Let \( p \) be a prime number, and let \( k \) and \( m \) be positive integers with \( k \geq 2 \). This paper studies the algebraic structure of \(\lambda\)-constacyclic codes of arbitrary length over the finite commutative ring \( R = \frac{\mathbb{F}_{p^m}[u, v]}{ \langle u^k, v^2, uv – vu \rangle } \), where \(\lambda\) is a unit in \( R \) given by \( \lambda = \sum\limits_{i=0}^{k-1} \lambda_i u^i + v\sum\limits_{i=0}^{k-1} \lambda_i’ u^i \), with \(\lambda_i, \lambda_i’ \in \mathbb{F}_{p^m}\) and \(\lambda_0, \lambda_1 \neq 0\). We provide a complete classification of these constacyclic codes, determine their dual structures, and compute their Hamming distances when the code length is \( p^s \).
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 \(\frac{R[x]}{\langle x^n – \lambda \rangle}\). 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 \(\bar{R}\). 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 \(R = \frac{\mathbb{F}_{p^{m}}[u]}{\langle u^{k}\rangle} = \mathbb{F}_{p^{m}} + u\mathbb{F}_{p^{m}} + \dots + u^{k-1}\mathbb{F}_{p^{m}}\). The structure of constacyclic codes over \(\frac{\mathbb{F}_{p^{m}}[u]}{\langle u^{k}\rangle}\) 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 \(k \geq 3\), Kai et al. [20] investigated \((1+\lambda u)\)-constacyclic codes of arbitrary length over \(\frac{\mathbb{F}_p[u]}{\langle u^k \rangle}\), where \(\lambda \in \mathbb{F}_p^{\times}\). Subsequently, Cao et al. [6,5] studied \((\delta+ \alpha u^2)\)-constacyclic codes of arbitrary length over \(\frac{\mathbb{F}_{p^{m}}[u]}{\langle u^{k}\rangle}\), 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 \(\frac{\mathbb{F}_{p^m}[u]}{\langle u^3 \rangle}\), 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 \(\frac{\mathbb{F}_{p^m}[u]}{\langle u^4 \rangle}\), where \(\alpha \in \mathbb{F}_{p^m}^{\times}\).
Another important ring is \(R = \frac{\mathbb{F}_{p^m}[u, v]}{\left\langle u^2, v^2, u v – v u \right\rangle},\) which is local but not a chain ring, with units of the form \[\lambda_1= \alpha, \quad \lambda_2 = \alpha + \delta_1 u v, \quad \lambda_3 = \alpha + \gamma v + \delta u v, \quad \lambda_4 = \alpha + \beta u + \delta u v, \quad \lambda_5 = \alpha + \beta u + \gamma v + \delta u v,\] where \(\alpha, \beta, \gamma, \delta_1 \in \mathbb{F}_{p^m}^*\) 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 \[R_{u,v}= \frac{\mathbb{F}_{p^m}[u, v]}{\left\langle u^k, v^2, u v – v u \right\rangle},\] with \(k \geq 2\), and \(\lambda\) given by \[\lambda = \sum\limits_{i=0}^{k-1} \lambda_i u^i + v\sum\limits_{i=0}^{k-1} \lambda_i' u^i,\] where \(\lambda_i, \lambda_i' \in \mathbb{F}_{p^m}\) and \(\lambda_0, \lambda_1 \neq 0\).
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 \(p^s\).
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 \(\operatorname{soc}(R)\), where \(J(R)\) denotes the Jacobson radical of \(R\) and \(\operatorname{soc}(R)\) 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, \(\overline{f}(x) \neq 0\) in \(\overline{R}[x]\) (see [22]).
The following result is a well-known property of finite commutative chain rings.
Proposition 2.1. [18] Let \(R\) be a finite commutative ring. The following conditions are equivalent:
\(R\) is a local ring whose maximal ideal \(M\) is principal (i.e., \(M = \langle \Gamma \rangle\) for some element \(\Gamma \in R).\)
\(R\) is a local principal ideal ring.
\(R\) is a chain ring whose ideals are \(\langle \Gamma^i \rangle\) for \(0 \le i \le t\), where \(t\) is the nilpotency index of \(\Gamma\).
Moreover, for each \(0 \le i \le t\), the cardinality of the ideal \(\langle \Gamma^i \rangle\) is given by \[\bigl|\langle \Gamma^i \rangle\bigr| \;=\; \left|\frac{R}{M}\right|^{\,t-i}\,.\]
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: \[(c_0, c_1, \ldots, c_{n-1}) \in C \implies (\lambda c_{n-1}, c_0, \ldots, c_{n-2}) \in C.\]
Each codeword \(c = (c_0, c_1, \ldots, c_{n-1}) \in C\) 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.
Proposition 2.2. [19] A code \(C\) of length \(n\) over \(R\) is a \(\lambda\)-constacyclic code if and only if \(C\) is an ideal of \(\frac{R[x]}{\langle x^n – \lambda \rangle}\).
For \(n\)-tuples \(a = (a_0, a_1, \ldots, a_{n-1})\) and \(b = (b_0, b_1, \ldots, b_{n-1})\) 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 \[C^{\perp} = \left\{ a \in R^n \mid a \cdot b = 0 \text{ for all } b \in C \right\}.\]
The following proposition is well known.
Proposition 2.3. [24] Consider a linear code \(C\) of length \(n\) over a finite Frobenius ring \(R\). Then, \[|C| \cdot |C^\perp| = |R|^n.\]
In general, the dual of a \(\lambda\)-constacyclic code satisfies the following proposition.
Proposition 2.4. [10] The dual of a \(\lambda\)-constacyclic code is a \(\lambda^{-1}\)-constacyclic code.
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 \(f(x) = a_0 + a_1 x + \ldots + a_l x^l \in R[x]\) with \(a_l \neq 0\), the reciprocal polynomial \(f^*(x)\) is defined as \[f^*(x) = x^l f(x^{-1}) = a_l + a_{l-1} x + \ldots + a_0 x^l.\]
The following proposition establishes a relationship between the dual and the annihilator of a \(\lambda\)-constacyclic code.
Proposition 2.5. [13] Let \(C\) be a \(\lambda\)-constacyclic code of length \(n\) over \(R\), i.e., an ideal of \(\frac{R[x]}{\langle x^n – \lambda \rangle}\). Then, \[C^{\perp} = \mathcal{A}(C)^* := \{ f^*(x) \mid f(x) \in \mathcal{A}(C) \}.\]
Throughout this paper, we consider the rings: \[R_{u,v} = \frac{\mathbb{F}_{p^m}[u,v]}{\langle u^k, v^2, uv – vu \rangle}, \quad T_u = \frac{\mathbb{F}_{p^m}[u]}{\langle u^k \rangle},\] 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: \[J(R_{u,v}) = \langle u, v \rangle, \quad \operatorname{soc}(R_{u,v}) = \langle uv \rangle.\]
Moreover, the quotient ring \(\frac{R_{u,v}}{ J(R_{u,v})}\) is isomorphic to \(\operatorname{soc}(R_{u,v})\) as an \(R_{u,v}\)-module via the natural isomorphism: \[r + J(R_{u,v}) \longmapsto r uv,\] for any \(r \in R_{u,v}\). Thus, \(R_{u,v}\) is a Frobenius ring.
