Codes over an algebra over ring

TL;DR

This paper explores the structure of linear codes over a specific algebraic ring constructed from a finite Frobenius ring and idempotent variables, expanding understanding of code properties in algebraic coding theory.

Contribution

It introduces new structures of linear codes over the algebra $\\mathcal{R}_k$ built from a finite Frobenius ring and idempotent variables, providing foundational insights.

Findings

Characterization of linear codes over the algebra \mathcal{R}_k

Analysis of code properties over the ring \mathcal{R}_k

Potential applications in coding theory and algebraic structures

Abstract

In this paper, we consider some structures of linear codes over the ring $\mathcal{R}_k=R[v_1,\dots,v_k],$ where $v_i^2=v_i$ forall $i=1,\dots,k),$ and $R$ is a finite commutative Frobenius ring.

Tables

Table 1. Table 1: Some examples of codes over ℤ 4 . subscript ℤ 4 \mathbb{Z}_{4}.

$n$	$C$	${\phi }_1$	$d_L\left({\phi }_1\left(C\right)\right)$	$\left|{\phi }_1\left(C\right)\right|$
2	$⟨1+v⟩$	$\alpha +v\beta \mapsto (\alpha ,2\alpha +\beta )$	2	8
2	$⟨2⟩$	$\alpha +v\beta \mapsto (\alpha ,\alpha +\beta )$	2	4
3	$⟨2⟩$	$\alpha +v\beta \mapsto (\alpha ,\beta ,\alpha +\beta )$	4	4
3	$⟨2+2v⟩$	$\alpha +v\beta \mapsto (\alpha ,\beta ,\alpha +\beta )$	4	2
3	$⟨2v⟩$	$\alpha +v\beta \mapsto (\alpha ,\beta ,\alpha +\beta )$	4	2
4	$⟨2v⟩$	$\alpha +v\beta \mapsto (\alpha ,\beta ,\alpha +\beta ,\alpha +\beta )$	6	2
5	$⟨2⟩$	$\alpha +v\beta \mapsto (\alpha ,\beta ,\alpha +\beta ,\alpha ,\alpha +\beta )$	6	4

Equations

$$\alpha =\sum _{i=1}^{2^k}{\alpha }_{S_i}{v}_{S_i},$$

$$\Theta_{i}(\alpha)=\left\{\begin{array}[]{ll}1-v_{i},&\text{if}\;\alpha=v_{i}\\ \alpha,&\text{otherwise}.\end{array}\right.$$

$${\Theta }_S=\prod _{i\in S}{\Theta }_i={\Theta }_{i_1}\circ {\Theta }_{i_2}\circ \cdots \circ {\Theta }_{i_{|S|}},$$

$${\Phi }_{S_1,S_2}(\alpha v_j)=\vartheta (\alpha )v_{{\varphi }_{S_1,S_2}(j)},$$

$$\begin{array}[]{llll}\varphi_{j}:&\mathcal{R}_{j}&\longrightarrow&\mathcal{R}_{j-1}^{l_{j}}\\ &\alpha_{1}+\alpha_{2}v_{j}&\longmapsto&\left(\alpha_{1},\beta_{1}\alpha_{1}+\beta_{1}^{\prime}\alpha_{2},\beta_{2}\alpha_{1}+\beta_{2}^{\prime}\alpha_{2},\dots,\beta_{l_{j}-1}\alpha_{1}+\beta_{l_{j}-1}^{\prime}\alpha_{2}\right).\end{array}$$

$$\begin{array}[]{lll}\varphi_{j}(\alpha+\alpha^{\prime})&=&\left(\alpha_{1}+\alpha_{1}^{\prime},\beta_{1}(\alpha_{1}+\alpha_{1}^{\prime})+\beta_{1}^{\prime}(\alpha_{2}+\alpha_{2}^{\prime}),\beta_{2}(\alpha_{1}+\alpha_{1}^{\prime})+\beta_{2}^{\prime}(\alpha_{2}+\alpha_{2}^{\prime}),\dots\right.\\ &&\left.\dots,\beta_{l_{j}-1}(\alpha_{1}+\alpha_{1}^{\prime})+\beta_{l_{j}-1}^{\prime}(\alpha_{2}+\alpha_{2}^{\prime})\right)\\ &=&\varphi_{j}(\alpha)+\varphi_{j}(\alpha^{\prime}),\end{array}$$

$$\begin{array}[]{lll}\varphi_{j}(\lambda\alpha)&=&\left(\lambda\alpha_{1},\beta_{1}\lambda\alpha_{1}+\beta_{1}^{\prime}\lambda\alpha_{2},\beta_{2}\lambda\alpha_{1}+\beta_{2}^{\prime}\lambda\alpha_{2},\dots,\beta_{l_{j}-1}\lambda\alpha_{1}+\beta_{l_{j}-1}^{\prime}\lambda\alpha_{2}\right)\\ &=&\lambda\varphi_{j}(\alpha).\end{array}$$

$$\begin{array}[]{llll}\varphi_{j-1}\circ\varphi_{j}:&\mathcal{R}_{j}&\longrightarrow&\mathcal{R}_{j-2}^{l_{j}\times l_{j-1}}\\ &\alpha_{1}+\alpha_{2}v_{j}&\longmapsto&\left(\varphi_{j-1}\left(\alpha_{1}\right),\varphi_{j-1}\left(\beta_{1}\alpha_{1}+\beta_{1}^{\prime}\alpha_{2}\right),\varphi_{j-1}\left(\beta_{2}\alpha_{1}+\beta_{2}^{\prime}\alpha_{2}\right),\dots\right.\\ &&&\left.\dots,\varphi_{j-1}(\beta_{l_{j}-1}\alpha_{1}+\beta_{l_{j}-1}^{\prime}\alpha_{2})\right)\end{array}$$

$$\begin{array}[]{llll}\overline{\varphi}_{j}:&\mathcal{R}_{j}^{n}&\longrightarrow&\mathcal{R}_{j-1}^{nl_{j}}\\ &(\alpha_{1,1}+\alpha_{1,2}v_{j},\dots,\alpha_{n,1}+\alpha_{n,2}v_{j})&\longmapsto&\left(\alpha_{1,1},\dots,\alpha_{n,1},\beta_{1}\alpha_{1,1}+\beta_{1}^{\prime}\alpha_{1,2},\right.\\ &&&\dots,\beta_{1}\alpha_{n,1}+\beta_{1}^{\prime}\alpha_{n,2},\dots\\ &&&\dots,\beta_{l_{j}-1}\alpha_{1,1}+\beta_{l_{j}-1}^{\prime}\alpha_{1,2},\dots\\ &&&\left.\dots,\beta_{l_{j}-1}\alpha_{n,1}+\beta_{l_{j}-1}^{\prime}\alpha_{n,2}\right)\end{array}$$

