcal {L}}({\mathscr {C}})$ for $a\in {\mathscr {C}}$. We show that ${\mathscr {C}}$ can be described as a union of the cosets of ${\mathcal {L}}({\mathscr {C}})$.

Theorem 4

Let ${\mathscr {C}}$ be a non-trivial additive conjucyclic code of length n over $GR(2^r,2)$. Then, ${\mathcal {L}}({\mathscr {C}})$ is a non-trivial cyclic code over ${\mathbb {Z}}_{2^r}$ and comprises of all codewords in ${\mathscr {C}}$ whose entries are in ${\mathbb {Z}}_{2^r}$.

Proof

Let ${\varvec{c}}=(c_0,\dots ,c_n)\in {\mathscr {C}}$, where $c_i=\beta _{i0}+2\beta _{i1}+\dots +2^{r-1}\beta _{i(r-1)}$ with $\beta _{ij}\in {\mathcal {T}}$, $0\le i\le n$ and $0\le j\le r-1.$ It is easy to see that for any $\beta \in {\mathcal {T}}$, we get $\beta +\theta (\beta )\in {\mathbb {Z}}_{2^r}$, and hence ${\varvec{c}}+\Theta ({\varvec{c}})\in {\mathbb {Z}}_{2^r}^n$. Since ${\mathscr {C}}$ is conjucyclic, we have that $\Theta ({\varvec{c}})=T^n({\varvec{c}})\in {\mathscr {C}}$. Again, since ${\mathscr {C}}$ is a additive group, we have that ${\varvec{c}}+\Theta ({\varvec{c}})\in {\mathscr {C}}$. This shows that ${\mathcal {L}}({\mathscr {C}})$ is non-trivial.

Moreover, for ${\varvec{c}}\in {\mathscr {C}} \cap {\mathbb {Z}}_{2^r}^n$, we have that $T({\varvec{c}})=\sigma ({\varvec{c}})\in {\mathscr {C}}$. Therefore, ${\varvec{c}}\in {\mathcal {L}}({\mathscr {C}})$. This completes the proof. $\square$

The next result follows immediately from Theorem 4 and the fact that $\Phi ^{-1}$ is a group isomorphism.

Theorem 5

Let ${\mathscr {C}}$ be an additive conjucyclic code of length n over $GR(2^r,2)$ such that $\Phi ({\mathscr {B}})={\mathscr {C}}$, where ${\mathscr {B}}$ is a linear cyclic code of length 2n over ${\mathbb {Z}}_{2^r}$. Then,

• 1. ${\mathscr {C}}$ consists of the cosets of ${\mathcal {L}}({\mathscr {C}})$.

• 2. ${\mathscr {B}}$ includes the subcode $\Phi ^{-1}({\mathcal {L}}({\mathscr {C}}))$ and ${\mathscr {B}}$ comprises of the cosets of $\Phi ^{-1}({\mathcal {L}}({\mathscr {C}}))$.

Theorem 6

Let ${\mathscr {C}}$ be an additive conjucyclic code of length n over $GR(2^r,2)$ such that $\Phi ({\mathscr {B}})={\mathscr {C}}$, where ${\mathscr {B}}$ is a linear cyclic code of length 2n over ${\mathbb {Z}}_{2^r}$. Then, ${\varvec{a}}=(a_0,a_1,\dots ,a_{n-1}, a_n,a_{n+1},\dots ,a_{2n-1})\in \Phi ^{-1}({\mathcal {L}}({\mathscr {C}}))$, where $a_i=c_{0i}+2c_{1i}+\dots +2^{r-1}c_{(r-1) i}$ and $c_{j i}\in \{0,1\}$ for $0\le j\le r-1$ and $0\le i\le 2n-1$, if and only if $c_{st}=c_{s(n+t)}$ for all $0\le t\le n-1$ and $0\le s\le r-1$.

Proof

Let ${\varvec{b}}\in GR(2^r,2)^n$ such that $\Phi ({\varvec{a}})={\varvec{b}}$. Then, the definition of $\Phi$ given in Eq. (2) gives the explicit form of ${\varvec{b}}$. Therefore, since $\omega ^2+\omega +1=0$, we have that ${\varvec{b}}\in {\mathcal {L}}({\mathscr {C}})$ if and only if $u_{st}\in {\mathbb {Z}}_{2^r}$ if and only if $c_{st}=c_{s(n+t)}$ for all $0\le t\le n-1$ and $0\le s\le r-1$. $\square$

Note that Theorem 6 suggests the form of the elements of $\Phi ^{-1}({\mathcal {L}}({\mathscr {C}}))$, and therefore we have the following result.