Each unit in \(R_{u,v}\) has the form: \[\lambda = \sum\limits_{i=0}^{k-1} \lambda_i u^i + v\sum\limits_{i=0}^{k-1} \lambda_i' u^i,\] which can be rewritten as: \[\lambda = \lambda_0 + \gamma u+ \mu v,\] where \(\lambda_i, \lambda_i' \in \mathbb{F}_{p^m}\) and \(\lambda_0, \lambda_1 \neq 0\), with: \[\gamma = \lambda_1 + \lambda_2 u + \cdots + \lambda_{k-1} u^{k-2}, \quad \mu = \lambda'_0 + \lambda'_1 u + \cdots + \lambda'_{k-1} u^{k-1}.\]
Throughout this article, we assume \(\lambda_1 \neq 0\), 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 \(0 \leq r_s < m\). 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: \[\mathcal{R}_\lambda := \frac{R_{u,v}[x]}{\langle x^{\eta p^s} – \lambda \rangle}.\]
The following ring homomorphisms will be used in subsequent sections: \[\Phi: R_{u,v}[x] \to T_u[x], \quad f(x) \mapsto f(x) \mod v,\] \[\overline{\cdot} : R_{u,v}[x] \to \mathbb{F}_{p^m}[x], \quad f(x) \mapsto f(x) \mod (u,v).\]
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 \(l_i(x), l^\prime_i(x) \in R_{u,v}[x]\) such that: \[l_i(x) \widehat{\omega_i}(x) + l^\prime_i(x) \omega_i(x) = 1.\]
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 \[\begin{aligned} \label{ceceeeeee} \sum\limits_{i=1}^r e_i(x) = 1, \quad e_i(x) e_j(x) = 0 \quad \text{for } i \neq j, \quad e_i(x)^2 = e_i(x). \end{aligned} \tag{1}\]
The ring \(\mathcal{R}_\lambda\) decomposes as \[\mathcal{R}_\lambda = \bigoplus_{i=1}^r e_i(x) \mathcal{R}_\lambda.\]
For each \(i\), the mapping \[\pi_i: \frac{R_{u,v}[x]}{\langle \omega_i(x) \rangle} \to e_i(x) \mathcal{R}_\lambda, \quad \pi_i(c(x)) = e_i(x) c(x),\] is a ring isomorphism.
As a consequence, we obtain the following theorem.
Theorem 3.1. A subset \(C\) of \(\mathcal{R}_\lambda\) is an ideal if and only if \[C = \bigoplus_{i=1}^r e_i(x) I_i,\] where \(I_i\) is an ideal of \(\frac{R_{u,v}[x]}{\langle \omega_i(x) \rangle}\).
Thus, to classify all \(\lambda\)-constacyclic codes over \(R_{u,v}\), we need only to study the ideals of \(\frac{R_{u,v}[x]}{\langle \omega_i(x) \rangle}\). To this end, for each \(1 \leq i \leq t\), we define:
\(\psi_i(x) = \Phi(\omega_i(x))\),
\(\mathcal{R}_i = \frac{R_{u,v}[x]}{\langle \omega_i(x) \rangle}\),
\(\mathcal{S}_i = \frac{T_u[x]}{\langle \psi_i(x) \rangle}\),
\(\hat{z}_i(x) = \frac{x^{\eta }-\theta}{z_i(x)}.\)
Since \(\psi_i(x) = \Phi(\omega_i(x))\), the universal property of quotient rings guarantees that \(\Phi\) induces a unique homomorphism \[\Phi_i : \mathcal{R}_i \to \mathcal{S}_i,\] satisfying \[\Phi_i(f(x) + \langle \omega_i(x) \rangle) = \Phi(f(x)) + \langle \psi_i(x) \rangle, \quad \forall f(x) \in R_{u,v}[x].\]
The following pairs of propositions will be used in the subsequent sections.
Proposition 3.2. The polynomial \(\hat{z}_i(x)\) is a unit in \(\mathcal{S}_i\) and \(\mathcal{R}_i\).
Proof. It suffices to prove that \(\hat{z}_i(x)\) is coprime to \(\omega_i(x)\) in \(R_{u,v}[x]\) and to \(\psi_i(x)\) in \(T_u[x]\).
Since \(\hat{z}_i(x)\) and \(z_i(x)^{p^s}\) are coprime in \(\mathbb{F}_{p^m}[x]\), there exist polynomials \(f_i(x), g_i(x) \in \mathbb{F}_{p^m}[x]\) such that \[f_i(x) z_i(x)^{p^s} + g_i(x) \hat{z}_i(x) = 1.\]
Given that \(\overline{\omega_i}(x) = z_i(x)^{p^s}\), we can write in \(R_{u,v}[x]\): \[\omega_i(x) = z_i(x)^{p^s} + u h_i(x) + v h^\prime_i(x),\] for some \(h_i(x), h^\prime_i(x) \in R_{u,v}[x]\). Substituting this into the previous relation, we obtain in \(R_{u,v}[x]\): \[\begin{aligned} \label{eq:coprime_relation} f_i(x) \omega_i(x) + g_i(x) \hat{z}_i(x) = 1 + u h_i(x) f_i(x) + v h^\prime_i(x) f_i(x) . \end{aligned} \tag{2}\]
Since \(u\) and \(v\) are nilpotent in \(R_{u,v}[x]\), the element \(1 + u h_i(x) f_i(x) + v h^\prime_i(x) f_i(x)\) is a unit in \(R_{u,v}[x]\). Consequently, \(\omega_i(x)\) and \(\hat{z}_i(x)\) are coprime in \(R_{u,v}[x]\).
Now, applying \(\Phi\) to Eq. (2), we get: \[\Phi(f_i(x)) \Phi(\omega_i(x)) + \Phi(g_i(x)) \Phi(\hat{z}_i(x)) = 1 + u\Phi(h(x) f_i(x) ).\] By definition of \(\Phi\), we have \(\Phi(\omega_i(x)) = \psi_i(x)\) and \(\Phi(\hat{z}_i(x)) = \hat{z}_i(x)\), leading to: \[\Phi(f_i(x)) \psi_i(x) + \Phi(g_i(x)) \hat{z}_i(x) = 1 + u \Phi(h(x) f_i(x) ).\]
Since \(u\) is nilpotent in \(T_u[x]\), the element \(1 + u \Phi(h^\prime_i(x) f_i(x) )\) is a unit in \(T_u[x]\). We conclude that \(\psi_i(x)\) and \(\hat{z}_i(x)\) are coprime in \(T_u[x]\). This proves that \(\hat{z}_i(x)\) is a unit in both \(\mathcal{S}_i\) and \(\mathcal{R}_i\), completing the proof. ◻
Proposition 3.3. [3] In \(\mathcal{S}_{i}\), we have the following properties:
\(\langle z_i(x)^{p^s} \rangle = \langle u\rangle\), and thus \(z_i(x)\) is nilpotent with nilpotency index \(k p^s\).
