Linear codes over the ring $\mathbb{Z}_4 + u\mathbb{Z}_4 + v\mathbb{Z}_4 + w\mathbb{Z}_4 + uv\mathbb{Z}_4 + uw\mathbb{Z}_4 + vw\mathbb{Z}_4 + uvw\mathbb{Z}_4$

TL;DR

This paper studies linear codes over a complex ring extension of our, analyzing their structure, weight enumerators, bounds, and special subclasses like cyclic codes, with new definitions and relations.

Contribution

It introduces the structure of linear codes over a multi-component ring extension of our, including weight, Gray map, and MacWilliams relations, and explores cyclic and quasi-cyclic codes.

Findings

Defined Lee weight and Gray map for these codes

Derived MacWilliams relations for various weight enumerators

Discussed bounds and examples of cyclic codes

Abstract

We investigate linear codes over the ring $\mathbb{Z}_4 + u\mathbb{Z}_4 + v\mathbb{Z}_4 + w\mathbb{Z}_4 + uv\mathbb{Z}_4 + uw\mathbb{Z}_4 + vw\mathbb{Z}_4 + uvw\mathbb{Z}_4$, with conditions $u^2=u$, $v^2=v$, $w^2=w$, $uv=vu$, $uw=wu$ and $vw=wv.$ We first analyze the structure of the ring and then define linear codes over this ring. Lee weight and Gray map for these codes are defined and MacWilliams relations for complete, symmetrized, and Lee weight enumerators are obtained. The Singleton bound as well as maximum distance separable codes are also considered. Furthermore, cyclic and quasi-cyclic codes are discussed, and some examples are also provided.

Equations

$${\eta }_1$$

$${\eta }_2$$

$${\eta }_3$$

$${\eta }_4$$

$${\eta }_5$$

$${\eta }_6$$

$${\eta }_7$$

$${\eta }_8$$

$$R=R{\eta }_1\oplus R{\eta }_2\oplus R{\eta }_3\oplus R{\eta }_4\oplus R{\eta }_5\oplus R{\eta }_6\oplus R{\eta }_7\oplus R{\eta }_8.$$

$$r$$

$$=r{\eta }_1+r{\eta }_2+r{\eta }_3+r{\eta }_4+r{\eta }_5+r{\eta }_6+r{\eta }_7+r{\eta }_8$$

$$=a{\eta }_1+(a+b){\eta }_2+(a+c){\eta }_3+(a+d){\eta }_4+(a+b+c+e){\eta }_5$$

$$+(a+b+d+f){\eta }_6+(a+c+d+g){\eta }_7+(a+b+c+d+e+f+g+h){\eta }_8$$

$$=r_1{\eta }_1+r_2{\eta }_2+r_3{\eta }_3+r_4{\eta }_4+r_5{\eta }_5+r_6{\eta }_6+r_7{\eta }_7+r_8{\eta }_8$$

$$r_1$$

$$r_2$$

$$r_3$$

$$r_4$$

$$r_5$$

$$r_6$$

$$r_7$$

$$r_8$$

$$r⟼(r_1,r_2,r_3,r_4,r_5,r_6,r_7,r_8).$$

$$\Phi :R^n$$

$$(c_0,c_1,\dots ,c_{n-1})$$

$$w_L(x):=\{\begin{array}{ll}0,& x=0,\\ 2,& x=2,\\ 1,& x=1\text{ or }3.\end{array}$$

$$⟨\mathbf{x},\mathbf{y}⟩:=\sum _{j=1}^n{x}_j{y}_j,$$

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

$$\mathbf{r}={\mathbf{r}}_1{\eta }_1+{\mathbf{r}}_2{\eta }_2+\cdots +{\mathbf{r}}_8{\eta }_8,$$

$$\mathbf{x}\cdot \mathbf{y}=({\mathbf{x}}_1\cdot {\mathbf{y}}_1){\eta }_1+({\mathbf{x}}_2\cdot {\mathbf{y}}_2){\eta }_2+\cdots +({\mathbf{x}}_8\cdot {\mathbf{y}}_8){\eta }_8,$$

$$C_1$$

$$C_2$$

$$⋮$$

$$C_8$$

$$G_j=\left(\begin{array}{ccc}{I}_{k_{j1}}& A_j& B_j\\ 0& 2I_{k_{j2}}& 2C_j\end{array}\right),$$

$$\left(\begin{array}{c}{\eta }_1{G}_1\\ {\eta }_2{G}_2\\ ⋮\\ {\eta }_8{G}_8\end{array}\right).$$

$$G_j^{\prime }=\left(\begin{array}{ccc}-B_i^T-C_i^T{A}_i^T& C_i^T& I_{n-k_{i1}-k_{i2}}\\ 2A_i^T& 2I_{k_{i2}}& 0\end{array}\right),\text{ for }1\le i\le 8,$$

$$\left(\begin{array}{c}{\eta }_1{G}_1^{\prime }\\ {\eta }_2{G}_2^{\prime }\\ ⋮\\ {\eta }_8{G}_8^{\prime }\end{array}\right),$$

$$G_j=\left(\begin{array}{cccc}1& 1& 1& 3\\ 0& 2& 0& 2\\ 0& 0& 2& 2\end{array}\right).$$

$$\left(\text{setcounter}MaxMatrixCols32\begin{array}{cccccccccccccccccccccccccccccccc}1& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 3& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 2& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 2& 0& 0& 0& 0& 0& 0& 0& 2& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 3& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 2& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 2& 0& 0& 0& 0& 0& 0& 0& 2& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 3& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 2& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 2& 0& 0& 0& 0& 0& 0& 0& 2& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 3& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 2& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 2& 0& 0& 0& 0& 0& 0& 0& 2& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 3& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 2& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 2& 0& 0& 0& 0& 0& 0& 0& 2& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 3& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 2& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 2& 0& 0& 0& 0& 0& 0& 0& 2& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 3& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 2& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 2& 0& 0& 0& 0& 0& 0& 0& 2& 0\\ 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 3\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 2\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 2& 0& 0& 0& 0& 0& 0& 0& 2\end{array}\right).$$

$$d_H(C)\le n-{\log }_{|R|}|C|+1,$$

Full text

Keywords: Linear codes, MacWilliams relations, Maximum distance separable codes, Cyclic codes, Quasi-cyclic codes.