$$\begin{array}[]{llll}\Psi:&\mathcal{R}_{k}&\longrightarrow&R^{2^{k}}\\ &\sum_{i=1}^{2^{k}}\alpha_{S_{i}}v_{S_{i}}&\longmapsto&\left(\sum_{S\subseteq S_{1}}\alpha_{S},\dots,\sum_{S\subseteq S_{2^{k}}}\alpha_{S}\right)\end{array}$$

$${\mathcal{R}}_k\cong \underset{2^k}{\underbrace{R\times R\times \cdots \times R}}.$$

$$\overline{\Psi }(a_1,\dots ,a_n)=\left(\Psi (a_1),\dots ,\Psi (a_n)\right).$$

$${\overline{\phi }}_1\circ {\overline{\phi }}_2\circ \cdots \circ {\overline{\phi }}_k(C)$$

$$[\mathbf{c},{\mathbf{c}}^{\prime }]=\sum _{i=1}^n{c}_i\overline{c_i^{\prime }}=\sum _{i=1}^n{c}_i{\Theta }_S(c_i^{\prime }).$$

$$\mathbf{c}\cdot {\mathbf{c}}^{\prime }=\sum _{i=1}^n{c}_i{c}_i^{\prime }.$$

$$C=\underset{n}{\underbrace{C_1\times C_1\times \cdots \times C_1}}.$$

$$\begin{array}[]{lll}[\mathbf{c},\mathbf{c}^{\prime}]&=&\sum_{i}c_{i}\overline{c_{i}^{\prime}}\\ &=&\sum_{i}\left(vc_{1i}+(1-v)c_{2i}\right)\overline{\left(vc_{1i}^{\prime}+(1-v)c_{2i}^{\prime}\right)}\\ &=&\sum_{i}\left(vc_{1i}+(1-v)c_{2i}\right)\left((1-v)c_{1i}^{\prime}+vc_{2i}^{\prime}\right)\\ &=&v\sum_{i}c_{1i}c_{2i}^{\prime}+(1-v)\sum_{i}c_{2i}c_{1i}^{\prime},\end{array}$$

$$C_1\times C_1^{\perp }\times \cdots \times C_{2^{k-1}}\times C_{2^{k-1}}^{\perp },$$

$$\begin{array}[]{lll}[\mathbf{c}_{1},\mathbf{c}_{2}]&=&\sum_{i}c_{1i}\overline{c_{2i}}\\ &=&\sum_{i}\left(v_{k}c_{1i}^{\prime}+(1-v_{k})c_{1i}^{\prime\prime}\right)\overline{\left(v_{k}c_{2i}^{\prime}+(1-v_{k})c_{2i}^{\prime\prime}\right)}\\ &=&\sum_{i}\left(v_{k}c_{1i}^{\prime}+(1-v_{k})c_{1i}^{\prime\prime}\right)\left((1-v_{k})\overline{c_{2i}^{\prime}}+v_{k}\overline{c_{2i}^{\prime\prime}}\right)\\ &=&v_{k}\sum_{i}c_{1i}^{\prime}\overline{c_{2i}^{\prime\prime}}+(1-v_{k})\sum_{i}c_{2i}^{\prime}\overline{c_{1i}^{\prime\prime}},\end{array}$$

$$\sum _i{c}_{1i}^{\prime }\overline{c_{2i}^{\prime \prime }}=0$$

$$\sum _i{c}_{2i}^{\prime }\overline{c_{1i}^{\prime \prime }}=0.$$

$$C_1\times C_1^{\perp }\times \cdots \times C_{2^{k-1}}\times C_{2^{k-1}}^{\perp },$$

$$d_H(C)=wt\left({\overline{\Psi }}^{-1}(0,\dots 0,{\mathbf{c}}_j,0,\dots ,0)\right)=d_H(C_j).$$

$$W_C(X,Y)=\sum _{\mathbf{c}\in C}{X}^{n-w{t}_H(\mathbf{c})}{Y}^{w{t}_H(\mathbf{c})},$$

$$W_{C^{\perp }}(X,Y)=\frac{1}{|C|}{W}_C\left(X+(|R{|}^{2^k}-1)Y,X-Y\right).$$

$$\mathrm{W}{\mathrm{t}}_{\mathrm{L}}(a)=\sum _{i=1}^{2^k}\mathrm{w}{\mathrm{t}}_{\mathrm{L}}\left(\sum _{S\subseteq S_i}{\alpha }_S\right)$$

$$\mathrm{W}{\mathrm{t}}_{\mathrm{L}}(\mathbf{a})=\sum _{j=1}^{n}W{t}_L(a_j).$$

$$d_L(C)=\underset{1\le i\le 2^k}{\min }{d}_L(C_i).$$

$$d_L(C)=\mathrm{W}{\mathrm{t}}_{\mathrm{L}}\left({\overline{\Psi }}^{-1}(0,\dots 0,{\mathbf{c}}_j,0,\dots ,0)\right)=d_L(C_j).$$

$$\chi (x)={\xi }^{w{t}_L(x)},$$

Full text

Keywords. Commutative Frobenius ring, Gray map, Euclidean self-dual, Hermitian self-dual, MacWilliams relation, cyclic code, quasi-cyclic code, skew-cyclic code, quasi-skew-cyclic code.

1 Introduction

Some special cases of codes over the ring of the form $\mathcal{R}_{k}=R[v_{1},\dots,v_{k}],$ where $v_{i}^{2}=v_{i}$ for all $i=1,\dots,k,$ and $R$ is a finite commutative Frobenius ring attract the attention of some researchers in coding theory. This is because codes over such kind of rings have a lot of nice structures. For example, in [1, 3, 6], they consider skew-cyclic codes over the ring $\mathbb{F}_{2}+v\mathbb{F}_{2},\mathbb{F}_{p}+v\mathbb{F}_{p}$ and $\mathbb{F}_{p^{r}}[v_{1},\dots,v_{k}],$ respectively. Moreover, in [2, 4, 5], they studied the structures of codes over $\mathbb{F}_{2}[v_{1},\dots,v_{k}],\mathbb{Z}_{4}+v\mathbb{Z}_{4},$ and $\mathbb{Z}_{9}+v\mathbb{Z}_{9},$ respectively, such as MacWilliams identity, self-dual codes, cyclic codes, constacyclic codes, etc. Also, we can find a constructon of good and new $\mathbb{Z}_{4}$-linear codes in [4].