Corollary 2

Let ${\mathscr {C}}$ be an additive conjucyclic code of length n over $GR(2^r,2)$ such that $\Phi ({\mathscr {B}})={\mathscr {C}}$, where ${\mathscr {B}}$ is a linear cyclic code of length 2n over ${\mathbb {Z}}_{2^r}$. Then, $\Phi ^{-1}({\mathcal {L}}({\mathscr {C}}))$ is a non-trivial subcode of ${\mathscr {B}}$ comprising all codewords of ${\mathscr {B}}$ satisfying $c_{st}=c_{s(n+t)}$ for all $0\le t\le n-1$ and $0\le s\le r-1$.

We now exhibit an example to show how conjucyclic code ${\mathscr {C}}$ can be described as a union of cosets of cyclic code ${\mathcal {L}}({\mathscr {C}})$.

Example 2

Let $r=2$ and ${\mathscr {C}}$ be an additive conjucyclic code of length 2 over ${\mathbb {Z}}_4[\omega ]$, where $\omega ^2+\omega +1=0$. Also, let ${\mathscr {C}}$ be generated by

$$\begin{aligned} \{(2,0), (0,2), (2\omega , 3+2\omega ), (3+2\omega , 2+2\omega )\}. \end{aligned}$$

Then, ${\mathscr {C}}$ has 16 elements and ${\mathcal {L}}({\mathscr {C}})=\{(0,0), (2,0), (0,2), (3,1),(3,3),(2,2), (1,1), (1,3)\}$. Note that ${\mathscr {C}}$ consists of two cosets ${\mathcal {L}}({\mathscr {C}})$ and ${\varvec{a}}+{\mathcal {L}}({\mathscr {C}})$, that is, ${\mathscr {C}}={\mathcal {L}}({\mathscr {C}})\bigcup {\varvec{a}}+{\mathcal {L}}({\mathscr {C}})$, where ${\varvec{a}}=(3+2\omega , 2+2\omega )$. In fact, we have that

$$\begin{aligned} {\begin{array}{ll} {\mathcal {L}}({\mathscr {C}})\\ \hline (0,0) \\ (2,0) \\ (0,2) \\ (3,1) \\ (3,3) \\ (2,2) \\ (1,1) \\ (1,3) \\ \end{array} } {\begin{array}{ll} {\varvec{a}}+{\mathcal {L}}({\mathscr {C}})\\ \hline (3+2\omega , 2+2\omega ) \\ (1+2\omega , 2+2\omega ) \\ (3+2\omega , 2\omega ) \\ (2+2\omega , 3+2\omega )\\ (2+2\omega , 1+2\omega ) \\ (1+2\omega , 2\omega ) \\ (2\omega , 3+2\omega ) \\ (2\omega , 1+2\omega ). \end{array} } \end{aligned}$$

Note that $\Phi ({\varvec{a}})={\varvec{b}}=(3,0,1,2)$. Also, we list all elements of linear cyclic code ${\mathscr {B}}=\Phi ^{-1}({\mathscr {C}})$ as cosets of $\Phi ^{-1}({\mathcal {L}}({\mathscr {C}}))$ below:

$$\begin{aligned} {\begin{array}{ll} \Phi ^{-1}\big ({\mathcal {L}}({\mathscr {C}})\big )\\ \hline (0,0,0,0) \\ (2,0,2,0) \\ (0,2,0,2) \\ (1,3,1,3) \\ (1,1,1,1) \\ (2,2,2,2) \\ (3,3,3,3) \\ (3,1,3,1) \\ \end{array} } {\begin{array}{ll} {\varvec{b}}+\Phi ^{-1}\big ({\mathcal {L}}({\mathscr {C}})\big ) \\ \hline (3,0,1,2) \\ (1,0,3,2) \\ (3,2,1,0) \\ (0,3,2,1) \\ (0,1,2,3) \\ (1,2,3,0) \\ (2,3,0,1) \\ (2,1,0,3). \end{array} } \end{aligned}$$