The ring \(\mathcal{S}_i\) is a chain ring with the following ideal chain: \[\begin{aligned} \label{eq:ideal_chain} \mathcal{S}_i = \langle 1 \rangle \supsetneq \langle z_i(x) \rangle \supsetneq \cdots \supsetneq \langle z_i(x)^{kp^s-1} \rangle \supsetneq \langle z_i(x)^{k p^s} \rangle = \langle 0 \rangle. \end{aligned} \tag{3}\]
Each ideal \(\langle z_i(x)^{j} \rangle\) contains \(p^{\deg z_i m(kp^s-j)}\) elements, for all \(0 \leq j \leq k p^s\).
In this section, we classify the ideals of the ring \(\mathcal{R}_i\) and determine their cardinalities. We start with the following proposition.
Proposition 4.1. Let \(I\) be an ideal of the ring \(\mathcal{R}_i\). Then, it can be expressed as \[\label{eq:ideal_decomposition} I = \langle z_i(x)^\alpha + v f(x) \rangle + v J, \tag{4}\] where \(0 \leq \alpha \leq kp^s\), \(\Phi_i(I) = \langle z_i(x)^\alpha \rangle\), and \(f(x)\) is a polynomial satisfying \(z_i(x)^\alpha + v f(x) \in I\). Moreover, the ideal \(J\) is defined as \[J = \{a(x) \in \mathcal{R}_i \mid va(x) \in I\}.\]
Proof. Let \(0 \leq \alpha \leq kp^s\) and \(f(x) \in \mathcal{R}_i\) such that \(\Phi_i(I) = \langle z_i(x)^\alpha \rangle\) and \(z_i(x)^\alpha + vf(x) \in I\). Then, applying \(\Phi_i\), \[\Phi_i(z_i(x)^\alpha + vf(x)) = z_i(x)^\alpha.\]
For any \(g(x) \in I\), there exists \(h(x) \in \mathcal{S}_i\) such that \(\Phi_i(g(x)) = h(x) z_i(x)^\alpha.\) By surjectivity of \(\Phi_i\), there exists \(h'(x) \in \mathcal{R}_i\) satisfying \(\Phi_i(h'(x)) = h(x).\) Thus, we obtain \(\Phi_i(g(x)) = \Phi_i(h'(x)) \Phi_i(z_i(x)^\alpha + vf(x)),\) which implies that \[g(x) – h'(x)(z_i(x)^\alpha + vf(x)) \in \ker(\Phi_i) \cap I.\]
Consequently, we deduce \[I \subseteq \langle z_i(x)^\alpha + vf(x) \rangle + \ker(\Phi_i) \cap I.\]
Finally, since \(\ker(\Phi_i) = v \mathcal{R}_i\), it follows that \[I = \langle z_i(x)^\alpha + vf(x) \rangle + vJ.\] ◻
The following theorem provides a complete classification of the distinct ideals of the ring \(\mathcal{R}_i\).
Theorem 4.2. The distinct ideals of \(\mathcal{R}_i\) are classified as follows:
Type 1: The trivial ideals \(\langle 0 \rangle\) and \(\langle 1 \rangle\).
Type 2: Ideals of the form \(\langle v z_i(x)^\alpha \rangle\), where \(0 \leq \alpha \leq kp^{s}-1\).
Type 3: Ideals of the form \(\langle z_i(x)^\alpha + v z_i(x)^\beta g(x) \rangle\), where \(1 \leq \alpha \leq kp^{s}-1\), \(0 \leq \beta < \Upsilon\), and \(g(x)\) is either \(0\) or a unit in \(\mathcal{S}_i\). The parameter \(\Upsilon\) is given by: \[\begin{aligned} \label{evececefe} \Upsilon = \min \left\{ t \mid v z_i(x)^t \in \langle z_i(x)^\alpha + v z_i(x)^\beta g(x) \rangle \right\}. \end{aligned} \tag{5}\]
Type 4: Ideals of the form \(\langle z_i(x)^\alpha + v z_i(x)^\beta g(x), v z_i(x)^\delta \rangle\), where \(0 \leq \beta < \delta < \Upsilon\), and \(g(x)\) is either \(0\) or a unit in \(\mathcal{S}_i\). The parameter \(\Upsilon\) is the same as given in (5).
Proof. The Type 1 ideals are trivial. Let \(I\) be a non-trivial ideal of \(\mathcal{R}_i\). We consider two cases:
If \(\Phi_i(I) = \{0\}\), then by Proposition 4.1, \(I\) can be expressed as \(I = v J\), where \[J = \{a(x) \in \mathcal{R}_i \mid va(x) \in I\}.\]
The set \(\Phi_i(J)\) forms an ideal of \(\mathcal{S}_i\), implying that \(\Phi_i(J) = \langle z_i(x)^\alpha \rangle\) for some \(0 \leq \alpha \leq kp^{s}-1\). Applying Proposition 4.1 again, we obtain: \[J = \langle z_i(x)^\alpha + v f(x) \rangle + v K,\] where \(K = \{a(x) \in \mathcal{R}_i \mid va(x) \in J\}\) and \(f(x) \in \mathcal{R}_i\). Given that \(v^2 = 0\), it follows that \[I = \langle v z_i(x)^\alpha \rangle.\]
This classifies \(I\) as Type 2.
If \(\Phi_i(I) \neq \{0\}\), we note that \(\Phi_i(I)\) is a non-trivial ideal of \(\mathcal{S}_i\), then \(\Phi_i(I) = \langle z_i(x)^\alpha \rangle,\) for some \(1 \leq \alpha \leq kp^{s}-1\). By Proposition 4.1 , the ideal \(I\) satisfies \[I = \langle z_i(x)^\alpha + v f(x) \rangle + v J,\] where \(f(x) \in \mathcal{R}_i\) and \[J = \{ a(x) \in \mathcal{R}_i \mid v a(x) \in I \}.\]
Since \(v J \subseteq \langle v \rangle\), it follows that \(\Phi_i(v J) = \{0\}\). As in the previous case, we have \[v J = \langle v z_i(x)^\delta \rangle,\] for some \(0 \leq \delta \leq kp^{s}\). Then \(I = \langle z_i(x)^\alpha + v f(x), v z_i(x)^\delta \rangle .\)
Considering \(\Phi_i(f(x)) \in \mathcal{S}_i\), if \(\Phi_i(f(x)) \neq 0\), there exists a maximal \(\beta\) such that \[\Phi_i(f(x)) \in \langle z_i(x)^\beta \rangle, \quad \Phi_i(f(x)) \notin \langle z_i(x)^{\beta+1} \rangle.\]
Thus, we write \[\Phi_i(f(x)) = z_i(x)^\beta d(x),\] where \(d(x) \in \mathcal{S}_i\) is a unit. In \(\mathcal{R}_i\), we have \[f(x) = v h(x) \quad \text{or} \quad f(x) = z_i(x)^\beta d(x) + v h(x),\] for some \(h(x) \in \mathcal{R}_i\). Since \(v^2 = 0\), we deduce \[v f(x) = 0 \quad \text{or} \quad v f(x) = v z_i(x)^\beta d(x).\]
Consequently, \[I = \langle z_i(x)^\alpha + v z_i(x)^\beta g(x), v z_i(x)^\delta \rangle,\] where \(g(x)\) is either \(0\) or a unit in \(\mathcal{S}_i\). Let \(\Upsilon\) be as defined in Eq. (5). If \(\delta \geq \Upsilon\), then \(I\) simplifies to \(\langle z_i(x)^\alpha + v z_i(x)^\beta g(x) \rangle\), corresponding to Type 3. Otherwise, if \(\delta < \Upsilon\), it remains \(I = \langle z_i(x)^\alpha + v z_i(x)^\beta g(x), v z_i(x)^\delta \rangle\), corresponding to Type 4.
