A class of constacyclic codes over $\frac{{\mathbb{F}}_{p^m}[u,v]}{⟨u^k,v^2,uv-vu⟩}$

1. Introduction

Constacyclic codes play a crucial role in the theory of error-correcting codes, serving as a natural extension of cyclic codes, which are among the most extensively studied code types. Let $R$ be a finite commutative ring, and let $\lambda$ be a unit in $R$. A $\lambda$-constacyclic code of length $n$ over the ring $R$ can be represented as an ideal in the quotient ring $\frac{R[x]}{⟨x^n–\lambda ⟩}$. This representation underscores the significance of constacyclic codes over rings as an important class of linear codes due to their rich algebraic structure.

In recent years, there has been a growing research interest in the study of constacyclic codes. However, the general classification of constacyclic codes remains a challenging task. To date, only a limited number of cases with specific lengths and defined over certain rings have been classified. Let $R$ be a commutative ring. Dinh and López-Permouth [18] studied the structures of cyclic and negacyclic codes of length $n$ when $n$ is not divisible by the characteristic of the residue field $\overline{R}$. Moreover, Cao [4] investigated the algebraic structure of $(1+w\gamma )$-constacyclic codes of arbitrary length over a finite commutative chain ring $R$, where $w\in R^{\times }$.

An important example of a finite ring is $R=\frac{{\mathbb{F}}_{p^m}[u]}{⟨u^k⟩}={\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}+\cdots +u^{k-1}{\mathbb{F}}_{p^m}$. The structure of constacyclic codes over $\frac{{\mathbb{F}}_{p^m}[u]}{⟨u^k⟩}$ has been extensively studied in the literature. For $k=2$, significant research has been devoted to cyclic and constacyclic codes of length $n$ over the ring ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$ for various prime numbers $p$ and positive integers $n$ (cf. [10,11,15,8,14,2]). For $k\ge 3$, Kai et al. [20] investigated $(1+\lambda u)$-constacyclic codes of arbitrary length over $\frac{{\mathbb{F}}_p[u]}{⟨u^k⟩}$, where $\lambda \in {\mathbb{F}}_p^{\times }$. Subsequently, Cao et al. [6,5] studied $(\delta +\alpha u^2)$-constacyclic codes of arbitrary length over $\frac{{\mathbb{F}}_{p^m}[u]}{⟨u^k⟩}$, focusing on the case where $k$ is an even integer and $\delta ,\alpha \in {\mathbb{F}}_{p^m}^{\times }$. Sobhani [23] determined the structure of $(\delta +\alpha u^2)$-constacyclic codes of length $p^s$ over $\frac{{\mathbb{F}}_{p^m}[u]}{⟨u^3⟩}$, where $\delta ,\alpha \in {\mathbb{F}}_{p^m}^{\times }$. Moreover, Mahmoodi and Sobhani [21] provided a complete classification of $(1+\alpha u^2)$-constacyclic codes of length $p^s$ over $\frac{{\mathbb{F}}_{p^m}[u]}{⟨u^4⟩}$, where $\alpha \in {\mathbb{F}}_{p^m}^{\times }$.

Another important ring is $R=\frac{{\mathbb{F}}_{p^m}[u,v]}{⟨u^2,v^2,uv–vu⟩},$ which is local but not a chain ring, with units of the form $${\lambda }_1=\alpha ,{\lambda }_2=\alpha +{\delta }_{1}uv,{\lambda }_3=\alpha +\gamma v+\delta uv,{\lambda }_4=\alpha +\beta u+\delta uv,{\lambda }_5=\alpha +\beta u+\gamma v+\delta uv,$$ where $\alpha ,\beta ,\gamma ,{\delta }_1\in {\mathbb{F}}_{p^m}^{\ast }$ and $\delta \in {\mathbb{F}}_{p^m}$.