This verifies that ${\mathscr {B}}$ consists of two cosets $\Phi ^{-1}\big ({\mathcal {L}}({\mathscr {C}})\big )$ and ${\varvec{b}}+\Phi ^{-1}\big ({\mathcal {L}}({\mathscr {C}})\big )$ of $\Phi ^{-1}({\mathcal {L}}({\mathscr {C}}))$. Thus, Theorem 5 is verified. Again, the set of all codewords of ${\mathscr {B}}$ satisfying the condition given in Corollary 2, that is, $\Phi ^{-1}({\mathcal {L}}({\mathscr {C}}))$ is

$$\begin{aligned}{} & {} \{(0,0,0,0),(2,0,2,0),(0,2,0,2),(1,3,1,3),\\{} & {} \quad (1,1,1,1),(2,2,2,2),(3,3,3,3), (3,1,3,1)\}, \end{aligned}$$

which is an additive cyclic subcode of ${\mathscr {B}}$. Moreover, note that $\Phi ^{-1}({\mathcal {L}}({\mathscr {C}}))$ is self-dual under the Euclidean inner product, and hence by Lemma 3, ${\mathcal {L}}({\mathscr {C}})$ is self-dual under the trace inner product.

5 Generator matrix

The description of additive generator matrices having a minimum number of rows for additive quaternary codes is given in [1]. Here, we derive the generator matrices for additive conjucyclic codes over $GR(2^r,2)$ by using our proposed commutative diagram. We start with the definition of generator matrices for additive codes.

Definition 1

[1] An additive generator matrix G for an additive code ${\mathscr {C}}$ is a matrix such that all codewords of ${\mathscr {C}}$ are formed by taking only the addition of its rows. Unlike linear codes, the scalar multiplication of its rows is not allowed.

Note that in the case of linear codes, all codewords are formed by taking a linear combination of its rows of the generator matrix G, in other words, by taking addition and scalar multiplication of its rows, whereas codewords for an additive code are formed by taking addition of its rows only. For linear codes, we simply use the term “generator matrix", but for an additive code, we use the extra term “additive" with the generator matrix. However, since additivity implies linearity in the case of additive codes over ${\mathbb {Z}}_{2^r}$, we have that an additive generator matrix is the same as a generator matrix for a linear code over ${\mathbb {Z}}_{2^r}$. In fact, these two notions of matrices are different in the case of codes over $GR(2^r,2)$.

For $g(x)=a_0+a_1x+a_2x^2+\dots +a_kx^k$, where $a_0,a_1,\dots ,a_k \in {\mathbb {Z}}_{2^r}$ with $a_0\ne 0$ and $a_k\ne 0$, we denote

$$\begin{aligned} \alpha _{g(x)}=(a_0, a_1,\dots ,a_k, 0,\dots ,0)\in {\mathbb {Z}}_{2^r}^{2n} \end{aligned}$$

and $\beta _{g(x)}=\Phi (\alpha _{g(x)}) \in GR(2^r,2)^{n}$. By the commutative diagram in Lemma 2, we have that

$$\begin{aligned} \Phi (\sigma (\alpha _{g(x)}))=T(\Phi (\alpha _{g(x)}))=T(\beta _{g(x)}) \end{aligned}$$

and

$$\begin{aligned} \Phi (\sigma ^i(\alpha _{g(x)}))=T^i(\beta _{g(x)}), ~\text {for all}~ i\ge 1. \end{aligned}$$

(5)

Now, we recall some known results related to the minimal spanning sets, or generator matrices for linear cyclic codes over ${\mathbb {Z}}_{2^r}$ (see [18, 23]).

Theorem 7

[23, Theorem 4.1] Let ${\mathscr {C}}_{r}$ be a cyclic code of length n over ${\mathbb {Z}}_{2^r}$.

• 1. If ${\mathscr {C}}_{r}=\Big \langle \gamma (x)\Big \rangle =\langle g(x)+2s_{1}(x)+\dots +2^{r-1}s_{r-1}(x)\rangle$ where $g(x), s_{i}(x)$ are polynomials in ${\mathbb {Z}}_{2}[x]$ with $g(x)\mid (x^{n}-1)$ and $\big (g(x)+2s_{1}(x)+\dots +2^{r-1}s_{r-1}(x)\big )\mid (x^{n}-1)$ in ${\mathbb {Z}}_{2^r}[x]/\langle x^{n}-1\rangle$, then

  $$\begin{aligned} \Gamma = \{ \gamma (x), x\gamma (x), \dots , x^{n-\epsilon -1}\gamma (x)\} \end{aligned}$$

  is a minimal spanning set for ${\mathscr {C}}_{r}$ and $rank({\mathscr {C}}_{r})=n-\epsilon$, where $\deg (\gamma (x))=\epsilon .$

