On a class of constacyclic codes over the non-principal ideal ring $\mathbb{Z}_{p^s}+u\mathbb{Z}_{p^s}$

TL;DR

This paper investigates the structure, enumeration, and duality of $(1+pw)$-constacyclic codes over a non-principal ideal ring, providing new insights into their properties and self-dual variants, especially over rings with characteristic 2.

Contribution

It characterizes the structure, counts, and dual codes of $(1+pw)$-constacyclic codes over $Z_{p^s}+uZ_{p^s}$, including self-dual codes over rings with characteristic 2.

Findings

Explicit structure of these constacyclic codes is provided.

Formulas for counting the number of codes and codewords are derived.

Self-dual codes over rings with characteristic 2 are characterized.

Abstract

$(1+pw)$-constacyclic codes of arbitrary length over the non-principal ideal ring $\mathbb{Z}_{p^s} +u\mathbb{Z}_{p^s}$ are studied, where $p$ is a prime, $w\in \mathbb{Z}_{p^s}^{\times}$ and $s$ an integer satisfying $s\geq 2$. First, the structure of any $(1+pw)$-constacyclic code over $\mathbb{Z}_{p^s} +u\mathbb{Z}_{p^s}$ are presented. Then enumerations for the number of all codes and the number of codewords in each code, and the structure of dual codes for these codes are given, respectively. Then self-dual $(1+2w)$-constacyclic codes over $\mathbb{Z}_{2^s} +u\mathbb{Z}_{2^s}$ are investigated, where $w=2^{s-2}-1$ or $2^{s-1}-1$ if $s\geq 3$, and $w=1$ if $s=2$.

Equations

$$\alpha +\beta =(a+b)+u(c+d)\mathrm{and}\alpha \beta =ac+u(ad+bc),$$

$$f(x{)}^{p^k}=p\vartheta (x)\mathrm{in}\mathcal{R},\mathrm{where}\vartheta (x)\in {\mathcal{R}}^{\times }$$

$$⟨p⟩=⟨f(x{)}^{p^k}⟩\subseteq ⟨f(x)⟩$$

$$\mathcal{R}/J$$

$$y^n-1=f_1(y)f_2(y)\dots f_r(y),$$

$${\Gamma }_i={\mathbb{Z}}_{p^s}[y]/⟨f_i(y)⟩\mathrm{and}{\mathcal{R}}_i={\mathbb{Z}}_{p^s}[x]/⟨f_i(x^{p^k}(1+p{w}_0{)}^{-1})⟩.$$

$$a_i(y)F_i(y)+b_i(y)f_i(y)=1.$$

$$(1+p{w}_0{)}^{-n}(x^{p^{k}n}-(1+pw))=(x^{p^k}(1+p{w}_0{)}^{-1}{)}^n-1=\prod _{i=1}^r{f}_i(\frac{x^{p^k}}{1+p{w}_0})$$

$${\theta }_i(x)\equiv a_i(\frac{x^{p^k}}{1+p{w}_0})F_i(\frac{x^{p^k}}{1+p{w}_0})=1-b_i(\frac{x^{p^k}}{1+p{w}_0})f_i(\frac{x^{p^k}}{1+p{w}_0})$$

$$\tau (a_1(x),\dots ,a_r(x))=\sum _{i=1}^r{\theta }_i(x)a_i(x)(\mathrm{mod}{x}^{p^{k}n}-(1+pw))$$

$$\alpha =\sum _{j=0}^{N-1}(a_j+b_{j}u)x^j,a_j,b_j\in {\mathbb{Z}}_{p^s},j=0,1,\dots ,N-1$$

$$\sigma :\alpha \mapsto \xi +\eta u,\forall \alpha \in ({\mathbb{Z}}_{p^s}+u{\mathbb{Z}}_{p^s})[x]/⟨x^N-(1+pw)⟩$$

$${\tau }_i(a(x)+b(x)u)$$

$${\Delta }_l^{(i)}=\{\sum _{j=0}^{l-1}{b}_j(x)f_i(x{)}^j\mid b_j(x)\in {\mathcal{T}}_i,b_0(x)\ne 0,0\le j\le l-1\}$$

$$\Omega (p^{d_i},p^{k}s)$$

$$N_{(p,d_{i},p^{k}s)}=\left\{\begin{array}[]{ll}\sum_{j=0}^{\lambda}(1+4j)p^{(\lambda-j)d_{i}}&{\rm if}\ p^{k}s=2\lambda;\cr&\cr\sum_{j=0}^{\lambda}(3+4j)p^{(\lambda-j)d_{i}}&{\rm if}\ p^{k}s=2\lambda+1.\end{array}\right.$$

$$\varrho (J)=f_i(x{)}^{l_i}{\mathcal{R}}_i\mathrm{and}\varrho (J:u)=f_i(x{)}^{m_i}{\mathcal{R}}_i.$$

$$\mathrm{Ker}(\varrho {|}_J)$$

$$|J|=p^{d_i(p^{k}s-l_i)}{p}^{d_i(p^{k}s-m_i)}=p^{d_i(2p^{k}s-(l_i+m_i))}.$$

$$u(f_i^{t_i}(x)h(x)-f_i(x{)}^{t_i^{\prime }}{h}^{\prime }(x))$$

$$N_1=\sum _{l_i=1}^{\lfloor \frac{p^{k}s+1}{2}\rfloor }\sum _{t_i=0}^{l_i-1}(p^{d_i}-1)p^{d_i(l_i-t_i-1)}=\frac{p^{d_i(\lfloor \frac{p^{k}s+1}{2}\rfloor +1)}-1}{p^{d_i}-1}-\left(\lfloor \frac{p^{k}s+1}{2}\rfloor +1\right).$$

$$N_2$$

$$\Psi (p^{d_i},p^{k}s)$$

$$p^{k}s+1+p^{k}s+\Omega (p^{d_i},p^{k}s)+\frac{1}{2}{p}^{k}s(p^{k}s-1)+(p^{d_i}-1)\Psi (p^{d_i},p^{k}s),$$

$$(1+p{\hat{w}}_0{)}^n=1+p\hat{w}\mathrm{and}(1+p{w}_0{)}^{-1}=1+p{\hat{w}}_0$$

$${\hat{\mathcal{R}}}_i={\mathbb{Z}}_{p^s}[x]/⟨f_i(x^{p^k}(1+p{\hat{w}}_0{)}^{-1})⟩$$

$$\frac{x^{p^{k}n}-(1+p\hat{w})}{(1+p{\hat{w}}_0{)}^n}={\left(\frac{x^{p^k}}{1+p{\hat{w}}_0}\right)}^n-1=\prod _{i=1}^r{f}_i(\frac{x^{p^k}}{1+p{\hat{w}}_0})$$

$${\hat{\theta }}_i(x)\equiv a_i(\frac{x^{p^k}}{1+p{\hat{w}}_0})F_i(\frac{x^{p^k}}{1+p{\hat{w}}_0})=1-b_i(\frac{x^{p^k}}{1+p{\hat{w}}_0})f_i(\frac{x^{p^k}}{1+p{\hat{w}}_0})$$

$$\hat{\tau }(a_1(x),\dots ,a_r(x))=\sum _{i=1}^r{\hat{\theta }}_i(x)a_i(x)(\mathrm{mod}{x}^{p^{k}n}-(1+p\hat{w}))$$

Full text

Keywords: Constacyclic code, Finite chain ring, Non-principal ideal ring, Dual code, Self-dual code.

MSC(2010): Primary 11T71; Secondary 94B15, 94B05.

∗Corresponding author

††copyright: ©0: Iranian Mathematical Society

1. Introduction

Algebraic coding theory deals with the design of error-correcting and error-detecting codes for the reliable transmission of information across noisy channel. The class of constacyclic codes play a very significant role in the theory of error-correcting codes. The most impotent classes of these codes are that of cyclic codes and negacyclic codes, which have been well studied since the late 1950s. Since 1999, special classes of constacyclic codes over certain classes of finite commutative chain rings have been studied by numerous authors. See [1-17], for example.

Let $A$ be a finite commutative ring with identity $1\neq 0$, $A^{\times}$ the multiplicative group of units of $A$ and $a\in A$. We denote by $\langle a\rangle_{A}$ or $\langle a\rangle$ the ideal of $A$ generated by $a$, i.e. $\langle a\rangle_{A}=aA$. For any ideal $I$ of $A$, we will identify the element $a+I$ of the residue class ring $A/I$ with $a$ (mod $I$) in this paper.

For any positive integer $N$, let $A^{N}=\{(a_{0},a_{1},\ldots,a_{N-1})\mid a_{i}\in A,\ 0\leq i\leq N-1\}$ which is an $A$-module with componentwise addition and scalar multiplication by elements of $A$. Then an $A$-submodule ${\mathcal{C}}$ of $A^{N}$ is called a linear code over $A$ of length $N$. For any vectors $a=(a_{0},a_{1},\ldots,a_{N-1}),b=(b_{0},b_{1},\ldots,b_{N-1})\in A^{N}$. The usual Euclidian inner product of $a$ and $b$ is defined by $[a,b]_{E}=\sum_{j=0}^{N-1}a_{j}b_{j}\in A$. Then $[-,-]_{E}$ is a symmetric and non-degenerate bilinear form on the $A$-module $A^{N}$. Let ${\mathcal{C}}$ be a linear code over $A$ of length $N$. The Euclidian dual code of ${\mathcal{C}}$ is defined by ${\mathcal{C}}^{\bot_{E}}=\{a\in A^{N}\mid[a,b]_{E}=0,\ \forall b\in{\mathcal{C}}\}$, and ${\mathcal{C}}$ is said to be self-dual if ${\mathcal{C}}={\mathcal{C}}^{\bot_{E}}$.

Let $\gamma\in A^{\times}$. Then a linear code ${\mathcal{C}}$ over $A$ of length $N$ is called a $\gamma$-constacyclic code if $(\gamma a_{N-1},a_{0},a_{1},\ldots,a_{N-2})\in{\mathcal{C}}$ for all $(a_{0},a_{1},\ldots,a_{N-1})\in{\mathcal{C}}$. Particularly, ${\mathcal{C}}$ is a negacyclic code if $\gamma=-1$, and ${\mathcal{C}}$ is a cyclic code if $\gamma=1$. For any $a=(a_{0},a_{1},\ldots,a_{N-1})\in A^{N}$, let $a(x)=a_{0}+a_{1}x+\ldots+a_{N-1}x^{N-1}\in A[x]/\langle x^{N}-\gamma\rangle$. We will identify $a$ with $a(x)$ in this paper. By [11] Propositions 2.2 and 2.3, we have the following conclusions.

Lemma 1.1 Let $\gamma\in A^{\times}$. Then ${\mathcal{C}}$ is a $\gamma$-constacyclic code of length $N$ over $A$ if and only if ${\mathcal{C}}$ is an ideal of the residue class ring $A[x]/\langle x^{N}-\gamma\rangle$.

Lemma 1.2 The dual code of a $\gamma$-constacyclic code of length $N$ over $A$ is a $\gamma^{-1}$-constacyclic code of length $N$ over $A$, i.e., an ideal of $A[x]/\langle x^{N}-\gamma^{-1}\rangle$.

Recent years, codes over finite non-principal ideal commutative rings have been studied by many authors. For example, in Yildiz et al [18], MacWilliams identities, projections, and formally self-dual codes for linear codes over the ring $\mathbb{Z}_{4}+u\mathbb{Z}_{4}$ and their application to real and complex lattices have been studied.

From now on, let $p$ be an arbitrary prime, $s$ an integer satisfying $s\geq 2$ and $\mathbb{Z}_{p^{s}}=\mathbb{Z}/\langle p^{s}\rangle=\{0,1,2,\ldots,p^{s}-1\}$. Denote $\mathbb{Z}_{p^{s}}[u]/\langle u^{2}\rangle$ by $\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}}=\{a+ub\mid a,b\in\mathbb{Z}_{p^{s}}\}$ ($u^{2}=0$), which is a non-principal ideal ring. The operations on $\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}}$ are defined by:

[TABLE]

for any $\alpha=a+bu,\beta=c+du\in\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}}$ with $a,b,c,d\in\mathbb{Z}_{p^{s}}$. For any fixed $w\in\mathbb{Z}_{p^{s}}^{\times}$, the following questions have not been investigated completely for $(1+pw)$-constacyclic codes over $\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}}$ of arbitrary length to the best of our knowledge:

(Q-1) Present precisely all distinct $(1+pw)$-constacyclic codes over $\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}}$ of arbitrary length $N$, and count the number of these codes.

(Q-2) For each code ${\mathcal{C}}$ presented above, determine the number of codewords contained in ${\mathcal{C}}$ and give the dual code of ${\mathcal{C}}$ precisely.

(Q-3) Determine the self-duality for $(1+pw)$-constacyclic codes over $\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}}$.

The present paper is organized as follows. In Section 2, we investigate the structure and properties of the ring $\mathbb{Z}_{p^{s}}[x]/\langle x^{p^{k}n}-(1+pw)\rangle$. In Section 3, we give a canonical form decomposition for any $(1+pw)$-constacyclic code of length $p^{k}n$ over $\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}}$ $(u^{2}=0)$, list all distinct codes by their generator sets and enumerate the number of all codes and the number of codewords in each code respectively. By use of the canonical form decomposition, we obtain the dual code of each code and investigate the self-duality of these codes in Section 4. In Section 5, we list all 6419 self-dual $3$-constacyclic code over $\mathbb{Z}_{8}+u\mathbb{Z}_{8}$ of length $14$.

2. Direct Sum Decomposition of The Ring $\mathbb{Z}_{p^{s}}[x]/\langle x^{p^{k}n}-(1+pw)\rangle$

From now on, let $N=p^{k}n$ where $p$ is a prime, $k,n$ are positive integers satisfying ${\rm gcd}(p,n)=1$. We consider how to decompose the ring $\mathbb{Z}_{p^{s}}[x]/\langle x^{N}-(1+pw)\rangle$ into a direct sum of finite chain rings where $s\geq 2$ and $w\in\mathbb{Z}_{p^{s}}^{\times}$. This decomposition will be used in the following sections.