Dinh et al. [16] determined the algebraic structures of all constacyclic codes of length $p^s$ over $R$, except for the ${\lambda }_2$-constacyclic codes, later studied in [12]. For $p=2$, Dinh et al. [17] classified all self-dual $\lambda$-constacyclic codes of length $2^s$ over $R$ corresponding to the units ${\lambda }_3,{\lambda }_4$, and ${\lambda }_5$. In our recent paper [1], we investigated the structure and duals of $\lambda$-constacyclic codes of length $2p^s$ over $R$ for the units ${\lambda }_3,{\lambda }_4$, and ${\lambda }_5$. Motivated by this, we aim to extend our previous results by classifying $\lambda$-constacyclic codes of arbitrary length over $$R_{u,v}=\frac{{\mathbb{F}}_{p^m}[u,v]}{⟨u^k,v^2,uv–vu⟩},$$ with $k\ge 2$, and $\lambda$ given by $$\lambda =\sum _{i=0}^{k-1}{\lambda }_i{u}^i+v\sum _{i=0}^{k-1}{\lambda }_i^{\prime }{u}^i,$$ where ${\lambda }_i,{\lambda }_i^{\prime }\in {\mathbb{F}}_{p^m}$ and ${\lambda }_0,{\lambda }_1\ne 0$.

The paper is structured as follows. Section 2 introduces essential preliminaries on finite rings and fundamental concepts in constacyclic codes. Section 3 establishes a decomposition of each $\lambda$-constacyclic code of arbitrary length over $R_{u,v}$ as a direct sum of ideals over certain rings. These rings are further examined in Section 4, where their ideals are classified into four types. Section 5 investigates the structure of dual $\lambda$-constacyclic codes of arbitrary length over over $R_{u,v}$. Finally, Section 6 determines their Hamming distance when the code length is $p^s$.

2. Preliminaries

In this paper, all rings considered are associative and commutative. An ideal $I$ of a ring $R$ is called principal if it is generated by a single element. A ring $R$ is a principal ideal ring if each of its ideals is principal. A ring $R$ is called local if it has a unique maximal ideal. A chain ring is a ring whose ideals are totally ordered by inclusion. A ring $R$ is said to be Frobenius if $\frac{R}{J(R)}$ is isomorphic (as an $R$-module) to $\mathrm{soc}(R)$, where $J(R)$ denotes the Jacobson radical of $R$ and $\mathrm{soc}(R)$ its socle.

Let $R$ be a finite local commutative ring with maximal ideal $M$ and residue field $\overline{R}=\frac{R}{M}$. For any polynomial $f(x)\in R[x]$, we denote by $\overline{f}(x)$ the polynomial in $\overline{R}[x]$ obtained by reducing each coefficient of $f(x)$ modulo $M$. Two polynomials $f_1(x),f_2(x)\in R[x]$ are said to be coprime if there exist polynomials $g_1(x),g_2(x)\in R[x]$ such that $$f_1(x)g_1(x)+f_2(x)g_2(x)=1.$$

A polynomial $f(x)\in R[x]$ is said to be regular if it is not a zero divisor; equivalently, $\overline{f}(x)\ne 0$ in $\overline{R}[x]$ (see [22]).

The following result is a well-known property of finite commutative chain rings.

A linear code $C$ of length $n$ over a ring $R$ is an $R$-submodule of $R^n$. Let $\lambda$ be a unit in $R$. A linear code $C$ of length $n$ over $R$ is called a $\lambda$-constacyclic code if it satisfies the following condition: $$(c_0,c_1,\dots ,c_{n-1})\in C⟹(\lambda c_{n-1},c_0,\dots ,c_{n-2})\in C.$$

Each codeword $c=(c_0,c_1,\dots ,c_{n-1})\in C$ can be represented as a polynomial $c(x)=c_0+c_{1}x+\cdots +c_{n-1}{x}^{n-1}$. This identification leads us to the following proposition.

For $n$-tuples $a=(a_0,a_1,\dots ,a_{n-1})$ and $b=(b_0,b_1,\dots ,b_{n-1})$ in $R^n$, their inner product is defined as $$a\cdot b=a_0{b}_0+a_1{b}_1+\cdots +a_{n-1}{b}_{n-1}.$$ Two $n$-tuples $a$ and $b$ are considered orthogonal if $a\cdot b=0$. The dual code $C^{\perp }$ of a linear code $C$ of length $n$ over a ring $R$ is defined as $$C^{\perp }=\{a\in R^n\mid a\cdot b=0\text{ for all }b\in C\}.$$

The following proposition is well known.

In general, the dual of a $\lambda$-constacyclic code satisfies the following proposition.