1 Introduction

Codes over rings have become an active research area in classical coding theory over the recent decades. In particular, after the appearance of the work of Hammons, Kumar, Calderbank, Sloane, and Solé [11], a lot of research went towards studying (linear) codes over $\mathbb{Z}_{4}.$ Although ”the results were generalized to many different types of rings, the codes over $\mathbb{Z}_{4}$ remain a special topic of interest in the field of algebraic coding theory because of their relation to lattices, designs, cryptography and their many applications”444[17], pp. 25. [17].

Recently, several new families of rings, namely the non-chain rings but are Frobenius, have been studied in connection with coding theory. These rings have rich mathematical theory, in particular algebraic structures. Yildiz and Karadeniz [17] derived algebraic structures related to linear codes over the ring $\mathbb{Z}_{4}+u\mathbb{Z}_{4},$ with $u^{2}=0.$ They [17] also have several good formally self-dual codes over $\mathbb{Z}_{4}$ from the codes over $\mathbb{Z}_{4}+u\mathbb{Z}_{4}.$ Bandi and Bhaintwal ([2], [3]) considered codes over the ring $\mathbb{Z}_{4}+v\mathbb{Z}_{4},$ with $v^{2}=v,$ and $\mathbb{Z}_{4}+w\mathbb{Z}_{4},$ with $w^{2}=1,2w,$ respectively and derived several algebraic structures including the MacWilliams relation with respect to Rosenbloom-Tsfasman metric over the ring $\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ and the properties as well as construction of self-dual codes over the ring $\mathbb{Z}_{4}+w\mathbb{Z}_{4}.$ Very recently, Li, Guo, Zhu, and Kai [13] generalized the ring considered by Bandi and Bhaintwal [2] by adding two new terms $u\mathbb{Z}_{4}$ and $uv\mathbb{Z}_{4},$ with the conditions $u^{2}=u,$ $v^{2}=v,$ and $uv=vu,$ and derived some properties corresponding to the linear codes over the ring $\mathbb{Z}_{4}+u\mathbb{Z}_{4}+v\mathbb{Z}_{4}+uv\mathbb{Z}_{4}.$

In this paper, we further generalized the ring considered by Li, Guo, Zhu, and Kai [13] to the ring $\mathbb{Z}_{4}+u\mathbb{Z}_{4}+v\mathbb{Z}_{4}+w\mathbb{Z}_{4}+uv\mathbb{Z}_{4}+uw\mathbb{Z}_{4}+vw\mathbb{Z}_{4}+uvw\mathbb{Z}_{4}$ with the conditions $u^{2}=u$, $v^{2}=v$, $w^{2}=w$, $uv=vu$, $uw=wu$ and $vw=wv$. We study linear codes over this ring and derive some corresponding properties. The paper is organized as follows. In Section 2, we study main properties of the ring $\mathbb{Z}_{4}+u\mathbb{Z}_{4}+v\mathbb{Z}_{4}+w\mathbb{Z}_{4}+uv\mathbb{Z}_{4}+uw\mathbb{Z}_{4}+vw\mathbb{Z}_{4}+uvw\mathbb{Z}_{4}.$ We then define linear codes, Lee weight, and also a Gray map for the linear codes over $\mathbb{Z}_{4}+u\mathbb{Z}_{4}+v\mathbb{Z}_{4}+w\mathbb{Z}_{4}+uv\mathbb{Z}_{4}+uw\mathbb{Z}_{4}+vw\mathbb{Z}_{4}+uvw\mathbb{Z}_{4}.$ Singleton bound as well as maximum distance separable codes are slightly considered. In Section 3, several kind of weight enumerators are defined and related MacWilliams relations are derived. Finally, in Section 4, cyclic and quasi-cyclic codes over $\mathbb{Z}_{4}+u\mathbb{Z}_{4}+v\mathbb{Z}_{4}+w\mathbb{Z}_{4}+uv\mathbb{Z}_{4}+uw\mathbb{Z}_{4}+vw\mathbb{Z}_{4}+uvw\mathbb{Z}_{4}$ are investigated and several examples are provided.

Throughout this paper, we follow standard definitions of undefined terms as used in many coding theory books (e.g. [12]).

2 Structures of linear codes over $R$

Throughout this paper, $R$ denotes the ring $\mathbb{Z}_{4}+u\mathbb{Z}_{4}+v\mathbb{Z}_{4}+w\mathbb{Z}_{4}+uv\mathbb{Z}_{4}+uw\mathbb{Z}_{4}+vw\mathbb{Z}_{4}+uvw\mathbb{Z}_{4}$ with $u^{2}=u$, $v^{2}=v$, $w^{2}=w$, $uv=vu$, $uw=wu$ dan $vw=wv$.