In this paper, we try to give general recipes for the structures of codes over such class of rings, including MacWilliams identities, self-dual codes, cyclic codes, quasi-cyclic codes, skew-cyclic codes, and quasi-skew-cyclic codes.

2 Automorphisms and Gray Map

Let $R$ be a finite Frobenius ring and $\mathcal{R}_{k}=R[v_{1},v_{2},\dots,v_{k}],$ for some $k\in\mathbb{N},$ where $v_{i}^{2}=v_{i},$ for all $i=1,2,\dots,k.$ The ring $\mathcal{R}_{k}$ can be viewed as a free module over $R$ with dimension $2^{k}.$ Let $w_{i}=\{1-v_{i},v_{i}\}$ and $w_{S}=\prod_{i\in S}w_{i}.$ Then, we have the following immediate properties.

Lemma 1**.**

The ring $\mathcal{R}_{k}$ has the cardinality $|R|^{2^{k}}$ and characteristic equals to $char(R).$

Proof.

As we can see, every element $\alpha\in\mathcal{R}_{k}$ can be written as

[TABLE]

for some $\alpha_{S_{i}}\in R,$ for all $1\leq i\leq 2^{k}.$ Therefore we have that $|\mathcal{R}_{k}|=|R|^{2^{k}}.$ ∎

Let $\Theta_{i}$ be a map on $\mathcal{R}_{k}$ such that

[TABLE]

Then define

[TABLE]

where $S\subseteq\{1,2,\dots,k\}.$

Also, let $S_{1},S_{2}\subseteq\{1,2,\dots,k\},$ where $|S_{1}|=|S_{2}|,$ and $\phi_{S_{1},S_{2}}:\{1,2,\dots,k\}\rightarrow\{1,2,\dots,k\}$ be a map such that it is a bijection from $S_{1}$ to $S_{2}$ and $\phi_{S_{1},S_{2}}(j)=j,$ for all $j\not\in S_{1}.$ Define a map $\Phi_{S_{1},S_{2}},$ where

[TABLE]

for some automorphism $\vartheta$ of $R.$

We have to note that the maps $\Theta_{S}$ and $\Phi_{S_{1},S_{2}}$ are automorphisms on the ring $\mathcal{R}_{k},$ so does their compositions. In this paper we consider automorphism $\theta$ as a composition of $\Theta_{S}$ or $\Phi_{S_{1},S_{2}}$ or both.

Now, we will define two Gray maps from the ring $\mathcal{R}_{k}.$ First, For any $j\geq 1,$ any element $\alpha$ in $\mathcal{R}_{j}$ can be written as $\alpha=\alpha_{1}+\alpha_{2}v_{j},$ for some $\alpha_{1},\alpha_{2}\in\mathcal{R}_{j-1}.$ For some $l_{j}\geq 2$ in $\mathbb{N},$ define a map

[TABLE]

where $\beta_{i},\beta_{i}^{\prime}$ are some elements in $\mathcal{R}_{j-1},$ for all $1\leq i\leq l_{j},$ with $\beta_{l_{j-1}}^{\prime}$ is a unit in $\mathcal{R}_{j-1}.$ The following lemma shows that $\varphi_{j}$ is an injective map and also a module homomorphism.

Lemma 2**.**

The map $\varphi_{j}$ is an injective and also a $\mathcal{R}_{j-1}$-module homomorphism from $\mathcal{R}_{j}$ to $\mathcal{R}_{j-1}^{l_{j}},$ for all $1\leq j\leq k.$

Proof.

For injectivity, take any $\alpha$ and $\alpha^{\prime}$ in $\mathcal{R}_{j},$ where $\varphi_{j}(\alpha)=\varphi_{j}(\alpha^{\prime}).$ Now, let $\alpha=\alpha_{1}+\alpha_{2}v_{j}$ and $\alpha^{\prime}=\alpha_{1}^{\prime}+\alpha_{2}^{\prime}v_{j},$ for some $\alpha_{1},\alpha_{2},\alpha_{1}^{\prime},$ and $\alpha_{2}^{\prime}$ in $\mathcal{R}_{j-1}.$ Since $\varphi_{j}(\alpha)=\varphi_{j}(\alpha^{\prime}),$ we have $\alpha_{1}=\alpha_{1}^{\prime}.$ Using the previous fact and by considering the last coordinate of the images under $\varphi_{j},$ we have $\beta_{l_{j}}^{\prime}\alpha_{2}=\beta_{l_{j}}^{\prime}\alpha_{2}^{\prime}.$ Since $\beta_{l_{j}}^{\prime}$ is a unit in $\mathcal{R}_{j-1},$ we also have $\alpha_{2}=\alpha_{2}^{\prime}$ as we hope.

Now, take any $\alpha$ and $\alpha^{\prime}$ in $\mathcal{R}_{j}$ and any $\lambda$ in $\mathcal{R}_{j-1}.$ Let $\alpha=\alpha_{1}+\alpha_{2}v_{j}$ and $\alpha^{\prime}=\alpha_{1}^{\prime}+\alpha_{2}^{\prime}v_{j},$ for some $\alpha_{1},\alpha_{2},\alpha_{1}^{\prime}$ and $\alpha_{2}^{\prime}$ in $\mathcal{R}_{j-1}.$ Consider

[TABLE]

and

[TABLE]

Therefore, the map $\varphi_{j}$ is a $\mathcal{R}_{j-1}$-module homomorphism for all $1\leq j\leq k.$ ∎

Note that, we can combine the maps $\varphi_{j}$ and $\varphi_{j-1}$ to get a map from $\mathcal{R}_{j}$ to $\mathcal{R}_{j-2}^{l_{j}\times l_{j-1}}$ as follows.

[TABLE]

By doing this inductively, we will have a map $\varphi_{1}\circ\varphi_{2}\circ\cdots\circ\varphi_{k}$ from $\mathcal{R}_{k}$ to $R^{l_{k}\times l_{k-1}\times\cdots\times l_{1}}.$

We can extend the map $\varphi_{j}$ to get a map from $\mathcal{R}_{j}^{n}$ to $\mathcal{R}_{j-1}^{nl_{j}}$ by the following way,

[TABLE]

We can combine $\overline{\varphi}_{j}$ and $\overline{\varphi}_{j-1}$ to get a map from $\mathcal{R}_{j}^{n}$ to $\mathcal{R}_{j-2}^{nl_{j}l_{j-1}},$ and inductively, to get a map from $\mathcal{R}_{k}^{n}$ to $R^{nl_{k}\cdots l_{1}}.$ The map $\varphi_{j}$ and its extensions are a generalization of Gray maps in [2, 6].