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Before computing the value of \(\Upsilon\), we first establish the following lemma.
Lemma 4.3. In \(\mathcal{R}_i\), we have \[z_i(x)^{kp^s} = k\mu \left( \widehat{z}(x)^{p^s} \right)^{-1} z_i(x)^{(k-1)p^s} v.\]
Proof. In \(R_{v, v}[x]\), with all computations carried out modulo \(x^{\eta p^s}-\lambda\), we obtain: \[\begin{aligned} z_i(x) ^{kp^s} \widehat{z}(x) ^{kp^s} &= \left( x^{\eta p^s}-\lambda_{0}\right) ^{k} \\ &= \left( \gamma u + \mu v \right)^k \\ &= \sum\limits_{l=0}^{k} \binom{k}{l} \left(\gamma u \right)^{k-l} (\mu v)^l \\ &= \left(\gamma u\right)^{k} + k\left(\gamma u \right)^{k-1}(\mu v) \\ &= k\mu\left(\gamma u \right)^{k-1} v \\ &= k\mu \left( x^{\eta p^s}-\lambda_{0}\right)^{(k-1)} v. \end{aligned}\]
The last step is justified by the fact that the \(v\)-terms in \(\left( x^{\eta p^s}-\lambda_{0}\right)^{(k-1)}\) vanish after multiplication by \(v\), since \(v^2 = 0\). On the other hand, by Proposition 3.2, \(\widehat{z}(x)\) is a unit in \(\mathcal{R}_i\). Thus, in \(\mathcal{R}_i\), we obtain: \[z_i(x) ^{kp^s} = k\mu \left( \widehat{z}(x) ^{kp^s} \right)^{-1} \left( x^{\eta p^s}-\lambda_{0}\right)^{(k-1)} v = k\mu \left( \widehat{z}(x) ^{p^s} \right)^{-1}z_i(x)^{(k-1)p^{s}} v.\] ◻
Theorem 4.4. Let \(I = \langle z_i(x)^\alpha + v z_i(x)^\beta g(x) \rangle\) and define \(\Upsilon = \min\left\{ t \mid v z_i(x)^t \in I \right\}.\) Then, \[\begin{aligned} \Upsilon = \begin{cases} \alpha, & \text{if } h(x) = 0, \\[8pt] \min\{\alpha, \varepsilon\}, & \text{if } h(x) \neq 0. \end{cases} \end{aligned} \tag{6}\] where \[\varepsilon = \max \left\{ 0\leq l\leq kp^s \mid z_i(x)^{kp^{s}+\beta -\alpha} g(x) + k\mu \left( \widehat{z}(x)^{p^s} \right)^{-1} z_i(x)^{(k-1)p^{s}}\in \left\langle z_i(x)^l\right\rangle \right\}.\]
Thus, we can write \[\begin{aligned} \label{reffcefcefce} z_i(x)^{kp^{s}+\beta -\alpha} g(x) + k\mu \left( \widehat{z}(x)^{p^s} \right)^{-1} z_i(x)^{(k-1)p^{s}} = z_i(x)^\varepsilon h(x), \end{aligned} \tag{7}\] where \(h(x) \in \mathcal{S}_i\) is either \(0\) or a unit in \(\mathcal{S}_i .\)
Proof. Consider \(v z_i(x)^t \in I\), which is equivalent to \[\begin{aligned} \label{eefefefce} v z_i(x)^t = f(x) \left( z_i(x)^\alpha + v z_i(x)^\beta g(x) \right), \end{aligned} \tag{8}\] for some \(f(x) \in \mathcal{R}_i\), applying \(\Phi_i\) yields \[\Phi_i(f(x)) z_i(x)^\alpha = 0 \quad \text{in } \mathcal{S}_i.\]
Since \(\mathcal{S}_i\) is a chain ring with maximal ideal \(\langle z_i(x) \rangle\) and nilpotency index \(kp^s\), we obtain \[\Phi_i(f(x))= z_i(x)^{kp^s – \alpha} f^\prime(x),\] with \(f^\prime(x) \in \mathcal{S}_i\). Consequently, in \(\mathcal{R}_i\), \[f(x) = z_i(x)^{kp^s – \alpha} f^\prime(x) + v f^{\prime\prime}(x),\] where \(f^{\prime\prime}(x) \in \mathcal{R}_i\). Substituting this into Eq. (8), we obtain: \[v z_i(x)^t = \left( z_i(x)^{kp^s – \alpha} f^\prime(x) + v f^{\prime\prime}(x) \right) \left( z_i(x)^\alpha + v z_i(x)^\beta g(x) \right).\]
Expanding and using \(v^2 = 0\), we get: \[\begin{aligned} \label{cececc} v z_i(x)^t = z_i(x)^{kp^s} f^\prime(x) + v f^{\prime\prime}(x) z_i(x)^\alpha + v z_i(x)^{kp^s – \alpha + \beta} f^\prime(x) g(x). \end{aligned} \tag{9}\]
Using Lemma 4.3, \[z_i(x)^{kp^s} = k\mu \left( \widehat{z}(x)^{p^s} \right)^{-1} z_i(x)^{(k-1)p^s} v.\]
Replacing this in Eq. (9), we get: \[v z_i(x)^t = v \left(z_i(x)^\alpha f^{\prime\prime}(x) + \left( z_i(x)^{kp^s – \alpha + \beta} g(x)+k\mu \left( \widehat{z}(x)^{p^s} \right)^{-1} z_i(x)^{(k-1)p^s}\right) f^\prime(x) \right).\]
Using Eq. (7), we obtain: \[v z_i(x)^t = v \left( z_i(x) ^\alpha f^{\prime\prime}(x) + z_i(x) ^{\varepsilon} h(x) f^\prime(x) \right).\]
We now consider two cases:
If \(h(x) = 0\), then \(\Upsilon \geq \alpha\). Since \[v z_i(x)^\alpha = v\left( z_i(x)^\alpha + v z_i(x)^\beta g(x) \right) \in I,\] it follows that \(\Upsilon = \alpha\).
If \(h(x)\neq0\), then \(\Upsilon\geq\min\{\alpha,\varepsilon\}\). Conversely, we have: \[v z_i(x) ^\alpha = v \left( z_i(x) ^\alpha + v z_i(x) ^\beta g(x)\right)\in I,\] and \[v z_i(x) ^{\varepsilon} = h(x)^{-1} z_i(x) ^{kp^{s}-\alpha }\left( z_i(x) ^\alpha + v z_i(x) ^\beta g(x)\right)\in I.\]
Therefore, \(\Upsilon = \min \left\{\alpha,\varepsilon\right\}\).