Lemma 2.1 ([10] Proposition 2.1) Let $A$ be a finite associative and commutative ring with identity. Then the following conditions are equivalent:

(i) $A$* is a local ring and the maximal ideal $M$ of $A$ is principal, i.e., $M=\langle\pi\rangle$ for some $\pi\in A$*;

(ii) $A$* is a local principal ideal ring*;

(iii) $A$* is a chain ring with ideals $\langle\pi^{i}\rangle$, $0\leq i\leq\nu$, where $\nu$ is the nilpotency index of $\pi$*.

Lemma 2.2 ([10] Proposition 2.2) Let $A$ be a finite commutative chain ring, with maximal ideal $M=\langle\pi\rangle$, and let $\nu$ be the nilpotency index of $\pi$. Then

(i) For some prime $p$ and positive integer $m$, $|A/\langle\pi\rangle|=q$ where $q=p^{m}$, $|A|=q^{\nu}$, and the characteristic of $A/\langle\pi\rangle$ and $A$ are powers of $p$;

(ii) For $i=0,1,\ldots,\nu$, $|\langle\pi^{i}\rangle|=q^{\nu-i}$.

Lemma 2.3 ([14] Lemma 2.4)Using the notations in Lemma 2.2, let $V\subseteq A$ be a system of representatives for the equivalence classes of $A$ under congruence modulo $\pi$. (Equivalently, we can define $V$ to be a maximal subset of $A$ with the property that $r_{1}-r_{2}\not\in\langle\pi\rangle$ for all $r_{1},r_{2}\in V$, $r_{1}\neq r_{2}$.) Then

(i) Every element $a$ of $A$ has a unique $\pi$-adic expansion: $a=\sum_{j=0}^{\nu-1}r_{j}\pi^{j}$, $r_{0},r_{1},\ldots$, $r_{\nu-1}\in V$.

(ii) $|A/\langle\pi\rangle|=|V|$* and $|\langle\pi^{i}\rangle|=|V|^{\nu-i}$ for $0\leq i\leq\nu-1$*.

Let $a\in\mathbb{Z}_{p^{s}}$. Then $a$ has a unique $p$-adic expansion:

$a=\sum_{j=0}^{s-1}p^{j}a_{j},\ a_{0},a_{1},\ldots,a_{s-1}\in\mathbb{F}_{p}=\{0,1,\ldots,p-1\},$

where we regard $\mathbb{F}_{p}$ as a subset of $\mathbb{Z}_{p^{s}}$ (but $\mathbb{F}_{p}$ is not a subring of $\mathbb{Z}_{p^{s}}$). It is well known that $a\in\mathbb{Z}_{p^{s}}^{\times}$ if and only if $a_{0}\neq 0$. Denote $\overline{a}=a_{0}\in\mathbb{F}_{p}$. Then ${}^{-}:a\mapsto\overline{a}$ ($\forall a\in\mathbb{Z}_{p^{s}}$) is a ring homomorphism from $\mathbb{Z}_{p^{s}}$ onto $\mathbb{F}_{p}$, and this homomorphism can be extended to a ring homomorphism from $\mathbb{Z}_{p^{s}}[y]$ onto $\mathbb{F}_{p}[y]$ by $\overline{f}(y)=\sum_{i=0}^{m}\overline{b}_{i}y^{i}$ for any $f(y)=\sum_{i=0}^{m}b_{i}y^{i}\in\mathbb{Z}_{p^{s}}[y]$. A monic polynomial $f(y)\in\mathbb{Z}_{p^{s}}[y]$ of positive degree is said to be basic irreducible if $\overline{f}(y)$ is an irreducible polynomial in $\mathbb{F}_{p}[y]$ ([16] Chapter 13, Page 328).

Lemma 2.4 Let $f(x)$ be a monic basic irreducible polynomial in $\mathbb{Z}_{p^{s}}[x]$ of degree $d$ and denote $\Gamma=\mathbb{Z}_{p^{s}}[x]/\langle f(x)\rangle$. Then

(i) ([16] Theorem 14.1]) $\Gamma$* is a Galois ring of characteristic $p^{s}$ and cardinality $p^{sd}$. Moreover, $\Gamma=\mathbb{Z}_{p^{s}}[\zeta]$ where $\zeta=x+\langle f(x)\rangle\in\Gamma$ satisfying $\zeta^{p^{d}-1}=1$*.

Denote $\overline{\Gamma}=\mathbb{F}_{p}[x]/\langle\overline{f}(x)\rangle$ and $\overline{\zeta}=x+\langle\overline{f}(x)\rangle\in\overline{\Gamma}$. Then $\overline{\Gamma}=\mathbb{F}_{p}[\overline{\zeta}]$ which is a finite field of cardinality $p^{d}$, $\overline{f}(x)=\prod_{i=0}^{d-1}(x-\overline{\zeta}^{p^{i}})$ and that - can be extended to a ring homomorphism from $\Gamma$ onto $\overline{\Gamma}$ by $\xi\mapsto\overline{\xi}=\sum_{j=0}^{d-1}\overline{a}_{j}\overline{\zeta}^{j}$, for all $\xi=\sum_{j=0}^{d-1}a_{j}\zeta^{j}\in\Gamma$ where $a_{0},a_{1},\ldots,a_{d-1}\in\mathbb{Z}_{p^{s}}$.

(ii) ([5] Lemma 2.3(ii)) $f(x)=\prod_{i=0}^{d-1}(x-\zeta^{p^{i}})$.

Theorem 2.5 Let $f(x)$ be a monic basic irreducible polynomial in $\mathbb{Z}_{p^{s}}[x]$ of degree $d$, $w_{0}\in\mathbb{Z}_{p^{s}}^{\times}$, denote ${\mathcal{R}}=\mathbb{Z}_{p^{s}}[x]/\langle f(x^{p^{k}}(1+pw_{0})^{-1})\rangle$ and set

${\mathcal{T}}=\{\sum_{j=0}^{d-1}a_{j}x^{j}\mid a_{0},a_{1},\ldots,a_{d-1}\in\mathbb{F}_{p}\}\subseteq{\mathcal{R}}$

in which we regard $\mathbb{F}_{p}$ as a subset of $\mathbb{Z}_{p^{s}}$. Then

(i) ${\mathcal{R}}$* is a finite chain ring with maximal ideal $\langle f(x)\rangle$ generated by $f(x)$, i.e., $\langle f(x)\rangle=f(x){\mathcal{R}}$, the nilpotency index of $f(x)$ is equal to $p^{k}s$ and ${\mathcal{R}}/\langle f(x)\rangle$ is a finite field of cardinality $p^{d}$*.

(ii) Each $\alpha\in{\mathcal{R}}$ has a unique $f(x)$-adic expansion: $\alpha=\sum_{j=0}^{p^{k}s-1}b_{j}(x)f(x)^{j}$, $b_{0}(x),b_{1}(x),\ldots,b_{p^{k}s-1}(x)\in{\mathcal{T}}.$

(iii) For each integer $l$, $1\leq l\leq p^{k}s$, let $\langle f(x)^{l}\rangle$ be the ideal of ${\mathcal{R}}$ generated by $f(x)^{l}$. Then $\langle f(x)^{l}\rangle=\{\sum_{j=l}^{p^{k}s-1}b_{j}(x)f(x)^{j}\mid b_{l}(x),\ldots,b_{p^{k}s-1}(x)\in{\mathcal{T}}\},$ and every element $\beta$ of the residue class ring ${\mathcal{R}}/\langle f(x)^{l}\rangle$ has a unique $f(x)$-adic expansion: $\beta=\sum_{j=0}^{l-1}b_{j}(x)f(x)^{j},\ b_{0}(x),\ldots,b_{l-1}(x)\in{\mathcal{T}}$.

Moreover, $\beta$ is an invertible element of ${\mathcal{R}}/\langle f(x)^{l}\rangle$, i.e. $\beta\in({\mathcal{R}}/\langle f(x)^{l}\rangle)^{\times}$, if and only if $b_{0}(x)\neq 0$. Hence $|\langle f(x)^{l}\rangle|=p^{d(p^{k}s-l)}$ and $|({\mathcal{R}}/\langle f(x)^{l}\rangle)^{\times}|=(p^{d}-1)p^{(l-1)d}$.

Proof Let ${\mathcal{R}}=\mathbb{Z}_{p^{s}}[x]/\langle f(x^{p^{k}}(1+pw_{0})^{-1})\rangle$. It is known that

[TABLE]

(A direct proof for this equation is given in Appendix). Then

[TABLE]

as ideals of ${\mathcal{R}}$, and hence $f(x)^{p^{k}s}=p^{s}\vartheta(x)^{s}=0$ in ${\mathcal{R}}$ by Equation (2.1).

(i) Let $J=\langle p,f(x)\rangle$ be the ideal of ${\mathcal{R}}$ generated by $p$ and $f(x)$. Then

[TABLE]

up to natural ring isomorphisms, where $\mathbb{F}_{p}[x]/\overline{f}(x)\rangle$ is a finite field of $p^{d}$ elements by Lemma 2.4(i). Hence $J$ is a maximal ideal of ${\mathcal{R}}$. Since $f(x)^{p^{k}s}=0$, both $p$ and $f(x)$ are nilpotent elements of ${\mathcal{R}}$. From this one can verify easily that every element in ${\mathcal{R}}\setminus J$ is invertible, which implies that ${\mathcal{R}}$ is a local ring with $J$ as its unique maximal ideal. Furthermore, by Equation (2.2) we conclude that $J=\langle f(x)\rangle$ and so ${\mathcal{R}}/\langle f(x)\rangle=\mathbb{F}_{p}[x]/\langle\overline{f}(x)\rangle$.

As stated above, by Lemma 2.1 we see that ${\mathcal{R}}$ is a finite chain ring with the unique maximal ideal generated by $f(x)$. Let $z$ be the nilpotency index of $f(x)$. By Lemma 2.2(i) it follows that $|{\mathcal{R}}|=|\mathbb{F}_{p}[x]/\overline{f}(x)\rangle|^{z}=p^{zd}$. On the other hand, by ${\deg}(f(x^{p^{k}}(1+pw_{0})^{-1}))=p^{k}d$ it follows that

$|{\mathcal{R}}|=|\mathbb{Z}_{p^{s}}[x]/\langle f(x^{p^{k}}(1+pw_{0})^{-1})\rangle|=(p^{s})^{p^{k}d}=p^{p^{k}sd}$.

Therefore, $z=p^{k}s$.

(ii) By ${\mathcal{R}}/\langle f(x)\rangle=\mathbb{F}_{p}[x]/\langle\overline{f}(x)\rangle=\{\sum_{j=0}^{d-1}a_{j}x^{j}\mid a_{0},a_{1},\ldots,a_{d-1}\in\mathbb{F}_{p}\}$ and $\mathbb{F}_{p}[x]/\langle\overline{f}(x)\rangle={\mathcal{T}}$ as sets, one can verify easily that ${\mathcal{T}}$ is a system of representatives for the equivalence classes of ${\mathcal{R}}$ under congruence modulo $f(x)$. Then the conclusion follows from Lemma 2.3(i) immediately.

(iii) It follows from (ii), Lemma 2.2 and the general finite chain ring theory (see [14], for example). $\Box$

Next, we consider how to decompose the ring $\mathbb{Z}_{p^{s}}[x]/\langle x^{N}-(1+pw)\rangle$ into a direct sum of finite chain rings, where $N=p^{k}n$ and ${\rm gcd}(p,n)=1$.

Lemma 2.6 Let $w\in\mathbb{Z}_{p^{s}}^{\times}$. Then ${\rm ord}(1+pw)=p^{v}$ for some positive integer $v$, and there exists a unique $w_{0}\in\mathbb{Z}_{p^{s}}^{\times}$ modulo $p^{s-1}$ such that $(1+pw_{0})^{n}=1+pw$.

Proof It is known that $1+p\mathbb{Z}_{p^{s}}$ is a multiplicative subgroup of $\mathbb{Z}_{p^{s}}^{\times}$ with order $p^{s-1}$ (cf. [16] Corollary 14.12]). Hence ${\rm ord}(1+pw)=p^{v}$ for some positive integer $v$. Since ${\rm gcd}(n,p)=1$, the mapping $1+pa\mapsto(1+pa)^{n}$ ($\forall a\in\mathbb{Z}_{p^{s}}$) is an automorphism of the multiplicative group $1+p\mathbb{Z}_{p^{s}}$. Hence there is a unique element $w_{0}\in\mathbb{Z}_{p^{s}}$ modulo $p^{s-1}$ such that $(1+pw_{0})^{n}=1+pw$.

As every element of $\mathbb{Z}_{p^{s}}$ has a unique $p$-expansion, there is a unique integer $k$, $0\leq k\leq s-1$, such that $w_{0}=p^{k}b$ for some $b\in\mathbb{Z}_{p^{s}}^{\times}$. Suppose that $k\geq 1$, then $1+pw_{0}=1+p^{k+1}b$, which implies $(1+pw_{0})^{n}=1+p^{k+1}bn+p^{2(k+1)}c=1+p(p^{k}bn+p^{2k+1}c)$ for some $c\in\mathbb{Z}_{p^{s}}$. From this and by $(1+pw_{0})^{n}=1+pw$ we deduce $pw=p(p^{k}bn+p^{2k+1}c)$, i.e., $w=p^{k}bn+p^{2k+1}c+p^{s-1}e$ for some $e\in\mathbb{Z}_{p^{s}}$, and hence $w^{s}=0$, which contradicts that $w\in\mathbb{Z}_{p^{s}}^{\times}$. Therefore, we conclude that $k=0$ and hence $w_{0}=b\in\mathbb{Z}_{p^{s}}^{\times}$. $\Box$

In the rest of this paper, we adopt the following notations.

