Explicit representation for a class of Type 2 constacyclic codes over the ring $\mathbb{F}_{2^m}[u]/\langle u^{2\lambda}\rangle$ with even length

TL;DR

This paper provides an explicit representation, enumeration, and structural description of a class of Type 2 constacyclic codes over a specific finite ring, enhancing understanding of their algebraic properties and counting methods.

Contribution

It introduces a new explicit representation and enumeration formula for $( ext{delta}+ ext{alpha} u^2)$-constacyclic codes over a finite ring of the form $rac{ ext{field}[u]}{ ext{ideal}}$, specifically for odd length codes.

Findings

Explicit formulas for code enumeration.

Representation of codes as ideals generated by at most two polynomials.

Complete classification of the codes over the specified ring.

Abstract

Let $\mathbb{F}_{2^m}$ be a finite field of cardinality $2^m$, $\lambda$ and $k$ be integers satisfying $\lambda,k\geq 2$ and denote $R=\mathbb{F}_{2^m}[u]/\langle u^{2\lambda}\rangle$. Let $\delta,\alpha\in \mathbb{F}_{2^m}^{\times}$. For any odd positive integer $n$, we give an explicit representation and enumeration for all distinct $(\delta+\alpha u^2)$-constacyclic codes over $R$ of length $2^kn$, and provide a clear formula to count the number of all these codes. As a corollary, we conclude that every $(\delta+\alpha u^2)$-constacyclic code over $R$ of length $2^kn$ is an ideal generated by at most $2$ polynomials in the residue class ring $R[x]/\langle x^{2^kn}-(\delta+\alpha u^2)\rangle$.

Equations

$$\gamma ={\alpha }_0+{\alpha }_k{u}^k+\dots +{\alpha }_{e-1}{u}^{e-1}$$

$$\Psi \left(x^i(x^{2^{k}n}{)}^l\right)=x^i(\delta +\alpha u^2{)}^l\mathrm{and}\Psi \left(u{x}^i(x^{2^{k}n}{)}^l\right)=u{x}^i(\delta +\alpha u^2{)}^l,$$

$$\Psi \left((x^n+{\delta }_0{)}^{i+l\cdot 2^{k}n}\right)={\alpha }^l{u}^{2l}(x^n+{\delta }_0{)}^i,\forall 0\le i\le 2^{k}n-1,\forall 0\le l\le \lambda -1.$$

$$(x^{2^{k}n}-\delta {)}^{\lambda }=(x^n+{\delta }_0{)}^{2^k\lambda }=f_1(x{)}^{2^k\lambda }\dots f_r(x{)}^{2^k\lambda }.$$

$$(x^{2^{k}n}-\delta {)}^{\lambda }=F_j(x{)}^{2^k\lambda }{f}_j(x{)}^{2^k\lambda }\mathrm{and}\gcd (F_j(x{)}^{2^k\lambda },f_j(x{)}^{2^k\lambda })=1.$$

$$g_j(x)F_j(x{)}^{\lambda p^k}+h_j(x)f_j(x{)}^{\lambda p^k}=1.$$

$${\epsilon }_j(x)\equiv g_j(x)F_j(x{)}^{2^k\lambda }=1-h_j(x)f_j(x{)}^{2^k\lambda }(\mathrm{mod}(x^n+{\delta }_0{)}^{2^k\lambda }).$$

$${\mathcal{F}}_j\subseteq {\mathcal{K}}_j\subseteq \mathcal{A}\mathrm{as}\mathrm{sets}.$$

$$\mathcal{A}={\epsilon }_1(x){\mathcal{K}}_1\oplus {\epsilon }_2(x){\mathcal{K}}_2\oplus \dots \oplus {\epsilon }_r(x){\mathcal{K}}_r(\mathrm{mod}(x^n+{\delta }_0{)}^{2^k\lambda }),$$