2.1 Structures of the ring $R$ and a Gray map

Let $R$ denote the ring $\mathbb{Z}_{4}+u\mathbb{Z}_{4}+v\mathbb{Z}_{4}+w\mathbb{Z}_{4}+uv\mathbb{Z}_{4}+uw\mathbb{Z}_{4}+vw\mathbb{Z}_{4}+uvw\mathbb{Z}_{4}$ with $u^{2}=u$, $v^{2}=v$, $w^{2}=w$, $uv=vu$, $uw=wu$ dan $vw=wv$. The ring $R$ can also be regarded as a quotient ring of a polynomial ring over $\mathbb{Z}_{4},$ namely $\mathbb{Z}_{4}[u,v,w]/\langle u^{2}-u,v^{2}-v,w^{2}-w\rangle$. $R$ is commutative with identity. The element $\eta\in R$ is called idempotent if $\eta^{2}=\eta.$ The elements $x$ and $y$ of $R$ is called orthogonal if $xy=0.$

We will define a Gray map from the ring $R$ to $\mathbb{Z}_{4}^{8}.$ For that purpose we consider first a decomposition of $R.$

Consider the idempotent elements of $R$ below

[TABLE]

The above eight elements are also pairwise orthogonal, since $\eta_{i}\eta_{j}=0$ for $i\neq j,$ and also satisfy $\sum\limits_{i=1}^{8}\eta_{i}=1$. Hence, by Chinese Remainder Theorem, we have

[TABLE]

Moreover, for any $r=a+bu+cv+dw+euv+fuw+gvw+huvw\in R$ with $a,b,c,d,e,f,g,h\in\mathbb{Z}_{4},$ we have

[TABLE]

with

[TABLE]

and hence $r_{1},r_{2},r_{3},r_{4},r_{5},r_{6},r_{7},r_{8}\in\mathbb{Z}_{4}$. It is clear that the expression $r=r_{1}\eta_{1}+r_{2}\eta_{2}+r_{3}\eta_{3}+r_{4}\eta_{4}+r_{5}\eta_{5}+r_{6}\eta_{6}+r_{7}\eta_{7}+r_{8}\eta_{8}$ is unique. Then we define the map $\phi$ from $R$ to $\mathbb{Z}_{4}^{8}$ by

[TABLE]

It is easy to see that $\phi$ defines an isomorphism. We define a Gray map as an extension of the map $\phi$ on $R^{n}$ as

[TABLE]

where $c_{i}\in R$ and $r_{ji}\in\mathbb{Z}_{4}$ satisfying $c_{i}=\sum\limits_{j=1}^{8}r_{ji}\eta_{j}.$

The Lee weight on $\mathbb{Z}_{4},$ denoted by $w_{L},$ is defined as

[TABLE]

Remembering the map $\phi:r\longmapsto(r_{1},r_{2},\ldots,r_{8}),$ we define the Lee weight on $R$ as $w_{L}(r)=\sum_{i=1}^{8}w_{L}(r_{i}).$ The Lee weight of a vector $\mathbf{c}=(c_{0},c_{1},\ldots,c_{n-1})\in R^{n}$ is defined to be a rational sum of the Lee weight of its components, that is $w_{L}(\mathbf{c})=\sum_{i=0}^{n-1}w_{L}(c_{i}).$ Moreover, for any $\mathbf{c},\mathbf{d}\in R^{n},$ the Lee distance between $\mathbf{c}$ and $\mathbf{d}$ is defined as $d_{L}(\mathbf{c},\mathbf{d})=w_{L}(\mathbf{c}-\mathbf{d}).$ Meanwhile, we have also another kind of weight and distance called Hamming weight and Hamming distance, that defined as $w_{H}(\mathbf{r})=|\{j:~{}r_{j}\neq 0,~{}0\leq j\leq n-1\}|$ and $d_{H}(\mathbf{r},\mathbf{s})=w_{H}(\mathbf{r}-\mathbf{s}),$ for all $\mathbf{r},\mathbf{s}\in R^{n},$ respectively.

2.2 Linear Codes over $R$

A nonempty subset $C\subseteq R^{n}$ is called linear code over $R$ if $C$ is a submodule of $R.$ To define a dual of the code $C,$ let us first define the Euclidean inner product on $R^{n}.$ Let $\mathbf{x}=(x_{0},x_{1},\ldots,x_{n-1})$ and $\mathbf{y}=(y_{0},y_{1},\ldots,y_{n-1})$ be two vectors in $R^{n}.$ The Euclidean inner product of $\mathbf{x}$ and $\mathbf{y}$ is defined as

[TABLE]

where the operations are performed in the ring $R.$

Dual of the code $C\subseteq R^{n}$ is the code

[TABLE]

Clearly, $C^{\perp}$ is also linear if $C$ is linear over $R.$ Since $R$ is a Frobenius ring, we also have $|C|\cdot|C^{\perp}|=4^{8n}.$

Denote $\mathbf{r}=(r^{(0)},r^{(1)},\ldots,r^{(n-1)})\in R^{n},$ and $\mathbf{r}^{(i)}=r_{i1}\eta_{1}+r_{i2}\eta_{2}+\cdots+r_{i8}\eta_{8},$ for $0\leq i\leq n-1.$ Then $\mathbf{r}$ can be uniquely expressed as

[TABLE]

where $\mathbf{r}_{j}=(r_{0j},r_{1j},\ldots,r_{n-1,j})\in\mathbb{Z}_{4}^{n},$ for $1\leq j\leq 8.$ By using this expression then the inner product of any two vectors $\mathbf{x},\mathbf{y}\in R^{n}$ can be written as

