The concatenated structure of cyclic codes over $\mathbb {Z}_{p^2}$

Abstract

Let $N=p^kn$ where p is a prime, and $k,\,n$ are positive integers satisfying $\mathrm{gcd}(p,\,n)=1.$ We present a canonical form decomposition for every cyclic code over $\mathbb {Z}_{p^2}$ of length N,  where each subcode is concatenated by a basic irreducible cyclic code over $\mathbb {Z}_{p^2}$ of length n as the inner code and a constacyclic code over a Galois extension ring of $\mathbb {Z}_{p^2}$ of length $p^k$ as the outer code. By determining their outer codes, we present a precise description for cyclic codes over $\mathbb {Z}_{p^2}$ when $p\ne 2,$ give precisely dual codes and investigate self-duality for cyclic codes over $\mathbb {Z}_{p^2}.$ We end by listing cyclic self-dual codes over $\mathbb {Z}_9$ of length 33.

1 Introduction

Abualrub and Oehmke determined the generators for cyclic codes over $\mathbb {Z}_4$ for lengths of the form $2^k$ in [1], and Blackford presented the generators for cyclic codes over $\mathbb {Z}_4$ of lengths of the form 2n where n is odd in [2]. The case for odd n follows from results in [3] and also appears in more detail in [6]. Dougherty and Ling [4] determined the structure of cyclic codes over $\mathbb {Z}_4$ of arbitrary even length by giving the generator polynomials for these codes, described the number and dual codes of cyclic codes for a given length and presented the form of cyclic codes that are self-dual. Moreover, [4] proposed an open problem: study the structure of cyclic codes of arbitrary lengths over $\mathbb {Z}_{p^e},$ where p is a prime and $e\ge 2$ is a positive integer.

Kiah et al. [5] derived a method of representing cyclic codes of length $p^k$ over $\mathrm{GR}(p^2,\,m),$ classified all cyclic codes and analysed the dual codes and self-duality. Then Sobhani and Esmaeili investigated cyclic and negacyclic codes over the Galois ring $\mathrm{GR}(p^2,\,m)$ in [7], and their main contribution is an expression for each cyclic code of length $p^k$ over $\mathrm{GR}(p^2,\,m)$ and an algorithm to find a unique set of generators for cyclic and negacyclic codes over the Galois ring $\mathrm{GR}(p^2,\,m).$ To the best of our knowledge, the problem of determining precise expressions for cyclic codes and their dual codes of arbitrary length over $\mathrm{GR}(p^2,\,m)$ has not been solved completely.

A code over a ring R of length N is a nonempty subset $\mathcal{C}$ of $R^N.$ The code $\mathcal{C}$ is said to be linear if $\mathcal{C}$ is an R-submodule. All codes in this paper are assumed to be linear unless otherwise specified. The ambient space $R^N$ is equipped with the usual Euclidean inner product, i.e., $[a,\,b]=\sum \nolimits _{j=0}^{N-1}a_jb_j,$ where $a=(a_0,\,a_1,\ldots ,a_{N-1}),\,b=(b_0,\,b_1,\ldots ,b_{N-1})\in R^N,$ and the dual code is defined by $\mathcal{C}^{\bot }=\{a\in R^N\mid [a,\,b]=0,\,\forall b\in \mathcal{C}\}.$ If $\mathcal{C}^{\bot }=\mathcal{C},$ then $\mathcal{C}$ is called a self-dual code over $R.\,\mathcal{C}$ is said to be $\zeta$-constacyclic if $(c_0,\,c_1,\ldots , c_{N-1})\in \mathcal{C}$ implies $(\zeta c_{N-1},\,c_0,\,c_1,\ldots , c_{N-2})\in \mathcal{C},$ where $\zeta$ is an invertible element of R. Especially, $\mathcal{C}$ is called a negacyclic code if $\zeta ={-}1,$ and $\mathcal{C}$ is called a cyclic code if $\zeta =1.$ We use the natural connection of $\zeta$-constacyclic codes to polynomial rings, where $c=(c_0,\,c_1,\ldots , c_{N-1})$ is viewed as $c(x)=\sum \nolimits _{j=0}^{N-1}c_jx^j$ and the $\zeta$-constacyclic code $\mathcal{C}$ is an ideal in the polynomial residue ring $R[x]/\langle x^N-\zeta \rangle .$

In this paper, let $N=p^kn$ where p is a prime, and $n,\,k$ are positive integers satisfying $\mathrm{gcd}(p,\,n)=1.$ Then cyclic codes over $\mathbb {Z}_{p^2}$ of length N are viewed as ideals of the ring $\mathbb {Z}_{p^2}[x]/\langle x^{N}-1\rangle .$ In this paper, following [7] we attempt to give a precise description for cyclic codes over $\mathbb {Z}_{p^2}$ of length N by use of concatenated structure of codes. It is clear that all the conclusions we obtained can be generalized to $\mathrm{GR}(p^2,\,m)$ directly.

The present paper is organized as follows. In Sect. 2, we overview properties for concatenated structure of codes over rings. In Sect. 3, we present a canonical form decomposition for every cyclic code over $\mathbb {Z}_{p^2}$ of length N,  where each subcode is concatenated by a basic irreducible cyclic code over $\mathbb {Z}_{p^2}$ of length n as the inner code and a constacyclic code over a Galois extension ring of $\mathbb {Z}_{p^2}$ of length $p^k$ as the outer code, and give a precise description for cyclic codes by determining their outer codes when $p\ne 2.$ Using the canonical form decomposition, we present precisely dual codes and investigate the self-duality of cyclic codes over $\mathbb {Z}_{p^2}$ in Sect. 4. Finally, we list all cyclic self-dual codes over $\mathbb {Z}_9$ of length 33.

2 Preliminaries

In this section, we overview properties for concatenated structure of codes.

Notation 2.1

In this paper, let n be a positive integer satisfying $\mathrm{gcd}(p,\,n)=1,$ and assume

$$\begin{aligned} y^n-1=f_1(y),\,f_2(y),\ldots ,f_r(y), \end{aligned}$$

(1)

where $f_1(y),\,f_2(y),\ldots , f_r(y)$ are pairwise coprime monic basic irreducible polynomials in $\mathbb {Z}_{p^2}[y].$ For each $i,\,1\le i\le r,$ we assume $\mathrm{deg}(f_i(y))=m_i,$ and denote $R_i=\mathbb {Z}_{p^2}[y]/\langle f_i(y)\rangle =\mathbb {Z}_{p^2}[\zeta _i]$ where $\zeta _i=y+\langle f_i(y)\rangle \in R_i$ satisfying $f_i(\zeta _i)=0.$

For each integer $i,\,1\le i\le r,$ It is known that $R_i$ is a GR of characteristic $p^2$ and cardinality $p^{2m_i}.$ The Teichmüller set of $R_i$ is

$$\begin{aligned} \mathcal{T}_i =\left\{ \sum \limits _{j=0}^{m_i-1}t_jy^j\mid t_0,\,t_1,\ldots ,t_{m_i-1}\in \mathbb {Z}_p\right\} =\left\{ \sum \limits _{j=0}^{m_i-1}t_j\zeta _i^j\mid t_0,\,t_1,\ldots ,t_{m_i-1}\in \mathbb {Z}_p\right\} , \end{aligned}$$

and every element $\alpha$ of $R_i$ has a unique p-adic expression: $\alpha =r_0+pr_1,\,r_0,\,r_1\in \mathcal{T}_i.$ Moreover, $\alpha$ is invertible if and only if $r_0\ne 0.$

Denote $F_i(y)=\frac{y^n-1}{f_i(y)}\in \mathbb {Z}_{p^2}[y]$ in the following. Since $F_i(y)$ and $f_i(y)$ are coprime, there are polynomials $a_i(y),\,b_i(y)\in \mathbb {Z}_{p^2}[y]$ such that

$$\begin{aligned} a_i(y)F_i(y)+b_i(y)f_i(y)=1. \end{aligned}$$

In the rest of this paper, we set

$$\begin{aligned} \varepsilon _i(y)\equiv a_i(y)F_i(y)=1-b_i(y)f_i(y) \, \left( \mathrm{mod} \, y^n-1\right) . \end{aligned}$$

(2)

Then using classical ring theory, we deduce the following lemma.

Lemma 2.2

Denote $\mathcal{A}=\mathbb {Z}_{p^2}[y]/\langle y^n-1\rangle .$ The following hold in $\mathcal{A}.$

(i) $\varepsilon _1(y)+\cdots +\varepsilon _r(y)=1,\, \varepsilon _i(y)^2=\varepsilon _i(y)$ and $\varepsilon _i(y)\varepsilon _j(y)=0$ for all $1\le i\ne j\le r.$

(ii) $\mathcal{A}=\mathcal{A}_1\oplus \cdots \oplus \mathcal{A}_r,$ where $\mathcal{A}_i=\varepsilon _i(y)\mathcal{A}$ and its multiplicative identity is $\varepsilon _i(y).$ Moreover, this decomposition is a direct sum of rings in that $\mathcal{A}_i\mathcal{A}_j=\{0\}$ for all i and $j,\,1\le i\ne j\le r.$

(iii) For each $1\le i\le r,$ define a mapping $\varphi _i{\text {:}}\, g(y)\mapsto \varepsilon _i(y)g(y)\,(\forall g(y)\in R_i).$ Then $\varphi _i$ is a ring isomorphism from $R_i$ onto $\mathcal{A}_i.$ Hence $|\mathcal{A}_i|=p^{2m_i}.$

(iv) For each $1\le i\le r,\,\mathcal{A}_i$ is a basic irreducible cyclic code over $\mathbb {Z}_{p^2}$ of length n having parity check polynomial $f_i(y)$ and generator polynomial $F_i(y).$

For convenience and self-sufficiency of the paper, we restate the concatenated structure of codes over rings.

Definition 2.3

Using notations above, let C be a linear code over $R_i$ of length l,  i.e., C is an $R_i$-submodule of $R_i^l=\{(r_0,\,r_1,\ldots , r_{l-1})\mid r_j\in R_i,\,j=0,\,1,\ldots ,l-1\}.$ The concatenated code of $\mathcal{A}_i$ and C is defined by

$$\begin{aligned} \mathcal{A}_i\Box _{\varphi _i}C=\left\{ \left( \varphi _i\left( c_0\right) ,\,\varphi _i\left( c_1\right) ,\ldots ,\varphi _{i}\left( c_{l-1}\right) \right) \mid \left( c_0,\,c_1,\ldots ,c_{l-1}\right) \in C\right\} , \end{aligned}$$

where the cyclic code $\mathcal{A}_i$ over $\mathbb {Z}_{p^2}$ of length n is called the inner code and C is called the outer code.

Lemma 2.4

$\mathcal{A}_i\Box _{\varphi _i}C$ is a linear code over $\mathbb {Z}_{p^2}$ of length nl. The number of codewords in this concatenated code is equal to $|\mathcal{A}_i\Box _{\varphi _i}C|=|C|$ and

$$\begin{aligned} d_{\mathrm{min}}\left( \mathcal{A}_i\Box _{\varphi _i}C\right) \ge d_{\mathrm{min}}\left( \mathcal{A}_i\right) d_{\mathrm{min}}(C), \end{aligned}$$

where $d_{\mathrm{min}}(\mathcal{A}_i)$ is the minimal distance of $\mathcal{A}_i$ as a linear code over $\mathbb {Z}_{p^2}$ of length n and $d_{\mathrm{min}}(C)$ is the minimal distance of C as a linear code over the GR $R_i$ of length l.

By the following theorem, we see that a generator matrix of the concatenated code $\mathcal{A}_i\Box _{\varphi _i}C$ as a $\mathbb {Z}_{p^2}$-submodule can be constructed from a generator matrix of the cyclic code $\mathcal{A}_i$ over $\mathbb {Z}_{p^2}$ and a generator matrix of the linear code C over the GR $R_i$ straightforwardly.

Theorem 2.5

Let $\varepsilon _i(y)=\sum \nolimits _{j=0}^{n-1}e_{i,j}y^j$ with $e_{i,j}\in \mathbb {Z}_{p^2},$ and C be a linear code over the GR $R_i$ of length l with a generator matrix $G_{C}\in \mathrm{M}_{t\times l}(R_i),$ i.e., C is an $R_i$-submodule of $R_i^l$ generated by the row vectors of $G_{C}.$ The following hold.

(i) A generator matrix of the cyclic code $\mathcal{A}_i$ is given by

$$\begin{aligned} G_{\mathcal{A}_i}=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c}e_{i,0} &{} e_{i,1} &{} \ldots &{} e_{i,n-2} &{} e_{i,n-1}\\ e_{i,n-1} &{} e_{i,0} &{} \ldots &{} e_{i,n-3} &{} e_{i,n-2}\\ \ldots &{}\ldots &{}\ldots &{}\ldots &{}\ldots \\ e_{i,n-m_i+1} &{} e_{i,n-m_i+2} &{} \ldots &{} e_{i,n-m_i-1} &{} e_{i,n-m_i} \end{array}\right) . \end{aligned}$$

(ii) Assume $f_i(y)=\sum \nolimits _{j=0}^{m_i}f_{i,j}y^j$ with $f_{i,j}\in \mathbb {Z}_{p^2}$ and $f_{i,m_i}=1,$ and let $M_{f_i}=\left( \begin{array}{cc}0&{} I_{m_i-1}\\ -f_{i,0} &{} V_i\end{array}\right)$ be the companion matrix of $f_i(y)$ where $I_{m_i-1}$ is the identity matrix of order $m_i-1$ and $V_i=(-f_{i,1},\ldots ,-f_{i,m_i-1}).$ For any $\alpha =\alpha (y)=\sum \nolimits _{j=0}^{m_i-1}r_jy^j\in R_i$ with $r_j\in \mathbb {Z}_{p^2},$ denote $A_{\alpha }=\alpha (M_{f_i})=\sum \nolimits _{j=0}^{m_i-1}r_jM_{f_i}^j\in \mathrm{M}_{m_i\times m_i}(\mathbb {Z}_{p^2})$ in the rest of the paper. Then

$$\begin{aligned} \alpha Y=A_{\alpha }Y, \quad \mathrm{where}\quad Y=\left( \begin{array}{c}1\\ y\\ \ldots \\ y^{m_i-1}\end{array}\right) . \end{aligned}$$

(iii) Let $G_C=(\alpha _{j,s})_{1\le j\le t,\,1\le s\le l}$ with $\alpha _{j,s}\in R_i.$ Then a generator matrix of the concatenated code $\mathcal{A}_i\Box _{\varphi _i}C$ is given by

$$\begin{aligned} G_{\mathcal{A}_i\Box _{\varphi _i}C}=\left( \begin{array}{c@{\quad }c@{\quad }c}A_{\alpha _{1,1}}G_{\mathcal{A}_i} &{} \ldots &{} A_{\alpha _{1,l}}G_{\mathcal{A}_i}\\ \ldots &{} \ldots &{} \ldots \\ A_{\alpha _{t,1}}G_{\mathcal{A}_i} &{} \ldots &{} A_{\alpha _{t,l}}G_{\mathcal{A}_i}\end{array}\right) . \end{aligned}$$