$$\xi =b_0(x)+b_1(x)f_j(x)+\dots +b_{2^k\lambda -1}{f}_j(x{)}^{2^k\lambda -1},$$

$$⟨f_j(x{)}^l⟩=f_j(x{)}^l{\mathcal{K}}_j,0\le l\le 2^k\lambda .$$

$${\mathcal{K}}_j/⟨f_j(x{)}^l⟩=\{\sum _{i=0}^{l-1}{b}_i(x)f(x{)}^i\mid b_i(x)\in {\mathcal{F}}_j,i=0,1,\dots ,l-1\}.$$

$$f_j(x{)}^l\cdot \frac{{\mathbb{F}}_{2^m}[x]}{⟨f_j(x{)}^t⟩}$$

$${\omega }_j^2{f}_j(x{)}^{2^k}\equiv {\alpha }^{-1}(x^n+{\delta }_0{)}^{2^k}(\mathrm{mod}{f}_j(x{)}^{2^k\lambda }).$$

$$\mathcal{A}+u\mathcal{A}=\underset{j=1}{\overset{r}{⨁}}{\epsilon }_j(x)({\mathcal{K}}_j+u{\mathcal{K}}_j)=\sum _{j=1}^r{\epsilon }_j(x)({\mathcal{K}}_j+u{\mathcal{K}}_j)(\mathrm{mod}(x^n+{\delta }_0{)}^{2^k\lambda }),$$

$${\omega }_j^2{f}_j(x{)}^{2^k}$$

$$\mathcal{A}+u\mathcal{A}$$

$$\frac{{\mathbb{F}}_{2^m}[x]}{⟨f_j(x{)}^l⟩}=\left\{\sum _{i=0}^{l-1}{a}_i(x)f_j(x{)}^i\mid a_0(x),a_1(x),\dots ,a_{l-1}(x)\in {\mathcal{F}}_j\right\},$$

$$\mathcal{C}=\underset{j=1}{\overset{r}{⨁}}{\epsilon }_j(x)C_j,$$

$$N_{(2^m,d_j,2^k\lambda )}$$

$$N_{(2^m,n,2^k\lambda )}$$

$$N_{(2^m,1,2^k\lambda )}$$

$$N_{(2^m,1,2^2\cdot 2)}=N_{(2^m,1,2^3)}=17+13\cdot 2^m+9\cdot (2^m{)}^2+5\cdot (2^m{)}^3+(2^m{)}^4.$$

$$\mathcal{C}=\underset{j=1}{\overset{r}{⨁}}{\epsilon }_j(x)C_j=\sum _{j=1}^r{\epsilon }_j(x)C_j(\mathrm{mod}(x^n+{\delta }_0{)}^{2^{k}n}).$$

$$\theta :{\mathcal{K}}_j^2\to {\mathcal{K}}_j+u{\mathcal{K}}_j\mathrm{via}(a_0,a_1)\mapsto a_0+u{a}_1(\forall a_0,a_1\in {\mathcal{K}}_j).$$

$$({\omega }_j^2{\pi }_j^{2^k}{a}_1,a_0)\in S,\forall (a_0,a_1)\in S$$

$$\Vert ({\beta }_1,{\beta }_2){\Vert }_{{\pi }_j}=\min \{\Vert {\beta }_1{\Vert }_{{\pi }_j},\Vert {\beta }_2{\Vert }_{{\pi }_j}\}.$$

$$\Vert G_j{\Vert }_{{\pi }_j}=t_i,\mathrm{where}0\le t_1\le \dots \le t_{\rho }\le 2^k\lambda -1.$$

$$({\omega }^2{\pi }^{2^k}\cdot {\pi }^{s}a,{\pi }^s)=b({\pi }^s,{\pi }^{s}a)=({\pi }^{s}b,{\pi }^{s}ab)\mathrm{for}\mathrm{some}b\in \mathcal{K},$$

$${\pi }^s=({\pi }^{s}b)a={\omega }^2{\pi }^{2^k+s}{a}^2={\pi }^{2^k+s}\cdot {\omega }^2{a}^2\mathrm{in}\mathcal{K}.$$

Full text

††journal: XX

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 plays a very significant role in the theory of error-correcting codes. It includes as a subclass of the important class of cyclic codes, which has been well studied since the late 1950’s. Constacyclic codes also have practical applications as they can be efficiently encoded with simple shift registers. This family of codes is thus interesting for both theoretical and practical reasons.