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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 \(I\).
\[\operatorname{Res}(I) = \Phi_i(I), \quad \operatorname{Tor}(I) = \{ c(x) \in \mathcal{S}_i \mid v c(x) \in I \}.\]
Clearly, \(\operatorname{Res}(I)\) and \(\operatorname{Tor}(I)\) are ideals of \(\mathcal{S}_i\). By Proposition 3.3, they can be expressed as \(\langle z_i(x)^l \rangle\), where \(0 \leq l \leq kp^s\). Now, consider the ring homomorphism \[\begin{array}{rrcl} T: & I & \longrightarrow & \Phi_i(I), \\ & c(x) & \longmapsto & \Phi_i(c(x)). \end{array}\]
Since \(\operatorname{Im} T = \operatorname{Res}(I)\) and \(\operatorname{ker} T \cong v\operatorname{Tor}(I)\), the first isomorphism theorem yields \[\label{cardtor} \lvert I \rvert = \lvert \operatorname{Res}(I) \rvert \cdot \lvert \operatorname{Tor}(I) \rvert. \tag{10}\]
The following lemma, whose proof is straightforward, provides explicit expressions for \(\operatorname{Res}(I)\) and \(\operatorname{Tor}(I)\) for any ideal \(I\) in \(\mathcal{R}_i\).
Lemma 4.5. With the notation of Theorem 4.2:
If \(I=\langle 0\rangle\), then \(\operatorname{Tor}(I)=\operatorname{Res}(I)=\langle 0\rangle\).
If \(I=\langle 1\rangle\), then \(\operatorname{Tor}(I)=\operatorname{Res}(I)=\langle 1\rangle\).
If \(I=\langle v z_i(x) ^\alpha \rangle\) is of type \(2\), then \(\operatorname{Tor}(I)=\langle z_i(x) ^\alpha \rangle\) and \(\operatorname{Res}(I)=\langle 0\rangle\).
If \(I=\langle z_i(x) ^\alpha + v z_i(x) ^\beta g(x)\rangle\) is of type \(3\), then \(\operatorname{Tor}(I)=\langle z_i(x) ^\Upsilon\rangle\) and \(\operatorname{Res}(I)=\langle z_i(x) ^\alpha \rangle\).
If \(I=\langle z_i(x) ^\alpha + v z_i(x) ^\beta g(x), v z_i(x) ^\delta\rangle\) is of type \(4\), then \(\operatorname{Tor}(I)=\langle z_i(x) ^\delta\rangle\) and \(\operatorname{Res}(I)=\langle z_i(x) ^\alpha \rangle\).
By determining \(\operatorname{Res}(I)\) and \(\operatorname{Tor}(I)\) in each case, we can compute the cardinalities of all ideals in \(\mathcal{R}_i\) using Eq. (10) and Proposition 3.3.
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, \[C = \bigoplus_{i=1}^r e_i(x) I_i,\] where \(I_i\) is an ideal of \(\mathcal{R}_i\) for \(1 \leq i \leq r\). By Eq. (1), we have \[\begin{array}{rcl}d(x)=\sum\limits_{i=1}^r e_i(x)\cdot d(x)\in \mathcal{A}(C) &\Leftrightarrow&\forall c(x)\in C,\left( \sum\limits_{i=1}^r e_i(x)\cdot d(x)\right) \cdot\left( \sum\limits_{i=1}^r e_i(x)\cdot c(x)\right)=0\\ &\Leftrightarrow&\forall c(x)\in C,\left( \sum\limits_{i=1}^r e_i(x)\cdot d(x)\cdot c(x)\right) =0 \\ &\Leftrightarrow&\forall c(x)\in C,\forall i\in \{1,…,r\},e_i(x)\cdot d(x)\cdot c(x) =0 \\ &\Leftrightarrow&\forall c(x)\in C,\forall i\in \{1,…,r\}, d(x)\cdot c(x) =0 \text{ in } \mathcal{R}_i\\ &\Leftrightarrow&\forall i\in \{1,…,r\}, d(x)\in \mathcal{A}(I_i). \end{array}\]
Therefore, \[\begin{aligned} \label{vdvedvdevdvd} \mathcal{A}(C) = \bigoplus_{i=1}^r e_i(x) \mathcal{A}(I_i). \end{aligned} \tag{11}\]
Thus, by Proposition 2.5, \[C^\perp = \mathcal{A}(C)^* = \bigoplus_{i=1}^r e_i^*(x) \mathcal{A}(I_i)^*.\]
We aim to determine \(\mathcal{A}(I)^*\), where \(I\) is an ideal of \(\mathcal{R}_i\). To this end, we first establish the following three lemmas.
Lemma 5.1. Let \(I\) be an ideal of \(\mathcal{R}_i\). Then, \(|I| \cdot |\mathcal{A}(I)| = | \mathcal{R}_i |.\)
Proof. Let \(C = \bigoplus_{j=1}^r e_j(x) I_j\) be an ideal of \(\mathcal{R}_\lambda\), where \[I_j = I \quad \text{if } j = i, \quad \text{and} \quad I_j = \langle 1 \rangle \quad \text{otherwise}.\]
Then, by Eq. (11), we obtain \[\mathcal{A}(C) = \bigoplus_{j=1}^r e_j(x) J_j,\] where \[J_j = \mathcal{A}(I) \quad \text{if } j = i, \quad \text{and} \quad J_j = \langle 0 \rangle \quad \text{otherwise}.\]
According to Propositions 2.3 and 2.5, we have \(|C| \cdot |\mathcal{A}(C)| = |R_{v,v}|^{\eta p^s}.\) Therefore, \[\big( \prod\limits_{\substack{1 \leq j \leq r \\ j \neq i}} |\mathcal{R}_j| \big) \cdot |I| \cdot |\mathcal{A}(I)| = |R_{v,v}|^{\eta p^s}.\]
It follows that \[|I| \cdot |\mathcal{A}(I)| = \frac{|R_{v,v}|^{\eta p^s}}{\prod\limits_{\substack{1 \leq j \leq r \\ j \neq i}} |\mathcal{R}_j|} = |\mathcal{R}_i|.\] ◻
Lemma 5.2. Let \(I\) be an ideal of \(\mathcal{R}_i\) and \(a, b\) two integers such that \[\operatorname{Res}(I) = \langle z_i(x)^a \rangle \quad \text{and} \quad \operatorname{Tor}(I) = \langle z_i(x)^b \rangle.\] Then, we have: \[\operatorname{Res}(\mathcal{A}(I)) = \langle z_i(x)^{kp^s – b} \rangle, \quad \operatorname{Tor}(\mathcal{A}(I)) = \langle z_i(x)^{kp^s – a} \rangle.\]
Proof. Let \[\operatorname{Res} (\mathcal{A}(I)) = \langle z_i(x)^{a^\prime} \rangle, \quad \operatorname{Tor}(\mathcal{A}(I)) = \langle z_i(x)^{b^\prime} \rangle,\] where \(a^\prime\) and \(b^\prime\) are two integers.