Hence $\mathcal{A}_i\Box _{\varphi _i}C=\{\underline{w}G_{\mathcal{A}_i\Box _{\varphi _i}C}\mid \underline{w}\in \mathbb {Z}_{p^2}^{m_it}\}.$

Proof

(i) Since $f_i(y)$ is a monic basic irreducible polynomial in $\mathbb {Z}_{p^2}[y]$ of degree $m_i,\,\{1,\,y,\ldots ,y^{m_i-1}\}$ is a $\mathbb {Z}_{p^2}$-basis of the GR $R_i=\mathbb {Z}_{p^2}[y]/\langle f_i(y)\rangle .$ As $\varphi _i$ is a $\mathbb {Z}_{p^2}$-module isomorphism from $R_i$ onto $\mathcal{A}_i$ by Lemma 2.2(iii), we conclude that $\{\varepsilon _i(y),\,y\varepsilon _i(y),\ldots ,y^{m_i-1}\varepsilon _i(y)\}$ is a $\mathbb {Z}_{p^2}$-basis of $\mathcal{A}_i.$ Hence $G_{\mathcal{A}_i}$ is a generator matrix of $\mathcal{A}_i$ as a $\mathbb {Z}_{p^2}$-submodule of $\mathbb {Z}_{p^2}^n.$

(ii) It is obvious that $yY=M_{f_i}Y,$ which then implies that $y^jY=M_{f_i}^jY$ for all $j=0,\,1,\ldots ,m_i-1.$ Hence $\alpha Y=\sum \nolimits _{j=0}^{m_i-1}r_j(y^jY)=A_\alpha Y.$

(iii) Let $\mathcal{C}$ be the $\mathbb {Z}_{p^2}$-submodule of $\mathbb {Z}_{p^2}^{nl}$ generated by the row vectors of $G_{\mathcal{A}_i\Box _{\varphi _i}C},$ i.e., $\mathcal{C}=\{\underline{w}G_{\mathcal{A}_i\Box _{\varphi _i}C}\mid \underline{w}\in \mathbb {Z}_{p^2}^{m_it}\}.$ By Definition 2.3, $\xi \in \mathcal{A}_i\Box _{\varphi _i}C$ if and only if there exists a unique codeword $c=(c_1,\ldots ,c_l)\in C$ such that $\xi =(\varphi _i(c_1),\ldots ,\varphi _i(c_l)).$ Since $G_C$ is a generator matrix of $C,\,c\in C$ if and only if c is an $R_i$-combination of the row vectors $(\alpha _{1,1},\ldots ,\alpha _{1,l}),\ldots ,(\alpha _{t,1},\ldots ,\alpha _{t,l})$ of $G_C,$ which is equivalent that there exist $\beta _1,\ldots ,\beta _t\in R_i$ such that

$$\begin{aligned} \xi= & {} \left( \varphi _i\left( \beta _1\alpha _{1,1}+\cdots +\beta _{t}\alpha _{t,1}\right) , \ldots , \varphi _i\left( \beta _1\alpha _{1,l}+\cdots +\beta _{t}\alpha _{t,l}\right) \right) \\= & {} \left( \varphi _i\left( \beta _1\alpha _{1,1}\right) +\cdots +\varphi _i\left( \beta _{t}\alpha _{t,1}\right) ,\ldots ,\varphi _i\left( \beta _1\alpha _{1,l}\right) +\cdots +\varphi _i\left( \beta _{t}\alpha _{t,l}\right) \right) , \end{aligned}$$

since $\varphi _i$ is a $\mathbb {Z}_{p^2}$-module isomorphism. For each integer $j,\,1\le j\le t,$ by $\beta _j\in R_i$ there is a unique row vector $\underline{b}_j\in \mathbb {Z}_{p^2}^{m_i}$ such that $\beta _j=\underline{b}_jY.$ From this and by (ii) we deduce that $\beta _j\alpha _{j,s}=\underline{b}_j(\alpha _{j,s}Y)=\underline{b}_jA_{\alpha _{j,s}}Y$ for all $s=1,\ldots ,l.$ Also, since $\varphi _i$ is a $\mathbb {Z}_{p^2}$-module isomorphism, we have

$$\begin{aligned} \xi= & {} \left( \underline{b}_1A_{\alpha _{1,1}}\varphi _i(Y)+\cdots +\underline{b}_tA_{\alpha _{t,1}}\varphi _i(Y), \ldots , \underline{b}_1A_{\alpha _{1,l}}\varphi _i(Y)+\cdots +\underline{b}_tA_{\alpha _{t,l}}\varphi _i(Y)\right) \\= & {} \underline{w}\left( \begin{array}{c@{\quad }c@{\quad }c}A_{\alpha _{1,1}}\varphi _i(Y) &{} \ldots &{} A_{\alpha _{1,l}}\varphi _i(Y)\\ \ldots &{} \ldots &{} \ldots \\ A_{\alpha _{t,1}}\varphi _i(Y) &{} \ldots &{} A_{\alpha _{t,l}}\varphi _i(Y)\end{array}\right) , \end{aligned}$$

where $\underline{w}=(\underline{b}_1,\ldots ,\underline{b}_t)\in \mathbb {Z}_{p^2}^{m_it}.$ Then by

$$\begin{aligned} \varphi _i(Y)=\left( \begin{array}{c}\varphi _i(1)\\ \varphi _i(y)\\ \ldots \\ \varphi _i(y^{m_i-1})\end{array}\right) =\left( \begin{array}{c}\varepsilon _i(y)\\ y\varepsilon _i(y)\\ \ldots \\ y^{m_i-1}\varepsilon _i(y)\end{array}\right) =G_{\mathcal{A}_i}\left( \begin{array}{c}1 \\ y\\ \ldots \\ y^{n-1}\end{array}\right) , \end{aligned}$$

and the identification of $\mathbb {Z}_{p^2}[y]\langle y^n-1\rangle$ with $\mathbb {Z}_{p^2}^n,$ we deduce $\xi =\underline{w}G_{\mathcal{A}_i\Box _{\varphi _i}C}\in \mathcal{C}.$ Therefore, $\mathcal{A}_i\Box _{\varphi _i}C=\mathcal{C}.$ $\square$

3 The concatenated structure of cyclic codes over $\mathbb {Z}_{p^2}$ of length $p^kn$

From now on, let $N=p^kn$ where k is a positive integer. As usual, we will identify $\mathbb {Z}_{p^2}^N$ with $\mathbb {Z}_{p^2}[x]/\langle x^N-1\rangle$ under the natural $\mathbb {Z}_{p^2}$-module isomorphism: $(c_0,\,c_1,\ldots ,c_{N-1})\mapsto c_0+c_1x+\cdots +c_{N-1}x^{N-1}$ ($\forall c_j\in \mathbb {Z}_{p^2},\,j=0,\,1,\ldots ,N-1$).

Using the notations of Lemma 2.2, each element of the ring $\mathcal{A}$ can be uniquely expressed as $a(y)=\sum \nolimits _{j=0}^{n-1}a_jy^j$ with $a_j\in \mathbb {Z}_{p^2}.$ Then each element of the quotient ring $\mathcal{A}[x]/\langle x^{p^k}-y\rangle$ can be uniquely expressed as

$$\begin{aligned} \alpha (x,\,y)=\left( 1,\,y,\ldots ,y^{n-1}\right) M \left( \begin{array}{c} 1 \\ x\\ \ldots \\ x^{p^k-1} \end{array}\right) , \end{aligned}$$

where M is a matrix over $\mathbb {Z}_{p^2}$ of size $n\times p^k.$ Now, define

$$\begin{aligned} \Psi (\alpha (x,\,y))=\alpha \left( x,\,x^{p^k}\right) =\left( 1,\,x^{p^k},\ldots ,x^{p^k(n-1)}\right) M\left( \begin{array}{c}1 \\ x\\ \ldots \\ x^{p^k-1} \end{array}\right) . \end{aligned}$$

It is clear that $\Psi$ is a ring isomorphism from $\mathcal{A}[x]/\langle x^{p^k}-y\rangle$ onto $\mathbb {Z}_{p^2}[x]/\langle x^N-1\rangle .$ In the rest of this paper, we will identify $\mathcal{A}[x]/\langle x^{p^k}-y\rangle$ with $\mathbb {Z}_{p^2}[x]/\langle x^N-1\rangle$ under this isomorphism $\Psi .$

Theorem 3.1

Using the notations in Notation 2.1 and Lemma 2.2, and let $\mathcal{C}\subseteq \mathbb {Z}_{p^2}[x]/\langle x^N-1\rangle .$ The following are equivalent:

(i) $\mathcal{C}$ is a cyclic code over $\mathbb {Z}_{p^2}$ of length N.

(ii) $\mathcal{C}$ is an ideal of the ring $\mathcal{A}[x]/\langle x^{p^k}-y\rangle .$

(iii) For each integer $i,\,1\le i\le r,$ there is a unique $\zeta _i$-constacyclic code $C_i$ over $R_i$ of length $p^k,$ i.e., an ideal $C_i$ of the ring $R_i[x]/\langle x^{p^k}-\zeta _i\rangle ,$ such that $\mathcal{C}=(\mathcal{A}_1\Box _{\varphi _1}C_1)\oplus \cdots \oplus (\mathcal{A}_r\Box _{\varphi _r}C_r).$

Proof

We only need to prove (ii)$\Leftrightarrow$(iii). By Lemma 2.2(ii) it follows that $\mathcal{A}[x]/\langle x^{p^k}-y\rangle =\oplus _{i=1}^r(\mathcal{A}_i[x]/\langle x^{p^k}-y\rangle ).$ As $\mathbb {Z}_{p^2}[x]/\langle x^N-1\rangle =\mathcal{A}[x]/\langle x^{p^k}-y\rangle ,$ we see that $\mathcal{C}$ is an ideal of the ring $\mathbb {Z}_{p^2}[x]/\langle x^N-1\rangle$ if and only if for each integer $i,\,1\le i\le r,$ there is a unique ideal $\mathcal{C}_i$ of the ring $\mathcal{A}_i[x]/\langle x^{p^k}-y\rangle$ such that $\mathcal{C}=\oplus _{i=1}^r\mathcal{C}_i.$

By Lemma 2.2(iii), $\varphi _i{\text {:}}\,g(y)\mapsto \varepsilon _i(y)g(y)$ $({\forall } g(y)\in R_i)$ is a ring isomorphism from $R_i$ onto $\mathcal{A}_i.$ As $R_i=\mathbb {Z}_{p^2}[y]/\langle f_i(y)\rangle =\mathbb {Z}_{p^2}[\zeta _i]$ where $\zeta _i=y+\langle f_i(y)\rangle ,$ the inverse isomorphism $\psi _i$ of $\varphi _i$ is given by

$$\begin{aligned} \psi _i(h(y))=h(y) \, \left( \mathrm{mod} \, f_i(y)\right) \quad \mathrm{or} \quad \psi _i(h(y))=h\left( \zeta _i\right) , \quad \forall h(y)\in \mathcal{A}_i. \end{aligned}$$

Then $\psi _i$ induces a ring isomorphism from $\mathcal{A}_i[x]/\langle x^{p^k}-y\rangle$ onto $R_i[x]/\langle x^{p^k}-\zeta _i\rangle$ in the natural way:

$$\begin{aligned} \psi _i\left( \sum \limits _{j=0}^{p^k-1}h_j(y)x^j\right) =\sum \limits _{j=0}^{p^k-1}h_j\left( \zeta _i\right) x^j, \quad {\forall } h_0(y),\,h_1(y),\ldots , h_{p^k-1}(y)\in \mathcal{A}_i. \end{aligned}$$

By $\varphi _i=\psi _i^{-1},\,\varphi _i$ induces a ring isomorphism from $R_i[x]/\langle x^{p^k}-\zeta _i\rangle$ onto $\mathcal{A}_i[x]/\langle x^{p^k}-y\rangle$ by the following: ${\forall } g_0,\,g_1,\ldots ,g_{p^k-1}\in R_i,$

$$\begin{aligned} \varphi _i\left( \sum \limits _{j=0}^{p^k-1}g_jx^j\right) =\sum \limits _{j=0}^{p^k-1}\varphi _i\left( g_j\right) x^j \leftrightarrow \left( \varphi _i\left( g_0\right) ,\,\varphi _i\left( g_1\right) ,\ldots ,\varphi _i\left( g_{p^k-1}\right) \right) \in \mathcal{A}_i^{p^k}. \end{aligned}$$

Therefore, for each integer $i,\,1\le i\le r,$ and the ideal $\mathcal{C}_i$ of $\mathcal{A}_i[x]/\langle x^{p^k}-y\rangle ,$ there is a unique ideal $C_i$ of $R_i[x]/\langle x^{p^k}-\zeta _i\rangle$ such that $\mathcal{C}_i=\varphi _i(C_i),$ which implies $\mathcal{C}_i=\mathcal{A}_i\Box _{\varphi _i}C_i$ by Definition 2.3. It is clear that $\mathcal{C}_i$ is a $\zeta _i$-constacyclic code over $R_i$ of length $p^k.$ $\square$

By Theorem 3.1, in order to present all cyclic codes over $\mathbb {Z}_{p^2}$ of length N it is sufficient to determine all ideals of the ring $R_i[x]/\langle x^{p^k}-\zeta _i\rangle ,$ where $R_i=\mathbb {Z}_{p^2}[\zeta _i]$ and $\zeta _i=y+\langle f_i(y)\rangle$ satisfies $f_i(\zeta _i)=0,$ for all $i=1,\ldots ,r.$

Since $\mathrm{gcd}(p,\,n)=1,$ there is a positive integer $v,\,1\le v< n,$ such that $p^kv\equiv 1$ (mod n). By Eq. (1) it follows that $\zeta _i^n=1.$ From this we deduce $(\zeta _i^v)^{p^k}=\zeta _i,$ which implies $(\zeta _i^e)^{p^k}=\zeta _i^{-1}$ where $e=n-v.$

Lemma 3.2

Using the notations above, define a mapping $\sigma _i{\text {:}}\,R_i[z]/\langle z^{p^k}-1\rangle \rightarrow R_i[x]/\langle x^{p^k}-\zeta _i\rangle$ by

$$\begin{aligned} \sigma _i(a(z))=a\left( \zeta _i^{e}x\right) , \quad {\forall } a(z)\in R_i[z]/\left\langle z^{p^k}-1\right\rangle . \end{aligned}$$

Then $\sigma _i$ is a ring isomorphism from $R_i[z]/\langle z^{p^k}-1\rangle$ onto $R_i[x]/\langle x^{p^k}-\zeta _i\rangle$ preserving Hamming weight.