The annihilator of an ideal $I$ in a ring $R$, denoted as $\mathcal{A}(I)$, is defined as $$\mathcal{A}(I)=\{a\in R\mid ab=0\text{ for all }b\in I\}.$$

For any polynomial $f(x)=a_0+a_{1}x+\dots +a_l{x}^l\in R[x]$ with $a_l\ne 0$, the reciprocal polynomial $f^{\ast }(x)$ is defined as $$f^{\ast }(x)=x^{l}f(x^{-1})=a_l+a_{l-1}x+\dots +a_0{x}^l.$$

The following proposition establishes a relationship between the dual and the annihilator of a $\lambda$-constacyclic code.

Throughout this paper, we consider the rings: $$R_{u,v}=\frac{{\mathbb{F}}_{p^m}[u,v]}{⟨u^k,v^2,uv–vu⟩},T_u=\frac{{\mathbb{F}}_{p^m}[u]}{⟨u^k⟩},$$ where $p$ is a prime number, and $m,k$ are positive integers. The Jacobson radical and the socle of $R_{u,v}$ are given by: $$J(R_{u,v})=⟨u,v⟩,\mathrm{soc}(R_{u,v})=⟨uv⟩.$$

Moreover, the quotient ring $\frac{R_{u,v}}{J(R_{u,v})}$ is isomorphic to $\mathrm{soc}(R_{u,v})$ as an $R_{u,v}$-module via the natural isomorphism: $$r+J(R_{u,v})⟼ruv,$$ for any $r\in R_{u,v}$. Thus, $R_{u,v}$ is a Frobenius ring.

Each unit in $R_{u,v}$ has the form: $$\lambda =\sum _{i=0}^{k-1}{\lambda }_i{u}^i+v\sum _{i=0}^{k-1}{\lambda }_i^{\prime }{u}^i,$$ which can be rewritten as: $$\lambda ={\lambda }_0+\gamma u+\mu v,$$ where ${\lambda }_i,{\lambda }_i^{\prime }\in {\mathbb{F}}_{p^m}$ and ${\lambda }_0,{\lambda }_1\ne 0$, with: $$\gamma ={\lambda }_1+{\lambda }_{2}u+\cdots +{\lambda }_{k-1}{u}^{k-2},\mu ={\lambda }_0^{\prime }+{\lambda }_1^{\prime }u+\cdots +{\lambda }_{k-1}^{\prime }{u}^{k-1}.$$

Throughout this article, we assume ${\lambda }_1\ne 0$, which ensures that $\gamma$ is a unit in $T_u$.

Let $s$ be a positive integer. Using the division algorithm, we write $s=q_{s}m+r_s$ with $0\le r_s<m$. Since ${\lambda }_0^{p^m}={\lambda }_0$, we define: $$\theta ={\lambda }_0^{p^{m-r_s}}={\lambda }_0^{p^{(q_s+1)m–s}}.$$ It follows that ${\theta }^{p^s}={\lambda }_0$.

Our objective is to study $\lambda$-constacyclic codes of length $\eta p^s$ over $R_{u,v}$, where $\eta$ is an integer coprime to $p$. By Proposition 2.2, these codes correspond to the ideals of the ring: $${\mathcal{R}}_{\lambda }:=\frac{R_{u,v}[x]}{⟨x^{\eta p^s}–\lambda ⟩}.$$

The following ring homomorphisms will be used in subsequent sections: $$\mathrm{\Phi }:R_{u,v}[x]\to T_u[x],f(x)\mapsto f(x)\mathrm{mod}v,$$ $$\overline{\cdot }:R_{u,v}[x]\to {\mathbb{F}}_{p^m}[x],f(x)\mapsto f(x)\mathrm{mod}(u,v).$$