For the second Gray map, any $\alpha$ in $\mathcal{R}_{k}$ can be written as $\alpha=\sum_{i=1}^{2^{k}}\alpha_{S_{i}}v_{S_{i}},$ for some $\alpha_{S_{i}}$ in $R,$ where $S_{i}\subseteq\{1,2,\dots,k\}$ and $v_{S_{i}}=\prod_{t\in S_{i}}v_{t},$ for all $1\leq i\leq 2^{k}.$ Define a map $\Psi$ as follows.

[TABLE]

We can check that the map $\Psi$ is a bijection map. Moreover, we can also check that the map $\Psi$ is an isomorphism, which implies

[TABLE]

This means $\mathcal{R}_{k}$ is also a Frobenius ring.

Let $\overline{\Psi}:\mathcal{R}_{k}^{n}\rightarrow R^{2^{k}\times n}$ be a map such that

[TABLE]

Then, we can see that $\overline{\Psi}$ is also a bijective map because $\Psi$ is bijective. Let $\Sigma_{S}$ and $\Gamma_{S_{1},S_{2}}$ be two maps such that $\overline{\Psi}\circ\Theta_{S}=\Sigma_{S}\circ\overline{\Psi}$ and $\overline{\Psi}\circ\Phi_{S_{1},S_{2}}=\Gamma_{S_{1},S_{2}}\circ\overline{\Psi}.$ As we can see, the maps $\Sigma_{S}$ and $\Gamma_{S_{1},S_{2}}$ are bijective maps induced by $\Theta_{S}$ and $\Phi_{S_{1},S_{2}},$ respectively.

3 Linear and Self-Dual Codes

In this part, we will describe linear codes over $\mathcal{R}_{k}$ using the gray map defined in Section 2. The following theorems describe the image of a linear code under the gray maps $\overline{\varphi}_{j}$ and $\overline{\Psi}.$ The following theorem describe the image of a linear code under the map $\overline{\varphi}_{j}.$

Theorem 3**.**

A code $C$ is a linear code of length $n$ over $\mathcal{R}_{j}$ if and only if the image $\overline{\varphi}_{j}(C)$ is a linear code of length $nl_{j}$ over $\mathcal{R}_{j-1}.$

We have the following consequence.

Corollary 4**.**

A code $C$ is a linear code of length $n$ over $\mathcal{R}_{k}$ if and only if the code

[TABLE]

is a linear code of length $nl_{1}\cdots l_{k}$ over $R.$

The following theorem describe the image of a linear code under the map $\overline{\Psi}.$

Theorem 5**.**

A code $C$ is a linear code of length $n$ over $\mathcal{R}_{k}$ if and only if there exist linear codes, $C_{1},C_{2},\dots,C_{2^{k}},$ of length $n$ over $R$ such that $C=\overline{\Psi}^{-1}(C_{1},C_{2},\dots,C_{2^{k}}).$

Proof.

Similar to the proof of [6, Lemma 16]. ∎

Now, we will describe Euclidean and Hermitian self-dual codes. Let $\Theta_{S}$ be an automorphism in the ring $\mathcal{R}_{k}$ as in Section 2, where $S=\{1,2,\dots,k\}.$ For any $\mathbf{c}=(c_{1},\dots,c_{n})$ and $\mathbf{c}^{\prime}=(c_{1}^{\prime},\dots,c_{n}^{\prime})$ in $\mathcal{R}_{k}^{n},$ define the Hermitian product as follows,

[TABLE]

Let $C^{H}=\{\mathbf{c}^{\prime}|[\mathbf{c},\mathbf{c}^{\prime}]=0\;\forall\mathbf{c}\in C\},$ then a code $C$ is called Hermitian self-orthogonal if $C\subseteq C^{H},$ and $C$ is called Hermitian self-dual if $C=C^{H}.$ Also, for any $\mathbf{c}=(c_{1},\dots,c_{n})$ and $\mathbf{c}^{\prime}=(c_{1}^{\prime},\dots,c_{n}^{\prime}),$ define the Euclidean product as the following rational sum,

[TABLE]

Let $C^{\bot}=\{\mathbf{c}^{\prime}|\mathbf{c}\cdot\mathbf{c}^{\prime}=0\;\forall\mathbf{c}\in C\},$ then a code $C$ is called self-orthogonal if $C\subseteq C^{\bot},$ and $C$ is called Euclidean self-dual if $C=C^{\bot}.$ The following theorem describe the existence of Hermitian self-dual codes over $\mathcal{R}_{k}.$

Theorem 6**.**

If $S\not=\emptyset,$ then there exist Hermitian self-dual codes over $\mathcal{R}_{k}$ for all length.

Proof.

Take $i$ in $S.$ Let $C_{1}=\langle v_{i}\rangle,$ then we have $C_{1}^{H}=\langle v_{i}\rangle=C_{1},$ because $v_{i}(1-v_{i})=0.$ So, Hermitian self-dual code of length $1$ over $\mathcal{R}_{k}$ exist. Now, for any length $n,$ define

[TABLE]

As we can see, $C^{H}=C,$ which means $C$ is an Hermitian self-dual code of length $n.$ ∎

Note that, the ring $\mathcal{R}_{k}$ can be written as $\mathcal{R}_{k}=v_{k}\mathcal{R}_{k-1}+(1-v_{k})\mathcal{R}_{k-1}.$ Consequently, any code $C$ of length $n$ over $\mathcal{R}_{k}$ can be written as $C=v_{k}C_{1}+(1-v_{k})C_{2},$ where $C_{1}$ and $C_{2}$ are codes of length $n$ over $\mathcal{R}_{k-1}.$

Proposition 7**.**

If $C$ is a Hermitian self-dual code of length $n$ over $\mathcal{R}_{1},$ then $C$ is isomorphic to $C_{1}\times C_{1}^{\bot},$ where $C_{1}$ is a code of length $n$ over $R.$

Proof.

Remember that $C$ can be written as $C=vC_{1}+(1-v)C_{2},$ where $C_{1}$ and $C_{2}$ are codes of length $n$ over $R.$ Consider

[TABLE]

where $(c_{j1},c_{j2},\dots,c_{jn})$ is in $C_{j},$ for $j=1,2.$ If the equation 1 is equal to $0,$ then it requires $\sum_{i}c_{1i}c_{2i}^{\prime}=0$ and $\sum_{i}c_{2i}c_{1i}^{\prime}=0.$ Since $C$ is self dual, we have $C_{1}=C_{2}^{\bot}$ and $C_{2}=C_{1}^{\bot}.$ Therefore, $C$ is isomorphic to $C_{1}\times C_{1}^{\bot}.$ ∎

Using the above property, we have the following theorem.