[TABLE]

where $\mathbf{x}=\mathbf{x}_{1}\eta_{1}+\mathbf{x}_{2}\eta_{2}+\cdots+\mathbf{x}_{8}\eta_{8},$ $\mathbf{x}_{j}=(x_{0j},x_{1j},\ldots,x_{n-1,j})\in\mathbb{Z}_{4}^{n},$ and $\mathbf{y}=\mathbf{y}_{1}\eta_{1}+\mathbf{y}_{2}\eta_{2}+\cdots+\mathbf{y}_{8}\eta_{8},$ $\mathbf{y}_{j}=(y_{0j},y_{1j},\ldots,y_{n-1,j})\in\mathbb{Z}_{4}^{n},$ and $\mathbf{x}_{j}\cdot\mathbf{y}_{j}=\sum_{k=0}^{n-1}x_{kj}y_{kj},$ for $1\leq j\leq n.$

Now, define the codes $C_{i},$ $1\leq i\leq 8,$ as follows:

[TABLE]

It is easy to see that $C_{i},$ $1\leq i\leq 8,$ is a linear code of length $n$ over $\mathbb{Z}_{n}.$ Moreover, $C$ can be uniquely decomposed into $C=C_{1}\eta_{1}\oplus C_{2}\eta_{2}\oplus\cdots\oplus C_{8}\eta_{8},$ and hence we have $|C|=\prod_{i=1}^{8}|C_{i}|.$ Furthermore, we have the following property.

Theorem 2.1**.**

Let $C$ be a linear code of length $n$ over $R.$ Then we have the following unique decomposition:

$C=C_{1}\eta_{1}\oplus C_{2}\eta_{2}\oplus\cdots\oplus C_{8}\eta_{8},$* a linear code of length $n$ over $\mathbb{Z}_{4}.$* 2. 2.

$C^{\perp}=C_{1}^{\perp}\eta_{1}\oplus C_{2}^{\perp}\eta_{2}\oplus\cdots\oplus C_{8}^{\perp}\eta_{8},$* for $1\leq i\leq 8.$*

Proof.

Similar to [13]. ∎

It is well-known (see for instance [11]) that the code $C_{j},$ $1\leq i\leq 8,$ is permutation-equivalent to a code generated by

[TABLE]

where $A_{j}$ and $C_{j}$ are $\mathbb{Z}_{2}$-matrices and $B$ is a $\mathbb{Z}_{4}$-matrix, and hence $C$ is permutation-equivalent to a linear code generated by

[TABLE]

Moreover, by the result in [11], the dual $C_{j}^{\perp}$ has generator

[TABLE]

and hence $C^{\perp}$ is permutation-equivalent to a linear code generated by

[TABLE]

a parity check matrix of the code $C.$

Example 2.2**.**

Let $C_{j}$ be a linear code over $\mathbb{Z}_{4}$ with generator matrix $G_{j},$ for $1\leq j\leq 8,$ as follows

[TABLE]

The code $C_{j}$ contains $2^{2k_{j1}+k_{j2}}=2^{2+2}=16.$ Since $4^{4}=|C|\cdot|C^{\perp}|=4^{2}\cdot|C^{\perp}|,$ then $|C^{\perp}|=4^{2}=|C|.$ The linear code $\displaystyle C=\bigoplus_{j=1}^{8}C_{j}\eta_{j}$ over $R$ is of cardinality $|C|=16^{8}=4^{2\cdot 8},$ while the generator matrix of $\Phi(C)$ is

[TABLE]

$\diamondsuit$

2.3 Singleton bound and MDS codes

Singleton bound is among the famous bound in Coding Theory. It is proven in 1964 by Singleton [16] that if $C\subseteq R^{n}$ is a code over $R,$ then we have

[TABLE]

where $d_{H}(C)=\text{min}\{w_{H}(\mathbf{x}-\mathbf{y}):~{}\mathbf{x},\mathbf{y}\in C,\mathbf{x}\neq\mathbf{y}\}.$ The code $C$ is called maximum distance separable (MDS) if it attains the Singleton bound mentioned above.

It has been proven by Guenda and Gulliver [10, Proposition 2.2] that the only MDS codes over $\mathbb{Z}_{4}$ is the trivial one. Moreover, it is also known that $C^{\perp}$ is an MDS code if $C$ is an MDS code (see [15, Theorem 1]). Hence, we have the following.

Lemma 2.3**.**

Let $C$ be a linear code of length $n$ over $\mathbb{Z}_{4}.$ Then $C$ is an MDS codes if and only if $C$ is either $\mathbb{Z}_{4}^{n}$ of parameters $[n,4^{n},1],$ $\langle\mathbb{1}\rangle$ of parameters $[n,4,n],$ or $\langle\mathbb{1}\rangle^{\perp}$ of parameters $[n,4^{n-1},2],$ where $\mathbb{1}$ denotes the all-one vector.

Let us look at the MDS codes over $R.$ By considering a linear code $C$ of length $n$ over $R$ as $C=C_{1}\eta_{1}\oplus C_{2}\eta_{2}\oplus\cdots\oplus C_{8}\eta_{8},$ where $C_{i},$ $1\leq i\leq 8,$ is a linear code of length $n$ over $\mathbb{Z}_{4},$ the Singleton bound can be written as

[TABLE]

where $d_{H}(C)=\text{min}\{d_{H}(C_{i}):~{}1\leq i\leq 8\},$ and $d_{H}(C_{i})$ is a Hamming distance of $C_{i},$ for $1\leq i\leq 8.$

Then we have the following theorem.