3. Direct sum decomposition of $\lambda$-constacyclic codes over $R_{u,v}$

Since $\eta$ is coprime with $p$, the polynomial $x^{\eta }–\theta$ has a unique factorization in ${\mathbb{F}}_{p^m}[x]$: $$x^{\eta }–\theta =z_1(x)z_2(x)\dots z_r(x),$$ where the factors are monic, irreducible, and pairwise coprime. Raising both sides to $p^s$ gives: $$x^{\eta p^s}–{\lambda }_0=z_1(x{)}^{p^s}{z}_2(x{)}^{p^s}\dots z_r(x{)}^{p^s}.$$ Since $R_{u,v}$ is a finite local ring and $x^{\eta p^s}–\lambda$ is a regular element in $R_{u,v}[x]$, Hensel’s Lemma [22] ensures the existence of a unique factorization $$x^{\eta p^s}–\lambda ={\omega }_1(x){\omega }_2(x)\dots {\omega }_r(x),$$ where ${\omega }_1(x),\dots ,{\omega }_r(x)$ are monic and pairwise coprime polynomials in $R_{u,v}[x]$. Moreover, for each $i=1,\dots ,r$, $$\overline{{\omega }_i}(x)=z_i(x{)}^{p^s}.$$

Defining $\widehat{{\omega }_i}(x)=\frac{x^{\eta p^s}–\lambda }{{\omega }_i(x)}$, it follows that $\widehat{{\omega }_i}(x)$ and ${\omega }_i(x)$ are coprime. Hence, there exist polynomials $l_i(x),l_i^{\mathrm{\prime }}(x)\in R_{u,v}[x]$ such that: $$l_i(x)\widehat{{\omega }_i}(x)+l_i^{\mathrm{\prime }}(x){\omega }_i(x)=1.$$

Setting $e_i(x)=l_i(x)\widehat{{\omega }_i}(x),$ it follows from [7, Theorem 3.2] that the following properties hold:

• The set $\{e_1(x),e_2(x),\dots ,e_r(x)\}$ forms a complete system of primitive pairwise orthogonal idempotents, meaning that $$\begin{array}{}\text{(1)}& \begin{array}{r}\sum _{i=1}^r{e}_i(x)=1,e_i(x)e_j(x)=0\text{for }i\ne j,e_i(x{)}^2=e_i(x).\end{array}\end{array}$$

• The ring ${\mathcal{R}}_{\lambda }$ decomposes as $${\mathcal{R}}_{\lambda }=\underset{i=1}{\overset{r}{⨁}}{e}_i(x){\mathcal{R}}_{\lambda }.$$

• For each $i$, the mapping $${\pi }_i:\frac{R_{u,v}[x]}{⟨{\omega }_i(x)⟩}\to e_i(x){\mathcal{R}}_{\lambda },{\pi }_i(c(x))=e_i(x)c(x),$$ is a ring isomorphism.

As a consequence, we obtain the following theorem.

Thus, to classify all $\lambda$-constacyclic codes over $R_{u,v}$, we need only to study the ideals of $\frac{R_{u,v}[x]}{⟨{\omega }_i(x)⟩}$. To this end, for each $1\le i\le t$, we define:

• ${\psi }_i(x)=\mathrm{\Phi }({\omega }_i(x))$,

• ${\mathcal{R}}_i=\frac{R_{u,v}[x]}{⟨{\omega }_i(x)⟩}$,

• ${\mathcal{S}}_i=\frac{T_u[x]}{⟨{\psi }_i(x)⟩}$,

• ${\hat{z}}_i(x)=\frac{x^{\eta }-\theta }{z_i(x)}.$

Since ${\psi }_i(x)=\mathrm{\Phi }({\omega }_i(x))$, the universal property of quotient rings guarantees that $\mathrm{\Phi }$ induces a unique homomorphism $${\mathrm{\Phi }}_i:{\mathcal{R}}_i\to {\mathcal{S}}_i,$$ satisfying $${\mathrm{\Phi }}_i(f(x)+⟨{\omega }_i(x)⟩)=\mathrm{\Phi }(f(x))+⟨{\psi }_i(x)⟩,\mathrm{\forall }f(x)\in R_{u,v}[x].$$

The following pairs of propositions will be used in the subsequent sections.

4. The ring ${\mathcal{R}}_i$ and its ideals

In this section, we classify the ideals of the ring ${\mathcal{R}}_i$ and determine their cardinalities. We start with the following proposition.

The following theorem provides a complete classification of the distinct ideals of the ring ${\mathcal{R}}_i$.

Before computing the value of $\mathrm{{\rm Y}}$, we first establish the following lemma.