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

A code over $\Gamma$ of length $N$ is a nonempty subset ${\cal C}$ of $\Gamma^{N}=\{(a_{0},a_{1},\ldots$, $a_{N-1})\mid a_{j}\in\Gamma,\ j=0,1,\ldots,N-1\}$. The code ${\cal C}$ is said to be linear if ${\cal C}$ is an $\Gamma$-submodule of $\Gamma^{N}$. All codes in this paper are assumed to be linear. The ambient space $\Gamma^{N}$ is equipped with the usual Euclidian inner product, i.e. $[a,b]=\sum_{j=0}^{N-1}a_{j}b_{j}$, where $a=(a_{0},a_{1},\ldots,a_{N-1}),b=(b_{0},b_{1},\ldots,b_{N-1})\in\Gamma^{N}$, and the dual code is defined by ${\cal C}^{\bot}=\{a\in\Gamma^{N}\mid[a,b]=0,\forall b\in{\cal C}\}$. If ${\cal C}^{\bot}={\cal C}$, ${\cal C}$ is called a self-dual code over $\Gamma$. Let $\gamma\in\Gamma^{\times}$. Then a linear code ${\cal C}$ over $\Gamma$ of length $N$ is called a $\gamma$-constacyclic code if $(\gamma c_{N-1},c_{0},c_{1},\ldots,c_{N-2})\in{\cal C}$ for all $(c_{0},c_{1},\ldots,c_{N-1})\in{\cal C}$. Particularly, ${\cal C}$ is called a negacyclic code if $\gamma=-1$, and ${\cal C}$ is called a cyclic code if $\gamma=1$.

For any $a=(a_{0},a_{1},\ldots,a_{N-1})\in\Gamma^{N}$, let $a(x)=a_{0}+a_{1}x+\ldots+a_{N-1}x^{N-1}\in\Gamma[x]/\langle x^{N}-\gamma\rangle$. We will identify $a$ with $a(x)$ in this paper. Then ${\cal C}$ is a $\gamma$-constacyclic code over $\Gamma$ of length $N$ if and only if ${\cal C}$ is an ideal of the residue class ring $\Gamma[x]/\langle x^{N}-\gamma\rangle$, and the dual code ${\cal C}^{\bot}$ of ${\cal C}$ is a $\gamma^{-1}$-constacyclic code of length $N$ over $\Gamma$, i.e. ${\cal C}^{\bot}$ is an ideal of the ring $\Gamma[x]/\langle x^{N}-\gamma^{-1}\rangle$ (cf. [12] Propositions 2.4 and 2.5). The ring $\Gamma[x]/\langle x^{N}-\gamma\rangle$ is usually called the ambient ring of $\gamma$-constacyclic codes over $\Gamma$ with length $N$. In addition, $\mathcal{C}$ is called a simple-root constacyclic code if ${\rm gcd}(q,N)=1$, and called a repeated-root constacyclic code otherwise.

Let $\mathbb{F}_{q}$ be a finite field of cardinality $q$, where $q$ is power of a prime, and denote $R=\mathbb{F}_{q}[u]/\langle u^{e}\rangle=\mathbb{F}_{q}+u\mathbb{F}_{q}+\ldots+u^{e-1}\mathbb{F}_{q}$ ($u^{e}=0$) where $e\geq 2$. Then $R$ is a finite chain ring. As in Dinh et al [12], if

[TABLE]

where $\alpha_{0},\alpha_{k},\ldots,\alpha_{e-1}\in\mathbb{F}_{q}$ satisfying $\alpha_{0}\alpha_{k}\neq 0$, then $\gamma$ is called a unit in $R$ to be of Type $k$. Especially, $\gamma$ is called a unit in $R$ to be of Type [math] if $\gamma=\alpha_{0}\in\mathbb{F}_{q}$. When $\gamma$ is a unit in $R$ of Type $k$, a $\gamma$-constacyclic code $\mathcal{C}$ of length $N$ over $R$ is said to be of Type $k$. Especially, cyclic codes and negacyclic codes are both constacyclic codes over $R$ of Type [math].