Theorem 2.4**.**

Let $C$ be an MDS codes of length $n$ over $R.$

• (1)

If $d_{H}=1,$ then $C_{i}$ is an MDS code of parameters $[n,4^{n},1],$ for $1\leq i\leq 8.$

• (2)

If $d_{H}=2,$ then $C_{i}$ is an MDS code of parameters $[n,4^{n-1},2],$ for $1\leq i\leq 8.$

• (3)

If $d_{H}=n,$ then $C_{i}$ is an MDS code of parameters $[n,4,n],$ for $1\leq i\leq 8.$

Proof.

Similar to the proof of Theorem 5 in [13]. ∎

Theorem 2.5**.**

$C$* is an MDS code of length $n$ over $R$ if and only if for $1\leq i\leq 8,$ $C_{i}$ is an MDS code over $\mathbb{Z}_{4}$ with the same parameters.*

Proof.

Similar to the proof of Theorem 6 in [13]. ∎

3 Weight enumerators and MacWilliams relations

In this section we consider several weight enumerators for a linear codes $C$ as well as the related MacWilliams relations.

3.1 The complete weight enumerator and MacWilliams relation

We knew that the number of elements of $R$ is $65536.$

The complete weight enumerator (CWE) of a linear code $C\subseteq R^{n}$ is defined as

[TABLE]

where $n_{a_{i}}(\mathbf{c})$ denotes the number of appearances of $a_{i}\in R$ in the vector $\mathbf{c}.$

Remark 1**.**

Note that $CWE_{C}(X_{0},X_{1},\ldots,X_{65535})$ is a homogeneous polynomial in $65536$ variables with total degree of each monomial being $n,$ the length of the code $C.$ Since the code $C$ is linear, then $C$ always contains the vector $\mathbf{0}.$ It implies that the term $X_{0}^{n}$ always appears in $CWE_{C}(X_{0},X_{1},\ldots,X_{65535}).$ From the complete weight enumerator we may obtain a lot of information related to the code, such as the size of the code:

[TABLE]

∎

Since the ring $R$ is a Frobenius ring, then the MacWilliams relation for the complete weight enumerator holds (see [19]). To find the exact relation we define the following character on $R.$

Let $I$ be a non-zero ideal in $R$. Define $\chi:I\rightarrow\mathbb{C}^{*}$ by

[TABLE]

with $\mathbb{C}^{*}$ is a unit group in complex number. We know that $\chi$ is a non-trivial character on $R.$

Defining the Hadamard transform by $\hat{f}(\textbf{c})=\sum\limits_{\textbf{d}\in R^{n}}\chi(\textbf{c}\cdot\textbf{d})f(\textbf{d}),$ we obtain the following equation

[TABLE]

We have the MacWilliams relation with respect to the complete weight enumerator as follows.

Theorem 3.1**.**

Let $C$ be a linear code of length $n$ over $R.$ Then

[TABLE]

with $M$ is a matrix of size $65536\times 65536$ defined by $M_{ij}=\chi(a_{i}a_{j})$.

Proof.

Let $f(x)=\prod\limits_{i=0}^{65535}X_{i}^{n_{a_{i}}(x)}$. The result follows from Theorem 8.1 in [19]. ∎

3.2 The Symmetrized Lee weight, Hamming weight and Lee weight enumerator

In the ring $\mathbb{Z}_{4},$ we know that $w_{L}(1)=1=w_{L}(3)$ and the symmetrized Lee weight enumerator for codes over $\mathbb{Z}_{4}$ is defined as

[TABLE]

Adopting the same idea, we will define the symmetrized Lee weight enumerator of codes over $R.$ For that purpose, we first decompose $R$ into $D_{i}=\{x\in R:~{}w_{L}(x)=i\},$ for $0\leq i\leq 16.$ Then we have

[TABLE]

By looking at the elements that have the same Lee weights, we can define the symmetrized Lee weight enumerator. Symmetrized Lee weight enumerator (SLWE) of a linear code $C$ over $R$ is defined as

[TABLE]

where $X_{0},X_{1},X_{2},\ldots,X_{16}$ denote the element of weight $0,1,2,3,\ldots,16,$ respectively. Then we have

[TABLE]

where

[TABLE]

The MacWilliams relation with respect to the symmetrized Lee weight enumerator is as follows.

Theorem 3.2**.**

Let $C$ be a linear code of length $n$ over $R$. Then

[TABLE]

where

[TABLE]

[TABLE]

Proof.

For $i,j=0,1,2,\ldots,16,$ we determine $\sum\limits_{s\in D_{j}}\chi(rs)$ for $r\in D_{i}$. By definition, we have

[TABLE]

Since for $a_{j},a_{k}\in D_{j}$ we have

[TABLE]

then

[TABLE]

By direct calculation, we obtain

[TABLE]

for $k=1,2,\ldots,16.$ Hence, we have

[TABLE]

∎

Another weight enumerator of a linear code $C,$ called Hamming weight enumerator, is defined as

[TABLE]

where $w_{H}(\mathbf{c})$ denotes the Hamming weight of the codeword $\mathbf{c}.$ Then we have the following theorem.

Theorem 3.3**.**

Let $C$ be a linear code of length $n$ over $R.$ Then

[TABLE]

Proof.

Similar to the proof of Theorem 9 in [13]. ∎

We also have the MacWilliams relation with respect to the Hamming weight enumerator.

Theorem 3.4**.**

Let $C$ be a linear code of length $n$ over $R.$ Then

[TABLE]

Proof.

Similar to the proof of Theorem 10 in [13]. ∎

Next, we consider the other weight enumerator with respect to the Lee weight, called Lee weight enumerator. For a linear code $C,$ define $A_{i}$ as a number of elements of $C$ having Lee weight $i$. The sequence $A_{0},A_{1},\ldots,A_{16n}$ is called weight distribution in $C$ with respect to the Lee weight. The Lee weight enumerator for $C$ is defined by

[TABLE]

Then we have the following property.

