# On $\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive cyclic and constacyclic
codes

Habibul Islam and Om Prakash111Corresponding author

Department of Mathematics
Indian Institute of Technology Patna
Patna- 801 106, India
E-mail<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
###### Abstract

Let $\mathbb{Z}_{p}$ be the ring of residue classes modulo a prime $p$. The
$\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive cyclic codes of length
$(\alpha,\beta)$ is identify as $\mathbb{Z}_{p}[u,v][x]$-submodule of
$\mathbb{Z}_{p}[x]/\langle
x^{\alpha}-1\rangle\times\mathbb{Z}_{p}[u,v][x]/\langle x^{\beta}-1\rangle$
where $\mathbb{Z}_{p}[u,v]=\mathbb{Z}_{p}+u\mathbb{Z}_{p}+v\mathbb{Z}_{p}$
with $u^{2}=v^{2}=uv=vu=0$. In this article, we obtain the complete sets of
generator polynomials, minimal generating sets for cyclic codes with length
$\beta$ over $\mathbb{Z}_{p}[u,v]$ and
$\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive cyclic codes with length
$(\alpha,\beta)$ respectively. We show that the Gray image of
$\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive cyclic code with length
$(\alpha,\beta)$ is either a QC code of length $4\alpha$ with index $4$ or a
generalized QC code of length $(\alpha,3\beta)$ over $\mathbb{Z}_{p}$.
Moreover, some structural properties like generating polynomials, minimal
generating sets of $\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive constacyclic
code with length $(\alpha,p-1)$ are determined.

Key Words : Additive code, Cyclic code, Constacyclic code, Local ring.

2010 MSC : 94B15, 94B05, 94B60.

##  1 Introduction

Linear codes over finite rings have been received remarkable attention of many
mathematicians after the works of Hammons et al. published in 1994 [Hammons].
Cyclic codes are the most important class of linear codes due to their nice
algebraic properties as well as for important generalizations, like
constacyclic, quasi-cyclic, skew cyclic etc. In 1998, Delsarte and Levenshtein
[Delsarte] introduced a generalized class of cyclic codes, namely additive
codes which are defined as the subgroup of a commutative group. In 2009,
Borges et al. [Borges09] studied $\mathbb{Z}_{2}\mathbb{Z}_{4}$-linear codes
and investigated their generator matrices and duality. Also, Abualrub et al.
[Abualrub14] characterized $\mathbb{Z}_{2}\mathbb{Z}_{4}$-additive cyclic
codes as $\mathbb{Z}_{4}[x]$-submodule of $\mathbb{Z}_{2}[x]/\langle
x^{r}-1\rangle\times\mathbb{Z}_{4}[x]/\langle x^{s}-1\rangle$ in the same year
and derived the unique set of generators as well as minimal generating set for
these codes where $s$ is odd. Further, in 2015, Aydogdu et al. [Aydogdu15a]
worked over $\mathbb{Z}_{2}\mathbb{Z}_{2}$-additive codes and later some good
techniques were introduced by researchers for additive codes [[Aydogdu15b,
Borges16, Srinivasulu, P.Li]]. Recently, Aydogdu et al. [Aydogdu16]
investigated the structure of $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-cyclic and
constacyclic codes. Here, they presented the complete structure of
constacyclic codes as $\mathbb{Z}_{2}[u][x]$-submodule of
$\mathbb{Z}_{2}[x]/\langle
x^{\alpha}-1\rangle\times\mathbb{Z}_{2}[u][x]/\langle x^{\beta}-(1+u)\rangle$
and obtained some optimal binary linear codes as the Gray images of
$\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-cyclic codes.
In this article for any prime $p$, we consider the finite local ring
$\mathbb{Z}_{p}[u,v]$ where $u^{2}=v^{2}=uv=vu=0$ and study the
$\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive cyclic and constacyclic codes.
The additive code of length $(\alpha,\beta)$ is define to be the subgroup of
the commutative group
$\mathbb{Z}_{p}^{\alpha}\times\mathbb{Z}_{p}[u,v]^{\beta}$. Note that additive
code is a linear code over $\mathbb{Z}_{p}$ if $\beta=0$ and over
$\mathbb{Z}_{p}[u,v]$ if $\alpha=0$. Therefore, it is a generalization of
linear code over $\mathbb{Z}_{p}$ as well as over $\mathbb{Z}_{p}[u,v]$ . The
main motive of this work is to find out the generator polynomials, minimal
spanning set of cyclic codes over $\mathbb{Z}_{p}[u,v]$,
$\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive cyclic and constacyclic codes
respectively.
The manuscript is organized as follows: In Section 2, we discuss some basic
facts and results. Section LABEL:sec3 discusses the structure of cyclic codes
over $\mathbb{Z}_{p}[u,v]$ while Section LABEL:sec4 finds generator
polynomials and minimal generating set of
$\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive cyclic codes. Also, Section
LABEL:sec4 includes an encoding method and the Gray images of
$\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive cyclic codes. Some properties of
constacyclic codes over $\mathbb{Z}_{p}[u,v]$ are obtained in Section
LABEL:sec5. Section LABEL:sec6 gives the structure of
$\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive constacyclic codes. Section
LABEL:sec7 concludes the manuscript.