Notation 2.7 Let $w,w_{0}\in\mathbb{Z}_{p^{s}}^{\times}$ satisfying $(1+pw_{0})^{n}=1+pw$, and assume

[TABLE]

where $f_{1}(y),\ldots,f_{r}(y)$ are pairwise coprime monic basic irreducible polynomials in $\mathbb{Z}_{p^{s}}[y]$. For each $i$, $1\leq i\leq r$, assume ${\rm deg}(f_{i}(y))=d_{i}$ and denote

[TABLE]

Let $1\leq i\leq r$. By Lemma 2.4(i) we know that $\Gamma_{i}$ is a Galois ring of characteristic $p^{s}$ and cardinality $p^{sd_{i}}$ and $\Gamma_{i}=\mathbb{Z}_{p^{s}}[\zeta_{i}]$, where $\zeta_{i}=y+\langle f_{i}(y)\rangle\in\Gamma_{i}$ satisfying $\zeta_{i}^{p^{d_{i}}-1}=1$. By Theorem 2.5 we have the following corollary.

Corollary 2.8 Denote ${\mathcal{T}}_{i}=\{\sum_{j=0}^{d-1}a_{j}x^{j}\mid a_{0},a_{1},\ldots,a_{d_{i}-1}\in\mathbb{F}_{p}\}\subseteq{\mathcal{R}}_{i}$ in which we regard $\mathbb{F}_{p}$ as a subset of $\mathbb{Z}_{p^{s}}$. Then

(i) ${\mathcal{R}}_{i}$* is a finite chain ring with maximal ideal $\langle f_{i}(x)\rangle$ generated by $f_{i}(x)$, the nilpotency index of $f_{i}(x)$ is equal to $p^{k}s$ and ${\mathcal{R}}_{i}/\langle f_{i}(x)\rangle$ is a finite field of cardinality $p^{d_{i}}$*.

(ii) Every element $\alpha$ of ${\mathcal{R}}_{i}$ has a unique $f_{i}(x)$-adic expansion:

$\alpha=\sum_{j=0}^{p^{k}s-1}b_{j}(x)f_{i}(x)^{j},\ b_{0}(x),b_{1}(x),\ldots,b_{p^{k}s-1}(x)\in{\mathcal{T}}_{i}.$

(iii) For each integer $l$, $1\leq l\leq p^{k}s$, let $\langle f_{i}(x)^{l}\rangle$ be the ideal of ${\mathcal{R}}_{i}$ generated by $f_{i}(x)^{l}$. Then $\langle f_{i}(x)^{l}\rangle=\{\sum_{j=l}^{p^{k}s-1}b_{j}(x)f_{i}(x)^{j}\mid b_{l}(x),\ldots,b_{p^{k}s-1}(x)\in{\mathcal{T}}_{i}\},$ and every element $\beta$ of the residue class ring ${\mathcal{R}}_{i}/\langle f_{i}(x)^{l}\rangle$ has a unique $f_{i}(x)$-adic expansion: $\beta=\sum_{j=0}^{l-1}b_{j}(x)f_{i}(x)^{j},\ b_{0}(x),\ldots,b_{l-1}(x)\in{\mathcal{T}}_{i}.$

Hence $\beta$ is an invertible element of ${\mathcal{R}}_{i}/\langle f_{i}(x)^{l}\rangle$, i.e. $\beta\in({\mathcal{R}}_{i}/\langle f_{i}(x)^{l}\rangle)^{\times}$, if and only if $b_{0}(x)\neq 0$. Moreover, $|f_{i}(x)^{l}{\mathcal{R}}_{i}|=|\langle f_{i}(x)^{l}\rangle|=p^{d_{i}(p^{k}s-l)}$ and $|({\mathcal{R}}_{i}/\langle f_{i}(x)^{l}\rangle)^{\times}|=(p^{d_{i}}-1)p^{(l-1)d_{i}}.$

Let $1\leq i\leq r$ and denote $F_{i}(y)=\frac{y^{n}-1}{f_{i}(y)}\in\mathbb{Z}_{p^{s}}[y]$. Since $F_{i}(y)$ and $f_{i}(y)$ are coprime, there are polynomials $a_{i}(y),b_{i}(y)\in\mathbb{Z}_{p^{s}}[y]$ such that

[TABLE]

Substituting $x^{p^{k}}(1+pw_{0})^{-1}=\frac{x^{p^{k}}}{1+pw_{0}}$ for $y$ in (2.3) and (2.4), by $(1+pw_{0})^{n}=1+pw$ we obtain

[TABLE]

and $a_{i}(\frac{x^{p^{k}}}{1+pw_{0}})F_{i}(\frac{x^{p^{k}}}{1+pw_{0}})+b_{i}(\frac{x^{p^{k}}}{1+pw_{0}})f_{i}(\frac{x^{p^{k}}}{1+pw_{0}})=1$ in the ring $\mathbb{Z}_{p^{s}}[x]$, respectively. It is clear that ${\rm deg}(f_{i}(x^{p^{k}}(1+pw_{0})^{-1}))=p^{k}d_{i}$ for $i=1,\ldots,r$. In the rest of this paper, we set

[TABLE]

(mod $x^{p^{k}n}-(1+pw)$). Then from Chinese Remainder Theorem for commutative rings with identity, we deduce that following theorem.

Theorem 2.9 Denote ${\mathcal{A}}=\mathbb{Z}_{p^{s}}[x]/\langle x^{p^{k}n}-(1+pw)\rangle$. We have the following:

(i) $\theta_{1}(x)+\ldots+\theta_{r}(x)=1$, $\theta_{i}(x)^{2}=\theta_{i}(x)$ and $\theta_{i}(x)\theta_{j}(x)=0$ in ${\mathcal{A}}$ for all $1\leq i\neq j\leq r$.

(ii) ${\mathcal{A}}={\mathcal{A}}_{1}\oplus\ldots\oplus{\mathcal{A}}_{r}$, where ${\mathcal{A}}_{i}=\theta_{i}(x){\mathcal{A}}$ and its multiplicative identity is $\theta_{i}(x)$. Moreover, this decomposition is a direct sum of rings in that ${\mathcal{A}}_{i}{\mathcal{A}}_{j}=\{0\}$ for all $i$ and $j$, $1\leq i\neq j\leq r$.

(iii) For each $1\leq i\leq r$, define a mapping $\tau_{i}:a(x)\mapsto\theta_{i}(x)a(x)$ $(\forall a(x)\in{\mathcal{R}}_{i}=\mathbb{Z}_{p^{s}}[x]/\langle f_{i}(x^{p^{k}}(1+pw_{0})^{-1})\rangle)$. Then $\tau_{i}$ is a ring isomorphism from ${\mathcal{R}}_{i}$ onto ${\mathcal{A}}_{i}$. Hence $|{\mathcal{A}}_{i}|=p^{p^{k}sd_{i}}$.

(iv) Define $\tau:(a_{1}(x),\ldots,a_{r}(x))\mapsto\tau_{1}(a_{1}(x))+\ldots+\tau_{r}(a_{r}(x))$, i.e.

[TABLE]

for all $a_{i}(x)\in{\mathcal{R}}_{i}$, $i=1,\ldots,r$. Then $\tau$ is a ring isomorphism from ${\mathcal{R}}_{1}\times\ldots$ $\times{\mathcal{R}}_{r}$ onto ${\mathcal{A}}$.

3. Structure of $(1+pw)$-Constacyclic Codes over $\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}}$

In this section, we list all distinct $(1+pw)$-constacyclic codes over the ring $\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}}$ ($u^{2}=0$) of length $N$, i.e., all distinct ideals of the ring $(\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}})[x]/\langle x^{N}-(1+pw)\rangle$ where $N=p^{k}n$ and $w\in\mathbb{Z}_{p^{s}}^{\times}$.

For any $\alpha\in(\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}})[x]/\langle x^{N}-(1+pw)\rangle$, $\alpha$ can be expressed as

[TABLE]

uniquely. Denote $\xi=\sum_{j=0}^{N-1}a_{j}x^{j}$ and $\eta=\sum_{j=0}^{N-1}b_{j}x^{j}$. Then $\xi,\eta\in{\mathcal{A}}=\mathbb{Z}_{p^{s}}[x]/\langle x^{N}-(1+pw)\rangle$ and the map

[TABLE]

is a ring isomorphism from $(\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}})[x]/\langle x^{N}-(1+pw)\rangle$ onto ${\mathcal{A}}+u{\mathcal{A}}$ ($u^{2}=0$). In the rest of this paper, we will identify $(\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}})[x]/\langle x^{N}-(1+pw)\rangle$ with ${\mathcal{A}}+u{\mathcal{A}}$ under this isomorphism $\sigma$. Moreover, for any $1\leq i\leq r$ let ${\mathcal{R}}_{i}+u{\mathcal{R}}_{i}={\mathcal{R}}_{i}[u]/\langle u^{2}\rangle$, where ${\mathcal{R}}_{i}=\mathbb{Z}_{p^{s}}[x]/\langle f_{i}(x^{p^{k}}(1+pw_{0})^{-1})\rangle$.

Lemma 3.1 Let $1\leq i\leq r$. Using the notations of Theorem 2.9, for any $a(x),b(x)\in{\mathcal{R}}_{i}$ we define

[TABLE]

Then $\tau_{i}$ is a ring isomorphism from ${\mathcal{R}}_{i}+u{\mathcal{R}}_{i}$ onto ${\mathcal{A}}_{i}+u{\mathcal{A}}_{i}$ ($u^{2}=0$).

Proof By Theorem 2.9(iii), the isomorphism $\tau_{i}:{\mathcal{R}}_{i}\rightarrow{\mathcal{A}}_{i}$ induces an isomorphism of polynomial rings from ${\mathcal{R}}_{i}[u]$ onto ${\mathcal{A}}_{i}[u]$ in the natural way that $\sum_{j}a_{j}(x)u^{j}\mapsto\sum_{j}\tau_{i}(a_{j}(x))u^{j}$ ($\forall a_{j}(x)\in{\mathcal{R}}_{i}$). Hence $\tau_{i}$ is a ring isomorphism from ${\mathcal{R}}_{i}+u{\mathcal{R}}_{i}={\mathcal{R}}_{i}[u]/\langle u^{2}\rangle$ onto ${\mathcal{A}}_{i}+u{\mathcal{A}}_{i}={\mathcal{A}}_{i}[u]/\langle u^{2}\rangle$. $\Box$

Lemma 3.2 Using the notations above and the notations of Section 2, The following statements are equivalent:

(i) ${\mathcal{C}}$* is a $(1+pw)$-constacyclic code over $\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}}$ of length $p^{k}n$*.

(ii) ${\mathcal{C}}$* is an ideal of ${\mathcal{A}}+u{\mathcal{A}}$*.

(iii) For each $1\leq i\leq r$, there is a unique ideal $C_{i}$ of the ring ${\mathcal{R}}_{i}+u{\mathcal{R}}_{i}$ $(u^{2}=0)$ such that ${\mathcal{C}}=\oplus_{i=1}^{r}\theta_{i}(x)C_{i}\ ({\rm mod}\ x^{p^{k}n}-(1+pw)).$

Proof (i)$\Leftrightarrow$ (ii) It follows from the identification of $(\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}})[x]/\langle x^{p^{k}n}-(1+pw)\rangle$ with ${\mathcal{A}}+u{\mathcal{A}}$.

(ii)$\Leftrightarrow$ (iii) By Theorem 2.9(ii) we have ${\mathcal{A}}=\oplus_{i=1}^{r}{\mathcal{A}}_{i}$. Hence ${\mathcal{A}}+u{\mathcal{A}}={\mathcal{A}}[u]/\langle u^{2}\rangle=\oplus_{i=1}^{r}({\mathcal{A}}_{i}[u]/\langle u^{2}\rangle)=\oplus_{i=1}^{r}({\mathcal{A}}_{i}+u{\mathcal{A}}_{i})$ and this decomposition is a direct sum of rings in that $({\mathcal{A}}_{i}+u{\mathcal{A}}_{i})({\mathcal{A}}_{j}+u{\mathcal{A}}_{j})=\{0\}$ for any $i,j$, $1\leq i\neq j\leq r$. Therefore, ${\mathcal{C}}$ is an ideal of ${\mathcal{A}}+u{\mathcal{A}}$ if and only if for each $1\leq i\leq r$, there is a unique ideal ${\mathcal{C}}_{i}$ of the ring ${\mathcal{A}}_{i}+u{\mathcal{A}}_{i}$ such that ${\mathcal{C}}=\oplus_{i=1}^{r}{\mathcal{C}}_{i}$. From this and by Lemma 3.1, we deduce that ${\mathcal{C}}_{i}$ is an ideal of ${\mathcal{A}}_{i}+u{\mathcal{A}}_{i}$ if and only if there is a unique ideal $C_{i}$ of ${\mathcal{R}}_{i}+u{\mathcal{R}}_{i}$ such that ${\mathcal{C}}_{i}=\tau_{i}(C_{i})=\theta_{i}(x)C_{i}=\{\theta_{i}(x)c_{i}(x)\mid c_{i}(x)\in C_{i}\}$ (mod $x^{p^{k}n}-(1+pw)$). $\Box$

In order to present all $(1+pw)$-constacyclic codes over $\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}}$ of length $p^{k}n$, by Lemma 3.2 it is sufficient to give all ideals of the ring ${\mathcal{R}}_{i}+u{\mathcal{R}}_{i}$ for all $i=1,\ldots,r$. Now, we give the following theorem.