Proof

It follows that $(\zeta _i^{e}x)^{p^k}-1=\zeta _i^{ep^k}x^{p^k}-1=\zeta _i^{-1}(x^{p^k}-\zeta _i).$ $\square$

Recall that ideals of the ring $R_i[z]/\langle z^{p^k}-1\rangle$ are in fact cyclic codes over the GR $R_i=\mathrm{GR}(p^2,\,m_i)$ of length $p^k.$ This kind of cyclic codes have been researched in many literatures, for example Kiah et al. [5] and Sobhani and Esmaeili [7]. For purpose of application in this paper, we list some conclusions.

Lemma 3.3

(cf. [7, Theorem 4.3]) The number of ideals of $R_i[z]/\langle z^{p^k}-1\rangle ,$ where $R_i=\mathrm{GR}(p^2,\,m_i),$ is equal to

$$\begin{aligned} N_{(p^2,m_i;k)}= & {} 4\left( \frac{p^{m_ip^{k-1}}-1}{p^{m_i}-1}\right) + \left( 2(p-2)p^{k-1}+1\right) p^{m_ip^{k-1}}\\&\quad +\left( p^{m_i}+3\right) \left( \frac{p^{m_ip^{k-1}}-1}{(p^{m_i}-1)^2}-\frac{p^{k-1}}{p^{m_i}-1}\right) \\&\quad +\,2(p-2)p^{k-1}\left( \frac{p^{m_ip^{k-1}}-1}{p^{m_i}-1}\right) +p^{k-1}. \end{aligned}$$

Especially, $N_{(p^2,m_i;k)}=1+2p+(2p-3)p^{m_i}$ when $k=1.$

Lemma 3.4

([7, Corollary 4.4]) Let $p\ne 2,\,\alpha =p-1$ and $\beta =p-2.$ Then all distinct cyclic codes $\mathcal{L}_i$ over the GR $R_i$ of length $p^k$ and their annihilating ideals $\mathrm{Ann}(\mathcal{L}_i)=\{\alpha \in R_i[z]/\langle z^{p^k}-1\rangle \mid \alpha \beta =0,\,{\forall } \beta \in \mathcal{L}_i\}$ are given by the following:

Cases	$\mathcal{L}_i$	Ann$(\mathcal{L}_i)$
(1)	$\langle 0\rangle$	$\langle 1\rangle$
(2)	$\langle 1\rangle$	$\langle 0\rangle$
(3)	$\langle p\rangle$	$\langle p\rangle$
(4)	$\langle p(z-1)^s\rangle \, (1\le s\le p^k-1)$	$\langle p,\,(z-1)^{p^k-s}\rangle$
(5)	$\langle (z-1)^s\rangle \, (1\le s\le p^{k-1})$	$\langle (z-1)^{p^k-s}+p(z-1)^{p^{k-1}-s}(-w(z))\rangle$
(6)	$\langle (z-1)^s\rangle \, (p^{k-1}+1\le s\le p^{k}-1)$	$\langle (z-1)^{\alpha p^{k-1}}+p(-w(z)),\,p(z-1)^{p^k-s}\rangle$
(7)	$\langle (z-1)^s+p(z-1)^{s-\alpha p^{k-1}}(-w(z))\rangle$	$\langle (z-1)^{p^k-s}\rangle$
	$(\alpha p^{k-1}\le s\le p^{k}-1)$
(8)	$\langle (z-1)^s+p(z-1)^{s-\alpha p^{k-1}}(-w(z)$	$\langle (z-1)^{p^k-s}+p(z-1)^{p^{k-1}+\nu -s}(-\widetilde{h}(z))\rangle$
	$+(z-1)^{\nu }\widetilde{h}(z))\rangle$
	$\quad (\alpha p^{k-1}\le s\le p^{k-1}+\nu , \, \nu \ge 1)$
(9)	$\langle (z-1)^s+p(z-1)^{s-\alpha p^{k-1}}(-w(z)$	$\langle (z-1)^{\alpha p^{k-1}-\nu }+p(-\widetilde{h}(z)),$
	$+(z-1)^{\nu }\widetilde{h}(z))\rangle$	$p(z-1)^{p^k-s}\rangle$
	$(p^{k-1}+\nu \le s\le p^{k}-1,$
	$s>\alpha p^{k-1}, \, \nu \ge 1)$
(10)	$\langle (z-1)^{\alpha p^{k-1}}+p(-w(z)$	$\langle (z-1)^{\alpha p^{k-1}-\nu }+p(-\widetilde{h}(z))\rangle$
	$+(z-1)^{\nu }\widetilde{h}(z))\rangle$
	$p^{k-1}+\nu <\alpha p^{k-1}, \, \nu \ge 1)$
(11)	$\langle (z-1)^{s}+p(z-1)^{s-\alpha p^{k-1}}h(z)\rangle$	$\langle (z-1)^{\alpha p^{k-1}}+p(1-h(z)),$
	$(\alpha p^{k-1}<s<p^k-1, \, h_0\ne 0,\,1)$	$p(z-1)^{p^k-s}\rangle$
(12)	$\langle (z-1)^{\alpha p^{k-1}}+ph(z)\rangle \, (h_0\ne 0,\,1)$	$\langle (z-1)^{\alpha p^{k-1}}+p(1-h(z))\rangle$
(13)	$\langle (z-1)^s+p(z-1)^{t}h(z)\rangle$	$\langle (z-1)^{p^{k}-s}+p(z-1)^{p^{k-1}-s}(-w(z)$
	$(p^k+t-s\ne p^{k-1},\,s\le p^{k-1},$	$+\,(z-1)^{\alpha p^{k-1}+t-s}(-h(z)))\rangle$
	$h(z)\ne 0)$

(14)	$\langle (z-1)^s+p(z-1)^{t}h(z)\rangle$	$\langle (z-1)^{\alpha p^{k-1}}+p(-w(z)$
	$(p^k+t-s\ne p^{k-1},$	$+\,(z-1)^{\alpha p^{k-1}+t-s}(-h(z))),$
	$p^{k-1}<s\le \alpha p^{k-1}+t,$	$p(z-1)^{p^k-s}\rangle$
	$t>0,\, h(z)\ne 0)$
(15)	$\langle (z-1)^s+ph(z)\rangle$	$\langle p(-w(z)+(z-1)^{\alpha p^{k-1}-s}(-h(z)))$
	$(p^{k-1}<s<\alpha p^{k-1},\,h(z)\ne 0)$	$\quad +\,(z-1)^{\alpha p^{k-1}}\rangle$
(16)	$\langle (z-1)^s+p(z-1)^th(z)\rangle$	$\langle p(-h(z)+(z-1)^{s-t-\alpha p^{k-1}})$
	$(p^k+t-s\ne p^{k-1},\, s>\alpha p^{k-1}+t,$	$+\,(z-1)^{s-t},\, p(z-1)^{p^k-s}\rangle$
	$h(z)\ne 0,\, t>0$)
(17)	$\langle (z-1)^s+ph(z)\rangle$	$\langle (z-1)^{s}+p(-h(z)+(z-1)^{s-\alpha p^{k-1}})\rangle$
	$(s>\alpha p^{k-1},\, h(z)\ne 0)$
(18)	$\langle (z-1)^s,\,p(z-1)^l\rangle$	$\langle (z-1)^{p^k-1}+p(z-1)^{p^{k-1}-l}(-w(z)),$
	$(1\le s\le p^k-1,$	$p(z-1)^{p^k-s}\rangle$
	$0\le l\le \mathrm{min}\{s,\,p^{k-1}\})$
(19)	$\langle p(z-1)^{s-\alpha p^{k-1}}(-w(z))$	$\langle (z-1)^{p^k-l},\,p(z-1)^{p^k-s}\rangle$
	$+(z-1)^s,\,p(z-1)^l\rangle$
	$(\alpha p^{k-1}\le s\le p^k-1,$
	$s-\alpha p^{k-1}<l<s)$
(20)	$\langle p(z-1)^{s-\alpha p^{k-1}}(-w(z)+\pi _i^{\nu }\widetilde{h}(z))$	$\langle (z-1)^{p^k-l}+p(z-1)^{p^{k-1}+\nu -l}(-\widetilde{h}(z)),$
	$+\,(z-1)^{s},\,p(z-1)^l\rangle$	$p(z-1)^{p^k-s}\rangle$
	$(\alpha p^{k-1}<s\le p^{k}-1,\,\nu \ge 1,$
	$s-\alpha p^{k-1}<l<\mathrm{min}\{s,\,p^{k-1}+\mu \})$
(21)	$\langle p(-w(z)+(z-1)^{\nu }\widetilde{h}(z))$	$\langle (z-1)^{p^k-l}+p(z-1)^{p^{k-1}+\nu -l}(-\widetilde{h}(z))\rangle$
	$+(z-1)^{\alpha p^{k-1}},\, p(z-1)^l\rangle$
	$(0<l<\mathrm{min}\{\alpha p^{k-1},\,p^{k-1}+\nu \},$
	$\nu \ge 1)$
(22)	$\langle (z-1)^s+p(z-1)^{s-\alpha p^{k-1}}h(z),$	$\langle (z-1)^{p^k-l}+p(z-1)^{p^{k-1}-l}(1-h(z)),$
	$p(z-1)^l\rangle$
	$(\alpha p^{k-1}<s\le p^k-1, \, h_0\ne 0,\,1,$	$p(z-1)^{p^k-s}\rangle$
	$s-\alpha p^{k-1}<l<p^{k-1})$
(23)	$\langle (z-1)^{\alpha p^{k-1}}+ph(z),\,p(z-1)^l\rangle$	$\langle (z-1)^{p^k-l}+p(z-1)^{p^{k-1}-l}(1-h(z))\rangle$
	$( h_0\ne 0,\,1, \, 0<l<p^{k-1})$
(24)	$\langle (z-1)^s+p(z-1)^th(z),\,p(z-1)^l\rangle$	$\langle (z-1)^{p^k-l}+p(z-1)^{p^{k-1}-l}(-w(z)$
	$(p^k+t-s\ne p^{k-1},\, 1\le s\le \alpha p^{k-1}+t,$	$+\,(z-1)^{\alpha p^{k-1}+t-s}(-h(z))),$
	$h(z)\ne 0,\, 0<t<l<\mathrm{min}\{s,\,p^{k-1}\})$	$p(z-1)^{p^k-s}\rangle$
(25)	$\langle (z-1)^s+ph(z),\,p(z-1)^l\rangle$	$\langle (z-1)^{p^k-l}+p(z-1)^{p^{k-1}-l}(-w(z)$
	$(1\le s<\alpha p^{k-1},\, h(z)\ne 0,$	$+\,(z-1)^{\alpha p^{k-1}+t-s}(-h(z)))\rangle$
	$0<l<\mathrm{min}\{s,\,p^{k-1}\})$
(26)	$\langle (z-1)^s+p(z-1)^th(z),\,p(z-1)^l\rangle$	$\langle (z-1)^{p^k-l}+p(z-1)^{p^{k}+t-s-l}(-h(z)$
	$(p^k+t-s\ne p^{k-1},\, h(z)\ne 0,\, t>0$	$+\,(z-1)^{s-t-\alpha p^{k-1}}),\,p(z-1)^{p^k-s}\rangle$
	$s>\alpha p^{k-1},\,0<t<l<p^k+t-s)$
(27)	$\langle (z-1)^s+ph(z),\,p(z-1)^l\rangle$	$\langle (z-1)^{p^k-l}+p(z-1)^{p^k+t-s-l}(-h(z)$
	$(s>\alpha p^{k-1},\, h(z)\ne 0,\, 0<l<p^k-s)$	$+\,(z-1)^{s-\alpha p^{k-1}}) \rangle$

Then by Theorem 3.1, Lemmas 3.2 and 3.3 we deduce the following corollary.

Corollary 3.5

Using the notations of Lemma 3.3, the number of all cyclic codes over $\mathbb {Z}_{p^2}$ of length $p^kn$ is equal to $\prod \nolimits _{i=1}^rN_{(p^2,m_i;k)}.$

Example 3.6

We calculate the number of cyclic codes over $\mathbb {Z}_9$ of length 33. In this case, we have $p=3,\,k=1$ and $n=11.$

Since {0, 1, 3, 9, 5, 4} and {2, 6, 7, 10, 8} are all distinct 3-cyclotomic cosets modulo 11, we have $y^{11}-1=f_1(y)f_2(y)f_3(y),$ where $f_1(y),\,f_2(y),\,f_3(y)$ are monic basic irreducible polynomials in $\mathbb {Z}_9[y]$ satisfying $m_1=\mathrm{deg}(f_1(y))=1$ and $m_i=\mathrm{deg}(f_1(y))=5$ for $i=2,\,3.$ By Corollary 3.5 and Lemma 3.3, the number of cyclic codes over $\mathbb {Z}_9$ of length 33 is equal to

$$\begin{aligned} \prod \limits _{i=1}^3N_{(3^2,m_i;1)}=\prod \limits _{i=1}^3\left( 1+2p+(2p-3)p^{m_i}\right) =16\cdot 736^2=8,667,136. \end{aligned}$$

For any ideal $C_i$ of the ring $R_i[x]/\langle x^{p^k}-\zeta _i\rangle ,$ the annihilating ideal of $C_i$ is defined as $\mathrm{Ann}(C_i)=\{\alpha \in R_i[x]/\langle x^{p^k}-\zeta _i\rangle \mid \alpha \beta =0,\,{\forall } \beta \in C_i \}.$ In the rest of this paper, we denote $w(z)=\sum \nolimits _{j=0}^{p-2}\left[ \frac{(-1)^{j+1}}{j+1}\right] _1 (z-1)^{jp^{k-1}},$ where $[a]_1$ denotes a $(\mathrm{mod} \ p)$ (cf. [7]), and

$$\begin{aligned} \pi _i=\zeta _i^ex-1\in R_i[x]/\left\langle x^{p^k}-\zeta _i\right\rangle , \quad \mathrm{where} \quad R_i=\mathbb {Z}_{p^2}\left[ \zeta _i\right] . \end{aligned}$$

Now, by Lemmas 3.2 and 3.3, we can list all distinct $\zeta _i$-constacyclic codes over the GR $R_i$ of length $p^k$ by the following theorem.