We now count the number of codewords in each ideal of ${\mathcal{R}}_i$. For this, we introduce two notions: the residue and the torsion of $I$.

$$\mathrm{Res}(I)={\mathrm{\Phi }}_i(I),\mathrm{Tor}(I)=\{c(x)\in {\mathcal{S}}_i\mid vc(x)\in I\}.$$

Clearly, $\mathrm{Res}(I)$ and $\mathrm{Tor}(I)$ are ideals of ${\mathcal{S}}_i$. By Proposition 3.3, they can be expressed as $⟨z_i(x{)}^l⟩$, where $0\le l\le k{p}^s$. Now, consider the ring homomorphism $$\begin{array}{rrcl}T:& I& ⟶& {\mathrm{\Phi }}_i(I),\\ & c(x)& ⟼& {\mathrm{\Phi }}_i(c(x)).\end{array}$$

Since $\mathrm{Im}T=\mathrm{Res}(I)$ and $\ker T\cong v\mathrm{Tor}(I)$, the first isomorphism theorem yields $$\begin{array}{}\text{(10)}& |I|=|\mathrm{Res}(I)|\cdot |\mathrm{Tor}(I)|.\end{array}$$

The following lemma, whose proof is straightforward, provides explicit expressions for $\mathrm{Res}(I)$ and $\mathrm{Tor}(I)$ for any ideal $I$ in ${\mathcal{R}}_i$.

By determining $\mathrm{Res}(I)$ and $\mathrm{Tor}(I)$ in each case, we can compute the cardinalities of all ideals in ${\mathcal{R}}_i$ using Eq. (10) and Proposition 3.3.

5. Dual codes of $\lambda$-constacyclic codes over $R_{v,v}$

In this section, we focus on the dual codes of $\lambda$-constacyclic codes of length $\eta p^s$ over the ring $R_{v,v}$. By Theorem 3.1, $$C=\underset{i=1}{\overset{r}{⨁}}{e}_i(x)I_i,$$ where $I_i$ is an ideal of ${\mathcal{R}}_i$ for $1\le i\le r$. By Eq. (1), we have $$\begin{array}{rcl}d(x)=\sum _{i=1}^r{e}_i(x)\cdot d(x)\in \mathcal{A}(C)& \iff & \mathrm{\forall }c(x)\in C,(\sum _{i=1}^r{e}_i(x)\cdot d(x))\cdot (\sum _{i=1}^r{e}_i(x)\cdot c(x))=0\\ & \iff & \mathrm{\forall }c(x)\in C,(\sum _{i=1}^r{e}_i(x)\cdot d(x)\cdot c(x))=0\\ & \iff & \mathrm{\forall }c(x)\in C,\mathrm{\forall }i\in \{1,\dots ,r\},e_i(x)\cdot d(x)\cdot c(x)=0\\ & \iff & \mathrm{\forall }c(x)\in C,\mathrm{\forall }i\in \{1,\dots ,r\},d(x)\cdot c(x)=0\text{ in }{\mathcal{R}}_i\\ & \iff & \mathrm{\forall }i\in \{1,\dots ,r\},d(x)\in \mathcal{A}(I_i).\end{array}$$

Therefore, $$\begin{array}{}\text{(11)}& \begin{array}{r}\mathcal{A}(C)=\underset{i=1}{\overset{r}{⨁}}{e}_i(x)\mathcal{A}(I_i).\end{array}\end{array}$$

Thus, by Proposition 2.5, $$C^{\perp }=\mathcal{A}(C{)}^{\ast }=\underset{i=1}{\overset{r}{⨁}}{e}_i^{\ast }(x)\mathcal{A}(I_i{)}^{\ast }.$$

We aim to determine $\mathcal{A}(I{)}^{\ast }$, where $I$ is an ideal of ${\mathcal{R}}_i$. To this end, we first establish the following three lemmas.

We now proceed to determine the annihilator of each type of ideal. For type 1 ideals, this is straightforward: if $I=⟨0⟩$, then $\mathcal{A}(I)=⟨1⟩$; if $I=⟨1⟩$, then $\mathcal{A}(I)=⟨0⟩$. For other types, we first identify two integers $a$ and $b$ such that $\mathrm{Res}(I)=⟨z_i(x{)}^a⟩$ and $\mathrm{Tor}(I)=⟨z_i(x{)}^b⟩$, as given in Lemma 4.3. Once these values are determined, we find a polynomial $f(x)$ satisfying $z_i(x{)}^{k{p}^s–b}+vf(x)\in \mathcal{A}(I)$. Finally, applying Lemma 5.3, we obtain $\mathcal{A}(I)$.