Theorem 8**.**

If $C$ is a Hermitian self-dual code of length $n$ over $\mathcal{R}_{k},$ then, with proper arrangement of indices, $C$ is isomorphic to

[TABLE]

where $C_{1},\dots,C_{2^{k-1}}$ are codes of length $n$ over $R.$

Proof.

We can write $C=v_{k}C^{\prime}+(1-v_{k})C^{\prime\prime},$ where $C^{\prime}$ and $C^{\prime\prime}$ are codes of length $n$ over $R_{k-1}.$ Consider

[TABLE]

where $(c_{j1}^{\prime},c_{j2}^{\prime},\dots,c_{jn}^{\prime})$ is in $C^{\prime}$ and $(c_{j1}^{\prime\prime},c_{j2}^{\prime\prime},\dots,c_{jn}^{\prime\prime})$ is in $C^{\prime\prime},$ for $j=1,2.$ If equation 2 is $0,$ then it requires

[TABLE]

and

[TABLE]

If we continue similar process on equation 3 and 4, we will have $2^{k}$ equations similar to equation 1 over $R.$ By Proposition 7, $2^{k}$ equations give $2^{k-1}$ pairs of Euclidean dual over $R.$ Therefore, we have $C$ is isomorphic to

[TABLE]

where $C_{1},C_{2},\dots,C_{2^{k-1}}$ are codes of length $n$ over $R.$ ∎

We have the following result.

Theorem 9**.**

A code $C$ is an Euclidean self-dual code of length $n$ over $\mathcal{R}_{k}$ if and only if $C=\overline{\Psi}^{-1}(C_{1},C_{2},\dots,C_{2^{k}}),$ where $C_{1},\dots,C_{2^{k}}$ are also Euclidean self-dual codes over $R.$

Proof.

Similar to the proof of [7, Proposition 4.1]. ∎

We have the following immediate consequence.

Corollary 10**.**

Eucidean self-dual codes of length $n$ over $\mathcal{R}_{k}$ exist if and only if Euclidean self-dual codes of length $n$ over $R$ exist.

4 Weights and MacWilliams Identities

Let $d_{H}(C)$ be the Hamming distance of a code $C.$ The following proposition gives the Hamming distance for codes over the ring $\mathcal{R}_{k}.$

Proposition 11**.**

If $C=\overline{\Psi}^{-1}(C_{1},\dots,C_{2^{k}}),$ is a code of length $n$ over $\mathcal{R}_{k},$ then $d_{H}(C)=\min_{1\leq i\leq 2^{k}}d_{H}(C_{i}).$

Proof.

Let $d_{H}(C_{j})=\min_{1\leq i\leq 2^{k}}d_{H}(C_{i}),$ for some $j.$ Also, let $\mathbf{c}_{j}$ be a codeword in $C_{j}$ such that $wt(\mathbf{c}_{j})=d_{H}(C_{j}).$ Then we have that

[TABLE]

∎

Let $wt_{H}(\mathbf{c})$ be a Hamming weight of codeword $\mathbf{c}.$ Also, let

[TABLE]

be the Hamming weight enumerator of code $C.$ We have the following relation between Hamming weight enumerator of a code $C$ and its dual.

Proposition 12**.**

If $C$ is a code of length $n$ over $\mathcal{R}_{k},$ then

[TABLE]

Proof.

Use the fact that $|\mathcal{R}_{k}|=|R|^{2^{k}}.$ ∎

Now, let $\operatorname{wt_{L}}(\alpha)$ be the Lee weight of any element $\alpha$ in $R.$ Let $a=\sum_{S\subseteq\{1,2,\dots,k\}}\alpha_{S}v_{S}$ be any element in $\mathcal{R}_{k}.$ Define

[TABLE]

be the Lee weight of $a.$ For any $\mathbf{a}=(a_{1},\dots,a_{n})$ in $\mathcal{R}_{k}^{n},$ define the Lee weight of $\mathbf{a}$ as follows,

[TABLE]

Then we have the following result.

Proposition 13**.**

If $C=\overline{\Psi}^{-1}(C_{1},\dots,C_{2^{k}})$ is a code of length $n$ over $\mathcal{R}_{k},$ then

[TABLE]

Proof.

Let $d_{L}(C_{j})=\min_{1\leq i\leq 2^{k}}d_{L}(C_{i}),$ for some $j,$ and let $\mathbf{c}_{j}$ be a codeword in $C_{j}$ such that $\operatorname{Wt_{L}}(\mathbf{c}_{j})=d_{L}(C_{j}).$ We have that

[TABLE]

∎

Since the ring $\mathcal{R}_{k}$ is isomorphic to $R^{2^{k}},$ the generating character for $\widehat{\mathcal{R}_{k}}$ is the product of generating character for $\widehat{R}.$ Now, if $\chi$ is a generating character for $R,$ such that

[TABLE]

for any $x\in R,$ then the generating character $\chi$ for $\mathcal{R}_{k}$ is

[TABLE]

for any $\beta\in\mathcal{R}_{k}.$

Define the matrix $T$ indexed by $\alpha,\beta\in\mathcal{R}_{k},$ as follows

[TABLE]

and the matrix $T_{H}$ as follows

[TABLE]

where $\overline{\beta}$ is the conjugate of $\beta$ induced by $\Theta_{S},$ for some $S\subseteq\{1,2,\dots,k\}.$

Also, define the complete weight enumerator for a code $C$ as follows,

[TABLE]

where $n_{b}(\mathbf{c})$ is the number of occurrences of the element $b$ in $\mathbf{c}.$ Then, we have the following result.

Theorem 14**.**

If $C$ is a linear code over $\mathcal{R}_{k},$ then

[TABLE]

and

[TABLE]

Proof.

This theorem is a consequence of [8, Corollary 8.2]. ∎

Note that $T$ is a $|R|^{2^{k}}$ by $|R|^{2^{k}}$ matrix indexed by the elements of $\mathcal{R}_{k}.$ Let $\mathcal{R}_{k}^{\times}$ be the group of units in the ring $\mathcal{R}_{k}$ and let $\alpha\sim\alpha^{\prime}$ if $\alpha^{\prime}=u\alpha,$ for some $u\in G,$ where $G$ is a subgroup of $\mathcal{R}_{k}^{\times}.$ It can be seen that the relation $\sim$ is an equivalence relation, so we define $\mathcal{A}=\{\alpha_{1},\dots,\alpha_{t}\}$ be the set of representatives. Let $S$ be the $t$ by $t$ matrix indexed by the elements in $\mathcal{A}.$ Also, define $S_{\alpha,\beta}=\sum_{\gamma\sim\beta}T_{\alpha,\gamma}.$ We have the following lemma.