Theorem 3.3 Using the notations above, in the rest of this paper for any $1\leq l\leq p^{k}s$ we denote $({\mathcal{R}}_{i}[x]/\langle f_{i}(x)^{l}\rangle)^{\times}$ by $\Delta_{l}^{(i)}$, i.e.,

[TABLE]

Then all distinct ideals $C_{i}$ of ${\mathcal{R}}_{i}+u{\mathcal{R}}_{i}$ are given by the following table:

[TABLE]

where

[TABLE]

if $p^{k}s$ is even;

[TABLE]

if $p^{k}s$ is odd, and $\Psi(p^{d_{i}},p^{k}s)$ can be calculated by the following recurrence formula:

$\Psi(p^{d_{i}},t)=0$* for $t=1,2,3$, $\Psi(p^{d_{i}},t)=1$ for $t=4$*;

$\Psi(p^{d_{i}},t)=\Psi(p^{d_{i}},t-1)+\sum_{j=1}^{\lfloor\frac{t}{2}\rfloor-1}(t-2j-1)p^{d_{i}(j-1)}$* for $t\geq 5$*.

Therefore, the number of all distinct ideals of the ring ${\mathcal{R}}_{i}+u{\mathcal{R}}_{i}$ is equal to

[TABLE]

Proof We define a map $\varrho:{\mathcal{R}}_{i}+u{\mathcal{R}}_{i}\rightarrow{\mathcal{R}}_{i}$ by $\varrho(\alpha+u\beta)=\alpha$ ($\forall\alpha,\beta\in{\mathcal{R}}_{i}$). Then $\varrho$ is a surjective ring homomorphism from ${\mathcal{R}}_{i}+u{\mathcal{R}}_{i}$ onto ${\mathcal{R}}_{i}$.

Let $J$ be an ideal of ${\mathcal{R}}_{i}+u{\mathcal{R}}_{i}$ and $\varrho|_{J}$ be the restriction of $\varrho$ to $J$. Then $\varrho|_{J}$ is a surjective ring homomorphism from $J$ onto $\varrho(J)=\{\varrho(\xi)\mid\xi\in J\}$, which implies $\varrho(J)\cong J/{\rm Ker}(\varrho|_{J})$ where ${\rm Ker}(\varrho|_{J})=\{\xi\in J\mid\varrho(\xi)=0\}$ is the kernel of $\varrho|_{J}$. Therefore, $|J|=|\varrho(J)||{\rm Ker}(\varrho|_{J})|$.

Let $(J:u)=\{\xi\in{\mathcal{R}}_{i}+u{\mathcal{R}}_{i}\mid u\xi\in J\}$. Then $(J:u)$ is an ideal of ${\mathcal{R}}_{i}+u{\mathcal{R}}_{i}$ satisfying $J\subseteq(J:u)$. Since $\varrho$ is a surjective ring homomorphism, we see that $\varrho(J)$ and $\varrho(J:u)$ are ideals of ${\mathcal{R}}_{i}$ satisfying $\varrho(J)\subseteq\varrho(J:u)$. As ${\mathcal{R}}_{i}$ is a finite chain ring described in Section 2, there is a unique pair $(l_{i},m_{i})$ of integers, $0\leq m_{i}\leq l_{i}\leq p^{k}s$, such that

[TABLE]

By the definition of $\varrho$, we have

[TABLE]

which implies $|{\rm Ker}(\varrho|_{J})|=|\varrho(J:u)|$. From this, by Equation (3.1) we deduce that $|J|=|f_{i}(x)^{l_{i}}{\mathcal{R}}_{i}||f_{i}(x)^{m_{i}}{\mathcal{R}}_{i}|$. Then by Corollary 2.8(iii) we have

[TABLE]

If $m_{i}=p^{k}s$, then $l_{i}=p^{k}s$ and $J=\{0\}=\langle f_{i}(x)^{p^{k}s}\rangle$. In the following, we assume $m_{i}\leq p^{k}s-1$. Then we have the following cases.

Case (i) $m_{i}=l_{i}$ where $0\leq l_{i}\leq p^{k}s-1$. In this case, we have $|J|=p^{d_{i}(2p^{k}s-2l_{i})}=p^{2d_{i}(p^{k}s-l_{i})}$ by Equation (3.2).

By $f_{i}(x)^{l_{i}}\in\varrho(J)$ there exists $\alpha\in{\mathcal{R}}_{i}$ such that $f_{i}(x)^{l_{i}}+u\alpha\in J$, which implies $\langle f_{i}(x)^{l_{i}}+u\alpha\rangle\subseteq J$. Conversely, let $\xi\in J$. By $\varrho(J)=f_{i}(x)^{l_{i}}{\mathcal{R}}_{i}$, there exist $\gamma,\beta\in{\mathcal{R}}_{i}$ such that $\xi=f_{i}(x)^{l_{i}}\gamma+u\beta$, which implies $u(\beta-\gamma\alpha)=\xi-\gamma(f_{i}(x)^{l_{i}}+u\alpha)\in J$, and hence $\beta-\gamma\alpha\in\varrho(J:u)$. But $\varrho(J:u)=f_{i}(x)^{l_{i}}{\mathcal{R}}_{i}$, there exists $\delta\in{\mathcal{R}}_{i}$ such that $\beta-\gamma\alpha=f_{i}(x)^{l_{i}}\delta$. Hence $\xi=\gamma(f_{i}(x)^{l_{i}}+u\alpha)+uf_{i}(x)^{l_{i}}\delta=(\gamma+u\delta)(f_{i}(x)^{l_{i}}+u\alpha)\in\langle f_{i}(x)^{l_{i}}+u\alpha\rangle$. Therefore, $J=\langle f_{i}(x)^{l_{i}}+u\alpha\rangle$.

If $l_{i}=0$, then $f_{i}(x)^{l_{i}}+u\alpha=1+u\alpha$ which is an invertible element of ${\mathcal{R}}_{i}+u{\mathcal{R}}_{i}$ ($u^{2}=0$). Hence $J={\mathcal{R}}_{i}+u{\mathcal{R}}_{i}=\langle f_{i}(x)^{0}\rangle$. In the following, we assume that $1\leq l_{i}\leq p^{k}s-1$.

Since every element of ${\mathcal{R}}_{i}$ has a unique $f_{i}(x)$-expansion, by $uf_{i}(x)^{l_{i}}=u(f_{i}(x)^{l_{i}}+u\alpha)\in J$ we may assume that $\alpha=0$ or $\alpha=f_{i}(x)^{t_{i}}h(x)$, where $0\leq t_{i}<l_{i}$ and $h(x)=\sum_{j=0}^{p^{k}s-t_{i}-1}h_{j}(x)f_{i}(x)^{j}$ with $h_{0}(x),\ldots,h_{p^{k}s-t_{i}-1}(x)\in{\mathcal{T}}_{i}$ satisfying $h_{0}(x)\neq 0$.

If $\alpha=0$, $J=\langle f_{i}(x)^{l_{i}}\rangle$ where $1\leq l_{i}\leq p^{k}s-1$.

Otherwise, we have $J=\langle f_{i}(x)^{l_{i}}+uf_{i}(x)^{t_{i}}h(x)\rangle$. As $h_{0}(x)\neq 0$, $h(x)$ is an invertible element of ${\mathcal{R}}_{i}$ and

$uf_{i}(x)^{p^{k}s-l_{i}+t_{i}}=f_{i}(x)^{p^{k}s-l_{i}}h(x)^{-1}(f_{i}(x)^{l_{i}}+uf_{i}(x)^{t_{i}}h(x))\in J$,

which implies $f_{i}(x)^{p^{k}s-l_{i}+t_{i}}\in\varrho(J:u)=f_{i}(x)^{l_{i}}{\mathcal{R}}_{i}$. Hence $p^{k}s-l_{i}+t_{i}\geq l_{i}$, i.e., $t_{i}\geq 2l_{i}-p^{k}s$.

Now, let $h^{\prime}(x)=\sum_{j=0}^{p^{k}s-t_{i}^{\prime}-1}h_{j}^{\prime}(x)f_{i}(x)^{j}$ satisfy $J=\langle f_{i}(x)^{l_{i}}+uf_{i}(x)^{t_{i}^{\prime}}h^{\prime}(x)\rangle$, where $0\leq t_{i}^{\prime}<l_{i}$, $h_{0}^{\prime}(x),\ldots,h_{p^{k}s-t_{i}^{\prime}-1}^{\prime}(x)\in{\mathcal{T}}_{i}$ and $h_{0}^{\prime}(x)\neq 0$. Then

[TABLE]

which implies $f_{i}^{t_{i}}(x)h(x)-f_{i}(x)^{t_{i}^{\prime}}h^{\prime}(x)\in f_{i}(x)^{l_{i}}{\mathcal{R}}_{i}$. As $t_{i},t_{i}^{\prime}<l_{i}$, the condition is equivalent to $t_{i}=t_{i}^{\prime}$ and $h(x)\equiv h^{\prime}(x)$ (mod $f_{i}(x)^{l_{i}-t_{i}}$). Therefore, all distinct ideals of ${\mathcal{R}}_{i}+u{\mathcal{R}}_{i}$ are given by: $J=\langle f_{i}(x)^{l_{i}}+uf_{i}(x)^{t_{i}}h(x)\rangle$, where $h(x)\in({\mathcal{R}}_{i}/\langle f_{i}(x)^{l_{i}-t_{i}}\rangle)^{\times}=\Delta_{l_{i}-t_{i}}^{(i)}$ and $t_{i}\geq 2l_{i}-p^{k}s$.

Case (ii) $l_{i}=p^{k}s$ and $0\leq m_{i}\leq p^{k}s-1$.

In this case, we have $|J|=p^{d_{i}(2p^{k}s-(l_{i}+m_{i}))}=p^{d_{i}(p^{k}s-m_{i})}$ by Equation (3.2) and $\varrho(J)=\{0\}$. By $\varrho(J)=\{0\}$ and $\varrho(J:u)=f_{i}(x)^{m_{i}}{\mathcal{R}}_{i}$, one can easily verify that $J=\langle uf_{i}(x)^{m_{i}}\rangle$.

Case (iii) $m_{i}=0$ and $1\leq l_{i}\leq p^{k}s-1$.

In this case, we have $|J|=p^{d_{i}(2p^{k}s-(l_{i}+m_{i}))}=p^{d_{i}(2p^{k}s-l_{i})}$ by Equation (3.2). Moreover, by $1\in f_{i}(x)^{0}{\mathcal{R}}_{i}=\varrho(J:u)$ we conclude that $u\in J$. Then by $\varrho(J)=f_{i}(x)^{l_{i}}{\mathcal{R}}_{i}$ it follows that $J=\langle f_{i}(x)^{l_{i}},u\rangle$ immediately.

Case (iv) $1\leq m_{i}<l_{i}\leq p^{k}s-1$.

In this case, we have $|J|=p^{d_{i}(2p^{k}s-(l_{i}+m_{i}))}$ by Equation (3.2). By $\varrho(J)=f_{i}(x)^{l_{i}}{\mathcal{R}}_{i}$ and $\varrho(J:u)=f_{i}(x)^{m_{i}}{\mathcal{R}}_{i}$ we have $uf_{i}(x)^{m_{i}}\in J$ and there exists $\alpha\in{\mathcal{R}}_{i}$ such that $f_{i}(x)^{l_{i}}+u\alpha\in J$. It is obvious that $\langle f_{i}(x)^{l_{i}}+u\alpha,uf_{i}(x)^{m_{i}}\rangle\subseteq J$. Conversely, let $\xi\in J$. By $\varrho(\xi)\in f_{i}(x)^{l_{i}}{\mathcal{R}}_{i}$ there exist $\gamma,\beta\in{\mathcal{R}}_{i}$ such that $\xi=f_{i}(x)^{l_{i}}\gamma+u\beta$. Then by $u(\beta-\gamma\alpha)=\xi-\gamma(f_{i}(x)^{l_{i}}+u\alpha)\in J$, it follows that $\beta-\gamma\alpha\in\varrho(J:u)=f_{i}(x)^{m_{i}}{\mathcal{R}}_{i}$, which implies $\beta-\gamma\alpha=f_{i}(x)^{m_{i}}\delta$ for some $\delta\in{\mathcal{R}}_{i}$. Hence $\xi=\gamma(f_{i}(x)^{l_{i}}+u\alpha)+\delta f_{i}(x)^{m_{i}}\in\langle f_{i}(x)^{l_{i}}+u\alpha,uf_{i}(x)^{m_{i}}\rangle$. Therefore, $J=\langle f_{i}(x)^{l_{i}}+u\alpha,uf_{i}(x)^{m_{i}}\rangle$.

If $\alpha=0$, we have $J=\langle f_{i}(x)^{l_{i}},uf_{i}(x)^{m_{i}}\rangle$ where $1\leq m_{i}<l_{i}\leq p^{k}s-1$.

Otherwise, an argument similar to the proof of Case (i) shows that: $J=\langle f_{i}(x)^{l_{i}}+uf_{i}(x)^{t_{i}}h(x),uf_{i}(x)^{m_{i}}\rangle$ where $h(x)\in({\mathcal{R}}_{i}/\langle f_{i}(x)^{m_{i}-t_{i}}\rangle)^{\times}=\Delta_{m_{i}-t_{i}}^{(i)}$ and $0\leq t_{i}\leq m_{i}-1$.

By $uf_{i}(x)^{p^{k}s-l_{i}+t_{i}}=f_{i}(x)^{p^{k}s-l_{i}}h(x)^{-1}(f_{i}(x)^{l_{i}}+uf_{i}(x)^{t_{i}}h(x))\in J$, we see that $f_{i}(x)^{p^{k}s-l_{i}+t_{i}}\in\varrho(J:u)=f_{i}(x)^{m_{i}}{\mathcal{R}}_{i}$, which implies $p^{k}s-l_{i}+t_{i}\geq m_{i}$. Hence we have one of the following two cases:

($\diamondsuit$-1) $p^{k}s-l_{i}+t_{i}=m_{i}$, i.e., $m_{i}-t_{i}=p^{k}s-l_{i}$ or $l_{i}+m_{i}=p^{k}s+t_{i}$.