Theorem 3.7

Let $p\ne 2,\,\alpha =p-1$ and $\beta =p-2.$ Then all distinct $\zeta _i$-constacyclic codes $C_i$ over the GR $R_i$ of length $p^k$ and their annihilating ideals are given by the following:

Cases	$C_i$	Ann$(C_i)$
(1)	$\langle 0\rangle$	$\langle 1\rangle$
(2)	$\langle 1\rangle$	$\langle 0\rangle$
(3)	$\langle p\rangle$	$\langle p\rangle$
(4)	$\langle p\pi _i^s\rangle$ $(1\le s\le p^k-1)$	$\langle p,\, \pi _i^{p^k-s}\rangle$
(5)	$\langle \pi _i^s\rangle \, (1\le s\le p^{k-1})$	$\langle \pi _i^{p^k-s}+p\pi _i^{p^{k-1}-s}(-w(\zeta _i^ex))\rangle$
(6)	$\langle \pi _i^s\rangle \, (p^{k-1}+1\le s\le p^{k}-1)$	$\langle \pi _i^{\alpha p^{k-1}}+p(-w(\zeta _i^ex)),\,p\pi _i^{p^k-s}\rangle$
(7)	$\langle \pi _i^s+p\pi _i^{s-\alpha p^{k-1}}(-w(\zeta _i^ex))\rangle$	$\langle \pi _i^{p^k-s}\rangle$
	$(\alpha p^{k-1}\le s\le p^{k}-1)$
(8)	$\langle \pi _i^s+p\pi _i^{s-\alpha p^{k-1}}(-w(\zeta _i^ex)+\pi _i^{\nu }\widetilde{h}(\zeta _i^ex))\rangle$	$\langle \pi _i^{p^k-s}+p\pi _i^{p^{k-1}+\nu -s}(-\widetilde{h}(\zeta _i^ex))\rangle$
	$(\alpha p^{k-1}\le s\le p^{k-1}+\nu , \, \nu \ge 1)$

(9)	$\langle \pi _i^s+p\pi _i^{s-\alpha p^{k-1}}(-w(\zeta _i^ex)+\pi _i^{\nu }\widetilde{h}(\zeta _i^ex))\rangle$	$\langle \pi _i^{\alpha p^{k-1}-\nu }+p(-\widetilde{h}(\zeta _i^ex)),\, p\pi _i^{p^k-s}\rangle$
	$(p^{k-1}+\nu \le s\le p^{k}-1,$
	$s>\alpha p^{k-1}, \, \nu \ge 1)$
(10)	$\langle \pi _i^{\alpha p^{k-1}}+p(-w(\zeta _i^ex)+\pi _i^{\nu }\widetilde{h}(\zeta _i^ex))\rangle$	$\langle \pi _i^{\alpha p^{k-1}-\nu }+p(-\widetilde{h}(\zeta _i^ex))\rangle$
	$p^{k-1}+\nu <\alpha p^{k-1}, \, \nu \ge 1)$
(11)	$\langle \pi _i^{s}+p\pi _i^{s-\alpha p^{k-1}}h(\zeta _i^ex)\rangle$	$\langle \pi _i^{\alpha p^{k-1}}+p(1-h(\zeta _i^ex)),\,p\pi _i^{p^k-s}\rangle$
	$(\alpha p^{k-1}<s<p^k-1, \, h_0\ne 0,1)$
(12)	$\langle \pi _i^{\alpha p^{k-1}}+ph(\zeta _i^ex)\rangle \, (h_0\ne 0,\,1)$	$\langle \pi _i^{\alpha p^{k-1}}+p(1-h(\zeta _i^ex))\rangle$
(13)	$\langle \pi _i^s+p\pi _i^{t}h(\zeta _i^ex)\rangle$	$\langle \pi _i^{p^{k}-s}+p\pi _i^{p^{k-1}-s}(-w(\zeta _i^ex)$
	$(p^k+t-s\ne p^{k-1},\, s\le p^{k-1},\, h(x)\ne 0)$	$+\,\pi _i^{\alpha p^{k-1}+t-s}(-h(\zeta _i^ex)))\rangle$
(14)	$\langle \pi _i^s+p\pi _i^{t}h(\zeta _i^ex)\rangle$	$\langle \pi _i^{\alpha p^{k-1}}+p(-w(\zeta _i^ex)$
	$(p^k+t-s\ne p^{k-1},\, p^{k-1}<s\le \alpha p^{k-1}+t,$	$+\,\pi _i^{\alpha p^{k-1}+t-s}(-h(\zeta _i^ex))),\, p\pi _i^{p^k-s}\rangle$
	$t>0,\, h(x)\ne 0)$
(15)	$\langle \pi _i^s+ph(\zeta _i^ex)\rangle$	$\langle p(-w(\zeta _i^ex)+\pi _i^{\alpha p^{k-1}-s}(-h(\zeta _i^ex)))$
	$(p^{k-1}<s<\alpha p^{k-1},\, h(x)\ne 0)$	$+\,\pi _i^{\alpha p^{k-1}}\rangle$
(16)	$\langle \pi _i^s+p\pi _i^th(\zeta _i^ex)\rangle$	$\langle \pi _i^{s-t}+p(-h(\zeta _i^ex)+\pi _i^{s-t-\alpha p^{k-1}}),$
	$(p^k+t-s\ne p^{k-1},\, s>\alpha p^{k-1}+t,$	$p\pi _i^{p^k-s}\rangle$
	$h(x)\ne 0,\, t>0$)
(17)	$\langle \pi _i^s+ph(\zeta _i^ex)\rangle$	$\langle \pi _i^{s}+p(-h(\zeta _i^ex)+\pi _i^{s-\alpha p^{k-1}})\rangle$
	$(s>\alpha p^{k-1},\, h(x)\ne 0)$
(18)	$\langle \pi _i^s,\,p\pi _i^l\rangle$	$\langle \pi _i^{p^k-1}+p\pi _i^{p^{k-1}-l}(-w(\zeta _i^ex)),$
	$(1\le s\le p^k-1,$	$p\pi _i^{p^k-s}\rangle$
	$0\le l\le \mathrm{min}\{s,\,p^{k-1}\})$
(19)	$\langle \pi _i^s+p\pi _i^{s-\alpha p^{k-1}}(-w(\zeta _i^ex)),\,p\pi _i^l\rangle$	$\langle \pi _i^{p^k-l},\,p\pi _i^{p^k-s}\rangle$
	$(\alpha p^{k-1}\le s\le p^k-1,\, s-\alpha p^{k-1}<l<s)$
(20)	$\langle \pi _i^s+p\pi _i^{s-\alpha p^{k-1}}(-w(\zeta _i^ex)+\pi _i^{\nu }\widetilde{h}(\zeta _i^ex)),$	$\langle \pi _i^{p^k-l}+p\pi _i^{p^{k-1}+\nu -l}(-\widetilde{h}(\zeta _i^ex)),$
	$p\pi _i^l\rangle$	$p\pi _i^{p^k-s}\rangle$
	$(\alpha p^{k-1}<s\le p^{k}-1,\,\nu \ge 1,$
	$s-\alpha p^{k-1}<l<\mathrm{min}\{s,\,p^{k-1}+\mu \})$
(21)	$\langle \pi _i^{\alpha p^{k-1}}+p(-w(\zeta _i^ex)+\pi _i^{\nu }\widetilde{h}(\zeta _i^ex)),\, p\pi _i^l\rangle$	$\langle \pi _i^{p^k-l}+p\pi _i^{p^{k-1}+\nu -l}(-\widetilde{h}(\zeta _i^ex))\rangle$
	$(0<l<\mathrm{min}\{\alpha p^{k-1},\,p^{k-1}+\nu \},\, \nu \ge 1)$
(22)	$\langle \pi _i^s+p\pi _i^{s-\alpha p^{k-1}}h(\zeta _i^ex),\,p\pi _i^l\rangle$	$\langle \pi _i^{p^k-l}+p\pi _i^{p^{k-1}-l}(1-h(\zeta _i^ex)),$
	$(\alpha p^{k-1}<s\le p^k-1, \, h_0\ne 0,\,1,$	$p\pi _i^{p^k-s}\rangle$
	$s-\alpha p^{k-1}<l<p^{k-1})$
(23)	$\langle \pi _i^{\alpha p^{k-1}}+ph(\zeta _i^ex),\,p\pi _i^l\rangle$	$\langle \pi _i^{p^k-l}+p\pi _i^{p^{k-1}-l}(1-h(\zeta _i^ex))\rangle$
	$( h_0\ne 0,\,1, \, 0<l<p^{k-1})$
(24)	$\langle \pi _i^s+p\pi _i^th(\zeta _i^ex),\,p\pi _i^l\rangle$	$\langle \pi _i^{p^k-l}+p\pi _i^{p^{k-1}-l}(-w(\zeta _i^ex)$
	$(p^k+t-s\ne p^{k-1},\, 1\le s\le \alpha p^{k-1}+t,$	$+\,\pi _i^{\alpha p^{k-1}+t-s}(-h(\zeta _i^ex))),\, p\pi _i^{p^k-s}\rangle$
	$h(x)\ne 0,\, 0<t<l<\mathrm{min}\{s,\,p^{k-1}\})$

(25)	$\langle \pi _i^s+ph(\zeta _i^ex),\,p\pi _i^l\rangle$	$\langle \pi _i^{p^k-l}+p\pi _i^{p^{k-1}-l}(-w(\zeta _i^ex)$
	$(1\le s<\alpha p^{k-1},\, h(x)\ne 0,$	$+\,\pi _i^{\alpha p^{k-1}+t-s}(-h(\zeta _i^ex)))\rangle$
	$0<l<\mathrm{min}\{s,\,p^{k-1}\})$
(26)	$\langle \pi _i^s+p\pi _i^th(\zeta _i^ex),\,p\pi _i^l\rangle$	$\langle \pi _i^{p^k-l}+p\pi _i^{p^{k}+t-s-l}(-h(\zeta _i^ex)$
	$(p^k+t-s\ne p^{k-1},\, h(x)\ne 0,\, t>0$	$+\,\pi _i^{s-t-\alpha p^{k-1}}),\,p\pi _i^{p^k-s}\rangle$
	$s>\alpha p^{k-1},\, 0<t<l<p^k+t-s)$
(27)	$\langle \pi _i^s+ph(\zeta _i^ex),\,p\pi _i^l\rangle$	$\langle \pi _i^{p^k-l}+p\pi _i^{p^k+t-s-l}(-h(\zeta _i^ex)$
	$(s>\alpha p^{k-1},\, h(x)\ne 0,\, 0<l<p^k-s)$	$+\,\pi _i^{s-\alpha p^{k-1}}) \rangle$

Finally, by Theorems 3.1 and 3.7 we deduce the following corollary.

Corollary 3.8

Every cyclic code $\mathcal{C}$ over $\mathbb {Z}_{p^2}$ of length $p^kn$ can be constructed by the following two steps:

(i) For each $i=1,\ldots ,r,$ choose a $\zeta _i$-constacyclic code $C_i$ over $R_i$ of length $p^k$ listed in Theorem 3.7.

(ii) Set $\mathcal{C}=\oplus _{i=1}^r\mathcal{C}_i$ with $\mathcal{C}_i=\mathcal{A}_i\Box _{\varphi _i}C_i.$

The number of codewords in $\mathcal{C}$ is equal to $|\mathcal{C}|=\prod \nolimits _{i=1}^r|C_i|$ and the minimal Hamming distance of $\mathcal{C}$ satisfies $d_{\mathrm{min}}(\mathcal{C})\le \mathrm{min}\{d_{\mathrm{min}}(\mathcal{A}_i)d_{\mathrm{min}}(C_i)\mid i=1,\ldots ,r \},$ where $d_{\mathrm{min}}(\mathcal{A}_i)$ is the minimal $\mathbb {Z}_{p^2}$-Hamming weight of $\mathcal{A}_i$ and $d_{\mathrm{min}}(C_i)$ is the minimal $R_i$-Hamming weight of $C_i.$ Moreover, a generator matrix of $\mathcal{C}$ is given by $G_{\mathcal{C}}=\left( \begin{array}{c}G_{\mathcal{A}_1\Box _{\varphi _1}C_1}\\ \ldots \\ G_{\mathcal{A}_r\Box _{\varphi _r}C_r}\end{array}\right) .$

Using the notations of Corollary 3.8(ii), $\mathcal{C}=\oplus _{i=1}^r\mathcal{C}_i$ with $\mathcal{C}_i=\mathcal{A}_i\Box _{\varphi _i}C_i$ is called the canonical form decomposition of the cyclic code $\mathcal{C}$ over $\mathbb {Z}_{p^2}.$

4 Dual codes of cyclic codes over $\mathbb {Z}_{p^2}$ of length $p^kn$

In this section, we give the dual code of each cyclic code over $\mathbb {Z}_{p^2}$ of length N and investigate the self-duality of these codes.

As usual, we will identify $a=(a_0,\,a_1,\ldots ,a_{N-1})\in \mathbb {Z}_{p^2}^N$ with $a(x)=\sum \nolimits _{j=0}^{N-1}a_jx^j\in \mathbb {Z}_{p^2}[x]/\langle x^N-1\rangle .$ In this paper, we define

$$\begin{aligned} \mu (a(x))=a\left( x^{-1}\right) =a_0+\sum \limits _{j=1}^{N-1}a_jx^{N-j}, \quad {\forall } a(x)\in \mathbb {Z}_{p^2}[x]/\left\langle x^N-1\right\rangle . \end{aligned}$$

Then $\mu$ is a ring automorphism of $\mathbb {Z}_{p^2}[x]/\langle x^N-1\rangle$ satisfying $\mu ^{-1}=\mu$ and $\mu (c)=c$ for all $c\in \mathbb {Z}_{p^2}.$ The following lemma is well known.