We now determine $\mathcal{A}(I{)}^{\ast }$ for any ideal in ${\mathcal{R}}_i$. If $I$ is of type 1, then $\mathcal{A}(I{)}^{\ast }=⟨0⟩$ when $I=⟨0⟩$, and $\mathcal{A}(I{)}^{\ast }=⟨1⟩$ when $I=⟨1⟩$. For other types, we introduce the following notation: Let $$\vartheta =max\{0\le l\le k{p}^s\mid x^{(k{p}^s-\epsilon )\mathrm{deg}{z}_i}h(x^{-1})\in ⟨z_i(x{)}^l⟩\}.$$

Then, we can write $$x^{(k{p}^s-\epsilon )\mathrm{deg}{z}_i}h(x^{-1})=z_i(x{)}^{\vartheta }{h}^{\mathrm{\prime }}(x),$$ where $h^{\mathrm{\prime }}(x)$ is either $0$ or a unit in ${\mathcal{S}}_i.$

6. Hamming distance of $\lambda$-constacyclic codes of length $p^s$ over $R_{u,v}$

The Hamming weight of a codeword $c$, denoted by $w{t}_H(c)$, represents the number of nonzero components in the vector $c$. The Hamming distance between two vectors $c$ and $c^{\prime }$, denoted by $d_H(c,c^{\prime })$, is defined as $w{t}_H(c–c^{\prime })$.

For a linear code $C$, the Hamming distance $d_H(C)$ is given by the minimum weight among all nonzero codewords in $C$.

In this section, we compute the Hamming distance of $\lambda$-constacyclic codes of length $p^s$ over the ring $R_{u,v}$. We begin by establishing the structure of these codes, which can be derived from Theorem 4.2.

The following lemma establishes a relationship between the Hamming distance of a $\lambda$-constacyclic code and its torsion $\mathrm{Tor}(C)$.

The previous lemma reduces the computation of the Hamming distance of a $\lambda$-constacyclic code $C$ of length $p^s$ over $R_{u,v}$ to that of its torsion $\mathrm{Tor}(C)$, which corresponds to a $({\lambda }_0+\gamma v)$-constacyclic code of the same length over $T_u$.

Let $C_{\tau }=⟨(x-\theta {)}^{\tau }⟩$ be a nonzero $({\lambda }_0+\gamma v)$-constacyclic code of length $p^s$ over $T_u$, where $0\le \tau \le p^{s}k$. We distinguish two cases:

• If $0\le \tau \le p^s(k-1)$, then the chain of inclusions $$C_{p^s(k-1)}\subset \cdots \subset C_{\tau }\subset \cdots \subset C_0=⟨1⟩$$ implies $$d_H(C_{p^s(k-1)})\ge \cdots \ge d_H(C_{\tau })\ge 1.$$

  By Proposition 3.3, $C_{p^s(k-1)}=⟨u^{k-1}⟩$. Then, $d_H(C_{p^s(k-1)})=1$, which implies that $d_H(C_{\tau })=1$ for all $0\le \tau \le p^s(k-1)$.

• If $p^s(k-1)+1\le \tau \le p^{s}k-1$, writing $\tau =p^s(k-1)+\varsigma$ with $1\le \varsigma \le p^s-1$, we obtain $$C_{\tau }=⟨v^{k-1}(x-\theta {)}^{\varsigma }⟩.$$ Thus, each $C_{\tau }$ corresponds to the ${\lambda }_0$-constacyclic code $⟨(x-\theta {)}^{\varsigma }⟩$ over ${\mathbb{F}}_{p^m}$, multiplied by $v^{k-1}$, leading to $$d_H(C_{\tau })=d_H(⟨(x-\theta {)}^{\varsigma }⟩).$$

The Hamming distance of constacyclic codes of length $p^s$ over ${\mathbb{F}}_{p^m}$ is determined by the following proposition.

Thus, we establish the following theorem.