In this case, $\Delta_{m_{i}-t_{i}}^{(i)}=\Delta_{p^{k}s-l_{i}}^{(i)}$. Now, by

$uf_{i}(x)^{m_{i}}=f_{i}(x)^{p^{k}s-l_{i}}h(x)^{-1}(f_{i}(x)^{l_{i}}+uf_{i}(x)^{t_{i}}h(x))$

it follows that $J=\langle f_{i}(x)^{l_{i}}+uf_{i}(x)^{t_{i}}h(x)\rangle$. Moreover, we have $2l_{i}>l_{i}+m_{i}=p^{k}s+t_{i}$, i.e., $t_{i}<2l_{i}-p^{k}s$. Therefore, $h(x)\in\Delta_{p^{k}s-l_{i}}^{(i)}$ and $|J|=p^{d_{i}(2p^{k}s-(l_{i}+m_{i}))}=p^{d_{i}(2p^{k}s-(p^{k}s+t_{i}))}=p^{d_{i}(p^{k}s-t_{i})}$.

($\diamondsuit$-2) $p^{k}s-l_{i}+t_{i}>m_{i}$, i.e., $l_{i}+m_{i}\leq p^{k}s+t_{i}-1$.

In this case, $J=\langle f_{i}(x)^{l_{i}}+uf_{i}(x)^{t_{i}}h(x),uf_{i}(x)^{m_{i}}\rangle$ where $h(x)\in\Delta_{m_{i}-t_{i}}^{(i)}$.

Therefore, all distinct ideals of ${\mathcal{R}}_{i}+u{\mathcal{R}}_{i}$ are given by (I)–(V) and the number of elements in each ideal is given at the right side of the table.

It is obvious that the numbers of ideals in cases (I), (II) and (IV) are equal to $p^{k}s+1$, $p^{k}s$ and $\frac{1}{2}p^{k}s(p^{k}s-1)$ respectively. Now, we count the number of ideals in (III) and (V), respectively.

First, we consider ideals in (III). Let $0\leq t_{i}<l_{i}\leq p^{k}s-1$. When $l_{i}\leq\lfloor\frac{p^{k}s+1}{2}\rfloor$, i.e., $2l_{i}\leq p^{k}s$, then $t_{i}$ satisfies $t_{i}\geq 2l_{i}-p^{k}s$ for all $0\leq t_{i}\leq l_{i}-1$. In this case, the number of ideals is equal to

[TABLE]

When $l_{i}\geq\lfloor\frac{p^{k}s+1}{2}\rfloor+1$, i.e., $2l_{i}>p^{k}s$, then $t_{i}$ satisfies $t_{i}\geq 2l_{i}-p^{k}s$ for all $2l_{i}-p^{k}s\leq t_{i}\leq l_{i}-1$, and $t_{i}$ satisfies $t_{i}<2l_{i}-p^{k}s$ for all $0\leq t_{i}\leq 2l_{i}-p^{k}s-1$. In this case, the number of ideals is equal to

[TABLE]

Hence the number of ideals in (III) is equal to $\Omega(p^{d_{i}},p^{k}s)=N_{1}+N_{2}$. Next, we denote $\Psi(p^{d_{i}},p^{k}s)=\sum_{t_{i}=0}^{p^{k}s-4}\sum_{m_{i}=t_{i}+1}^{\frac{p^{k}s+t_{i}}{2}-1}\sum_{l_{i}=m_{i}+1}^{p^{k}s+t_{i}-1-m_{i}}p^{d_{i}(m_{i}-t_{i}-1)}.$ Then the number of ideals in (V) is equal to $(p^{d_{i}}-1)\Psi(p^{d_{i}},p^{k}s)$, where

[TABLE]

for all $s\geq 5$.

If $1\leq p^{k}s\leq 3$, there is no triple $(t_{i},m_{i},l_{i})$ of integers satisfying $0\leq t_{i}<m_{i}<l_{i}\leq p^{k}s-1$ and $l_{i}+m_{i}\leq p^{k}s+t_{i}-1$. In this case, the number of ideals in (V) is equal to [math]. Then we set $\Psi(p^{d_{i}},p^{k}s)=0$ for all $p^{k}s\leq 3$.

If $p^{k}s=4$, there is a unique triple $(t_{i},m_{i},l_{i})=(0,1,2)$ of integers satisfying $0\leq t_{i}<m_{i}<l_{i}\leq p^{k}s-1$ and $l_{i}+m_{i}\leq p^{k}s+t_{i}-1$. In this case, all distinct ideals in (V) are given by $\langle f_{i}(x)^{2}+uh(x),uf_{i}(x)\rangle$, where $h(x)\in\Delta_{1}^{(i)}={\mathcal{T}}_{i}\setminus\{0\}$ and $|{\mathcal{T}}_{i}\setminus\{0\}|=p^{d_{i}}-1$. Then we set $\Psi(p^{d_{i}},4)=1$.

Therefore, the number $N_{i}$ of ideals of ${\mathcal{R}}_{i}+u{\mathcal{R}}_{i}$ is equal to

[TABLE]

i.e., $N_{i}=1+\frac{1}{2}p^{k}s(p^{k}s+3)+\Omega(p^{d_{i}},p^{k}s)+(p^{d_{i}}-1)\Psi(p^{d_{i}},p^{k}s)$. $\Box$

Then from Lemma 3.2 and Theorem 3.3 we deduce the following corollary.

Corollary 3.4 Every $(1+pw)$-constacyclic code ${\mathcal{C}}$ over $\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}}$ of length $p^{k}n$ can be constructed by the following two steps:

(i) For each $i=1,\ldots,r$, choose an ideal $C_{i}$ of ${\mathcal{R}}_{i}+u{\mathcal{R}}_{i}$ listed in Theorem 3.3.

(ii) Set ${\mathcal{C}}=\oplus_{i=1}^{r}\theta_{i}(x)C_{i}=\sum_{i=1}^{r}\theta_{i}(x)C_{i}$ $({\rm mod}\ x^{p^{k}n}-(1+pw))$. Then the codewords in ${\mathcal{C}}$ is equal to $|{\mathcal{C}}|=\prod_{i=1}^{r}|C_{i}|$.

Using the notations of Theorem 3.3, the number of $(1+pw)$-constacyclic codes over $\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}}$ of length $p^{k}n$ is equal to $\prod_{i=1}^{r}N_{i}$.

Using the notations of Corollary 3.4, ${\mathcal{C}}=\oplus_{i=1}^{r}\theta_{i}(x)C_{i}$ is called the canonical form decomposition of the $(1+pw)$-constacyclic code ${\mathcal{C}}$ over $\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}}$.

4. Dual Codes of $(1+pw)$-Constacyclic Codes

In this section, we give the dual code of every $(1+pw)$-constacyclic code over $\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}}$ of length $N$ and investigate the self-duality of these codes.

Lemma 4.1 Let $w\in\mathbb{Z}_{p^{s}}^{\times}$ and denote $\widehat{w}=-w(1+pw)^{-1}$. Then $\widehat{w}\in\mathbb{Z}_{p^{s}}^{\times}$ and $(1+pw)^{-1}=1+p\widehat{w}$.

Proof Obviously, we have $\widehat{w}=-w(1+pw)^{-1}\in\mathbb{Z}_{p^{s}}^{\times}$ and $(1+pw)^{-1}=(1+pw-pw)(1+pw)^{-1}=1+p(w(1+pw)^{-1})$. $\Box$

Using the notations of Lemma 4.1, by Lemma 1.2 we know that the dual code of each $(1+pw)$-constacyclic code over $\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}}$ of length $p^{k}n$ is a $(1+p\widehat{w})$-constacyclic code over $\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}}$ of length $p^{k}n$, i.e., an ideal of the ring $(\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}})[x]/\langle x^{p^{k}n}-(1+p\widehat{w})\rangle\rangle$. In the following, we will determining the dual code of each $(1+p\widehat{w})$-constacyclic code over $\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}}$ of length $p^{k}n$ from its canonical form decomposition given by Theorem 3.3 and Corollary 3.4.

By Lemmas 2.6 and 4.1, we can select a fixed $\widehat{w}_{0}\in\mathbb{Z}_{p^{s}}^{\times}$ satisfying

[TABLE]

in the rest of the paper. First, from Theorem 2.5 we deduce

Corollary 4.2 Using the notations of Notation 2.7, we denote

[TABLE]

and ${\mathcal{T}}_{i}=\{\sum_{j=0}^{d_{i}-1}a_{j}x^{j}\mid a_{0},a_{1},\ldots,a_{d_{i}-1}\in\mathbb{F}_{p}\}\subseteq\widehat{{\mathcal{R}}}_{i}$.

(i) $\widehat{{\mathcal{R}}}_{i}$* is a finite chain ring with maximal ideal $\langle f_{i}(x)\rangle$ generated by $f_{i}(x)$, the nilpotency index of $f_{i}(x)$ is equal to $p^{k}s$ and $\widehat{{\mathcal{R}}}_{i}/\langle f_{i}(x)\rangle$ is a finite field of cardinality $p^{d_{i}}$*.

(ii) Every element $\alpha$ of $\widehat{{\mathcal{R}}}_{i}$ has a unique $f_{i}(x)$-adic expansion:

$\alpha=\sum_{j=0}^{p^{k}s-1}b_{j}(x)f_{i}(x)^{j},\ b_{0}(x),b_{1}(x),\ldots,b_{p^{k}s-1}(x)\in{\mathcal{T}}_{i}.$

Substituting $\frac{x^{p^{k}}}{1+p\widehat{w}_{0}}$ for $y$ in (2.3) and (2.4), we obtain

[TABLE]

and $a_{i}(\frac{x^{p^{k}}}{1+p\widehat{w}_{0}})F_{i}(\frac{x^{p^{k}}}{1+p\widehat{w}_{0}})+b_{i}(\frac{x^{p^{k}}}{1+p\widehat{w}_{0}})f_{i}(\frac{x^{p^{k}}}{1+p\widehat{w}_{0}})=1$ in the ring $\mathbb{Z}_{p^{s}}[x]$, respectively. It is clear that ${\rm deg}(f_{i}(x^{p^{k}}(1+p\widehat{w}_{0})^{-1}))=p^{k}d_{i}$ for $i=1,\ldots,r$. In the rest of this paper, we set

[TABLE]

(mod $x^{p^{k}n}-(1+p\widehat{w})$). Then by Chinese remainder theorem for commutative rings with identity, paralleling to Theorem 2.9 we have the following

Corollary 4.3 Denote $\widehat{{\mathcal{A}}}=\mathbb{Z}_{p^{s}}[x]/\langle x^{p^{k}n}-(1+p\widehat{w})\rangle$. We have the following:

(i) $\widehat{\theta}_{1}(x)+\ldots+\widehat{\theta}_{r}(x)=1$, $\widehat{\theta}_{i}(x)^{2}=\widehat{\theta}_{i}(x)$ and $\widehat{\theta}_{i}(x)\widehat{\theta}_{j}(x)=0$ in $\widehat{{\mathcal{A}}}$ for all $1\leq i\neq j\leq r$.

(ii) $\widehat{{\mathcal{A}}}=\widehat{{\mathcal{A}}}_{1}\oplus\ldots\oplus\widehat{{\mathcal{A}}}_{r}$, where $\widehat{{\mathcal{A}}}_{i}=\widehat{\theta}_{i}(x)\widehat{{\mathcal{A}}}$ and its multiplicative identity is $\widehat{\theta}_{i}(x)$. Moreover, this decomposition is a direct sum of rings in that $\widehat{{\mathcal{A}}}_{i}\widehat{{\mathcal{A}}}_{j}=\{0\}$ for all $i$ and $j$, $1\leq i\neq j\leq r$.

(iii) For each $1\leq i\leq r$, define a mapping $\widehat{\tau}_{i}:a(x)\mapsto\widehat{\theta}_{i}(x)a(x)$ $(\forall a(x)\in\widehat{{\mathcal{R}}}_{i}=\mathbb{Z}_{p^{s}}[x]/\langle f_{i}(x^{p^{k}}(1+p\widehat{w}_{0})^{-1})\rangle)$. Then $\widehat{\tau}_{i}$ is a ring isomorphism from $\widehat{{\mathcal{R}}}_{i}$ onto $\widehat{{\mathcal{A}}}_{i}$ and can be extended to a ring isomorphism from $\widehat{{\mathcal{R}}}_{i}+u\widehat{{\mathcal{R}}}_{i}$ onto $\widehat{{\mathcal{A}}}_{i}+u\widehat{{\mathcal{A}}}_{i}$ in the natural way.

(iv) Define $\widehat{\tau}:(a_{1}(x),\ldots,a_{r}(x))\mapsto\widehat{\tau}_{1}(a_{1}(x))+\ldots+\widehat{\tau}_{r}(a_{r}(x))$, i.e.

[TABLE]

for all $a_{i}(x)\in\widehat{{\mathcal{R}}}_{i}$, $i=1,\ldots,r$. Then $\widehat{\tau}$ is a ring isomorphism from $\widehat{{\mathcal{R}}}_{1}\times\ldots\times\widehat{{\mathcal{R}}}_{r}$ onto $\widehat{{\mathcal{A}}}$.

As in Section 3, we can identify $(\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}})[x]/\langle x^{N}-(1+p\widehat{w})\rangle$ with $\widehat{{\mathcal{A}}}+u\widehat{{\mathcal{A}}}$ ($u^{2}=0$). Paralleling to Sections 3, we can obtain conclusions for $(1+p\widehat{w})$-constacyclic codes over $\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}}$ of length $N$, i.e., ideals of the ring $\widehat{{\mathcal{A}}}+u\widehat{{\mathcal{A}}}$. We omit these conclusions here for space saving.