Lemma 4.1

Let $a,\,b\in \mathbb {Z}_{p^2}^N.$ Then $[a,\,b]=0$ if $a(x)\mu (b(x))=0$ in the ring $\mathbb {Z}_{p^2}[x]/\langle x^N-1\rangle .$

Using the notations of Sect. 3, we have $\mathbb {Z}_{p^2}[x]/\langle x^N-1\rangle =\mathcal{A}[x]/\langle x^{p^k}-y\rangle$ under the substitution $y=x^{p^k},$ where $\mathcal{A}=\mathbb {Z}_{p^2}[y]/\langle y^n-1\rangle .$ Hence

$$\begin{aligned} \mu (y)=\left( x^{-1}\right) ^{p^k}=y^{-1} \quad \mathrm{in} \quad \mathcal{A}[x]/\left\langle x^{p^k}-y\right\rangle . \end{aligned}$$

Therefore, the restriction of $\mu$ to $\mathcal{A}$ is given by

$$\begin{aligned} \mu (f(y))=f\left( y^{-1}\right) \quad ({\forall } f(y)\in \mathcal{A}), \end{aligned}$$

which is a ring automorphism of $\mathcal{A}.$ For notations simplicity, we still denote this restriction by $\mu .$ From this and by Notation 2.1, we deduce

$$\begin{aligned} \mu \left( \varepsilon _i(y)\right) =a_i\left( y^{-1}\right) F_i\left( y^{-1}\right) =1-b_i\left( y^{-1}\right) f_i\left( y^{-1}\right) \, \mathrm{in} \, \mathcal{A}. \end{aligned}$$

(3)

Let $f(y)=\sum \nolimits _{j=0}^mc_jy^j$ be a polynomial in $\mathbb {Z}_{p^2}[y]$ of degree $m\ge 1.$ Recall that the reciprocal polynomial of f(y) is defined by $\widetilde{f}(y)=y^mf\left( \frac{1}{y}\right) =\sum \nolimits _{j=0}^m c_jy^{m-j}.$ Especially, f(y) is said to be self-reciprocal if $\widetilde{f}(y)=\delta f(y)$ for some invertible element $\delta$ in $\mathbb {Z}_{p^2},$ i.e., $\delta \in \mathbb {Z}_{p^2}^{\times }.$ Then by Eq. (1) in Sect. 2, we have

$$\begin{aligned} y^n-1=-\widetilde{f}_1(y),\widetilde{f}_2(y),\ldots ,\widetilde{f}_r(y). \end{aligned}$$

Since $f_1(y),\,f_2(y),\ldots ,f_r(y)$ are pairwise coprime monic basic polynomials in $\mathbb {Z}_{p^2}[y],$ for each $1\le i\le r$ there is a unique integer $i^{\prime },\,1\le i^{\prime }\le r,$ such that $\widetilde{f}_i(y)=\delta _if_{i^{\prime }}(y)$ where $\delta _i\in \mathbb {Z}_{p^2}^{\times }.$ Then by (3) and $y^n=1$ in $\mathcal{A},$ we have

$$\begin{aligned} \mu \left( \varepsilon _i(y)\right)= & {} 1-y^{n-\mathrm{deg}(b_i(y))-m_i}\left( y^{\mathrm{deg}(b_i(y))}b_i\left( y^{-1}\right) \right) \left( y^{m_i}f_i\left( y^{-1}\right) \right) \\= & {} 1-y^{n-\mathrm{deg}(b_i(y))-m_i}\widetilde{b}_i(y)\widetilde{f}_i(y)\\= & {} 1-h_i(y)f_{i^{\prime }}(y), \end{aligned}$$

where $h_i(y)=\delta _iy^{n-\mathrm{deg}(b_i(y))-m_i}\widetilde{b}_i(y)\in \mathcal{A}.$ Similarly, by (3) it follows that $\mu (\varepsilon _i(y))=g_i(y)F_{i^{\prime }}(y)$ for some $g_i(y)\in \mathcal{A}.$ Then from these and by Eq.  (2) we deduce that $\mu (\varepsilon _i(y))=\varepsilon _{i^{\prime }}(y).$

As stated above, we see that for each $1\le i\le r$ there is a unique integer $i^{\prime },\,1\le i^{\prime }\le r,$ such that $\mu (\varepsilon _i(y))= \varepsilon _{i^{\prime }}(y).$ We still use $\mu$ to denote this map $i\mapsto i^{\prime };$ i.e., $\mu (\varepsilon _i(y))=\varepsilon _{\mu (i)}(y).$ Whether $\mu$ denotes the automorphism of $\mathcal{A}$ 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 4.2

With the notations above, the following hold.

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

(ii) After a rearrangement of $\varepsilon _1(y),\ldots ,\varepsilon _r(y)$ there are integers $\lambda ,\,\rho$ such that $\mu (i)=i$ for all $i=1,\ldots ,\lambda$ and $\mu (\lambda +j)=\lambda +\rho +j$ for all $j=1,\ldots ,\rho ,$ where $\lambda \ge 1,\,\rho \ge 0$ and $\lambda +2\rho =r.$

(iii) For each integer $i,\,1\le i\le r,$ there is a unique invertible element $\delta _i$ of $\mathbb {Z}_{p^2}$ such that $\widetilde{f}_i(y)=\delta _i f_{\mu (i)}(y).$

(iv) For any integer $i,\,1\le i\le r,\, \mu (\varepsilon _i(y))=\varepsilon _{\mu (i)}(y)$ in the ring $\mathcal{A},$ and $\mu (\mathcal{A}_{i})=\mathcal{A}_{\mu (i)}.$ Then $\mu$ induces a ring isomorphism from $\mathcal{A}_{i}$ onto $\mathcal{A}_{\mu (i)}.$

Proof

(i)–(iii) follow from the definition of the map $\mu ,$ and (iv) follows from that $\mathcal{A}_i=\varepsilon _i(y)\mathcal{A}$ immediately. $\square$

Lemma 4.3

Using the notations above, the following hold for any $1\le i\le r.$

(i) $\mu$ induces a ring isomorphism $\varphi _i^{-1}\mu \varphi _i$ from $R_i$ onto $R_{\mu (i)}.$ We still denote this isomorphism by $\mu$ for notations simplicity. Then the following diagram commutes

$$\begin{aligned} \begin{array}{c@{\quad }c@{\quad }c} R_i=\mathbb {Z}_{p^2}[y]/\left\langle f_i(y)\right\rangle &{} \mathop {\longrightarrow }\limits ^{\mu } &{} R_{\mu (i)}=\mathbb {Z}_{p^2}[y]/\left\langle f_{\mu (i)}(y)\right\rangle \\ \varphi _i \downarrow &{} &{} \downarrow \varphi _i \\ \mathcal{A}_i &{} \mathop {\longrightarrow }\limits ^{\mu } &{} \mathcal{A}_{\mu (i)}. \end{array}\end{aligned}$$

Specifically, $\mu (a(y))=a(y^{-1})\in R_{\mu (i)}$ for any $a(y)\in R_i.$

(ii) Using the notations in $(\mathrm{i}),\,\mu (\zeta _i)=\zeta _{\mu (i)}^{-1}$ and $\mu$ induces a ring isomorphism from $R_i[x]/\langle x^{p^k}-\zeta _i\rangle$ onto $R_{\mu (i)}[x]/\langle x^{p^k}-\zeta _{\mu (i)}\rangle$ given by

$$\begin{aligned} \alpha (x)=\sum \limits _{j=0}^{p^k-1}\alpha _jx^j\mapsto \widehat{\alpha }\left( x^{-1}\right) :=\mu \left( \alpha _0\right) +\zeta _{\mu (i)}^{-1}\sum \limits _{j=1}^{p^k-1}\mu \left( \alpha _j\right) x^{p^k-j}, \end{aligned}$$

where $\widehat{\alpha }(x)=\sum \nolimits _{j=0}^{p^k-1}\mu (\alpha _j)x^j,\, {\forall } \alpha _0,\,\alpha _1,\ldots ,\alpha _{p^k-1}\in R_i.$

Proof

(i) It follows from Lemma 2.2(iii) and Lemma 4.2(iii) and (iv).

(ii) From $\zeta _i=y+\langle f_i(y)\rangle \in R_i$ and $\zeta _{\mu (i)}=y+\langle f_{\mu (i)}(y)\rangle \in R_{\mu (i)},$ by (i) we deduce that $\mu (\zeta _i)=\zeta _{\mu (i)}^{-1}\in R_{\mu (i)}.$ Since x and $\zeta _{\mu (i)}$ are invertible elements of $R_{\mu (i)}[x]/\langle x^{p^k}-\zeta _{\mu (i)}\rangle ,$ from $(x^{-1})^{p^k}-\zeta _{\mu (i)}^{-1}=-x^{-p^k}\zeta _{\mu (i)}^{-1}(x^{p^k}-\zeta _{\mu (i)})$ we deduce that $\mu$ induces a ring isomorphism from $R_i[x]/\langle x^{p^k}-\zeta _i\rangle$ onto $R_{\mu (i)}[x]/\langle x^{p^k}-\zeta _{\mu (i)}\rangle$ given by $\alpha (x)=\sum \nolimits _{j=0}^{p^k-1}\alpha _jx^j\mapsto \mu (\alpha (x))=\widehat{\alpha }(x^{-1})=\sum \nolimits _{j=0}^{p^k-1}\mu (\alpha _j)x^{-j}, \, {\forall }\alpha _0,\ldots ,\alpha _{2^k-1}\in R_i.$ Finally, by $x^{p^k}=\zeta _{\mu (i)}$ in $R_{\mu (i)}[x]/\langle x^{p^k}-\zeta _{\mu (i)}\rangle$ it follows that $\widehat{\alpha }(x^{-1})=\mu (\alpha _0)+\zeta _{\mu (i)}^{-1}\sum \nolimits _{j=1}^{p^k-1}\mu (\alpha _j)x^{p^k-j}$ as required. $\square$

Corollary 4.4

For each integer $i,\,1\le i\le r,$ denote $\pi _i=\zeta _i^ex-1\in R_i[x]/\langle x^{p^k}-\zeta _i\rangle ,$ where $R_i=\mathbb {Z}_{p^2}[\zeta _i].$ Then $\mu (\pi _i^l)=(-1)^l\zeta _{\mu (i)}^{-el}x^{-l}\pi _{\mu (i)}^l\in R_{\mu (i)}[x]/\langle x^{p^k}-\zeta _{\mu (i)}\rangle ,$ for any integer $l,\,1\le l\le p^{k}-1.$

Proof

By the proof of Lemma 4.3(ii), we have $\mu (\pi _i^l)=(\mu (\zeta _i^ex-1))^l=((\zeta _{\mu (i)}^{-1})^ex^{-1}-1)^l=(-1)^l\zeta _{\mu (i)}^{-el}x^{-l}(\zeta _{\mu (i)}^ex-1)^l =(-1)^l\zeta _{\mu (i)}^{-el}x^{-l}\pi _{\mu (i)}^l.$ $\square$

Lemma 4.5

Let $a(x)=\sum \nolimits _{i=1}^ra_i(x),\,b(x)=\sum \nolimits _{i=1}^rb_i(x)\in \mathcal{A}[x]/\langle x^{p^k}-y\rangle ,$ with $a_i(x),\,b_i(x)\in \mathcal{A}_i[x]/\langle x^{p^k}-y\rangle .$ Then $a(x)\mu (b(x))=\sum \nolimits _{i=1}^ra_i(x)\mu (b_{\mu (i)}(x)).$

Proof

By Lemma 4.2 we have $\mu (b_{\mu (i)}(x))\in \mu (\mathcal{A}_{\mu (i)}[x]/\langle x^{p^k}-y\rangle )$ and $\mu (\mathcal{A}_{\mu (i)}[x]/\langle x^{p^k}-y\rangle )=\mathcal{A}_i[x]/\langle x^{p^k}-y\rangle .$ Hence $a_i(x)\mu (b_{\mu (i)}(x))\in \mathcal{A}_i[x]/\langle x^{p^k}-y\rangle$ for all i. If $j\ne \mu (i),$ then $i\ne \mu (j),$ which implies $a_i(x)\mu (b_j(x))\in (\mathcal{A}_i[x]/\langle x^{p^k}-y\rangle )(\mathcal{A}_{\mu (j)}[x]/\langle x^{p^k}-y\rangle ) =\{0\}$ by Lemma 2.2(ii). Therefore, $a(x)\mu (b(x))=\sum \nolimits _{i=1}^r\sum \nolimits _{j=1}^ra_i(x)\mu (b_{j}(x))=\sum \nolimits _{i=1}^ra_i(x)\mu (b_{\mu (i)}(x)).$ $\square$

Now, we can determine the dual code of each cyclic code over $\mathbb {Z}_{p^2}.$

Theorem 4.6

Let $\mathcal{C}$ be a cyclic code over $\mathbb {Z}_{p^2}$ of length N with concatenated structure $\mathcal{C}=\oplus _{i=1}^r(\mathcal{A}_i\Box _{\varphi _i}C_i),$ where $C_i$ is an ideal of the ring $R_i[x]/\langle x^{p^k}-\zeta _i\rangle$ for all $i=1,\ldots ,r.$ Using the notations of Theorem 3.7 and Lemma 4.3 $(\mathrm{ii}),$ the dual code $\mathcal{C}^{\bot }$ is given by

$$\begin{aligned} \mathcal{C}^{\bot }=\oplus _{i=1}^r\left( \mathcal{A}_{\mu (i)}\Box _{\varphi _{\mu (i)}}D_{\mu (i)}\right) , \end{aligned}$$

where $D_{\mu (i)}$ is an ideal of the ring $R_{\mu (i)}[x]/\langle x^{p^k}-\zeta _{\mu (i)}\rangle$ given by one of the following cases $(1\le i\le r){\text {:}}$

Cases	$C_i$	$D_{\mu (i)}$
(1)	$\langle 0\rangle$	$\langle 1\rangle$
(2)	$\langle 1\rangle$	$\langle 0\rangle$
(3)	$\langle p\rangle$	$\langle p\rangle$
(4)	$\langle p\pi _i^s\rangle \, (1\le s\le p^k-1)$	$\langle p,\, \pi _{\mu (i)}^{p^k-s}\rangle$
(5)	$\langle \pi _i^s\rangle \, (1\le s\le p^{k-1})$	$\langle \pi _{\mu (i)}^{p^k-s}+p\pi _{\mu (i)}^{p^{k-1}-s}(-\widehat{w}(\zeta _{\mu (i)}^{-e}x^{-1}))\omega \rangle$
		$\omega =(-1)^{p^{k-1}-p^k}\zeta _{\mu (i)}^{e(p^k-p^{k-1})}x^{p^k-p^{k-1}}$
(6)	$\langle \pi _i^s\rangle \, (p^{k-1}+1\le s\le p^{k}-1)$	$\langle \pi _{\mu (i)}^{\alpha p^{k-1}}+p(-\widehat{w}(\zeta _{\mu (i)}^{-e}x^{-1}))\omega ,\,p\pi _{\mu (i)}^{p^k-s}\rangle$
		$\omega =(-1)^{-\alpha p^{k-1}}\zeta _{\mu (i)}^{e\alpha p^{k-1}}x^{\alpha p^{k-1}}$
(7)	$\langle \pi _i^s+p\pi _i^{s-\alpha p^{k-1}}(-w(\zeta _i^ex))\rangle$	$\langle \pi _{\mu (i)}^{p^k-s}\rangle$
	$(\alpha p^{k-1}\le s\le p^{k}-1)$
(8)	$\langle \pi _i^s+p\pi _i^{s-\alpha p^{k-1}}(-w(\zeta _i^ex)+\pi _i^{\nu }\widetilde{h}(\zeta _i^ex))\rangle$	$\langle \pi _{\mu (i)}^{p^k-s}+p\pi _{\mu (i)}^{p^{k-1}+\nu -s}(-\widehat{\widetilde{h}}(\zeta _{\mu (i)}^{-e}x^{-1}))\omega \rangle$
	$(\alpha p^{k-1}\le s\le p^{k-1}+\nu , \, \nu \ge 1)$	$\omega =(-1)^{p^{k-1}+\nu -p^k}\zeta _{\mu (i)}^{e(p^k-p^{k-1}-\nu )}$
		$\cdot x^{p^k-p^{k-1}-\nu }$