##  2 Preliminary

Throughout the manuscript, for a prime $p$, $\mathbb{Z}_{p}$ represents the
ring of residue classes modulo $p$ and $\mathbb{Z}_{p}[u,v]$ denotes the ring
$\mathbb{Z}_{p}+u\mathbb{Z}_{p}+v\mathbb{Z}_{p}$ where $u^{2}=v^{2}=uv=vu=0$.
Clearly, $\mathbb{Z}_{p}[u,v]$ is a finite commutative ring with
characteristic $p$ and cardinality $p^{3}$. Moreover, $\mathbb{Z}_{p}[u,v]$ is
a local ring with unique maximal ideal $\langle u,v\rangle$ and quotient ring
$\mathbb{Z}_{p}[u,v]/\langle u,v\rangle\cong\mathbb{Z}_{p}$. We consider
$\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]=\\{(c,d)\mid
c\in\mathbb{Z}_{p},d\in\mathbb{Z}_{p}[u,v]\\}$. Then
$\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$ is a commutative group with respect to
component wise addition. Also, for two positive integers $\alpha$ and $\beta$,
it is easy to verify that
$\mathbb{Z}_{p}^{\alpha}\times\mathbb{Z}_{p}[u,v]^{\beta}=\\{(c,d)~{}|~{}c=(c_{1},c_{2},\dots,c_{\alpha})\in\mathbb{Z}_{p}^{\alpha},d=(d_{1},d_{2},\dots,d_{\beta})\in\mathbb{Z}_{p}[u,v]^{\beta}\\}$
is an additive group. We define each subgroup of
$\mathbb{Z}_{p}^{\alpha}\times\mathbb{Z}_{p}[u,v]^{\beta}$ as a
$\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive code of length $(\alpha,\beta)$.
Let $\mathcal{C}$ be a $\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive code of
length $(\alpha,\beta)$. Then $\mathcal{C}$ being subgroup of
$\mathbb{Z}_{p}^{\alpha}\times\mathbb{Z}_{p}[u,v]^{\beta}$, it is isomorphic
to a commutative structure like
$\mathbb{Z}_{p}^{k_{0}}\times\mathbb{Z}_{p}^{3k_{1}}\times\mathbb{Z}_{p}^{2k_{2}}\times\mathbb{Z}_{p}^{k_{3}}$
and contains $p^{k_{0}+3k_{1}+2k_{2}+k_{3}}$ codewords. Therefore,
$\mathcal{C}$ is consider as $\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive code
of type $(\alpha,\beta;k_{0},k_{1},k_{2},k_{3})$. Now, we define a ring
homomorphism $\rho:\mathbb{Z}_{p}[u,v]\rightarrow\mathbb{Z}_{p}$ by
$\rho(a+ub+vc)=a$ for $a,b,c\in\mathbb{Z}_{p}$ and multiplication
$\ast:\mathbb{Z}_{p}[u,v]\times\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]\rightarrow\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$
by $z\ast(c,d)=(\rho(z)c,zd)$, for $z\in\mathbb{Z}_{p}[u,v]$. The extension of
above multiplication over
$\mathbb{Z}_{p}^{\alpha}\times\mathbb{Z}_{p}[u,v]^{\beta}$ is as follows: for
any $z\in\mathbb{Z}_{p}[u,v]$, we have
$z\ast(c,d)=(\rho(z)c_{1},\rho(z)c_{2},\dots,\rho(z)c_{\alpha},zd_{1},zd_{2},\dots,zd_{\beta})$
where
$c=(c_{1},c_{2},\dots,c_{\alpha})\in\mathbb{Z}_{p}^{\alpha},d=(d_{1},d_{2},\dots,d_{\beta})\in\mathbb{Z}_{p}[u,v]^{\beta}$.
Under this extended multiplication, one can verify easily the following lemma.

######  Lemma 1.

The additive group $\mathbb{Z}_{p}^{\alpha}\times\mathbb{Z}_{p}[u,v]^{\beta}$
is a $\mathbb{Z}_{p}[u,v]$-module under the above multiplication.