Since \(v z_i(x)^b \in I\) and \(v z_i(x)^{b^\prime} \in \mathcal{A}(I)\), there exist two polynomials \(g(x)\) and \(f(x)\) in \(\mathcal{R}_i\) such that \[z_i(x)^a + v g(x) \in I, \quad z_i(x)^{a^\prime} + v f(x) \in \mathcal{A}(I).\]
By the definition of \(\mathcal{A}(I)\), we obtain:
\[0 = v z_i(x)^b \left( z_i(x)^{a^\prime} + v f(x) \right) = v z_i(x)^{b + a^\prime}. \tag{12}\]
Similarly, we have:
\[0 = v z_i(x)^{b^\prime} \left( z_i(x)^a + v g(x) \right) = v z_i(x)^{b^\prime + a}. \tag{13}\]
Therefore, we necessarily have: \[a^\prime \geq kp^s – b, \quad b^\prime \geq kp^s – a.\]
By Lemma 5.1, \(|I| \cdot |\mathcal{A}(I)| = |\mathcal{R}_i|\), and by Proposition 3.3, we obtain
\[p^{m2k\deg z_i} =|\mathcal{R}_i| = p^{\deg z_i m(kp^s-(a+b+a^\prime+b^\prime))}\leq p^{m2k\deg z_i}\]
It follows that: \[b^\prime = kp^s – a, \quad a^\prime = kp^s – b.\] ◻
Lemma 5.3. With the previous notation, we have: \[\mathcal{A}(I) = \langle z_i(x)^{kp^{s}-b} + v f(x), v z_i(x)^{kp^{s}-a} \rangle.\]
Proof. Firstly, it is clear that \(v z_i(x)^{kp^{s}-a} \in \mathcal{A}(I)\) and \(z_i(x)^{kp^{s}-b} + v f(x) \in \mathcal{A}(I)\).
Now, let \(c(x) \in \mathcal{A}(I)\). Then, we have \(\Phi_i(c(x)) \in \operatorname{Res}(\mathcal{A}(I))\), which implies that there exists \(c_0(x) \in \mathcal{S}_i\) such that: \[\Phi_i(c(x)) = c_0(x) z_i(x)^{kp^{s}-b} = \Phi_i\left( c_0(x) \left( z_i(x)^{kp^{s}-b} + v f(x) \right) \right).\]
Therefore, we obtain: \[c(x) – c_0(x) \left( z_i(x)^{kp^{s}-b} + v f(x) \right) \in \ker \Phi_i.\]
This implies that: \[\mathcal{A}(I) \ni c(x) – c_0(x) \left( z_i(x)^{kp^{s}-a} + v f(x) \right) = v c_1(x),\] where \(c_1(x) \in \mathcal{S}_i\). Since \(c_1(x) \in \operatorname{Tor}(\mathcal{A}(I)) = \langle z_i(x)^{kp^{s}-a} \rangle\), we deduce that: \[c(x) \in \langle z_i(x)^{kp^{s}-b} + v f(x), v z_i(x)^{kp^{s}-a} \rangle.\]
This shows that: \[\mathcal{A}(I) = \langle z_i(x)^{kp^{s}-b} + v f(x), v z_i(x)^{kp^{s}-a} \rangle.\] ◻
We now proceed to determine the annihilator of each type of ideal. For type 1 ideals, this is straightforward: if \(I = \langle 0 \rangle\), then \(\mathcal{A}(I) = \langle 1 \rangle\); if \(I = \langle 1 \rangle\), then \(\mathcal{A}(I) = \langle 0 \rangle\). For other types, we first identify two integers \(a\) and \(b\) such that \(\operatorname{Res}(I) = \langle z_i(x)^a \rangle\) and \(\operatorname{Tor}(I) = \langle z_i(x)^b \rangle\), as given in Lemma 4.3. Once these values are determined, we find a polynomial \(f(x)\) satisfying \(z_i(x)^{kp^s – b} + v f(x) \in \mathcal{A}(I)\). Finally, applying Lemma 5.3, we obtain \(\mathcal{A}(I)\).
Proposition 5.4. If \(I=\langle v z_i(x) ^\alpha\rangle\) is an ideal of type \(2\), then \(\mathcal{A}(I)=\langle z_i(x) ^{kp^{s}-\alpha}, v \rangle\).
Proof. It is clear that \(z_i(x) ^{kp^{s}-\alpha} \in \mathcal{A}(I)\). Therefore, we have \(\mathcal{A}(I) = \langle z_i(x) ^{kp^{s}-\alpha}, v \rangle\). ◻
Proposition 5.5. If \(I=\langle z_i(x) ^\alpha+ v z_i(x) ^\beta g(x)\rangle\) is an ideal of type \(3\), then:
\[\mathcal{A}(I)=\begin{cases} \left\langle z_i(x) ^{kp^{s}-\alpha}\right\rangle,&\text{ if } h(x)=0, \\ \langle z_i(x)^{kp^{s}-\Upsilon} – v z_i(x)^{\varepsilon-\Upsilon}h(x), v z_i(x)^{kp^{s}-\alpha} \rangle,&\text{ if }h(x)\neq0 . \end{cases}\]
Proof. Since \(\operatorname{Tor}(I) = \langle z_i(x)^\Upsilon \rangle\), it suffices to determine a polynomial \(f(x) \in \mathcal{S}_i\) satisfying \[z_i(x)^{kp^{s}-\Upsilon} + v f(x) \in \mathcal{A}(I).\]
By Theorem 4.4, we have \[\label{eq20} \begin{aligned} 0 &= \left( z_i(x)^{kp^{s}-\Upsilon} + v f(x)\right)\left( z_i(x) ^\alpha+ v z_i(x) ^\beta g(x)\right) \\ &= z_i(x)^{kp^{s}-\Upsilon+\alpha} + v \left( z_i(x)^{kp^{s}-\Upsilon+\beta} g(x) + z_i(x)^\alpha f(x) \right) \\ &= z_i(x)^{\alpha-\Upsilon} z_i(x)^{kp^{s}} + v \left( z_i(x)^{kp^{s}-\Upsilon+\beta} g(x) + z_i(x)^\alpha f(x) \right) \\ &= v \left( k\mu \left( \widehat{z}(x)^{p^s} \right)^{-1}z_i(x)^{(k-1)p^{s}+\alpha-\Upsilon} + z_i(x)^{kp^{s}-\Upsilon+\beta} g(x) + z_i(x)^\alpha f(x) \right) \\ &= v \left( z_i(x)^{\alpha-\Upsilon+\varepsilon} h(x) + z_i(x)^\alpha f(x) \right). \end{aligned} \tag{14}\]
We now distinguish two cases:
If \(h(x) = 0\), then \(\Upsilon = \alpha\), and Eq. (14) simplifies to \[0 = v z_i(x)^\alpha f(x).\]
In this case, choosing \(f(x) = 0\) leads to \[\mathcal{A}(I) = \langle z_i(x)^{kp^{s}-\alpha}, v z_i(x)^{kp^{s}-\alpha} \rangle = \langle z_i(x)^{kp^{s}-\alpha} \rangle.\]
If \(h(x) \neq 0\), we take \(f(x) = -z_i(x)^{\varepsilon-\Upsilon}h(x)\), yielding \[\mathcal{A}(I) = \langle z_i(x)^{kp^{s}-\Upsilon} – v z_i(x)^{\varepsilon-\Upsilon}h(x), v z_i(x)^{kp^{s}-\alpha} \rangle.\]
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Proposition 5.6. If \(I = \langle z_i(x) ^\alpha + v z_i(x) ^\beta g(x), v z_i(x) ^\delta \rangle\), an ideal of type \(4\), then:
\[\mathcal{A}(I) =\begin{cases}\left\langle z_i(x) ^{kp^{s}-\delta}, v z_i(x) ^{kp^{s}-\alpha}\right\rangle ,& \text{ if }h(x) = 0,\\ \left\langle z_i(x) ^{kp^{s}-\delta}- v z_i(x) ^{\varepsilon-\delta} h(x) , v z_i(x) ^{kp^{s}-\alpha}\right\rangle,& \text{ if }h(x) \neq 0. \end{cases}\]
Proof. Since \(\operatorname{Tor}(I) = \langle z_i(x)^\delta \rangle\), it suffices to determine a polynomial \(f(x) \in \mathcal{S}_i\) satisfying \[z_i(x)^{kp^{s}-\delta} + v f(x) \in \mathcal{A}(I).\]
Given that \[\left( z_i(x)^{kp^{s}-\delta} + v f(x) \right) \left( v z_i(x)^{kp^{s}-\alpha} \right) = 0,\] it suffices that \[\left( z_i(x)^{kp^{s}-\delta} + v f(x) \right)\left( z_i(x)^\alpha + v z_i(x)^\beta g(x) \right) = 0.\]
By Theorem 4.4, this is equivalent to: \[\begin{aligned} \label{eq30} \begin{array}{rcl} 0 &=& z_i(x)^{kp^{s}-\delta+\alpha} + v \left( z_i(x)^{kp^{s}-\delta+\beta} g(x) + z_i(x)^\alpha f(x) \right) \\ &=& z_i(x)^{\alpha-\delta} z_i(x)^{kp^{s}} + v \left( z_i(x)^{kp^{s}-\delta+\beta} g(x) + z_i(x)^\alpha f(x) \right) \\ &=& v \left( k\mu \left( \widehat{z}(x)^{p^s} \right)^{-1}z_i(x)^{(k-1)p^{s}+\alpha-\delta} + z_i(x)^{kp^{s}-\delta+\beta} g(x) + z_i(x)^\alpha f(x) \right) \\ &=& v \left( z_i(x)^{\alpha-\delta+\varepsilon} h(x) + z_i(x)^\alpha f(x) \right). \end{array} \end{aligned} \tag{15}\]
Then, we choose \(f(x) = -z_i(x)^{\varepsilon-\delta}h(x)\) if \(h(x) \neq 0\) and \(f(x) = 0\) if \(h(x) = 0\), resulting in:
If \(h(x) = 0\), \[\mathcal{A}(I) = \left\langle z_i(x)^{kp^{s}-\delta}, v z_i(x)^{kp^{s}-\alpha} \right\rangle.\]
If \(h(x) \neq 0\), \[\mathcal{A}(I) = \left\langle z_i(x)^{kp^{s}-\delta} – v z_i(x)^{\varepsilon-\delta}h(x), v z_i(x)^{kp^{s}-\alpha} \right\rangle.\]
◻
We now determine \(\mathcal{A}(I)^*\) for any ideal in \(\mathcal{R}_i\). If \(I\) is of type 1, then \(\mathcal{A}(I)^* = \langle 0 \rangle\) when \(I = \langle 0 \rangle\), and \(\mathcal{A}(I)^* = \langle 1 \rangle\) when \(I = \langle 1 \rangle\). For other types, we introduce the following notation: Let \[\vartheta = \max \left\{ 0 \leq l \leq kp^s \mid x^{(kp^s-\varepsilon)\deg z_i} h(x^{-1}) \in \left\langle z_i(x)^l \right\rangle \right\}.\]
Then, we can write \[x^{(kp^s-\varepsilon)\deg z_i} h(x^{-1}) = z_i(x)^\vartheta h^\prime(x),\] where \(h^\prime(x)\) is either \(0\) or a unit in \(\mathcal{S}_i .\)
Corollary 5.7. If \(I=\langle v z_i(x) ^\alpha\rangle\) is an ideal of type \(2\), then \(\mathcal{A}(I)^*=\langle z_i^*(x) ^{kp^{s}-\alpha}, v \rangle\).
Corollary 5.8. If \(I=\langle z_i(x) ^\alpha+ v z_i(x) ^\beta g(x)\rangle\) is an ideal of type \(3\), then:
\[\mathcal{A}(I)^*=\begin{cases} \left\langle z_i^*(x) ^{kp^{s}-\alpha}\right\rangle,&\text{ if } h(x)=0, \\ \left\langle z_i^*(x) ^{kp^{s}-\Upsilon}- v z_i(x) ^{\varepsilon-\Upsilon+\vartheta} h^\prime(x) , v z_i^*(x) ^{kp^{s}-\alpha}\right\rangle,&\text{ if }h(x)\neq0. \end{cases}\]
Proof. The result is immediate when \(h(x) = 0\). If \(h(x) \neq 0\), we have
\[\begin{array}{rcl} \mathcal{A}(I)^* & = & \left\langle z_i^*(x) ^{kp^{s}-\Upsilon}- v x^{(kp^s-\Upsilon)\deg z_i} z(x^{-1}) ^{\varepsilon-\Upsilon} h(x^{-1}) , v z_i^*(x) ^{kp^{s}-\alpha}\right\rangle\\ & = & \left\langle z_i^*(x) ^{kp^{s}-\Upsilon}- v x^{(kp^s-\varepsilon)\deg z_i} z_i^*(x) ^{\varepsilon-\Upsilon} h(x^{-1}) , v z_i^*(x) ^{kp^{s}-\alpha}\right\rangle \\ & = & \left\langle z_i^*(x) ^{kp^{s}-\Upsilon}- v z_i(x) ^{\varepsilon-\Upsilon+\vartheta} h^\prime(x) , v z_i^*(x) ^{kp^{s}-\alpha}\right\rangle. \end{array}\] ◻
Corollary 5.9. If \(I = \langle z_i(x) ^\alpha + v z_i(x) ^\beta g(x), v z_i(x) ^\delta \rangle\), an ideal of type \(4\), then:
\[\mathcal{A}(I)^* =\begin{cases}\left\langle z_i^*(x) ^{kp^{s}-\delta}, v z_i^*(x) ^{kp^{s}-\alpha}\right\rangle ,& \text{ if }h(x) = 0,\\ \left\langle z_i^*(x) ^{kp^{s}-\delta}- v z_i(x) ^{\varepsilon-\delta+\vartheta} h^\prime(x) , v z_i^*(x) ^{kp^{s}-\alpha}\right\rangle,& \text{ if }h(x) \neq 0. \end{cases}\]
Proof. It is similar to Corollary 5.8; it suffices to substitute \(\Upsilon\) with \(\delta\). ◻
The Hamming weight of a codeword \(c\), denoted by \(wt_H(c)\), represents the number of nonzero components in the vector \(c\). The Hamming distance between two vectors \(c\) and \(c'\), denoted by \(d_H(c, c')\), is defined as \(wt_H(c – c')\).
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.
Corollary 6.1. \(\lambda\)-constacyclic codes of length \(p^s\) over the ring \(R_{u,v}\), i.e., ideals of \[\frac{R_{u,v}[x]}{\langle x^{p^s} – \lambda \rangle},\] can be classified as follows:
Type 1: The trivial ideals \(\langle 0 \rangle\) and \(\langle 1 \rangle\).
Type 2: Ideals of the form \(\langle v (x-\theta)^\alpha \rangle\), where \(0 \leq \alpha \leq kp^{s}-1\).