(9)	$\langle \pi _i^s +p\pi _i^{s-\alpha p^{k-1}}(-w(\zeta _i^ex)+\pi _i^{\nu }\widetilde{h}(\zeta _i^ex))\rangle$	$\langle \pi _{\mu (i)}^{\alpha p^{k-1}-\nu }+p(-\widehat{\widetilde{h}}(\zeta _{\mu (i)}^{-e}x^{-1}))\omega , p\pi _{\mu (i)}^{p^k-s}\rangle$
	$(p^{k-1}+\nu \le s\le p^{k}-1,$
	$s>\alpha p^{k-1}, \, \nu \ge 1)$	$\omega =(-1)^{\nu -\alpha p^{k-1}}\zeta _{\mu (i)}^{e(\alpha p^{k-1}-\nu )}x^{\alpha p^{k-1}-\nu }$
(10)	$\langle \pi _i^{\alpha p^{k-1}}+p(-w(\zeta _i^ex)+\pi _i^{\nu }\widetilde{h}(\zeta _i^ex))\rangle$	$\langle \pi _{\mu (i)}^{\alpha p^{k-1}-\nu }+p(-\widehat{\widetilde{h}}(\zeta _{\mu (i)}^{-e}x^{-1}))\omega \rangle$
	$p^{k-1}+\nu <\alpha p^{k-1}, \, \nu \ge 1)$	$\omega =(-1)^{\nu -\alpha p^{k-1}}\zeta _{\mu (i)}^{e(\alpha p^{k-1}-\nu )}x^{\alpha p^{k-1}-\nu }$
(11)	$\langle \pi _i^{s}+p\pi _i^{s-\alpha p^{k-1}}h(\zeta _i^ex)\rangle$	$\langle \pi _{\mu (i)}^{\alpha p^{k-1}}+p(1-\widehat{h}(\zeta _{\mu (i)}^{-e}x^{-1}))\omega ,\,p\pi _{\mu (i)}^{p^k-s}\rangle$
	$(\alpha p^{k-1}<s<p^k-1, \, h_0\ne 0,\,1)$	$\omega =(-1)^{-\alpha p^{k-1}}\zeta _{\mu (i)}^{e\alpha p^{k-1}}x^{\alpha p^{k-1}}$
(12)	$\langle \pi _i^{\alpha p^{k-1}}+ph(\zeta _i^ex)\rangle \, (h_0\ne 0,\,1)$	$\langle \pi _{\mu (i)}^{\alpha p^{k-1}}+p(1-\widehat{h}(\zeta _{\mu (i)}^{-e}x^{-1}))\omega \rangle$
		$\omega =(-1)^{-\alpha p^{k-1}}\zeta _{\mu (i)}^{e\alpha p^{k-1}}x^{\alpha p^{k-1}}$
(13)	$\langle \pi _i^s+p\pi _i^{t}h(\zeta _i^ex)\rangle$	$\langle \pi _{\mu (i)}^{p^{k}-s}+p\pi _{\mu (i)}^{p^{k-1}-s}(-\widehat{w}(\zeta _{\mu (i)}^{-e}x^{-1})$
	$(p^k+t-s\ne p^{k-1},\, s\le p^{k-1},$	$+\,\pi _{\mu (i)}^{\alpha p^{k-1}+t-s}(-\widehat{h}(\zeta _{\mu (i)}^{-e}x^{-1}))\omega _1)\omega _2\rangle$
	$h(x)\ne 0)$	$\omega _1=(-1)^{\alpha p^{k-1}+t-s}\zeta _{\mu (i)}^{e(s-\alpha p^{k-1}-t)}$
		$\cdot x^{s-\alpha p^{k-1}-t}$
		$\omega _2=(-1)^{p^{p-1}-p^k}\zeta _{\mu (i)}^{e(p^k-p^{k-1})}x^{p^k-p^{k-1}}$
(14)	$\langle \pi _i^s+p\pi _i^{t}h(\zeta _i^ex)\rangle$	$\langle \pi _{\mu (i)}^{\alpha p^{k-1}}+p(-\widehat{w}(\zeta _{\mu (i)}^{-e}x^{-1})$
	$(p^k+t-s\ne p^{k-1},$	$+\,\pi _{\mu (i)}^{\alpha p^{k-1}+t-s}(-\widehat{h}(\zeta _{\mu (i)}^{-e}x^{-1}))\omega _1)\omega _2,\, p\pi _{\mu (i)}^{p^k-s}\rangle$
	$p^{k-1}<s\le \alpha p^{k-1}+t,$	$\omega _1=(-1)^{\alpha p^{k-1}+t-s}\zeta _{\mu (i)}^{e(s-\alpha p^{k-1}-t)}$
	$t>0,\,h(x)\ne 0)$	$\cdot x^{s-\alpha p^{k-1}-t}$
		$\omega _2=(-1)^{-\alpha p^{k-1}}\zeta _{\mu (i)}^{e\alpha p^{k-1}}x^{\alpha p^{k-1}}$
(15)	$\langle \pi _i^s+ph(\zeta _i^ex)\rangle$	$\langle \pi _{\mu (i)}^{\alpha p^{k-1}}+p(-\widehat{w}(\zeta _{\mu (i)}^{-e}x^{-1})$
	$(p^{k-1}<s<\alpha p^{k-1},\,h(x)\ne 0)$	$+\,\pi _{\mu (i)}^{\alpha p^{k-1}-s}(-\widehat{h}(\zeta _{\mu (i)}^{-e}x^{-1}))\omega _1)\omega _2\rangle$
		$\omega _1=(-1)^{\alpha p^{k-1}-s}\zeta _{\mu (i)}^{e(s-\alpha p^{k-1})}x^{s-\alpha p^{k-1}}$
		$\omega _2=(-1)^{-\alpha p^{k-1}}\zeta _{\mu (i)}^{e\alpha p^{k-1}}x^{\alpha p^{k-1}}$
(16)	$\langle \pi _i^s+p\pi _i^th(\zeta _i^ex)\rangle$	$\langle p(-\widehat{h}(\zeta _{\mu (i)}^{-e}x^{-1})+\pi _{\mu (i)}^{s-t-\alpha p^{k-1}}\omega _1)\omega _2$
	$(p^k+t-s\ne p^{k-1},$	$+\,\pi _{\mu (i)}^{s-t},\, p\pi _{\mu (i)}^{p^k-s}\rangle$
	$s>\alpha p^{k-1}+t,$	$\omega _1=(-1)^{s-t-\alpha p^{k-1}}\zeta _{\mu (i)}^{e(\alpha p^{k-1}+t-s)}$
	$h(x)\ne 0,\, t>0$)	$\cdot x^{\alpha p^{k-1}+t-s}$
		$\omega _2=(-1)^{t-s}\zeta _{\mu (i)}^{e(s-t)}x^{s-t}$
(17)	$\langle \pi _i^s+ph(\zeta _i^ex)\rangle$	$\langle p(-\widehat{h}(\zeta _{\mu (i)}^{-e}x^{-1})+\pi _{\mu (i)}^{s-\alpha p^{k-1}}\omega _1)\omega _2$
	$(s>\alpha p^{k-1},\,h(x)\ne 0)$	$+\,\pi _{\mu (i)}^{s}\rangle$
		$\omega _1=(-1)^{s-\alpha p^{k-1}}\zeta _{\mu (i)}^{e(\alpha p^{k-1}-s)}x^{\alpha p^{k-1}-s}$
		$\omega _2=(-1)^{-s}\zeta _{\mu (i)}^{es}x^{s}$

(18)	$\langle \pi _i^s,\,p\pi _i^l\rangle$	$\langle \pi _{\mu (i)}^{p^k-1}+p\pi _{\mu (i)}^{p^{k-1}-l}(-\widehat{w}(\zeta _{\mu (i)}^{-e}x^{-1})),$
	$(1\le s\le p^k-1,$	$p\pi _{\mu (i)}^{p^k-s}\rangle$
	$0\le l\le \mathrm{min}\{s,\,p^{k-1}\})$	$\omega =(-1)^{p^{p-1}-p^k-l+1}\zeta _{\mu (i)}^{e(p^k-p^{k-1}+l-1)}$
		$\cdot x^{p^k-p^{k-1}+l-1}$
(19)	$\langle \pi _i^s+p\pi _i^{s-\alpha p^{k-1}}(-w(\zeta _i^ex)),\,p\pi _i^l\rangle$	$\langle \pi _{\mu (i)}^{p^k-l},\,p\pi _{\mu (i)}^{p^k-s}\rangle$
	$(\alpha p^{k-1}\le s\le p^k-1,$
	$s-\alpha p^{k-1}<l<s)$
(20)	$\langle p\pi _i^{s-\alpha p^{k-1}}(-w(\zeta _i^ex)+\pi _i^{\nu }\widetilde{h}(\zeta _i^ex))$	$\langle p\pi _{\mu (i)}^{p^{k-1}+\nu -l}(-\widehat{\widetilde{h}}(\zeta _{\mu (i)}^{-e}x^{-1}))\omega$
	$+\pi _i^s,\,p\pi _i^l\rangle$	$+\,\pi _{\mu (i)}^{p^k-l},\,p\pi _{\mu (i)}^{p^k-s}\rangle$
	$(\alpha p^{k-1}<s\le p^{k}-1,\,\nu \ge 1,$	$\omega =(-1)^{p^{k-1}+\nu -p^k}\zeta _{\mu (i)}^{-e(p^k-p^{k-1}-\nu )}$
	$s-\alpha p^{k-1}<l<\mathrm{min}\{s,\,p^{k-1}+\mu \})$	$\cdot x^{p^k-p^{k-1}-\nu }$
(21)	$\langle \pi _i^{\alpha p^{k-1}}+p(-w(\zeta _i^ex)+\pi _i^{\nu }\widetilde{h}(\zeta _i^ex)),$	$\langle \pi _{\mu (i)}^{p^k-l}+p\pi _{\mu (i)}^{p^{k-1}+\nu -l}(-\widehat{\widetilde{h}}(\zeta _{\mu (i)}^{-e}x^{-1}))\omega \rangle$
	$p\pi _i^l\rangle$	$\omega =(-1)^{p^{k-1}+\nu -p^k}\zeta _{\mu (i)}^{-e(p^k-p^{k-1}-\nu )}$
	$(0<l<\mathrm{min}\{\alpha p^{k-1},\,p^{k-1}+\nu \},$	$\cdot x^{p^k-p^{k-1}-\nu }$
	$\nu \ge 1)$
(22)	$\langle \pi _i^s+p\pi _i^{s-\alpha p^{k-1}}h(\zeta _i^ex),\,p\pi _i^l\rangle$	$\langle \pi _{\mu (i)}^{p^k-l}+p\pi _{\mu (i)}^{p^{k-1}-l}(1-\widehat{h}(\zeta _{\mu (i)}^{-e}x^{-1}))\omega ,$
	$(\alpha p^{k-1}<s\le p^k-1, \, h_0\ne 0,\,1,$	$p\pi _{\mu (i)}^{p^k-s}\rangle$
	$s-\alpha p^{k-1}<l<p^{k-1})$	$\omega =(-1)^{p^{k-1}-p^k}\zeta _{\mu (i)}^{-e(p^k-p^{k-1})}x^{p^k-p^{k-1}}$
(23)	$\langle \pi _i^{\alpha p^{k-1}}+ph(x),\,p\pi _i^l\rangle$	$\langle \pi _{\mu (i)}^{p^k-l}+p\pi _{\mu (i)}^{p^{k-1}-l}(1-\widehat{h}(\zeta _{\mu (i)}^{-e}x^{-1}))\omega \rangle$
	$( h_0\ne 0,\,1, \, 0<l<p^{k-1})$	$\omega =(-1)^{p^{k-1}-p^k}\zeta _{\mu (i)}^{-e(p^k-p^{k-1})}x^{p^k-p^{k-1}}$
(24)	$\langle \pi _i^s+p\pi _i^th(\zeta _i^ex),\,p\pi _i^l\rangle$	$\langle \pi _{\mu (i)}^{p^k-l}+p\pi _{\mu (i)}^{p^{k-1}-l}(-\widehat{w}(\zeta _{\mu (i)}^{-e}x^{-1})$
	$(p^k+t-s\ne p^{k-1},\, 1\le s\le \alpha p^{k-1}+t,$	$+\,\pi _{\mu (i)}^{\alpha p^{k-1}+t-s}(-\widehat{h}(\zeta _{\mu (i)}^{-e}x^{-1}))\omega _1)\omega _2,$
	$h(x)\ne 0,\, 0<t<l<\mathrm{min}\{s,\,p^{k-1}\})$	$p\pi _{\mu (i)}^{p^k-s}\rangle$
		$\omega _1=(-1)^{\alpha p^{k-1}+t-s}\zeta _{\mu (i)}^{-e(\alpha p^{k-1}+t-s)}$
		$\cdot x^{s-t-\alpha p^{k-1}}$
		$\omega _2(-1)^{p^{k-1}-p^k}\zeta _{\mu (i)}^{-e(p^k-p^{k-1})}x^{p^k-p^{k-1}}$
(25)	$\langle \pi _i^s+ph(\zeta _i^ex),\,p\pi _i^l\rangle$	$\langle \pi _{\mu (i)}^{p^k-l}+p\pi _{\mu (i)}^{p^{k-1}-l}(-\widehat{w}(\zeta _{\mu (i)}^{-e}x^{-1})$
	$(1\le s<\alpha p^{k-1},\, h(x)\ne 0,$	$+\,\pi _{\mu (i)}^{\alpha p^{k-1}+t-s}(-\widehat{h}(\zeta _{\mu (i)}^{-e}x^{-1}))\omega _1)\omega _2\rangle$
	$0<l<\mathrm{min}\{s,\,p^{k-1}\})$	$\omega _1=(-1)^{\alpha p^{k-1}+t-s}\zeta _{\mu (i)}^{-e(\alpha p^{k-1}+t-s)}$
		$\cdot x^{s-t-\alpha p^{k-1}}$
		$\omega _2(-1)^{p^{k-1}-p^k}\zeta _{\mu (i)}^{-e(p^k-p^{k-1})}x^{p^k-p^{k-1}}$
(26)	$\langle \pi _i^s+p\pi _i^th(\zeta _i^ex),\,p\pi _i^l\rangle$	$\langle \pi _{\mu (i)}^{p^k-l}+p\pi _{\mu (i)}^{p^{k}+t-s-l}(-\widehat{h}(\zeta _{\mu (i)}^{-e}x^{-1})$
	$(p^k+t-s\ne p^{k-1},\,h(x)\ne 0,\,t>0$	$+\,\pi _{\mu (i)}^{s-t-\alpha p^{k-1}}\omega _1)\omega _2,\,p\pi _{\mu (i)}^{p^k-s}\rangle$

	$s>\alpha p^{k-1},\,0<t<l<p^k+t-s)$	$\omega _1=(-1)^{s-t-\alpha p^{k-1}}\zeta _{\mu (i)}^{-e(s-t-\alpha p^{k-1})}$
		$\cdot x^{\alpha p^{k-1}+t-s}$
		$\omega _2=(-1)^{t-s}\zeta _{\mu (i)}^{e(s-t)}x^{s-t}$
(27)	$\langle \pi _i^s+ph(\zeta _i^ex),\,p\pi _i^l\rangle$	$\langle \pi _{\mu (i)}^{p^k-l}+p\pi _{\mu (i)}^{p^k+t-s-l}(-\widehat{h}(\zeta _{\mu (i)}^{-e}x^{-1})$
	$(s>\alpha p^{k-1},\, h(x)\ne 0,\, 0<l<p^k-s)$	$+\,\pi _{\mu (i)}^{s-\alpha p^{k-1}}\omega _1)\omega _2 \rangle$
		$\omega _1=(-1)^{s-\alpha p^{k-1}}\zeta _{\mu (i)}^{e(\alpha p^{k-1}-s)}x^{\alpha p^{k-1}-s}$
		$\omega _2=(-1)^{t-s}\zeta _{\mu (i)}^{e(s-t)}x^{s-t}$