Theorem 3.5**.**

Let $C$ be a linear code of length $n$ over $R$. Then

[TABLE]

Proof.

Denote $w_{L}(\mathbf{c})=\sum\limits_{i=0}^{16}in_{i}(\mathbf{c})$. Then we have

[TABLE]

By definition, we obtain

[TABLE]

∎

The following theorem give us a MacWilliams relation with respect to the Lee weight enumerator.

Theorem 3.6**.**

Let $C$ be a linear code of length $n$ over $R$. Then

[TABLE]

Proof.

By Theorem 3.2 dan Theorem 3.5, we obtain

[TABLE]

with

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Hence, we have

[TABLE]

∎

4 Cyclic and quasi-cyclic Codes

Now, let us look at an important class of linear codes, namely cyclic codes. In this section we mainly consider the structural properties of cyclic codes over the ring $R.$

The notion of cyclic codes is standard for codes over all rings. A cyclic shift on $R^{n}$ is a permutation $T$ such that

[TABLE]

A linear code $C$ over $R$ is called cyclic code if $C$ is invariant under the cyclic shift $T,$ namely $T(C)=C.$ We use the usual ideas of identifying vectors in $R^{n}$ and polynomials in the residue class ring $R[x]/\langle x^{n}-1\rangle$ as follows:

[TABLE]

With this identification we see that $T(\mathbf{c})$ is identified by $x\cdot c(x)\in R[x]/\langle x^{n}-1\rangle.$ It implies that cyclic codes over $R$ are identified by ideals in the residue class ring $R[x]/\langle x^{n}-1\rangle.$ Hence, in order to understand cyclic codes over the ring $R$ it is essential for us to understand the structure of the residue class ring $R[x]/\langle x^{n}-1\rangle.$

The first theorem below is a straightforward generalization of Theorem 13 proven by Li, Guo, Zhu, and Kai [13].

Theorem 4.1**.**

Let $\displaystyle C=C_{1}\eta_{1}\oplus C_{2}\eta_{2}\oplus\cdots\oplus C_{8}\eta_{8}.$ Then $C$ is a cyclic code over $R$ if and only if one of following three conditions is satisfied:

1. (1)

For $t\in\{1,2,\ldots,8\},$ $C_{t}$ is a cyclic code over $\mathbb{Z}_{4}.$ 2. (2)

For $t\in\{1,2,\ldots,8\},$ $C_{t}^{\perp}$ is a cyclic code over $\mathbb{Z}_{4}.$ 3. (3)

$C^{\perp}$* is a cyclic code over $R$.*

Proof.

Let $\displaystyle\mathbf{c}=\sum_{t=1}^{8}\eta_{t}\mathbf{c}_{t}\in C$, and write $\mathbf{c}_{t}=(c_{t,0},c_{t,1},\ldots,c_{t,n-1})\in C_{t},$ for $1\leq t\leq 8.$ Since $C$ is a cyclic code, we also have

[TABLE]

So, $(c_{t,n-1},c_{t,0},\dots,c_{t,n-2})\in C_{t},$ for $1\leq t\leq 8,$ and hence, $C_{t}$ is cyclic for $1\leq t\leq 8.$ The reverse also holds, so the first condition is proven.

If $C_{t}$ is cyclic over $\mathbb{Z}_{4},$ then $C_{t}^{\perp}$ is also cyclic ([18], Proposition 7.9). From $(1),$ we have $C^{\perp}$ is a cyclic code over $R,$ so we have $C$ is a cyclic code over $R.$ ∎

We start to observe the generator polynomials of cyclic code and its dual over $R.$ For that purpose, we need the following theorem proven by Li, Guo, Zhu, and Kai [13].

Theorem 4.2** ([13], Theorem 15).**

Let $C=\langle f(x)+2p(x),~{}2g(x)\rangle$ be a cyclic code over $\mathbb{Z}_{4}.$ Then

[TABLE]

with $\displaystyle\widehat{f}(x):=\left(\frac{x^{n}-1}{f(x)}\right),$ $\displaystyle\widehat{g}(x):=\left(\frac{x^{n}-1}{g(x)}\right),$ and $\displaystyle f(x)^{*}:=x^{\deg(f(x))}f\left(\frac{1}{x}\right)$.

The following two theorems provide generator polynomials of cyclic code and its dual over $R.$

Theorem 4.3**.**

Let $\displaystyle C=C_{1}\eta_{1}\oplus C_{2}\eta_{2}\oplus\cdots\oplus C_{8}\eta_{8}$ be a cyclic code of length $n$ over $R$. If for every $t\in\{1,2,\dots,8\}$, there exist polynomials $f_{t}(x),g_{t}(x),p_{t}(x)\in\mathbb{Z}_{4}[x]$ such that $C_{t}=\langle f_{t}(x)+2p_{t}(x),~{}2g_{t}(x)\rangle,$ then

[TABLE]

Furthermore, if $n$ is odd, then $\displaystyle C=\left\langle\sum_{t=1}^{8}\eta_{t}f_{t}(x)+2\sum_{t=1}^{8}\eta_{t}g_{t}(x)\right\rangle.$

Proof.