• 2. If

  $$\begin{aligned} {\mathscr {C}}_{r}&=\Big \langle g(x)+2e_{1}(x)+\dots +2^{r-1}e_{r-1}(x), 2a_{1}(x)+2^{2}s_{1}(x)\\&\quad +\dots +2^{r-1}s_{r-2}(x), 2^{2}a_{2}(x)+2^{3}l_{1}(x)+\dots +2^{r-1}l_{r-3}(x),\dots ,\\&\quad 2^{r-2}a_{r-2}(x)+2^{r-1}t_{1}(x), 2^{r-1}a_{r-1}(x)\Big \rangle , \end{aligned}$$

  where $a_{r-1}(x)\mid a_{r-2}(x)\mid \dots \mid a_{1}(x)\mid g(x)\mid (x^{n}-1) \pmod {2}$ and $a_{r-1}(x)\mid e_{1}(x)(\frac{x^{n}-1}{g(x)}),\dots ,a_{r-1}(x)\mid t_{1}(x)(\frac{x^{n}-1}{a_{r-2}(x)}),\dots ,a_{r-1}(x)\mid e_{r-1}(x)(\frac{x^{n}-1}{g(x)})\cdots (\frac{x^{n}-1}{a_{r-2}(x)}) \pmod {2}$, then

  $$\begin{aligned} \Gamma&= \Bigg \{ g(x)+2e_{1}(x)+\dots +2^{r-1}e_{r-1}(x), x\big (g(x)+2e_{1}(x)+\dots \\&\quad +2^{r-1}e_{r-1}(x)\big ),\dots , x^{n-\tau _{1}-1}\big (g(x)+2e_{1}(x)+\dots +2^{r-1}e_{r-1}(x)\big ), 2a_{1}(x)\\&\quad +2^{2}s_{1}(x)+\dots + 2^{r-1}s_{r-2}(x), x\big (2a_{1}(x)+2^{2}s_{1}(x)+\dots \\&\quad +2^{r-1}s_{r-2}(x)\big ),\dots , x^{\tau _{1}-\tau _{2}-1}\big (2a_{1}(x)+2^{2}s_{1}(x)+\dots +2^{r-1}s_{r-2}(x)\big ),\dots ,\\&\quad 2^{r-1}a_{r-1}(x), x\big (2^{r-1}a_{r-1}(x)\big ),\dots , x^{\tau _{r-1}-\tau _{r}-1}\big (2^{r-1}a_{r-1}(x)\big )\Bigg \} \end{aligned}$$

  is a minimal spanning set for ${\mathscr {C}}_{r}$ and $rank({\mathscr {C}}_{r})=n-\tau _{r}$, where $\deg (g(x))=\tau _{1}$ and $\deg (a_{i}(x))=\tau _{i+1}$ for $i=1,2,\dots ,r-1$.

Let ${\mathscr {B}}$ be a linear cyclic code of length 2n over ${\mathbb {Z}}_{2^r}$. Then, $\Phi ({\mathscr {B}})={\mathscr {C}}$ is an additive conjucyclic code of length n over $GR(2^r,2)$. By using Eq. (5) and Theorem 7, we propose the additive generator matrix for ${\mathscr {C}}$. We explicitly give the additive generator matrix for ${\mathscr {C}}$ for the first case of Theorem 7, where cyclic codes are principally generated, and for the other case, it follows similarly. In fact, based on the above discussion and Theorem 7, we have that the additive generator matrix for ${\mathscr {B}}$ is

$$\begin{aligned} G'=\left( \begin{array}{c} \alpha _{\gamma (x)} \\ \sigma (\alpha _{\gamma (x)}) \\ \vdots \\ \sigma ^{2n-\epsilon -1}(\alpha _{\gamma (x)})\\ \end{array} \right) _{(2n-\epsilon )\times 2n}. \end{aligned}$$

(6)