Proof

For any integer $i,\,1\le i\le r,$ let $D_{\mu (i)}=\mu (\mathrm{Ann}(C_i)).$ Then $D_{\mu (i)}$ is an ideal of the ring $R_{\mu (i)}[x]/\langle x^{p^k}-\zeta _{\mu (i)}\rangle .$ Set $\mathcal{D}=\oplus _{i=1}^r(\mathcal{A}_{\mu (i)}\Box _{\varphi _{\mu (i)}}D_{\mu (i)}).$ Then $\mathcal{D}$ is an ideal of $\mathcal{A}[x]/\langle x^{p^k}-y\rangle$ and satisfies

$$\begin{aligned} \mathcal{C}\cdot \mu (\mathcal{D})= & {} \sum \limits _{i=1}^r\left( \mathcal{A}_i\Box _{\varphi _i}C_i\right) \cdot \mu \left( \mathcal{A}_{\mu (i)}\Box _{\varphi _{\mu (i)}}D_{\mu (i)}\right) \\= & {} \sum \limits _{i=1}^r\left( \mathcal{A}_i\Box _{\varphi _i}C_i\right) \cdot \left( \mathcal{A}_i\Box _{\varphi _{i}}\mathrm{Ann}\left( C_i\right) \right) \\= & {} \sum \limits _{i=1}^r\varepsilon _i(y)\left( C_i\cdot \mathrm{Ann}\left( C_i\right) \right) \\= & {} \{0\}, \end{aligned}$$

by Lemma 4.5. From this and by Lemma 4.1 we deduce $\mathcal{D}\subseteq \mathcal{C}^{\bot }.$

On the other hand, by [5, Theorem 3.5] and Lemma 3.2 we see that $|C_i||D_{\mu (i)}|=|C_i||\mathrm{Ann}(C_i)|=p^{2p^km_i}$ for all $i=1,\ldots ,r,$ which then implies

$$\begin{aligned} |\mathcal{C}||\mathcal{D}|= & {} \prod \limits _{i=1}^r\left| \mathcal{A}_i\Box _{\varphi _i}C_i\right| \left| \mathcal{A}_{\mu (i)}\Box _{\varphi _{\mu (i)}}D_{\mu (i)}\right| =\prod \limits _{i=1}^r\left| C_i\right| \left| D_{\mu (i)}\right| \\= & {} p^{2p^k\sum \nolimits _{i=1}^rm_i}=p^{2p^kn} =\left| \mathbb {Z}_{p^2}[x]/\left\langle x^{p^kn}-1\right\rangle \right| . \end{aligned}$$

As stated above, we conclude that $\mathcal{C}^{\bot }=\mathcal{D}$ since $\mathbb {Z}_{p^2}$ is a finite chain ring. Finally, the conclusions follow from Theorem 3.7 and Corollary 4.4 immediately. $\square$

Finally, by Theorem 4.6 and [7, Lemma 4.5] we deduce the following corollary for cyclic self-dual codes over $\mathbb {Z}_{p^2}.$

Corollary 4.7

Using the notations in Theorem 4.6 and Lemma 4.2 $(\mathrm{ii}),$ let $\mathcal{C}$ be a cyclic code over $\mathbb {Z}_{p^2}$ of length N with $\mathcal{C}=\oplus _{i=1}^r(\mathcal{A}_i\Box _{\varphi _i}C_i),$ where $C_i$ is an ideal of $R_i[x]/\langle x^{p^k}-\zeta _i\rangle .$ Then $\mathcal{C}$ is self-dual if and only if for each integer $i,\,1\le i\le r,\,C_i$ satisfies one of the following conditions:

(i) If $1\le i\le \lambda ,\,C_i$ is given by one of the following six cases:

(i-1) $C_i=\langle p\rangle .$

(i-2) $C_i=\langle \pi _i^{\alpha p^{k-1}}+ph(\zeta _i^ex)\rangle ,$ where h(z) satisfies $h_0\ne 0,\,1$ and $h(\zeta _i^ex)-(1-\widehat{h}(\zeta _i^{-e}x^{-1}))(-1)^{-\alpha p^{k-1}}\zeta _i^{e\alpha p^{k-1}}x^{\alpha p^{k-1}}=0.$

(i-3) $C_i=\langle \pi _i^{s}+ph(\zeta _i^ex)\rangle ,$ where $s>\alpha p^{k-1},\,h(z)\ne 0$ and $h(\zeta _i^ex)-(\pi _{i}^{s-\alpha p^{k-1}}\omega _1-\widehat{h}(\zeta _{i}^{-e}x^{-1}))\omega _2=0$ with $\omega _1=(-1)^{s-\alpha p^{k-1}}\zeta _{i}^{e(\alpha p^{k-1}-s)}x^{\alpha p^{k-1}-s}$ and $\omega _2=(-1)^{-s}\zeta _{i}^{es}x^{s}.$

(i-4) $C_i=\langle \pi _i^{s}+p\pi _i^{p^k-s}\rangle ,$ where $2s\ge \alpha p^{k-1}+p^k.$

(i-5) $C_i=\langle \pi _i^{s}+p\pi _i^{s-\alpha p^{k-1}}h(\zeta _i^ex),\,p\pi _i^{p^k-s}\rangle ,$ where $s\ge \alpha p^{k-1}+p^k,\,h_0\ne 0,\,1$ and $h(\zeta _i^ex)-(1-\widehat{h}(\zeta _{i}^{-e}x^{-1}))\omega =0$ with $\omega =(-1)^{p^{k-1}-p^k}\zeta _{i}^{-e(p^k-p^{k-1})}x^{p^k-p^{k-1}}.$

(i-6) $C_i=\langle \pi _i^{s}+p\pi _i^{t}h(\zeta _i^ex),\, p\pi _i^{p^k-s}\rangle ,$ where $0<t<p^k-s,\,s>\alpha p^{k-1}+t,\, h(\zeta _i^ex)-(\pi _{i}^{s-t-\alpha p^{k-1}}\omega _1-\widehat{h}(\zeta _{i}^{-e}x^{-1}))(-1)^{t-s}\zeta _{i}^{e(s-t)}x^{s-t}=0$ with $\omega _1=(-1)^{s-t-\alpha p^{k-1}}\zeta _{i}^{-e(s-t-\alpha p^{k-1})}x^{\alpha p^{k-1}+t-s}.$

(ii) If $i=\lambda +j$ where $1\le j\le \rho ,$ then $\mu (i)=i+\rho ,\,C_{\mu (i)}=D_{\mu (i)}$ and $(C_i,\,D_{\mu (i)})$ is given by Theorem 4.6.

5 Cyclic self-dual codes over $\mathbb {Z}_{9}$ of length 33

In this section, we consider to present all cyclic self-dual codes over $\mathbb {Z}_{9}$ of length 33. In this case, we have $N=33=3^kn$ where $k=1$ and $n=11.$

It is known that $y^{11}-1=f_1(y)f_2(y)f_3(y),$ where $f_1(y)=y-1,\,f_2(y)= y^5 + 3y^4 + 8y^3 + y^2 + 2y + 8$ and $f_3(y)=y^5 + 7y^4 + 8y^3 + y^2 + 6y + 8$ are pairwise coprime monic basic irreducible polynomials in $\mathbb {Z}_{9}[y].$ Obviously, $\widetilde{f}_1(y)=\delta _1f_1(y)$ and $\widetilde{f}_2(y)=\delta _2f_3(y)$ where $\delta _1=\delta _2=-1,$ which implies that $\mu (1)=1$ and $\mu (2)=3.$ Hence $m_1=1,\, m_2=m_3=5,\,r=3$ and $\lambda =\rho =1.$

Using the notations in Sect. 2, for each integer $i,\,1\le i\le 3,$ we denote $F_i(y)=\frac{y^{11}-1}{f_i(y)},$ and find polynomials $a_i(y),\,b_i(y)\in \mathbb {Z}_{9}[y]$ satisfying $a_i(y)F_i(y)+b_i(y)f_i(y)=1.$ Then set $\varepsilon _i(y)\equiv a_i(y)F_i(y)$ (mod $y^{11}-1$). Precisely, we have

$$\begin{aligned} \varepsilon _1(y)= & {} 5y^{10}+5y^9+5y^8+5y^7+5y^6+5y^5+5y^4+5y^3+5y^2+5y+5;\\ \varepsilon _2(y)= & {} 3y^{10}+y^9+3y^8+3y^7+3y^6+y^5+y^4+y^3+3y^2+y+7;\\ \varepsilon _3(y)= & {} y^{10}+3y^9+y^8+y^7+y^6+3y^5+3y^4+3y^3+y^2+3y+7. \end{aligned}$$

Let $\mathcal{A}=\mathbb {Z}_9[y]/\langle y^{11}-1\rangle$ and $\mathcal{A}_i=\mathcal{A}\varepsilon _i(y).$ Then $\mathcal{A}_i$ is a cyclic code over $\mathbb {Z}_9$ of length 11 with parity check polynomial $f_i(y)$ for $i=1,\,2,\,3.$ Therefore,

$\diamond$ $\mathcal{A}_1$ is a free $\mathbb {Z}_9$-submodule of $\mathbb {Z}_9^{11}$ with $\mathrm{rank}_{\mathbb {Z}_9}(\mathcal{A}_1)=1.$

$\diamond$ $\mathcal{A}_i$ is a free $\mathbb {Z}_9$-submodule of $\mathbb {Z}_9^{11}$ with $\mathrm{rank}_{\mathbb {Z}_9}(\mathcal{A}_i)=5$ for $i=2,\,3.$

Precisely, a generator matrix $G_{\mathcal{A}_i}$ of the cyclic code $\mathcal{A}_i$ over $\mathbb {Z}_9$ is given by: $G_{\mathcal{A}_1}=(5,\,5,\,5,\,5,\,5,\,5,\,5,\,5,\,5,\,5,\,5),$

$$\begin{aligned} G_{\mathcal{A}_2}= & {} \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 7 &{} 1 &{} 3 &{} 1 &{}1 &{} 1 &{}3 &{} 3 &{} 3 &{} 1 &{}3\\ 3 &{} 7 &{} 1 &{} 3 &{} 1 &{}1 &{} 1 &{}3 &{} 3 &{} 3 &{} 1\\ 1 &{} 3 &{} 7 &{} 1 &{} 3 &{} 1 &{}1 &{} 1 &{}3 &{} 3 &{} 3\\ 3 &{} 1 &{} 3 &{} 7 &{} 1 &{} 3 &{} 1 &{}1 &{} 1 &{}3 &{} 3\\ 3 &{} 3 &{} 1 &{} 3 &{} 7 &{} 1 &{} 3 &{} 1 &{}1 &{} 1 &{}3\end{array}\right) \quad \mathrm{and} \quad \nonumber \\ G_{\mathcal{A}_3}= & {} \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 7 &{} 3 &{} 1 &{} 3 &{}3 &{} 3 &{} 1 &{}1 &{}1 &{} 3 &{} 1 \\ 1 &{} 7 &{} 3 &{} 1 &{} 3 &{}3 &{} 3 &{} 1 &{}1 &{}1 &{} 3\\ 3 &{} 1 &{} 7 &{} 3 &{} 1 &{} 3 &{}3 &{} 3 &{} 1 &{}1 &{}1 \\ 1 &{} 3 &{} 1 &{} 7 &{} 3 &{} 1 &{} 3 &{}3 &{} 3 &{} 1 &{}1 \\ 1 &{} 1 &{} 3 &{} 1 &{} 7 &{} 3 &{} 1 &{} 3 &{}3 &{} 3 &{} 1\end{array}\right) , \end{aligned}$$

respectively. Hence $\mathcal{A}_1=\{(a,\,a,\,a,\,a,\,a,\,a,\,a,\,a,\,a,\,a,\,a)\mid a\in \mathbb {Z}_9\}$ with $d_{\mathrm{min}}(\mathcal{A}_1)=11,$ and $\mathcal{A}_i=\{wG_{\mathcal{A}_i}\mid w\in \mathbb {Z}_9^5\}$ with $d_{\mathrm{min}}(\mathcal{A}_i)=6$ for $i=2,\,3.$

Denote $\zeta _i=y+\langle f_i(y)\rangle \in R_i$ where $R_i=\mathbb {Z}_9[y]/\langle f_i(y)\rangle$ for $i=1,\,2,\,3.$ Obviously, $3^k\cdot 7\equiv -1$ (mod 11), which implies $(\zeta _i^7)^3=\zeta _i^{-1}$ by $\zeta _i^{11}=1,$ for all $i=1,\,2,\,3.$ Using the notations in Sect. 3, we have $e=7.$ Therefore,

$\diamondsuit$ $\pi _1=\zeta _1^7x-1=x-1\in R_1[x]/\langle x^3-1\rangle$ where $R_1=\mathbb {Z}_9[y]/\langle f_1(y)\rangle =\mathbb {Z}_9.$

$\diamondsuit$ $\pi _2=\zeta _2^7x-1=(2y^4+2y^3+6y^2+4y+1)x-1\in R_2[x]/\langle x^3-y\rangle$ where $R_2=\mathbb {Z}_9[y]/\langle f_2(y)\rangle ,$ since $y^7\equiv 2y^4+2y^3+6y^2+4y+1$ (mod $f_2(y)$).

$\diamondsuit$ $\pi _3=\zeta _3^7x-1=(2y^4+6y^3+2y^2+8y+5)x-1\in R_3[x]/\langle x^3-y\rangle$ where $R_3=\mathbb {Z}_9[y]/\langle f_3(y)\rangle ,$ since $y^7\equiv 2y^4+6y^3+2y^2+8y+5$ (mod $f_3(y)$).