Let $\displaystyle D=\left\langle\sum_{t=1}^{8}\eta_{t}f_{t}(x)+2\sum_{t=1}^{8}\eta_{t}p_{t}(x),~{}2\sum_{t=1}^{8}\eta_{t}g_{t}(x)\right\rangle.$ It is obvious that $D\subseteq C$. Let $c(x)\in C$. Because $\displaystyle C=\oplus_{t=1}^{8}\eta_{t}C_{t}$ and $C_{t}=\langle f_{t}(x)+2p_{t}(x),~{}2g_{t}(x)\rangle,$ then there exist $u_{t}(x),v_{t}(x)\in\mathbb{Z}_{2}[x]$ such that

[TABLE]

So we have $C\subseteq D$ and hence $C=D$. ∎

By using Theorem 4.2 and the similar technique as in proof of Theorem 4.3, we obtain generator polynomials for the dual of cyclic codes as given in the theorem below.

Theorem 4.4**.**

Let $C=\langle f(x)+2p(x),~{}2g(x)\rangle$ be a cyclic code over $\mathbb{Z}_{4}.$ Then

[TABLE]

Now, let us turn to the special class of cyclic codes called quasi-cyclic codes.

Let $\sigma$ be a cyclic shift operator over $\mathbb{Z}_{4}^{n}.$ For any positive integer $s,$ let $\sigma_{s}$ be the quasi-shift defined by

[TABLE]

with $a^{(1)},a^{(2)},\ldots,a^{(s)}\in\mathbb{Z}_{4}^{n}$ and $"|"$ is a vector concatenation. A quaternary quasi-cyclic code $C$ of index $s$ and length $ns$ is a subset of $(\mathbb{Z}_{4}^{n})^{s}$ such that $\displaystyle\sigma_{s}(C)=C$. If $\displaystyle R=\oplus_{t=1}^{8}\eta_{t}R_{t},$ we can write any $r\in R$ as $\displaystyle r=\sum_{t=1}^{8}\eta_{t}r_{t}$ with $r_{t}\in R_{t},$ for $1\leq t\leq 8.$ We define the mapping

$\Phi:R^{n}\longrightarrow\left(\mathbb{Z}_{4}^{2^{3}}\right)^{n}$

$~{}~{}\displaystyle\times_{i=0}^{n}r_{i}\longmapsto\times_{t=1}^{8}\times_{i=0}^{n-1}r_{t,i}$

with $\displaystyle r_{i}=\times_{t=1}^{8}r_{t,i}$ for $i=0,1,\ldots,n-1$ and $r_{t,i}\in R_{t}.$

Then we have a similar theorem of Theorem 17 in [13].

Theorem 4.5**.**

Let $\displaystyle C=C_{1}\eta_{1}\oplus C_{2}\eta_{2}\oplus\cdots\oplus C_{8}\eta_{8}$ be a cyclic code of length $n$ over $R.$ Then $\Phi(C)$ is a quasi-cyclic code of index $8$ and length $8n$ over $\mathbb{Z}_{4}.$

Proof.

Let $\times_{i=0}^{n-1}c_{i}\in C.$ Let $c_{i}=\times_{t=1}^{8}c_{t,i}$ for $i=0,1,\dots,n-1$ and $c_{t,i}\in C_{t}$. Since $C$ is cyclic code, we have $C_{t}$ is cyclic for $1\leq t\leq 8.$ This means that for every $t\in\{1,2,\dots,8\},$ we have $\sigma(\times_{i=0}^{n-1}r_{t,i})\in C_{t},$ if $\times_{i=0}^{n-1}r_{t,i}\in C_{t}.$ Write $\Phi(\times_{i=0}^{n-1}c_{i})=\times_{t=1}^{8}\times_{i=0}^{n-1}r_{t,i}.$ Then

[TABLE]

So we have $\Phi(C)$ is a quasi-cyclic code $C$ of index $8$ and length $8n$ over $\mathbb{Z}_{4}.$ ∎

Furthermore, by using the Theorem 18 of [13] below, we obtain directly the type of $\Phi(C)$ as given in Corollary 4.7.

Theorem 4.6** ([13], Theorem 18).**

Let $C_{t},t\in\{1,2,\dots,8\}$ be a cyclic code of length $n$ ($n$ is odd) over $\mathbb{Z}_{4}.$ Write $C_{t}=\langle f_{1,t}(x)+2f_{2,t}(x)\rangle$ with $f_{1,t}(x)$ and $f_{2,t}(x)$ are monic factors of $x^{n}-1$ over $\mathbb{Z}_{4}$ and $f_{2,t}(x)\mid f_{1,t}(x).$ Then the cardinality of $C_{t},$ for $1\leq t\leq 8,$ is

[TABLE]

The corollary below follows directly.

Corollary 4.7**.**

Let $\displaystyle\Phi(C)=\prod_{t=1}^{8}C_{t}$ be a linear code of length $8n$ ($n$ is odd) over $\mathbb{Z}_{4}$ and $C_{t}$ is a cyclic code over $\mathbb{Z}_{4}$ for every $t\in\{1,2,\ldots,8\}.$ Then the cardinality of $\Phi(C)$ is

[TABLE]

4.1 Some examples

Here we provide some examples of cyclic codes of odd length over $R$ and their $\mathbb{Z}_{4}$-images.

Example 4.8**.**

Let $n=3$. In $\mathbb{Z}_{4}[x],$ $x^{3}-1=(x-1)(x^{2}+x+1)$.