Type 3: Ideals of the form \(\langle (x-\theta)^\alpha + v (x-\theta)^\beta g(x) \rangle\), where \(1 \leq \alpha \leq kp^{s}-1\), \(0 \leq \beta < \Upsilon\), and \(g(x)\) is either \(0\) or a unit in \(\frac{T_{u}[x]}{\langle x^{p^s} – \lambda_0-\gamma u \rangle},\).
Type 4: Ideals of the form \(\langle (x-\theta)^\alpha + v (x-\theta)^\beta g(x), v (x-\theta)^\delta \rangle\), where \(0 \leq \beta < \delta < \Upsilon\), and \(g(x)\) is either \(0\) or a unit in \(\frac{T_{u}[x]}{\langle x^{p^s} – \lambda_0-\gamma u \rangle}\).
Moreover, \[\Upsilon = \begin{cases} \alpha, & \text{if } h(x) = 0, \\ \min\{\alpha, \varepsilon\}, & \text{if } h(x) \neq 0. \end{cases}\]
where \[\varepsilon = \max \left\{0\leq l\leq kp^s \mid (x-\theta)^{kp^{s}+\beta -\alpha} g(x) + k\mu (x-\theta)^{(k-1)p^{s}}\in \left\langle (x-\theta)^l\right\rangle \right\}.\]
The following lemma establishes a relationship between the Hamming distance of a \(\lambda\)-constacyclic code and its torsion \(\operatorname{Tor}(C)\).
Lemma 6.2. For any \(\lambda\)-constacyclic code \(C\) of length \(p^s\) over \(R_{u,v}\), we have \[d_H(C) = d_H(\operatorname{Tor}(C)).\]
Proof. We first prove that \(d_H(C) \leq d_H(\operatorname{Tor}(C))\). Let \(a(x)\) be a nonzero polynomial in \(\operatorname{Tor}(C)\), so that \(v a(x) \in C\). Since \(a(x)\) does not involve \(v\), both \(a(x)\) and \(v a(x)\) share the same nonzero coefficients, which implies \[wt_H(v a(x)) = wt_H(a(x)) \neq 0.\]
Hence, we obtain \[d_H(C) \leq d_H(\operatorname{Tor}(C)).\]
To prove the reverse inequality, let \(f(x) \in C\) be a nonzero polynomial, and decompose it as \[f(x) = a(x) + v b(x),\] where \(a(x), b(x)\) are polynomials that do not involve \(v\).
If \(a(x) = 0\), then \(b(x) \in \operatorname{Tor}(C)\), leading to \[d_H(\operatorname{Tor}(C)) \leq wt_H(f(x)).\]
If \(a(x) \neq 0\), then \(v a(x) = v f(x)\) is a nonzero element of \(C\). Hence, \(a(x) \in \operatorname{Tor}(C)\). Moreover, \(a(x)\) and \(v a(x)\) share the same nonzero components, while \(v f(x)\) has more zero components than \(f(x)\). It follows that
\[d_H(\operatorname{Tor}(C)) \leq wt_H(a(x)) =wt_H(va(x)) \leq wt_H(f(x)).\]
Thus, we conclude that \(d_H(\operatorname{Tor}(C)) \leq d_H(C)\), completing the proof. ◻
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 \(\operatorname{Tor}(C)\), which corresponds to a \((\lambda_0+\gamma v)\)-constacyclic code of the same length over \(T_u\).
Let \(C_{\tau}=\langle (x-\theta)^{\tau}\rangle\) be a nonzero \((\lambda_0+\gamma v)\)-constacyclic code of length \(p^s\) over \(T_u\), where \(0 \leq \tau \leq p^{s}k\). We distinguish two cases:
If \(0 \leq \tau \leq p^{s}(k-1)\), then the chain of inclusions \[C_{p^s (k-1)} \subset \cdots \subset C_\tau \subset \cdots \subset C_0 = \langle 1\rangle\] implies \[d_H(C_{p^s (k-1)}) \geq \cdots \geq d_H(C_\tau) \geq 1.\]
By Proposition 3.3, \(C_{p^s( k-1)} = \langle u^{k-1} \rangle\). Then, \(d_H(C_{p^s( k-1)}) = 1\), which implies that \(d_H(C_\tau) = 1\) for all \(0 \leq \tau \leq p^s(k-1)\).
If \(p^s(k-1)+1 \leq \tau \leq p^s k-1\), writing \(\tau = p^s(k-1) + \varsigma\) with \(1 \leq \varsigma \leq p^s-1\), we obtain \[C_\tau = \langle v^{k-1}(x-\theta )^\varsigma \rangle.\] Thus, each \(C_\tau\) corresponds to the \(\lambda_{0}\)-constacyclic code \(\langle (x-\theta)^\varsigma \rangle\) over \(\mathbb{F}_{p^m}\), multiplied by \(v^{k-1}\), leading to \[d_H(C_{\tau}) = d_H(\langle (x-\theta )^{\varsigma} \rangle).\]
The Hamming distance of constacyclic codes of length \(p^s\) over \(\mathbb{F}_{p^m}\) is determined by the following proposition.
Proposition 6.3. [9] Let \(C_\varsigma = \langle (x-\theta)^\varsigma \rangle\) be a \(\lambda_{0}\)-constacyclic code of length \(p^s\) over \(\mathbb{F}_{p^m}\), where \(\varsigma \in \{0,1, \ldots, p^s\}\). The Hamming distance \(d_H(C_\varsigma)\) is given by \[d_H(C_\varsigma) = \begin{cases} 1, & \text{if } \varsigma = 0, \\[8pt] \varpi+2, & \text{if } \varpi p^{s-1}+1 \leq \varsigma \leq (\varpi+1) p^{s-1}, \quad 0 \leq \varpi \leq p-2, \\[8pt] (q+1) p^j, & \text{if } p^s-p^{s-j}+(q-1) p^{s-j-1}+1 \leq \varsigma \leq p^s-p^{s-j}+q p^{s-j-1}, \\ & \quad 1 \leq q \leq p-1, \quad 1 \leq j\leq s-1, \\[8pt] 0, & \text{if } \varsigma = p^s. \end{cases}\]
Thus, we establish the following theorem.
Theorem 6.4. Let \(C\) be a \(\lambda\)-constacyclic code of length \(p^s\) over \(R_{u,v}\). Then, its Hamming distance is given by \[d_H(C) = \begin{cases} 1, & \text{if } 0 \leq \tau \leq p^{s}(k-1) \text{ or } \varsigma = 0, \\[8pt] \varpi+2, & \text{if } \varpi p^{s-1}+1 \leq \varsigma \leq (\varpi+1) p^{s-1}, \quad 0 \leq \varpi \leq p-2, \\[8pt] (q+1) p^j, & \text{if } p^s-p^{s-j}+(q-1) p^{s-j-1}+1 \leq \varsigma \leq p^s-p^{s-j}+q p^{s-j-1}, \\ & \quad 1 \leq q \leq p-1, \quad 1 \leq j\leq s-1, \\[8pt] 0, & \text{if } \varsigma = p^s, \end{cases}\] where \(0\leq \tau \leq kp^s\) satisfies \(\operatorname{Tor}(C) = \langle (x-\theta)^\tau \rangle\), and if \(\tau \geq p^s(k-1)\), then \(\varsigma = \tau – p^s(k-1)\).