Now, let $\alpha=(\alpha_{0},\alpha_{1},\ldots,\alpha_{N-1}),\beta=(\beta_{0},\beta_{1},\ldots,\beta_{N-1})\in(\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}})^{N}$, where $N=p^{k}s$ and $\alpha_{j},\beta_{j}\in\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}}$ for all $j=0,1\ldots,N-1$. In the rest of this paper, we denote

[TABLE]

[TABLE]

Recall that the usual Euclidian inner product of $\alpha$ and $\beta$ is defined by $[\alpha,\beta]_{E}=\sum_{j=0}^{N-1}\alpha_{j}\beta_{j}\in\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}}$. Let ${\mathcal{C}}$ be a linear code over $\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}}$ of length $N$, i.e., a $(\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}})$-submodule of $(\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}})^{N}$. The Euclidian dual code of ${\mathcal{C}}$ is defined by ${\mathcal{C}}^{\bot_{E}}=\{\alpha\in(\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}})^{N}\mid[\alpha,\beta]_{E}=0,\ \forall\beta\in{\mathcal{C}}\}$, and ${\mathcal{C}}$ is said to be self-dual if ${\mathcal{C}}={\mathcal{C}}^{\bot_{E}}$.

From now on, we define a mapping $\mu:{\mathcal{A}}\rightarrow\widehat{{\mathcal{A}}}$ by

[TABLE]

Then one can easily verify that $\mu$ is a ring isomorphism from ${\mathcal{A}}$ onto $\widehat{{\mathcal{A}}}$. Precisely, by $x^{N}=(1+p\widehat{w})$ in $\widehat{{\mathcal{A}}}$ and Lemma 4.1 it follows that

[TABLE]

and the inverse $\mu^{-1}:\widehat{{\mathcal{A}}}\rightarrow{\mathcal{A}}$ of $\mu$ is given by

[TABLE]

for all $b(x)=\sum_{j=0}^{N-1}b_{j}x^{j}\in\widehat{{\mathcal{A}}}$ where $b_{j}\in\mathbb{Z}_{p^{s}}$.

For notations simplicity, we still denote $\mu^{-1}$ by $\mu$. Now, $\mu$ can be extended to a ring isomorphism between ${\mathcal{A}}+u{\mathcal{A}}$ and $\widehat{{\mathcal{A}}}+u\widehat{{\mathcal{A}}}$ in the natural way that

[TABLE]

and $\mu:\eta_{1}+u\eta_{2}\mapsto\mu(\eta_{1})+u\mu(\eta_{2}),\ \forall\eta_{1},\eta_{2}\in\widehat{{\mathcal{A}}}$, respectively.

Using the notations above, by a direct calculation we get the following

Lemma 4.4 Let $\alpha,\beta\in(\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}})^{N}$. Then $[\alpha,\beta]_{E}=0$ if $\alpha(x)\mu(\beta(x))=0$ in $(\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}})[x]/\langle x^{N}-(1+pw)\rangle$, where $\alpha(x)\in{\mathcal{A}}+u{\mathcal{A}}$ and $\beta(x)\in\widehat{{\mathcal{A}}}+u\widehat{{\mathcal{A}}}$.

For any polynomial $f(y)=\sum_{j=0}^{d}c_{j}y^{j}\in\mathbb{Z}_{p^{s}}[y]$ of degree $d\geq 1$, recall that the reciprocal polynomial of $f(y)$ is defined as $\widetilde{f}(y)=y^{d}f(\frac{1}{y})=\sum_{j=0}^{d}c_{j}y^{d-j}$, and $f(y)$ is said to be self-reciprocal if $\widetilde{f}(y)=\delta f(y)$ for some $\delta\in\mathbb{Z}_{p^{s}}^{\times}$. Then by Equation (2.3) in Section 2, we have

[TABLE]

Since $f_{1}(y),f_{2}(y),\ldots,f_{r}(y)$ are pairwise coprime monic basic irreducible polynomials in $\mathbb{Z}_{p^{s}}[y]$, it is known that $\widetilde{f}_{1}(y),\widetilde{f}_{2}(y),\ldots,\widetilde{f}_{r}(y)$ are pairwise coprime monic basic polynomials in $\mathbb{Z}_{p^{s}}[y]$ as well. Hence for each integer $i$, $1\leq i\leq r$, there is a unique integer $i^{\prime}$, $1\leq i^{\prime}\leq r$, such that

[TABLE]

Since $x^{p^{k}n}=1+p\widehat{w}=(1+p\widehat{w}_{0})^{n}$ in $\widehat{{\mathcal{A}}}$, we see that

[TABLE]

Then by Equation (2.5) in Section 2 and $(1+pw_{0})^{-1}=1+p\widehat{w}_{0}$, we have

[TABLE]

where $g_{i}(z)=\delta_{i}z^{n-{\rm deg}(b_{i}(y))-d_{i}}\widetilde{b}_{i}(z)\in\mathbb{Z}_{p^{s}}[z]$. Similarly, we can prove that $\mu(\theta_{i}(x))=h_{i}(x^{-p^{k}}(1+p\widehat{w}_{0})^{-1})F_{i}(x^{-p^{k}}(1+p\widehat{w}_{0})^{-1})$ for some $h_{i}(z)\in\mathbb{Z}_{p^{s}}[z]$. From these and by Equation (11), we deduce that $\mu(\theta_{i}(x))=\widehat{\theta}_{i^{\prime}}(x).$

Therefore, for each $1\leq i\leq r$ there is a unique integer $i^{\prime}$, $1\leq i^{\prime}\leq r$, such that $\mu(\theta_{i}(x))=\widehat{\theta}_{i^{\prime}}(x)$. We still use $\mu$ to denote this map $i\mapsto i^{\prime}$; i.e.,

[TABLE]

Whether $\mu$ denotes the ring isomorphism between ${\mathcal{A}}+u{\mathcal{A}}$ and $\widehat{{\mathcal{A}}}+u\widehat{{\mathcal{A}}}$ or this map on the set $\{1,\ldots,r\}$ is determined by context. The next lemma shows the compatibility of the two uses of $\mu$.

Lemma 4.5 With the notations above, we have the following:

(i) $\mu$* is a permutation on $\{1,\ldots,r\}$ satisfying $\mu^{-1}=\mu$*.

(ii) After a rearrangement of $\theta_{1}(x),\ldots,\theta_{r}(x)$ there are integers $\lambda,\epsilon$ such that $\mu(i)=i$ for all $i=1,\ldots,\lambda$ and $\mu(\lambda+j)=\lambda+\epsilon+j$ for all $j=1,\ldots,\epsilon$, where $\lambda\geq 1,\epsilon\geq 0$ and $\lambda+2\epsilon=r$.

(iii) For each integer $i$, $1\leq i\leq r$, there is a unique invertible element $\delta_{i}$ of $\mathbb{Z}_{p^{s}}$ such that $\widetilde{f}_{i}(y)=\delta_{i}f_{\mu(i)}(y)$.

(iv) For any integer $i$, $1\leq i\leq r$, $\mu(\theta_{i}(x))=\widehat{\theta}_{\mu(i)}(x)$, $\mu(\widehat{\theta}_{i}(x))=\theta_{\mu(i)}(x)$, $\mu({\mathcal{A}}_{i})=\widehat{{\mathcal{A}}}_{\mu(i)}$ and $\mu(\widehat{{\mathcal{A}}}_{i})={\mathcal{A}}_{\mu(i)}$. Then $\mu$ induces a ring isomorphism between ${\mathcal{A}}_{i}+u{\mathcal{A}}_{i}$ and $\widehat{{\mathcal{A}}}_{\mu(i)}+u\widehat{{\mathcal{A}}}_{\mu(i)}$.

Proof (i)–(iii) follow from the definition of the map $\mu$.

(iv) By the definition of $\mu$, we have $\mu(\theta_{i}(x))=\widehat{\theta}_{\mu(i)}(x)$ and $\mu(\widehat{\theta}_{i}(x))=\theta_{\mu(i)}(x)$. Then the other conclusions follow from that ${\mathcal{A}}_{i}=\theta_{i}(x){\mathcal{A}}$, $\widehat{{\mathcal{A}}}_{\mu(i)}=\widehat{\theta}_{\mu(i)}(x)\widehat{{\mathcal{A}}}$, $\widehat{{\mathcal{A}}}_{i}=\widehat{\theta}_{i}(x)\widehat{{\mathcal{A}}}$, ${\mathcal{A}}_{\mu(i)}=\theta_{\mu(i)}(x){\mathcal{A}}$, ${\mathcal{A}}_{i}+u{\mathcal{A}}_{i}={\mathcal{A}}_{i}[u]/\langle u^{2}\rangle$ and $\widehat{{\mathcal{A}}}_{\mu(i)}+u\widehat{{\mathcal{A}}}_{\mu(i)}=\widehat{{\mathcal{A}}}_{\mu(i)}[u]/\langle u^{2}\rangle$, immediately. $\Box$

Lemma 4.6 Let $1\leq i\leq r$. Denote by $\mu|_{{\mathcal{A}}_{i}}$ the restriction of $\mu$ to ${\mathcal{A}}_{i}$ and define $\mu_{i}:{\mathcal{R}}_{i}\rightarrow\widehat{{\mathcal{R}}}_{\mu(i)}$ by $\mu_{i}(c(x))=c(x^{-1})=\sum_{j=0}^{p^{k}d_{i}-1}c_{j}x^{-j}$ (for all $c(x)=\sum_{j=0}^{p^{k}d_{i}-1}c_{j}x^{j}$ with $c_{j}\in\mathbb{Z}_{p^{s}}$). Then $\mu_{i}$ is a ring isomorphism from ${\mathcal{R}}_{i}$ onto $\widehat{{\mathcal{R}}}_{\mu(i)}$ such that the following diagram commutes

[TABLE]

Moreover, $\mu_{i}$ can be extended to a ring isomorphism from ${\mathcal{R}}_{i}+u{\mathcal{R}}_{i}$ onto $\widehat{{\mathcal{R}}}_{\mu(i)}+u\widehat{{\mathcal{R}}}_{\mu(i)}$ by the natural way: $\mu_{i}:\alpha+u\beta\mapsto\mu_{i}(\alpha)+u\mu_{i}(\beta)$ for all $\alpha,\beta\in{\mathcal{R}}_{i}$.

Proof For any $c(x)\in{\mathcal{R}}_{i}$, by Theorem 2.9(iii), Corollary 4.3(iii) and $\mu(\theta_{i}(x))$ $=\widehat{\theta}_{\mu(i)}(x)=1-g_{i}(x^{p^{k}}(1+p\widehat{w}_{0})^{-1})f_{\mu(i)}(x^{p^{k}}(1+p\widehat{w}_{0})^{-1}),$ it follows that

[TABLE]

which implies $\mu_{i}(c(x))=((\widehat{\tau}_{\mu(i)})^{-1}\mu|_{{\mathcal{A}}_{i}}\tau_{i})(c(x))$ for all $c(x)\in{\mathcal{R}}_{i}$. Hence $\mu_{i}=(\widehat{\tau}_{\mu(i)})^{-1}\mu|_{{\mathcal{A}}_{i}}\tau_{i}$, which is a ring isomorphism from ${\mathcal{R}}_{i}$ onto $\widehat{{\mathcal{R}}}_{\mu(i)}$ such that the diagram commutes.

Obviously, $\mu_{i}$ can be extended to a ring isomorphism from ${\mathcal{R}}_{i}[u]$ onto $\widehat{{\mathcal{R}}}_{\mu(i)}[u]$ in the natural way that $\mu_{i}:\sum_{j}\alpha_{j}u^{j}\mapsto\sum_{j}\mu_{i}(\alpha_{j})u^{j}$ ($\forall\alpha_{j}\in{\mathcal{R}}_{i}$). Therefore, $\alpha+u\beta\mapsto\mu_{i}(\alpha)+u\mu_{i}(\beta)$ $(\forall\alpha,\beta\in{\mathcal{R}}_{i})$ is a ring isomorphism from ${\mathcal{R}}_{i}[u]/\langle u^{2}\rangle={\mathcal{R}}_{i}+u{\mathcal{R}}_{i}$ onto $\widehat{{\mathcal{R}}}_{\mu(i)}[u]/\langle u^{2}\rangle=\widehat{{\mathcal{R}}}_{\mu(i)}+u\widehat{{\mathcal{R}}}_{\mu(i)}$. $\Box$

Lemma 4.7 Using the notations of Theorem 2.9(iv)* and Corollary 4.3*(iv), let $\alpha(x)=\sum_{i=1}^{r}\theta_{i}(x)a_{i}(x)\in{\mathcal{A}}+u{\mathcal{A}}$ and $\beta(x)=\sum_{i=1}^{r}\widehat{\theta}_{i}(x)b_{i}(x)\in\widehat{{\mathcal{A}}}+u\widehat{{\mathcal{A}}}$, where $a_{i}(x)\in{\mathcal{R}}_{i}+u{\mathcal{R}}_{i}$ and $b_{i}(x)\in\widehat{{\mathcal{R}}}_{i}+u\widehat{{\mathcal{R}}}_{i}$. Then

[TABLE]

where $\mu$ is the ring isomorphism from $\widehat{{\mathcal{A}}}+u\widehat{{\mathcal{A}}}$ onto ${\mathcal{A}}+u{\mathcal{A}}$ and $\mu_{i}^{-1}:\widehat{{\mathcal{R}}}_{\mu(i)}+u\widehat{{\mathcal{R}}}_{\mu(i)}\rightarrow{\mathcal{R}}_{i}+u{\mathcal{R}}_{i}$ being the inverse of $\mu_{i}$ defined in Lemma 4.6 for all $i=1,\ldots,r$.