Choose $C_{i}=\langle(x^{2}+x+1)+2\rangle$ for $i=1,2,\ldots,8.$ We have $\displaystyle C=\oplus_{t=1}^{8}C_{t}\eta_{t}.$ Parameters of $\Phi(C)$ is $[24,4^{8}2^{16},2].$ $\blacktriangleleft$

Example 4.9**.**

Let $n=5$. In $\mathbb{Z}_{4}[x],$ $x^{5}-1=(x-1)(x^{4}+x^{3}+x^{2}+x+1).$ Choose $C_{i}=\langle(x^{4}+x^{3}+x^{2}+x+1)+2\rangle$ for $i=1,2,\ldots,8$. We have $\displaystyle C=\oplus_{t=1}^{8}C_{t}\eta_{t}.$ Parameters of $\Phi(C)$ is $[40,4^{8}2^{32},2].$ $\blacktriangleleft$

Example 4.10**.**

Let $n=7$. In $\mathbb{Z}_{4}[x],$ $x^{7}-1=(x+1)(x^{3}+x+1)(x^{3}+x^{2}+1)$. Choose

[TABLE]

Then we have $\displaystyle C=\oplus_{t=1}^{8}C_{t}\eta_{t}$ is a cyclic code over $R.$ Parameters of $\Phi(C)$ is $[56,4^{32}2^{24},2].$

If we choose another set of $C_{i}$ with

[TABLE]

then we have $\displaystyle C=\oplus_{t=1}^{8}C_{t}\eta_{t}$ is also a cyclic code over $R.$ Parameters of $\Phi(C)$ is $[56,4^{8}2^{48},6].$ Let us choose

[TABLE]

We have $\displaystyle C=\oplus_{t=1}^{8}C_{t}\eta_{t}$ is also a cyclic code over $R.$ Parameters of $\Phi(C)$ is $[56,4^{23}2^{27},2].$ $\blacktriangleleft$

Example 4.11**.**

Let $n=9.$ In $\mathbb{Z}_{4}[x]$, $x^{9}-1=(x+1)(x^{2}+x+1)(x^{6}+x^{3}+1).$ Choose

$C_{i}=\langle(x^{2}+x+1)(x^{6}+x^{3}+1)+2(x^{6}+x^{3}+1)\rangle,~{}i=1,2,\ldots,8,$

then $\displaystyle C=\oplus_{t=1}^{8}C_{t}\eta_{t}$ is a cyclic code over $R$. Parameters of $\Phi(C)$ is $[72,4^{8}2^{16},6].$

If we choose

[TABLE]

We have $\displaystyle C=\oplus_{t=1}^{8}C_{t}\eta_{t}$ is a cyclic code over $R.$ Parameters of $\Phi(C)$ is $[72,4^{8}2^{32},3].$ $\blacktriangleleft$

Example 4.12**.**

Let $n=15.$ In $\mathbb{Z}_{4}[x],$ $x^{15}-1=(x+1)(x^{2}+x+1)(x^{4}+x+1)(x^{4}+x^{3}+1)(x^{4}+x^{3}+x^{2}+1).$

Choose

[TABLE]

We have $\displaystyle C=\oplus_{t=1}^{8}C_{t}\eta_{t}$ is a cyclic code over $R.$ Parameters of $\Phi(C)$ is $[120,4^{24}2^{32},10].$

If we choose

[TABLE]

then $\displaystyle C=\oplus_{t=1}^{8}C_{t}\eta_{t}$ is a cyclic code over $R.$ Parameters of $\Phi(C)$ is $[120,4^{40}2^{16},8].$ $\blacktriangleleft$

Example 4.13**.**

Let $n=31.$ In $\mathbb{Z}_{4}[x],$ $x^{31}-1=F_{1}(x)F_{2}(x)F_{3}(x)F_{4}(x)F_{5}(x)F_{6}(x)F_{7}(x)$ with

[TABLE]

Choose

[TABLE]

We have $\displaystyle C=\oplus_{t=1}^{8}C_{t}\eta_{t}$ is a cyclic code over $R.$ Parameters of $\Phi(C)$ is $[248,4^{160}2^{40},8].$

For another set of $C_{i}$ such as

[TABLE]

we have $\displaystyle C=\oplus_{t=1}^{8}C_{t}\eta_{t}$ is a cyclic code over $R$. Parameters of $\Phi(C)$ is $[248,4^{80}2^{80},12].$

Let us choose another set of $C_{i}:$

[TABLE]

We have $\displaystyle C=\oplus_{t=1}^{8}C_{t}\eta_{t}$ is a cyclic code over $R.$ Parameters of $\Phi(C)$ is $[248,4^{48}2^{32},22].$ $\blacktriangleleft$

Remark 2**.**

We compare our results on linear codes over $\mathbb{Z}_{4}$ with the database of $\mathbb{Z}_{4}$ codes available online [4]. We conclude that the resulting linear codes are all new with the highest known minimum distances.

5 Conclusion

In this paper we derive structural properties of linear codes over the ring $R:=\mathbb{Z}_{4}+u\mathbb{Z}_{4}+v\mathbb{Z}_{4}+w\mathbb{Z}_{4}+uv\mathbb{Z}_{4}+uw\mathbb{Z}_{4}+vw\mathbb{Z}_{4}+uvw\mathbb{Z}_{4}.$ We also obtained some new and optimal linear codes having parameters which are unknown to exist before.

There are several direction to further research on the codes over the ring. We are now observing the self-duality as well as polycyclic codes over the ring $R.$ We obtained structural properties regarding self-dual codes as well as constacyclic codes over $R.$ The results, which are not included here, will be published elsewhere in separate papers.

Acknowledgement

This research is supported by Riset P3MI ITB 2018. A part of this work was done while the third author visited Research Center for Pure and Applied Mathematics (RCPAM), Tohoku University, Japan on July 2018 under the financial support from Penelitian Berbasis Kompetensi Kemenristekdikti. The third author thanks Prof. Hajime Tanaka for warm hospitality.