Theorem 8

Let ${\mathscr {C}}$ be an additive conjucyclic code of length n such that ${\mathscr {C}}=\Phi ({\mathscr {B}})$, where ${\mathscr {B}}$ is a linear cyclic code of length 2n over ${\mathbb {Z}}_{2^r}$, whose additive generator matrix is given by Eq. (6). Then, the additive generator matrix for ${\mathscr {C}}$ is

$$\begin{aligned} G=\left( \begin{array}{c} \beta _{\gamma (x)} \\ T\big (\beta _{\gamma (x)}\big ) \\ \vdots \\ T^{2n-\epsilon -1}\big (\beta _{\gamma (x)}\big )\\ \end{array} \right) _{(2n-\epsilon )\times n}. \end{aligned}$$

Proof

It follows from Eqs. (5) and (6). $\square$

Example 3

Let $r=2$ and ${\mathscr {B}}$ be a linear cyclic code of length 6 over ${\mathbb {Z}}_4$. Also, let

$$\begin{aligned} {\mathscr {B}}=&\langle g(x)+2e_1(x), 2a_1(x)\rangle , \text { where } a_1(x)\mid g(x)\mid x^6-1 \pmod {2} \\&\text { and } a_1(x)\mid e_1(x)\frac{x^6-1}{g(x)} \pmod {2}. \end{aligned}$$

Let $g(x)=(x+1)(x^2+x+1)=x^3+1$, $a_1(x)=x^2+x+1$ and $e_1(x)=x+1$. Then, $g(x)+2e_1(x)=x^3+2x+3$ and $\alpha _{g(x)+2e_1(x)}=(3,2,0,1,0,0), \alpha _{2a_1(x)}=(2,2,2,0,0,0)\in {\mathbb {Z}}_4^6$. So, the generator matrix for ${\mathscr {B}}$ is

$$\begin{aligned} G'=\left( \begin{array}{c} \alpha _{g(x)+2e_1(x)} \\ \sigma (\alpha _{g(x)+2e_1(x)}) \\ \sigma ^2(\alpha _{g(x)+2e_1(x)}) \\ \alpha _{2a_1(x)} \end{array} \right) _{4\times 6}= \begin{pmatrix} 3 &{} 2&{} 0 &{} 1&{}0&{}0 \\ 0&{}3 &{} 2&{} 0 &{} 1&{}0 \\ 0&{} 0&{}3 &{} 2&{} 0 &{} 1\\ 2&{}2 &{} 2&{} 0 &{} 0&{}0 \end{pmatrix}_{4\times 6}. \end{aligned}$$

Hence, $\Phi ({\mathscr {B}})={\mathscr {C}}$ is an additive conjucyclic code of length 3 over GR(4, 2) having the additive generator matrix

$$\begin{aligned} G=\begin{pmatrix} 3+2\omega &{} 2\omega &{} 0 \\ 0&{}3+2\omega &{} 2\omega \\ 2+2\omega &{} 0&{}3+2\omega \\ 2\omega &{}2\omega &{} 2\omega \end{pmatrix}_{4\times 3}. \end{aligned}$$

6 Conclusion

In this paper, we derive the algebraic properties of additive conjucyclic codes over $GR(2^r,2)$, and this is the first attempt to study these codes over finite rings. We propose a correspondence which helps to analyze all algebraic behaviour of additive conjucyclic codes via known linear cyclic codes over ${\mathbb {Z}}_{2^r}$. The defining polynomial $x^2+x+1$ has an impact on the proof of Lemmas 2 and 3, and also we frequently use the calculations done under modulo 2 related to the term tr(a) for any $a\in {\mathbb {F}}_4$. This simplifies our calculation in several places and leads to the restriction of the class of Galois rings $GR(2^r,2)$. It would be interesting to generalize the works to $GR(p^r,2)$. In fact, due to [13, Lemma 4.1], we have a monic irreducible polynomial $f(x)=x^2+x+(p-1)$ in ${\mathbb {Z}}_p[x]$ for $p\equiv 2$ or 3 (mod 5). Based on this defining polynomial f(x) for $GR(p^r,2)$, in the future, one can investigate additive conjucyclic codes over $GR(p^r,2)$. Furthermore, it would be interesting to explore these codes over most general Galois rings $GR(p^r,m)$, where $\deg f(x)=m$ and p prime. In that case, we believe that a different setup would be required for establishing a connection between additive conjucyclic codes over $GR(p^r,m)$ and cyclic codes over ${\mathbb {Z}}_{p^r}$.