Proof Since $\mu$ is the ring isomorphism from $\widehat{{\mathcal{A}}}+u\widehat{{\mathcal{A}}}$ onto ${\mathcal{A}}+u{\mathcal{A}}$, by Lemma 4.5(iv) and Lemma 4.6 it follows that

[TABLE]

where $\mu_{\mu(j)}^{-1}$ is the inverse of the ring isomorphism $\mu_{\mu(j)}$ from ${\mathcal{R}}_{\mu(j)}+u{\mathcal{R}}_{\mu(j)}$ onto $\widehat{{\mathcal{R}}}_{j}+u\widehat{{\mathcal{R}}}_{j}$. Hence $\mu_{\mu(j)}^{-1}(b_{j}(x))\in{\mathcal{R}}_{\mu(j)}+u{\mathcal{R}}_{\mu(j)}$ for all $j$. When $j=\mu(i)$, we have $i=\mu(j)$ by Lemma 4.5(i), and so $\mu_{i}^{-1}(b_{\mu(i)}(x))\in{\mathcal{R}}_{i}+u{\mathcal{R}}_{i}$, which implies $a_{i}(x)\mu_{i}^{-1}(b_{\mu(i)}(x))\in{\mathcal{R}}_{i}+u{\mathcal{R}}_{i}$ for all $i$.

If $j\neq\mu(i)$, then $i\neq\mu(j)$ and hence $\theta_{i}(x)\theta_{\mu(j)}(x)=0$ by Theorem 2.9(i). From these, by $\mu(\mu(i))=i$ and $\theta_{i}(x)^{2}=\theta_{i}(x)$ by Theorem 2.9(i) for all $i$, we deduce that

[TABLE]

i.e. $\alpha(x)\mu(\beta(x))=\sum_{i=1}^{r}\theta_{i}(x)\left(a_{i}(x)\mu_{i}^{-1}(b_{\mu(i)}(x)\right)$. $\Box$

Lemma 4.8 Using the notations of Lemma 4.6, we have that $\mu_{i}(f_{i}(x)^{l})=\delta_{i}^{l}x^{-ld_{i}}f_{\mu(i)}(x)^{l}\in\widehat{{\mathcal{R}}}_{\mu(i)}$ for all $1\leq l\leq p^{k}s$.

Proof By Lemma 4.5(iii), we have $\widetilde{f}_{i}(x)=\delta_{i}f_{\mu(i)}(x)$. From this and by the definition of $\mu_{i}$ in Lemma 4.6, we deduce that $\mu_{i}(f_{i}(x)^{l})=f_{i}(x^{-1})^{l}=x^{-ld_{i}}\left(x^{d_{i}}f_{i}(x^{-1})\right)^{l}=x^{-ld_{i}}\widetilde{f}_{i}(x)^{l}=\delta_{i}^{l}x^{-ld_{i}}f_{\mu(i)}(x)^{l}.$ $\Box$

Now, we give the dual code of each $(1+pw)$-constacyclic code over the ring $\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}}$ of length $N$ where $N=p^{k}n$.

Theorem 4.9 Let ${\mathcal{C}}$ be a $(1+pw)$-constacyclic code over $\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}}$ of length $N$ with ${\mathcal{C}}=\oplus_{i=1}^{r}\theta_{i}(x)C_{i}$, where $C_{i}$ is an ideal of ${\mathcal{R}}_{i}+u{\mathcal{R}}_{i}$. Then the dual code ${\mathcal{C}}^{\bot_{E}}$ is a $(1+p\widehat{w})$-constacyclic code over $\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}}$ of length $N$. Precisely, ${\mathcal{C}}^{\bot_{E}}$ is given by ${\mathcal{C}}^{\bot_{E}}=\oplus_{j=1}^{r}\widehat{\theta}_{j}(x)D_{j},$ where $D_{j}$ is an ideal of $\widehat{{\mathcal{R}}}_{j}+u\widehat{{\mathcal{R}}}_{j}$ given by the following table:

[TABLE]

Proof For each integer $i$, $1\leq i\leq r$, let $J_{i}$ be an ideal of ${\mathcal{R}}_{i}+u{\mathcal{R}}_{i}$ given by one of the following eight cases:

(i) $J_{i}=\langle f_{i}(x)^{p^{k}s-l_{i}}\rangle$, if $C_{i}=\langle f_{i}(x)^{l_{i}}\rangle$, where $0\leq l_{i}\leq p^{k}s$.

(ii) $J_{i}=\langle f_{i}(x)^{p^{k}s-m_{i}},u\rangle$, if $C_{i}=\langle uf_{i}(x)^{m_{i}}\rangle$, where $0\leq m_{i}\leq p^{k}s-1$.

(iii) $J_{i}=\langle f_{i}(x)^{p^{k}s-l_{i}}-uf_{i}(x)^{p^{k}s+t_{i}-2l_{i}}h(x)\rangle$, if $C_{i}=\langle f_{i}(x)^{l_{i}}+uf_{i}(x)^{t_{i}}h(x)\rangle$ where $h(x)\in\Delta_{l_{i}-t_{i}}^{(i)}$, $0\leq t_{i}<l_{i}\leq p^{k}s-1$ and $t_{i}\geq 2l_{i}-p^{k}s$.

(iv) $J_{i}=\langle f_{i}(x)^{l_{i}}-uh(x)\rangle$, if $C_{i}=\langle f_{i}(x)^{l_{i}}+uh(x)\rangle$ where $h(x)\in\Delta_{p^{k}s-l_{i}}^{(i)}$, $0<l_{i}\leq p^{k}s-1$ and $2l_{i}>p^{k}s$.

(v) $J_{i}=\langle f_{i}(x)^{l_{i}-t_{i}}-uh(x),uf_{i}(x)^{p^{k}s-l_{i}}\rangle$, if $C_{i}=\langle f_{i}(x)^{l_{i}}+uf_{i}(x)^{t_{i}}h(x)\rangle$ where $h(x)\in\Delta_{p^{k}s-l_{i}}^{(i)}$, $1\leq t_{i}<l_{i}\leq p^{k}s-1$ and $t_{i}<2l_{i}-p^{k}s$.

(vi) $J_{i}=\langle f_{i}(x)^{p^{k}s-m_{i}},uf_{i}(x)^{p^{k}s-l_{i}}\rangle$, if $C_{i}=\langle f_{i}(x)^{l_{i}},uf_{i}(x)^{m_{i}}\rangle$, where $0\leq m_{i}<l_{i}\leq p^{k}s-1$.

(vii) $J_{i}=\langle f_{i}(x)^{p^{k}s-m_{i}}-uf_{i}(x)^{p^{k}s-l_{i}-m_{i}}h(x)\rangle$, when $C_{i}=\langle f_{i}(x)^{l_{i}}+uh(x)$, $uf_{i}(x)^{m_{i}}\rangle$ in which $h(x)\in\Delta_{m_{i}}^{(i)}$, $1\leq m_{i}<l_{i}\leq p^{k}s-1$ and $l_{i}+m_{i}\leq p^{k}s-1$.

(viii) $J_{i}=\langle f_{i}(x)^{p^{k}s-m_{i}}-uf_{i}(x)^{p^{k}s+t_{i}-l_{i}-m_{i}}h(x),uf_{i}(x)^{p^{k}s-l_{i}}\rangle$, if $C_{i}=\langle f_{i}(x)^{l_{i}}+uf_{i}(x)^{t_{i}}h(x),uf_{i}(x)^{m_{i}}\rangle$, where $h(x)\in\Delta_{m_{i}-t_{i}}^{(i)}$, $1\leq t_{i}<m_{i}<l_{i}\leq p^{k}s-1$, $l_{i}+m_{i}\leq p^{k}s+t_{i}-1$.

Then it follows that $C_{i}\cdot J_{i}=\{0\}$ by a straightforward computation. Furthermore, by Theorem 3.3 we see that $|C_{i}||J_{i}|=p^{2p^{k}sd_{i}}$.

Let $1\leq i\leq r$ and denote $D_{\mu(i)}=\mu_{i}(J_{i})$. By Lemma 4.6 we conclude that $D_{\mu(i)}$ is an ideal of $\widehat{{\mathcal{R}}}_{\mu(i)}+u\widehat{{\mathcal{R}}}_{\mu(i)}$, $|D_{\mu(i)}|=|J_{i}|$. and $\mu_{i}^{-1}(D_{\mu(i)})=J_{i}$. We set ${\mathcal{D}}=\oplus_{i=1}^{r}\widehat{\theta}_{\mu(i)}(x)D_{\mu(i)}=\sum_{j=1}^{r}\widehat{\theta}_{j}(x)D_{j}$ (mod $x^{p^{k}n}-(1+p\widehat{w})$). Then by the conclusion paralleling to Lemma 3.2, we see that ${\mathcal{D}}$ is a $(1+p\widehat{w})$-constacyclic code over $\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}}$ of length $p^{k}n$. Moreover, by Lemma 4.7 and $C_{i}\cdot\mu_{i}^{-1}(D_{\mu(i)})=C_{i}\cdot J_{i}=\{0\}$ it follows that

[TABLE]

which implies ${\mathcal{D}}\subseteq{\mathcal{C}}^{\bot_{E}}$ by Lemma 4.4. On the other hand, by Lemma 3.1 and Corollary 4.3(iii) we have

[TABLE]

Since both ${\mathcal{C}}$ and ${\mathcal{D}}$ are linear codes over $\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}}$ of length $p^{k}n$ and $\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}}$ is a finite Frobenius ring, from the theory of linear codes over Frobenius rings (see [13], for example) we deduce that ${\mathcal{C}}^{\bot_{E}}={\mathcal{D}}$.

Finally, we give the precise representation of each $D_{\mu(i)}$, $1\leq i\leq r$.

Case 1. Let $J_{i}=\langle f_{i}(x)^{p^{k}s-l_{i}}\rangle$ as in (i). Since $x$ is an invertible element of $\widehat{{\mathcal{R}}}_{\mu(i)}+u\widehat{{\mathcal{R}}}_{\mu(i)}$, by Lemmas 4.6 and 4.8 we have

[TABLE]

Cases 2 and 6 can be obtained similarly as Case 1.

Case 3. Let $J_{i}=\langle f_{i}(x)^{p^{k}s-l_{i}}-uf_{i}(x)^{p^{k}s+t_{i}-2l_{i}}h(x)\rangle$ as in (iii). By Lemmas 4.6 and 4.8 it follows that

[TABLE]

where $\widehat{h}(x)=-\delta_{i}^{t_{i}-l_{i}}x^{(l_{i}-t_{i})d_{i}}h(x^{-1})\in\widehat{{\mathcal{A}}}_{\mu(i)}^{\times}$.

Case 4. Let $J_{i}=\langle f_{i}(x)^{l_{i}}-uw(x)\rangle$ as in (iv). By Lemmas 4.6 and 4.8 we have

[TABLE]

where $\widehat{h}(x)=-\delta_{i}^{-l_{i}}x^{l_{i}d_{i}}h(x^{-1})\in\widehat{{\mathcal{A}}}_{\mu(i)}^{\times}$.

Cases 5, 7 and 8 can be obtained similarly as Cases 1, 3 and 4. $\Box$

Finally, we consider the self-duality of the constacyclic codes over $\mathbb{Z}_{p^{s}}+u\mathbb{Z}_{p^{s}}$. When $p$ is odd, by Lemma 2.6 we know that $(1+pw)^{-1}\neq 1+pw$ for all $w\in\mathbb{Z}_{p^{s}}^{\times}$. Let $p=2$ and $w\in\mathbb{Z}_{2^{s}}^{\times}$. Then it is clear that $(1+2w)^{-1}=1+2w$ if and only if $w=2^{s-2}-1$ or $2^{s-1}-1$ when $s\geq 3$, and $w=1$ when $s=2$.

Now, let $p=2$, and $w=2^{s-2}-1$ or $2^{s-1}-1$ if $s\geq 3$, and $w=1$ if $s=2$. We assume $N=2^{k}n$ where $n$ is an odd positive integer. Then $(1+2w)^{2}=1\in\mathbb{Z}_{2^{s}}^{\times}$. In this case, Using the notations of Lemma 2.6 and Lemma 4.1 we have $1+2w_{0}=1+2w=1+2\widehat{w}=1+2\widehat{w}_{0}$. Furthermore, using Equations (2.5) and (4.1) it follows that

[TABLE]

which implies ${\mathcal{A}}_{i}=\widehat{{\mathcal{A}}}_{i}$ for all $i=1,\ldots,r$. Then as a corollary of Lemma 3.2, Theorems 3.3 and 4.9, we can present all distinct self-dual $(1+2w)$-constacyclic codes over the ring $\mathbb{Z}_{2^{s}}+u\mathbb{Z}_{2^{s}}$ by the following theorem.

Theorem 4.10 Let $w=2^{s-2}-1$ or $2^{s-1}-1$ when $s\geq 3$, and $w=1$ when $s=2$. Using the notations in Theorem 4.9, Lemma 4.5$({\rm ii})$ and Equation $(12)$, let ${\mathcal{C}}$ be a $(1+2w)$-constacyclic code over $\mathbb{Z}_{2^{s}}+u\mathbb{Z}_{2^{s}}$ of length $N$ with ${\mathcal{C}}=\oplus_{i=1}^{r}\theta_{i}(x)C_{i}$, where $C_{i}$ is an ideal of ${\mathcal{R}}_{i}+u{\mathcal{R}}_{i}$. Then ${\mathcal{C}}$ is self-dual if and only if for each integer $i$, $1\leq i\leq r$, $C_{i}$ satisfies one of the following conditions:

$\bullet$ If $i=\lambda+j$ where $1\leq j\leq\epsilon$, $(C_{i},C_{i+\epsilon})$ are given by the following table:

[TABLE]

$\bullet$ If $1\leq i\leq\lambda$, $C_{i}$ is given by one of the following six cases:

(i) $C_{i}=\langle f_{i}(x)^{2^{k-1}s}\rangle$.  (ii) $C_{i}=\langle u\rangle$.

(iii) $C_{i}=\langle f_{i}(x)^{2^{k-1}s}+uf_{i}(x)^{t_{i}}h(x)\rangle$, where $0\leq t_{i}\leq 2^{k-1}s-1$ and $h(x)\in\triangle_{2^{k-1}s-t_{i}}^{(i)}$ satisfying $h(x)+\delta_{i}^{t_{i}-2^{k-1}s}x^{(2^{k-1}s-t_{i})d_{i}}h(x^{-1})\equiv 0$ $($mod f_{i}(x)^{2^{k-1}s-t_{i}}$$).