Lemma 15**.**

If $\alpha\sim\alpha^{\prime}$ then the row $S_{\alpha}$ is equal to the row $S_{\alpha^{\prime}}.$

Proof.

If $\alpha\sim\alpha^{\prime}$ then for any column $\beta$ we have

[TABLE]

Since $\overline{\Psi}(\alpha\gamma)=\overline{\Psi}(\alpha)\overline{\Psi}(\gamma),$ where the multiplication in the right side of equal sign carried out coordinate-wise, we have that

[TABLE]

Therefore, $S_{\alpha}=S_{\alpha^{\prime}}$ when $\alpha\sim\alpha^{\prime}.$ ∎

Now, define the symmetrized weight enumerator for a code $C$ to be

[TABLE]

where $\operatorname{swc}_{\alpha}(\mathbf{c})=\sum_{\alpha^{\prime}\sim\alpha}n_{\alpha^{\prime}}(\mathbf{c}).$ Then, we have the following theorem.

Theorem 16**.**

If $C$ is a linear code over $\mathcal{R}_{k},$ then

[TABLE]

Proof.

Apply [8, Theorem 8.4]. ∎

5 Cyclic and Quasi-Cyclic Codes

Let $C$ be a linear code of length $n$ over the ring $R.$ In this paper, we use the following definition of quasi-cyclic codes.

Definition 17**.**

Let $n=md,$ for some $m$ and $d$ in $\mathbb{N}.$ Also, let $\mathbf{c}\in R^{n},$ with $\mathbf{c}=\left(\mathbf{c}^{(1)}|\mathbf{c}^{(2)}|\cdots|\mathbf{c}^{(d)}\right),$ where $\mathbf{c}^{(i)}\in R^{m},$ for all $i=1,2,\dots,d.$ Let $\sigma_{d}$ be a map from $R^{n}$ to $R^{n}$ such that $\sigma_{d}(\mathbf{c})=\left(\sigma\left(\mathbf{c}^{(1)}\right)|\sigma\left(\mathbf{c}^{(2)}\right)|\cdots|\sigma\left(\mathbf{c}^{(d)}\right)\right),$ where $\sigma$ is a cyclic shift from $R^{m}$ to $R^{m}.$ A code $C$ of length $n$ over ring $R$ is said to be a quasi-cyclic code with index $d$ if $\sigma_{d}(C)=C.$

Note that, Definition 17 is permutation equivalent to the usual definition of quasi-cyclic codes. Also, a code $C$ is said to be cyclic if its a quasi-cyclic code of index $d=1.$ We have the following characterization for quasi-cyclic codes over the ring $\mathcal{R}_{k}.$

Theorem 18**.**

A code $C$ of length $n$ over $\mathcal{R}_{k}$ is a quasi-cyclic code with index $d$ if and only if $C=\overline{\Psi}^{-1}(C_{1},\dots,C_{2^{k}}),$ where $C_{1},\dots,C_{2^{k}}$ are quasi-cyclic codes of length $n$ with index $d$ over $R.$

Proof.

($\Longrightarrow$) For any $i,$ take any $\mathbf{c}\in C_{i}.$ Since $C$ is a quasi-cyclic code of index $d,$, we have that

[TABLE]

is also in $C.$ This gives $\sigma_{d}(\mathbf{c})\in C_{i}$ as we hope.

($\Longleftarrow$) For any $\mathbf{w}$ in $C,$ there exist codewords $\mathbf{w}_{1},\mathbf{w}_{2},\dots,\mathbf{w}_{2^{k}},$ where $\mathbf{w}_{i}\in C_{i},$ for all $1\leq i\leq 2^{k},$ such that $\mathbf{w}=\overline{\Psi}^{-1}(\mathbf{w}_{1},\dots,\mathbf{w}_{2^{k}}).$ Also, we have that

[TABLE]

Since $C_{i}$ is a quasi-cyclic code of index $d,$ we have $\sigma_{d}(\mathbf{w}_{i})$ is in $C_{i},$ for all $i=1,2,\dots,2^{k}.$ So, $(\sigma_{d}(\mathbf{w}_{1}),\dots,\sigma_{d}(\mathbf{w}_{2^{k}}))$ is in $\overline{\Psi}(C).$ This means $\sigma_{d}(\mathbf{w})$ is in $C.$ ∎

Theorem 19**.**

A code $C$ of length $n$ over $\mathcal{R}_{k}$ is cyclic if and only if $C=\overline{\Psi}^{-1}(C_{1},\dots,C_{2^{k}}),$ where $C_{1},\dots,C_{2^{k}}$ are cyclic codes of length $n$ over $R.$

Proof.

Apply Theorem 18 with $d=1.$ ∎

We also have the following characterization of quasi-cyclic codes.

Theorem 20**.**

A code $C$ of length $n$ over $\mathcal{R}_{j}$ is a quasi-cyclic code with index $d$ if and only if $\overline{\varphi}_{j}(C)$ is a quasi-cyclic code of length $nl_{j}$ with index $l_{j}d$ over $\mathcal{R}_{j-1}.$

Proof.

For any $\mathbf{c}^{\prime}$ in $\overline{\varphi}_{j}(C),$ there exists $\mathbf{c}$ in $C$ such that $\overline{\varphi}_{j}(\mathbf{c})=\mathbf{c}^{\prime}.$ Now, let $\mathbf{c}=\left(\alpha^{(1)}|\cdots|\alpha^{(d)}\right),$ where $\alpha^{(i)}=(\alpha_{i1}+\alpha_{i1}^{\prime}v_{j},\dots,\alpha_{im}+\alpha_{im}^{\prime}v_{j}),$ for all $1\leq i\leq d.$ So, we have

[TABLE]

where $\beta_{0}^{(i)}=(\alpha_{i1},\dots,\alpha_{im}),$ for all $1\leq i\leq d,$ and

[TABLE]

for all $r=1,\dots,l_{j}-1,$ $i=1,\dots,d.$ Consider,

[TABLE]

Therefore, $\sigma_{d}(\mathbf{c})\in C$ if and only if $\sigma_{l_{j}d}(\mathbf{c}^{\prime})\in\overline{\varphi}_{j}(C).$ ∎

The following results are direct consequences of Theorem 20.