Moreover, by $\zeta _3^{11}=1,\,x^{33}=1$ and $x^3=\zeta _3=y$ in $R_3[x]/\langle x^3-\zeta _3\rangle$ we have $\zeta _3^{-7}x^{-1}=\zeta _3^4x^{32}=y^4y^{10}x^2=y^3x^2$ and $\zeta _3^{14}x^{2}=y^3x^2.$

Now, by Corollary 4.7 we conclude that all distinct cyclic self-dual codes over $\mathbb {Z}_{9}$ of length 33 are given by

$$\begin{aligned} \mathcal{C}=\left( \mathcal{A}_1\Box _{\varphi _1}C_1\right) \oplus \left( \mathcal{A}_2\Box _{\varphi _2}C_2\right) \oplus \left( \mathcal{A}_3\Box _{\varphi _3}C_3\right) , \end{aligned}$$

where $\varphi _i{\text {:}}\,R_i\rightarrow \mathcal{A}_i$ is given by the following

$\diamond$ $\varphi _1(a)=a\varepsilon _1(y)$ for all $a\in R_1;$

$\diamond$ $\varphi _i(a(y))=a(y)\varepsilon _i(y)$ for all $a(y)\in R_i,\,i=2,\,3,$

and $C_i$ is a $\zeta _i$-constacyclic code over $R_i$ of length 3, i.e., an ideal of the ring $R_i[x]/\langle x^3-y\rangle ,$ satisfying the following conditions:

$\bullet$ $C_1$ is an ideal of $\mathbb {Z}_9/\langle x^3-1\rangle$ given by one of the following two cases:

$$\begin{aligned} \langle 3\rangle ; \, \left\langle (x-1)^2+6,\,3(x-1)\right\rangle . \end{aligned}$$

$\bullet$ $(C_2,\,C_3)$ is given by one of the following 736 cases:

Cases	$C_2$	$C_3$
(1)	$\langle 0\rangle$	$\langle 1\rangle$
(2)	$\langle 1\rangle$	$\langle 0\rangle$
(3)	$\langle 3\rangle$	$\langle 3\rangle$
(4)	$\langle 3\pi _2\rangle$	$\langle \pi _3^2,\,3\rangle$
(5)	$\langle 3\pi _2^2\rangle$	$\langle \pi _3,3\rangle$
(6)	$\langle \pi _2+3h\rangle$ ($h\in \mathcal{T}_2$)	$\langle \pi _3^2+3(1+\pi _3(\widehat{h}-1)y^3x^2)y^3x^2\rangle$
(7)	$\langle \pi _2^2+3(1+\pi _2h)\rangle$ ($h\in \mathcal{T}_2$)	$\langle \pi _3+3(\widehat{h}-1)y^3x^2\rangle$
(8)	$\langle \pi _2,\,3\rangle$	$\langle 3\pi _3^2\rangle$
(9)	$\langle \pi _2^2,\,3\rangle$	$\langle 3\pi _3\rangle$
(10)	$\langle \pi _2^2+3h,\,3\pi _2\rangle$ ($h\in \mathcal{T}_2$)	$\langle \pi _3^2+3(1-\widehat{h})y^3x^2,\,3\pi _3\rangle$

in which $\mathcal{T}_2=\{\sum \nolimits _{j=0}^4t_jy^j\mid t_j\in \{0,\,1,\,2\}, \, 0\le j\le 4\}$ and $\hat{h}=t_0+t_1y^{10}+t_2y^9+t_3y^8+t_4y^7$ (mod $f_3(y)$) for any $h=\sum \nolimits _{j=0}^{4}t_jy^{j}\in \mathcal{T}_2.$ Hence the number of cyclic self-dual codes over $\mathbb {Z}_{9}$ of length 33 is equal to $2\times 736=1472.$

Finally, we consider how to give an encode for each self-dual code listed above. For $i=2,\,3,$ we have $R_i[x]/\langle x^3-y\rangle =\{b_0(y)+b_1(y)x+b_2(y)x^2\mid b_0(y),\,b_1(y),\,b_2(y)\in R_i\}.$ If $\beta (x)=b_0(y)+b_1(y)x+b_2(y)x^2\in R_i[x]/\langle x^3-y\rangle ,$ the ideal $\langle \beta (x)\rangle$ of $R_i[x]/\langle x^3-y\rangle$ is a y-constacyclic code over $R_i$ of length 3 having an $R_i$-generator matrix given by $\left( \begin{array}{c@{\quad }c@{\quad }c}b_0(y) &{} b_1(y)&{} b_2(y) \\ yb_2(y) &{} b_0(y) &{} b_1(y)\\ yb_1(y) &{}yb_2(y) &{} b_0(y) \end{array}\right) .$

For example, we choose $\mathcal{C}=(\mathcal{A}_1\Box _{\varphi _1}C_1)\oplus (\mathcal{A}_2\Box _{\varphi _2}C_2)\oplus (\mathcal{A}_3\Box _{\varphi _3}C_3),$ where $C_1=\langle (x-1)^2+6,\,3(x-1)\rangle ,\,C_2=\langle \pi _2+3h\rangle$ and $C_3=\langle \pi _3^2+3(1+\pi _3(\widehat{h}-1)y^3x^2)y^3x^2\rangle$ with $h=1+2y^2.$

$\diamond$ Since the companion matrix of $f_1(y)=y-1$ is $M_{f_1}=(1)$ and $C_1=\langle 7+7x+x^2\rangle \oplus \langle 6+3x\rangle ,$ a generator matrix of the cyclic code $C_1$ over $R_1$ is $G_{C_1}=\left( \begin{array}{c}P\\ Q\end{array}\right)$ where $P=\left( \begin{array}{c@{\quad }c@{\quad }c} 7 &{} 7 &{} 1\\ 1 &{} 7 &{} 7\\ 7 &{} 1 &{} 7\end{array}\right)$ and $Q=\left( \begin{array}{c@{\quad }c@{\quad }c} 6 &{} 3 &{} 0\\ 0 &{} 6 &{} 3\\ 3 &{} 0 &{}6\end{array}\right) .$ Then by Theorem 2.5, a generator matrix of $\mathcal{A}_1\Box _{\varphi _1}C_1$ is given by

$$\begin{aligned} G_{\mathcal{A}_1\Box _{\varphi _1}C_1}=\left( \begin{array}{c@{\quad }c@{\quad }c} 7G_{\mathcal{A}_1} &{} 7G_{\mathcal{A}_1} &{} G_{\mathcal{A}_1}\\ G_{\mathcal{A}_1} &{} 7G_{\mathcal{A}_1} &{} 7G_{\mathcal{A}_1}\\ 7G_{\mathcal{A}_1} &{} G_{\mathcal{A}_1} &{} 7G_{\mathcal{A}_1}\\ 6G_{\mathcal{A}_1} &{} 3G_{\mathcal{A}_1} &{} 0\\ 0 &{} 6G_{\mathcal{A}_1} &{} 3G_{\mathcal{A}_1}\\ 3G_{\mathcal{A}_1} &{} 0 &{}6G_{\mathcal{A}_1}\end{array}\right) . \end{aligned}$$

$\diamond$ Since the companion matrix of $f_2(y)$ is $M_{f_2}=\left( \begin{array}{cc} 0 &{} I_4 \\ 1 &{} V_2\end{array}\right) ,$ where $V_2=(7,\,8,\,1,\,6),$ and $C_2=\langle (2+6y^2)+(1+4y+6y^2+2y^3+2y^4)x\rangle ,$ a generator matrix of the y-constacyclic code $C_2$ over $R_2$ is $G_{C_2}=\left( \begin{array}{c@{\quad }c@{\quad }c}\alpha _2 &{} \beta _2 &{} 0\\ 0 &{} \alpha _2 &{} \beta _2\\ y\beta _2 &{}0 &{} \alpha _2\end{array}\right) ,$ where $\alpha _2=2+6y^2,\,\beta _2=1+4y+6y^2+2y^3+2y^4$ and $y\beta _2=2+6y+2y^2+8y^3+5y^4.$ Using the notations of Theorem 2.5, we have $A_{\alpha _2}=2I_5+6M_{f_2}^2,\, A_{\beta _2}=I_5+4M_{f_2}+6M_{f_2}^2+2M_{f_2}^3+2M_{f_2}^4$ and $A_{y\beta _2}=2I_5+6M_{f_2}+2M_{f_2}^2+8M_{f_2}^3+5M_{f_2}^4.$ Specifically, we obtain

$$\begin{aligned} A_{\alpha _2}=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c}2 &{} 0&{} 6 &{} 0 &{} 0 \\ 0 &{} 2 &{} 0&{} 6 &{} 0 \\ 0 &{} 0 &{} 2 &{} 0&{} 6 \\ 6 &{} 6 &{} 3 &{} 8 &{} 0\\ 0 &{} 6 &{} 6 &{} 3&{} 8 \end{array}\right) ,\quad \!\! A_{\beta _2}=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c}1 &{} 4&{} 6 &{} 2 &{} 2 \\ 2 &{} 6 &{} 2&{} 8 &{} 5 \\ 5 &{} 1 &{} 1 &{} 7&{} 2 \\ 2 &{} 1 &{} 8 &{} 3 &{} 1\\ 1 &{} 0 &{} 0 &{} 0&{} 0 \end{array}\right) ,\quad \!\! A_{y\beta _2}=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c}2 &{} 6&{} 2 &{} 8 &{} 5 \\ 5 &{} 1 &{} 1&{} 7 &{} 2 \\ 2 &{} 1 &{} 8 &{} 3&{} 1 \\ 1 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 1 &{} 0 &{} 0&{} 0 \end{array}\right) . \end{aligned}$$

Then by Theorem 2.5, a generator matrix of $\mathcal{A}_2\Box _{\varphi _2}C_2$ is given by

$$\begin{aligned} G_{\mathcal{A}_2\Box _{\varphi _2}C_2}=\left( \begin{array}{c@{\quad }c@{\quad }c}A_{\alpha _2}G_{\mathcal{A}_2} &{} \quad A_{\beta _2}G_{\mathcal{A}_2} &{} \quad 0\\ 0 &{} \quad A_{\alpha _2}G_{\mathcal{A}_2} &{} \quad A_{\beta _2}G_{\mathcal{A}_2} \\ A_{y\beta _2}G_{\mathcal{A}_2} &{}\quad 0 &{} \quad A_{\alpha _2}G_{\mathcal{A}_2}\end{array}\right) . \end{aligned}$$

$\diamond$ Since the companion matrix of $f_3(y)$ is $M_{f_3}=\left( \begin{array}{cc} 0 &{} I_4 \\ 1 &{} V_3\end{array}\right) ,$ where $V_3=(3,\,8,\,1,\,2),$ and $C_3=\langle 1+(2+2y+2y^2+2y^4)x+(6y+4y^3)x^2\rangle ,$ a generator matrix of the y-constacyclic code $C_3$ over $R_3$ is $G_{C_3}=\left( \begin{array}{c@{\quad }c@{\quad }c}\alpha _3 &{} \beta _3 &{} \gamma _3\\ y\gamma _3 &{} \alpha _3 &{} \beta _3\\ y\beta _3 &{}y\gamma _3 &{} \alpha _3\end{array}\right) ,$ where $\alpha _3=1,\, \beta _3=2+2y+2y^2+2y^4,\,\gamma _3=6y+4y^3,\, y\beta _3=2+8y+4y^3+4y^4$ and $y\gamma _3=6y^2+4y^4.$ Using the notations of Theorem 2.5, we have $A_{\alpha _3}=I_5,\, A_{\beta _3}=2I_5+2M_{f_3}+2M_{f_3}^2+2M_{f_3}^4,\, A_{\gamma _3}=6M_{f_3}+4M_{f_3}^3,\, A_{y\beta _3}=2I_5+8M_{f_3}+4M_{f_3}^3+4M_{f_3}^4$ and $A_{y\gamma _3}=6M_{f_3}^2+4M_{f_3}^4.$ Specifically, we obtain

$$\begin{aligned}&A_{\beta _3}=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c}2 &{} 2&{} 2 &{} 0 &{} 2 \\ 2 &{} 8 &{} 0&{} 4 &{} 4 \\ 4 &{} 5 &{} 4 &{} 4&{} 3 \\ 3 &{} 4 &{} 2 &{} 7 &{} 1\\ 1 &{} 6 &{} 3 &{} 3&{} 0 \end{array}\right) , \quad \!\! A_{\gamma _3}=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c}0 &{} 6&{} 0 &{} 4 &{} 0 \\ 0 &{} 0 &{} 6&{} 0 &{} 4 \\ 4 &{} 3 &{}5 &{} 1&{} 8 \\ 8 &{} 1 &{} 4 &{} 4 &{} 8\\ 8 &{} 5 &{} 2 &{} 3&{} 2 \end{array}\right) , \quad \!\! A_{y\beta _3}=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c}2 &{} 8&{} 0 &{} 4 &{} 4 \\ 4 &{} 5 &{} 5&{} 4 &{} 3 \\ 3 &{} 4 &{} 2 &{} 7&{} 1 \\ 1 &{} 6 &{} 3 &{} 3 &{} 0\\ 0 &{} 1 &{} 6 &{} 3&{} 3 \end{array}\right) , \\&\quad A_{y\gamma _3}=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c}0 &{} 0&{} 6 &{} 0 &{} 4 \\ 4 &{} 3 &{} 5&{} 1 &{} 8 \\ 8 &{} 1 &{} 4 &{} 4&{} 8 \\ 8 &{} 5 &{} 2 &{} 3 &{} 2\\ 2 &{} 5 &{} 3 &{} 4&{} 7 \end{array}\right) . \end{aligned}$$

Then by Theorem 2.5 a generator matrix of $\mathcal{A}_3\Box _{\varphi _3}C_3$ is given by

$$\begin{aligned} G_{\mathcal{A}_3\Box _{\varphi _3}C_3}=\left( \begin{array}{c@{\quad }c@{\quad }c}G_{\mathcal{A}_3} &{} A_{\beta _3}G_{\mathcal{A}_3} &{} A_{\gamma _3}G_{\mathcal{A}_3}\\ A_{y\gamma _3}G_{\mathcal{A}_3} &{} G_{\mathcal{A}_3} &{} A_{\beta _3}G_{\mathcal{A}_3} \\ A_{y\beta _3}G_{\mathcal{A}_3} &{}A_{y\gamma _3}G_{\mathcal{A}_3} &{} G_{\mathcal{A}_3}\end{array}\right) . \end{aligned}$$

Now, by Corollary 3.8 a generator matrix of the self-dual cyclic code $\mathcal{C}$ over $\mathbb {Z}_9$ of length 33 is given by $G_{\mathcal{C}}=\left( \begin{array}{c}G_{\mathcal{A}_1\Box _{\varphi _1}C_1}\\ G_{\mathcal{A}_2\Box _{\varphi _2}C_2} \\ G_{\mathcal{A}_3\Box _{\varphi _3}C_3}\end{array}\right) .$ Hence $\mathcal{C}=\{uG_{\mathcal{C}}\mid u\in \mathbb {Z}_9^{36}\}.$

6 Conclusions

We present a canonical form decomposition for every cyclic code over $\mathbb {Z}_{p^2}$ of length $p^kn\,(k\ge 1$ and $\mathrm{gcd}(p,\,n)=1$), where each subcode is concatenated by a basic irreducible cyclic code over $\mathbb {Z}_{p^2}$ of length n as the inner code and a constacyclic code over a Galois extension ring of $\mathbb {Z}_{p^2}$ of length $p^k$ as the outer code. By determining their outer codes, we present a precise description for cyclic codes over $\mathbb {Z}_{p^2}$ when $p\ne 2,$ give precisely dual codes and investigate self-duality for cyclic codes over $\mathbb {Z}_{p^2}.$ These codes enjoy a rich algebraic structure compared to arbitrary linear codes (which makes the search process much simpler). Obtaining some bounds for minimal distance such as BCH-like of a cyclic code over the ring $\mathbb {Z}_{p^2}$ by just looking at the concatenated structure would be rather interesting.