(iv) $C_{i}=\langle f_{i}(x)^{l_{i}}+uh(x)\rangle$, where $2^{k-1}s+1\leq l_{i}\leq 2^{k}s-1$ and $h(x)\in\Delta_{2^{k}s-l_{i}}^{(i)}$ satisfying $h(x)+\delta_{i}^{-l_{i}}x^{l_{i}d_{i}}h(x^{-1})\equiv 0$ $($mod f_{i}(x)^{2^{k}s-l_{i}}$$).

(v) $C_{i}=\langle f_{i}(x)^{l_{i}},uf_{i}(x)^{2^{k}s-l_{i}}\rangle$, where $2^{k-1}s+1<l_{i}\leq 2^{k}s-1$.

(vi) $C_{i}=\langle f_{i}(x)^{l_{i}}+uf_{i}(x)^{t_{i}}w(x),uf_{i}(x)^{2^{k}s-l_{i}}\rangle$, where $1\leq t_{i}<2^{k}s-l_{i}$, $2^{k-1}s+1\leq l_{i}\leq 2^{k}s-1$ and $h(x)\in\Delta_{2^{k}s-l_{i}-t_{i}}^{(i)}$ satisfying

$h(x)+\delta_{i}^{t_{i}-l_{i}}x^{(l_{i}-t_{i})d_{i}}h(x^{-1})\equiv 0$ (mod $f_{i}(x)^{2^{k}s-l_{i}-t_{i}}$).

5. An Example

In this section, we consider how to construct all distinct self-dual $3$-constacyclic codes over $\mathbb{Z}_{8}+u\mathbb{Z}_{8}$ ($u^{2}=0$) of length $14$. In this case, we have $N=2^{k}n$ with $k=1$, $n=7$, $p=2$, $s=3$ and $1+pw=3$ satisfying $3^{2}=1$ in $\mathbb{Z}_{8}$. It is known that $y^{7}-1=f_{1}(y)f_{2}(y)f_{3}(y),$ where $f_{1}(y)=y-1$, $f_{2}(y)=y^{3}+6y^{2}+5y+7$ and $f_{3}(y)=y^{3}+3y^{2}+2y+7$ are pairwise coprime minic basic irreducible polynomials in $\mathbb{Z}_{8}[y]$.

As $d_{1}=1$, $d_{2}=d_{3}=3$ and $r=3$, by Theorem 3.3 and Corollary 3.4, the number $N$ of $3$-constacyclic codes over $\mathbb{Z}_{8}+u\mathbb{Z}_{8}$ of length $14$ is equal to $N=\prod_{i=1}^{3}(2^{3d_{i}}+5\cdot 2^{2d_{i}}+9\cdot 2^{d_{i}}+13)=59\cdot 917^{2}=49,612,451$.

Obviously, $\widetilde{f}_{1}(y)=\delta_{1}f_{1}(y)$ and $\widetilde{f}_{2}(y)=\delta_{2}f_{3}(y)$ where $\delta_{1}=\delta_{2}=-1$, which implies that $\mu(1)=1$ and $\mu(2)=3$. Hence $\lambda=\epsilon=1$.

For each integer $i$, $1\leq i\leq 3$, we denote $F_{i}(y)=\frac{y^{7}-1}{f_{i}(y)}$, and find polynomials $a_{i}(y),b_{i}(y)\in\mathbb{Z}_{8}[y]$ satisfying $a_{i}(y)F_{i}(y)+b_{i}(y)f_{i}(y)=1$. Then set $\varrho_{i}(y)\equiv a_{i}(y)F_{i}(y)$ (mod $y^{7}-1$). By Equation (4.2) in Section 4, it follows that $\theta_{i}(x)\equiv\varrho_{i}(3x^{2})$ (mod $x^{14}+1$) in $\mathbb{Z}_{8}[x]$. Precisely, we have

$\theta_{1}(x)=7+5x^{2}+7x^{4}+5x^{6}+7x^{8}+5x^{10}+7x^{12}$;

$\theta_{2}(x)=5+x^{2}+3x^{4}+2x^{6}+3x^{8}+2x^{10}+6x^{12}$;

$\theta_{3}(x)=5+2x^{2}+6x^{4}+x^{6}+6^{8}+x^{10}+3x^{12}$.

Using the notations in Section 3, for each integer $j$, $1\leq j\leq 4$ we have that ${\mathcal{R}}_{i}=\mathbb{Z}_{8}[x]/\langle f_{i}(3x^{2})\rangle$, which is a finite chain ring with the maximal ideal $f_{i}(x)$ and the nilpotency index of $f_{i}(x)$ ie equal to $2\cdot 3=6$.

Now, by Theorem 4.10 all distinct self-dual $3$-constacyclic codes over $\mathbb{Z}_{8}+u\mathbb{Z}_{8}$ ($u^{2}=0$) of length $14$ are given by

${\mathcal{C}}=\theta_{1}(x)C_{1}\oplus\theta_{2}(x)C_{2}\oplus\theta_{3}(x)C_{3}=\sum_{i=1}^{3}\theta_{i}(x)C_{i}$ (mod $x^{14}-3$),

where $C_{i}$ is an ideal of the ring ${\mathcal{R}}_{i}+u{\mathcal{R}}_{i}$ satisfying one of the following conditions:

$\bullet$ $C_{1}$ is one of the following seven ideals:

$\langle(x-1)^{3}\rangle$, $\langle u\rangle$, $\langle(x-1)^{3}+u(x-1)\rangle$, $\langle(x-1)^{5}+u\rangle$, $\langle(x-1)^{4},u(x-1)^{2}\rangle$,

$\langle(x-1)^{5},u(x-1)\rangle$, $\langle(x-1)^{4}+u(x-1),u(x-1)^{2}\rangle$.

$\bullet$ $(C_{2},C_{3})$ is given by the following table, where $C_{i}$ is an ideal of ${\mathcal{R}}_{i}+u{\mathcal{R}}_{i}$ for $i=2,3$, and $L_{(C_{2},C_{3})}$ is the number of pairs $(C_{2},C_{3})$ on the same line.

[TABLE]

where $\Delta^{(2)}_{k}=\{\sum_{j=0}^{k-1}b_{j}(x)f_{2}(x)^{j}\mid b_{j}(x)\in{\mathcal{T}}_{2},b_{0}(x)\neq 0,0\leq j\leq k-1\}$ with ${\mathcal{T}}_{2}=\{a_{0}+a_{1}x+a_{2}x^{2}\mid a_{0},a_{1},a_{2}\in\{0,1\}\}$, for $k=1,2,3$.

Therefore, the number of self-dual $3$-constacyclic codes over $\mathbb{Z}_{8}+u\mathbb{Z}_{8}$ of length $14$ is equal to $7\cdot 917=6419$.

Acknowledgments

Part of this work was done when Yonglin Cao was visiting Chen Institute of Mathematics, Nankai University, Tianjin, China. Yonglin Cao would like to thank the institution for the kind hospitality. This research is supported in part by the National Natural Science Foundation of China (Grant Nos. 11671235, 11471255).

Appendix A A direct proof for Equation (2.1)

Denote $\eta=1+pw_{0}\in\mathbb{Z}_{p^{s}}^{\times}$. By Lemma 2.4(ii), we have $f(\eta^{-1}x^{p^{k}})=\prod_{i=0}^{d-1}(\eta^{-1}x^{p^{k}}-\zeta^{p^{i}})=\eta^{-d}\prod_{i=0}^{d-1}(x^{p^{k}}-\eta\zeta^{p^{i}})$. As $x^{p^{k}}-\eta\zeta^{p^{i}}=(x^{p^{k}}-\zeta^{p^{i}})-pw_{0}\zeta^{p^{i}}$, we have $f(\eta^{-1}x^{p^{k}})=\eta^{-d}\prod_{i=0}^{d-1}(x^{p^{k}}-\zeta^{p^{i}})-pg_{1}(x)+p^{2}g_{2}(x)$ where

[TABLE]

and $g_{2}(x)\in\Gamma[x]$. By Lemma 2.4(ii) we have

[TABLE]

where $a_{i}(x)\in\Gamma[x]$ satisfies $(x-\zeta^{p^{i}})^{p^{k}}=x^{p^{k}}-(\zeta^{p^{i}})^{p^{k}}+pa_{i}(x)$,

[TABLE]

and $h_{2}(x)\in\Gamma[x]$. In fact, we have a_{i}(x)=\sum_{t=1}^{p^{k}-1}\frac{1}{p}\left(\begin{array}[]{c}p^{k}\cr t\end{array}\right)(-\zeta^{p^{i}})^{t}x^{p^{k}(p^{k}-t)} where \frac{1}{p}\left(\begin{array}[]{c}p^{k}\cr t\end{array}\right)=\frac{1}{p}\frac{p^{k}!}{(p^{k}-t)!t!}\in\mathbb{Z} for all $t=1,\ldots,p^{k}-1$, when $p$ is odd. Let $p=2$. Then a_{i}(x)=\zeta^{2^{i+k}}+\sum_{t=1}^{2^{k}-1}\frac{1}{2}\left(\begin{array}[]{c}2^{k}\cr t\end{array}\right)(-\zeta^{2^{i}})^{t}x^{2^{k}(2^{k}-t)} where \frac{1}{2}\left(\begin{array}[]{c}2^{k}\cr t\end{array}\right)=\frac{1}{2}\frac{2^{k}!}{(2^{k}-t)!t!}\in\mathbb{Z} for all $t=1,\ldots,2^{k}-1$.

By Lemma 2.4(i), we know that $\zeta^{p^{d}}=\zeta=\zeta^{p^{0}}$, which implies $\prod_{i=0}^{d-1}(x^{p^{k}}-\zeta^{p^{i}})=\prod_{i=0}^{d-1}(x^{p^{k}}-\zeta^{p^{i+k}})$, and hence

[TABLE]

Now, denote $\vartheta(x)=h_{1}(x)+\eta^{d}g_{1}(x)+p(h_{2}(x)-\eta^{d}g_{2}(x))$. Then

[TABLE]

which implies $\vartheta(x)\in\mathbb{Z}_{p^{s}}[x]$, hence we see that

[TABLE]

by $(1+pw_{0})^{-1}\in\mathbb{Z}_{p^{s}}=\{0,1,\ldots,p^{s}-1\}$. Moreover, we have

[TABLE]

By Lemma 2.4(i), we know that $\zeta^{p^{d}}=\zeta=\zeta^{p^{0}}$ in the Galois ring $\Gamma$, which implies $\overline{\zeta}^{p^{d}}=\overline{\zeta}^{p^{0}}$ in the finite field $\overline{\Gamma}$. From this and by Equations (A.1) and (A.2), we have

[TABLE]

[TABLE]

By Lemma 2.4(i), we know that $\overline{\zeta},\overline{\zeta}^{p},\ldots,\overline{\zeta}^{p^{d-1}}$ are all distinct roots of $\overline{f}(x)$ in $\overline{\Gamma}$. Then for any integer $i$, $0\leq i\leq d-1$, we have $\prod_{0\leq j\leq d-1,j\neq i}(\overline{\zeta}^{p^{i}}-\overline{\zeta}^{p^{j}})^{p^{k}}=\left(\prod_{0\leq j\leq d-1,j\neq i}(\overline{\zeta}^{p^{i}}-\overline{\zeta}^{p^{j}})\right)^{p^{k}}\neq 0$ and $\prod_{0\leq j\leq d-1,j\neq i}(\overline{\zeta}^{p^{t}}-\overline{\zeta}^{p^{j}})^{p^{k}}=0$ for all $t\neq i$. Therefore,

[TABLE]

By $(x-\zeta^{p^{i}})^{p^{k}}=x^{p^{k}}-(\zeta^{p^{i}})^{p^{k}}+pa_{i}(x)$ in $\Gamma[x]$, we have

[TABLE]

Since $\Gamma$ is a finite chain with the unique maximal ideal generated by $p$ and $s\geq 2$ is the nilpotency index of $p$, we conclude that $a_{i}(\zeta^{p^{i}})=p^{s-1}\alpha$ for some $\alpha\in\Gamma$ (see [14], for example), which then implies $\overline{a}_{i}(\overline{\zeta}^{p^{i}})=\overline{a_{i}(\zeta^{p^{i}})}=0$ and hence $\overline{h}_{1}(\overline{\zeta}^{p^{i}})=0$ in $\overline{\Gamma}$.

As state above, we conclude that $\overline{\vartheta}(\overline{\zeta}^{p^{i}})=\overline{g}_{1}(\overline{\zeta}^{p^{i}})\neq 0$ for all $i=0,1,\ldots,d-1$. From this and by $\overline{f(\eta^{-1}x^{p^{k}})}=\overline{f}(x^{p^{k}})=\overline{f}(x)^{p^{k}}=\prod_{i=0}^{d-1}(x-\overline{\zeta}^{p^{i}})^{p^{k}}$, we deduce that $\overline{\vartheta}(x)$ and $\overline{f(\eta^{-1}x^{p^{k}})}$ are coprime polynomials in $\mathbb{F}_{p}[x]$. Hence $\vartheta(x)$ and $f(\eta^{-1}x^{p^{k}})$ are coprime polynomials in $\mathbb{Z}_{p^{s}}[x]$ by [16] Lemma 13.5. So there exist $a(x),b(x)\in\mathbb{Z}_{p^{s}}[x]$ such that $a(x)\vartheta(x)+b(x)f(\eta^{-1}x^{p^{k}})=1,$ which implies that $\vartheta(x)$ is an invertible element of ${\mathcal{R}}=\mathbb{Z}_{p^{s}}[x]/\langle f(\eta^{-1}x^{p^{k}})\rangle$. By Equation (A.3), we have $f(x)^{p^{k}}=p\vartheta(x)$ in ${\mathcal{R}}$ where $\vartheta(x)\in{\mathcal{R}}^{\times}$.