Theorem 21**.**

A code $C$ of length $n$ over $\mathcal{R}_{j}$ is a cyclic code if and only if $\overline{\varphi}_{j}(C)$ is a quasi-cyclic code of length $nl_{j}$ with index $l_{j}$ over $\mathcal{R}_{j-1}.$

Corollary 22**.**

A code $C$ of length $n$ over $\mathcal{R}_{k}$ is a quasi-cyclic code with index $d$ if and only if $\overline{\varphi}_{1}\circ\cdots\circ\overline{\varphi}_{k}(C)$ is a quasi-cyclic code of length $nl_{1}\cdots l_{k}$ with index $d\cdot l_{1}\cdots l_{k}$ over $R.$

Proof.

Apply Theorem 20 repeatedly while considering the image of $\overline{\varphi}_{1}\circ\cdots\circ\overline{\varphi}_{k}.$ ∎

Corollary 23**.**

A code $C$ of length $n$ over $\mathcal{R}_{k}$ is a cyclic code if and only if $\overline{\varphi}_{1}\circ\cdots\circ\overline{\varphi}_{k}(C)$ is a quasi-cyclic code of length $nl_{1}\cdots l_{k}$ with index $l_{1}\cdots l_{k}$ over $R.$

6 Skew-Cyclic and Quasi-Skew-Cyclic Codes

Let $C$ be a code of length $n$ over the ring $\mathcal{R}_{k}.$ Given an atomorphism on the ring $\mathcal{R}_{k},$ say $\theta,$ then $C$ is said to be a $\theta$-cyclic code or skew-cyclic code if

• (1)

$C$ is a linear code over $\mathcal{R}_{k},$ and

• (2)

For any $c=(c_{0},\dots,c_{n-1})$ in $C,$ we have that $T_{\theta}(c)=\left(\theta(c_{n-1}),\theta(c_{0}),\dots,\theta(c_{n-2})\right)$ is also in $C.$

Also, $C$ is said to be a quasi-$\theta$-cyclic code of index $d$ if

• (1)

$C$ is a linear code over $\mathcal{R}_{k},$ and

• (2)

For any $c=(c_{0},\dots,c_{n-1})$ in $C,$ we have that $T^{d}_{\theta}(c)=\left(\theta(c_{n-d}),\theta(c_{n-d+1}),\dots,\theta(c_{n-d-1})\right)$ is also in $C.$

Let $T$ be a cyclic-shift operator on $R^{n2^{k}}.$ We have the following characterizations.

Theorem 24**.**

A code $C$ over $\mathcal{R}_{k}$ is a quasi-$\theta$-cyclic code of index $d$ if and only if $T^{d2^{k}}\circ\Sigma_{S}\circ\Phi_{S_{1},S_{2}}(\overline{\Psi}(C))\subseteq\overline{\Psi}(C),$ for some $S,S_{1},S_{2}\subseteq\{1,2,\dots,k\},$ where $|S_{1}|=|S_{2}|.$

Proof.

Let $c=(c_{0},c_{1},\dots,c_{n-1})$ be any element in $C.$ We can see that

[TABLE]

Since $\theta$ is a composition of $\Theta_{S}$ and $\Phi_{S_{1},S_{2}},$ for some $S,S_{1},S_{2}\subseteq\{1,2,\dots,k\},$ we have that

[TABLE]

Therefore, $C$ is invariant under the action of $T^{d}_{\theta}$ if and only if $\overline{\Psi}(C)$ invariant under the action of $T^{d2^{k}}\circ\Sigma_{S}\circ\Gamma_{S_{1},S_{2}}.$ ∎

Theorem 25**.**

A code $C$ over $\mathcal{R}_{k}$ is a $\theta$-cyclic code if and only if $T^{2^{k}}\circ\Sigma_{S}\circ\Phi_{S_{1},S_{2}}(\overline{\Psi}(C))\subseteq\overline{\Psi}(C),$ for some $S,S_{1},S_{2}\subseteq\{1,2,\dots,k\},$ where $|S_{1}|=|S_{2}|.$

Proof.

Apply Theorem 24 with $d=1.$ ∎

We can also have more technical characterizations as follow.

Theorem 26**.**

A linear code $C$ over $\mathcal{R}_{k}$ is quasi-$\theta$-cyclic of index $d$ and length $n$ if and only if there exist quasi-$\vartheta$-cyclic codes $C_{1},C_{2},\dots,C_{2^{k}}$ of length $n$ over $R$ with index $d\cdot\operatorname{Ord}(\phi_{S_{1},S_{2}}),$ such that

[TABLE]

where $\vartheta$ is an automorphism in $R,$ and $T^{d}_{\tilde{\theta}}(C_{i})\subseteq C_{j},$ where $j\in S\cup S_{2},$ for all $i=1,2,\dots,2^{k}.$

Proof.

$(\Longrightarrow)$ Remember that there exist codes over $R,$ $C_{1},C_{2},\dots,C_{2^{k}},$ such that,

[TABLE]

For any $c_{i}\in C_{i},$ let $c_{i}=(\alpha_{1},\dots,\alpha_{n}).$ If, $c=\overline{\Psi}_{k}^{-1}(0,\dots,0,c_{i},0,\dots,0),$ then

[TABLE]

So, if we consider

[TABLE]

then we have $T_{\vartheta}^{d}(c_{i})$ is in $C_{j},$ where $j\in S\cup S_{2}.$ By continuing this process, we have $T^{d\cdot\operatorname{Ord}(\phi_{S_{1},S_{2}})}_{\vartheta}(c_{i})\in C_{i},$ which means, $C_{i}$ is quasi-$\vartheta$-cyclic code over $R$ with index $d\cdot\operatorname{Ord}(\phi_{S_{1},S_{2}}),$ for all $i=1,\dots,2^{k}.$

$(\Longleftarrow)$ For any $c\in C,$ we can see that $\overline{\Psi}_{k}(c)\in(C_{1},\dots,C_{2^{k}}).$ Since $C_{i}$ is quasi-$\vartheta$-cyclic code over $R$ with index $d\cdot\operatorname{Ord}(\phi_{S_{1},S_{2}}),$ for all $i=1,\dots,2^{k},$ $C_{1},$ and $T_{\vartheta}^{dt_{1}}(C_{i})\subseteq C_{j},$ where $j\in S\cup S_{2},$ for all $i=1,2,\dots,2^{k},$ where $1\leq t_{1}\leq 2^{k}.$ Then we have $T_{\theta}^{d}(c)=\overline{\Psi}_{k}^{-1}(T_{\vartheta}(\Psi_{k}(c)))\in C,$ as we hope. ∎

Theorem 27**.**

A linear code $C$ over $\mathcal{R}_{k}$ is $\theta$-cyclic of length $n$ if and only if there exist quasi-$\vartheta$-cyclic codes $C_{1},C_{2},\dots,C_{2^{k}}$ of length $n$ over $R$ with index $\operatorname{Ord}(\phi_{S_{1},S_{2}}),$ such that

[TABLE]

where $\vartheta$ is an automorphism in $R,$ and $T_{\tilde{\theta}}(C_{i})\subseteq C_{j},$ where $j\in S\cup S_{2},$ for all $i=1,2,\dots,2^{k}.$

Proof.

Apply Theorem 26 with $d=1.$ ∎

Theorem 26 gives us an algorithm to construct quasi-skew-cyclic codes over the ring $B_{k}$ as follows.

Algorithm 28**.**

Given $n,d,$ the ring $\mathcal{R}_{k},$ and an automorphism $\theta.$

• (1)

Decompose $\theta$ to be $\theta=\Theta_{S}\circ\Phi_{S_{1},S_{2}}.$

• (2)

Determine $\operatorname{Ord}(\phi_{S_{1},S_{2}})$ and $\vartheta.$

• (3)

Choose quasi-$\vartheta$-cyclic codes over $R,$ say $C_{1},\dots,C_{2^{k}},$ such that

[TABLE]

where $j\in S\cup S_{2},$ for all $i=1,2,\dots,2^{k}.$

• (4)

Calculate $C=\overline{\Psi}_{k}^{-1}(C_{1},\dots,C_{2^{k}}).$

• (5)

$C$ is a quasi-$\theta$-cyclic code of index $d$ over the ring $\mathcal{R}_{k}.$

Note that Algorithm 28 can be used to construct skew-cyclic code over $\mathcal{R}_{k}$ by choosing $d=1.$

7 Examples

7.1 Examples using the map $\Psi$

As a direct consequence of Theorem 5, we have that for any code $C$ of length $n$ over $\mathcal{R}_{k}=\mathbb{Z}_{m}[v_{1},v_{2},\dots,v_{k}],$ where $v_{i}^{2}=v_{i},$ for all $i=1,2,\dots,k,$ there exist codes $C_{1},C_{2},\dots,C_{2^{k}}$ of length $n$ over $\mathbb{Z}_{m}$ such that $C=\overline{\Psi}^{-1}(C_{1},C_{2},\dots,C_{2^{k}}).$

Example 29**.**

Let $\mathcal{R}_{1}=\mathbb{Z}_{4}[v],$ where $v^{2}=v.$ Also, let $C=\langle(1\;v\;1+v\;3)\rangle.$ We can check that

[TABLE]

Then, if we choose $C_{1}=\langle(1\;0\;1\;3)\rangle$ and $C_{2}=\langle(1\;1\;2\;3)\rangle,$ we have $C=\overline{\Psi}^{-1}(C_{1},C_{2}).$

Moreover, we can have more explicit example for Hermitian self-dual codes as follow.

Example 30**.**

Let $\mathcal{R}_{1}=\mathbb{Z}_{4}[v],$ where $v^{2}=v.$ In this ring, $\Theta_{1}(v)=1-v.$ Let $C=\langle(v\;v\;v)\rangle$ be a code over $\mathcal{R}_{1}.$ By Proposition 6, $C$ is a Hermitian self-dual code. Since

[TABLE]

we have that $C=\overline{\Psi}^{-1}(C_{1},C_{2}),$ where $C_{1}=C_{2}=\langle(1\;1\;1)\rangle.$ We can check that $C_{1}$ is an Euclidean self-dual code over $\mathbb{Z}_{4}.$ Therefore, we have $C_{2}=C_{1}^{\perp},$ as stated in Proposition 7 and Theorem 8.

Also, we have the following example for Euclidean self-dual codes.

Example 31**.**

Let $\mathcal{R}_{1}=\mathbb{Z}_{4}[v],$ where $v^{2}=v.$ Take $C=\langle(v\;1-v),(1-v\;v)\rangle.$ We can see that $C$ is an Euclidean self-dual code over $\mathcal{R}_{1}.$ Also, we know that

[TABLE]

and

[TABLE]

If we take $C_{1}=C_{2}=\langle(1\;0),(0\;1)\rangle,$ then we have $C=\overline{\Psi}^{-1}(C_{1},C_{2}).$ We can check that $C_{1}$ and $C_{2}$ are Euclidean self-dual codes over $\mathbb{Z}_{4}$ also, as stated in Theorem 9.

7.2 Codes over $\mathbb{Z}_{4}$

In this part, we will use the map $\varphi_{1}$ to get codes over $\mathbb{Z}_{4}$ from codes over $\mathcal{R}_{1}=\mathbb{Z}_{4}+v\mathbb{Z}_{4},$ where $v^{2}=v.$ For any element $\mathbf{x}=(x_{1},\dots,x_{n})$ in $\mathbb{Z}_{4}^{n},$ Lee weight of $\mathbf{x},$ denoted by $w_{L}(\mathbf{x}),$ as

[TABLE]

Using the above weight, we define Lee distance of a code $C$ as

[TABLE]

We will give some examples of codes over $\mathbb{Z}_{4}$ with maximum Lee distance so far, as in http://www.asamov.com/Z4Codes/CODES/ShowCODESTablePage.aspx, constructed using the map $\varphi_{1}.$

Example 32**.**

Define a map $\varphi_{1}$ as follows.

[TABLE]

Let $C=\langle 1+v\rangle=\{0,1+v,2+2v,3+3v,2v,2,1+3v,3+v\}$ be a code of length 1 over $\mathcal{R}_{1}=\mathbb{Z}_{4}+v\mathbb{Z}_{4},$ where $v^{2}=v.$ We have,

[TABLE]

[TABLE]

We can see that $d_{L}(\varphi_{1}(C))=2$ and $|\varphi_{1}(C)|=8.$

Example 33**.**

Define a map $\varphi_{1}$ as follows.

[TABLE]

Let $C=\langle 2\rangle=\{0,2,2v,2+2v\}.$ We have that

[TABLE]

So, $d_{L}(\varphi_{1}(C))=4$ and $|\varphi_{1}(C)|=4.$

Example 34**.**

Define a map $\varphi_{1}$ as follows.

[TABLE]

Let $C=\langle 2\rangle.$ We can see that,

[TABLE]

Therefore, we have $d_{L}(\varphi_{1}(C))=6$ and $|\varphi_{1}(C)|=4.$

The following table gives some examples of codes over $\mathbb{Z}_{4}$ obtained by a similar way as in Example 32-34.