For examples, let $e\geq 3$ and $\delta,\alpha\in\mathbb{F}_{q}^{\times}$. Then $\delta+\alpha u^{2}$ is a a unit in $R$ of Type 2. Hence $(\delta+\alpha u^{2})$-constacyclic codes form a typical subclass of the class of all Type 2 constacyclic codes over the finite chain ring $R$.

$\diamondsuit$ When $e=2$, there were a lot of literatures on linear codes, Type $1$ constacyclic codes and some special class of Type [math] constacyclic codes of length $N$ over rings $\mathbb{F}_{p^{m}}[u]/\langle u^{2}\rangle=\mathbb{F}_{p^{m}}+u\mathbb{F}_{p^{m}}$ for various prime $p$ and positive integers $m$ and $N$. See [1], [2], [13–19], [21], [25] and [27], for examples. In particular, we [3] gave an explicit representation and a complete classification for all Type [math] repeated-root constacyclic codes over $\mathbb{F}_{p^{m}}+u\mathbb{F}_{p^{m}}$ and their dual codes for any prime number $p$ and positive integer $m$.

$\diamondsuit$ When $e\geq 3$, the structures for repeated-root constacyclic codes of Type 1 over $R$ had been studied by many literatures. For examples, Kai et al. [22] investigated $(1+\lambda u)$-constacyclic codes of arbitrary length over $\mathbb{F}_{p}[u]/\langle u^{k}\rangle$, where $\lambda\in\mathbb{F}_{p}^{\times}$. Cao [4] generalized these results to $(1+w\gamma)$-constacyclic codes of arbitrary length over an arbitrary finite chain ring $\Gamma$, where $w$ is a unit of $\Gamma$ and $\gamma$ generates the unique maximal ideal of $\Gamma$ with nilpotency index $e\geq 2$. Hence every Type 1 constacyclic code over any finite chain ring is a one-generator ideal of the ambient ring.

$\diamondsuit$ When $e\geq 3$, we known the following literatures for the structures of $(\delta+\alpha u^{2})$-constacyclic codes over $R$, where $\delta,\alpha\in\mathbb{F}_{p^{m}}^{\times}$:

Let $e=3$.

$\triangleright$ Sobhani [26] determined the structure of $(\delta+\alpha u^{2})$-constacyclic codes of length $p^{k}$ over $\mathbb{F}_{p^{m}}[u]/\langle u^{3}\rangle$.

$\triangleright$ On the basis of [7], using methods different from [26] we gave a complete description for $(\delta+\alpha u^{2})$-constacyclic codes over the ring $\mathbb{F}_{2^{m}}[u]/\langle u^{3}\rangle$ of length $2n$ and determine explicitly the self-dual codes among them for any odd positive integer $n$ [10].

Let $e=4$.

$\triangleright$ When ${\rm gcd}(q,n)=1$, in [5] for any $\delta,\alpha\in\mathbb{F}_{q}^{\times}$, an explicit representation for all distinct $(\delta+\alpha u^{2})$-constacyclic codes over the ring $\mathbb{F}_{q}[u]/\langle u^{4}\rangle$ of length $n$ is given, and the dual code for each of these codes is determined. For the case of $q=2^{m}$ and $\delta=1$, all self-dual $(1+\alpha u^{2})$-constacyclic codes over $R$ of length $n$ are provided.

$\triangleright$ When $p=3$, in [6] an explicit representation for all distinct $(\delta+\alpha u^{2})$-constacyclic codes over $\mathbb{F}_{3^{m}}[u]/\langle u^{4}\rangle$ of length $3n$ was given, where ${\rm gcd}(3,n)=1$. Formulas for the number of all such codes and the number of codewords in each code are provided respectively, and the dual code for each of these codes was determined explicitly.

$\triangleright$ When $p=2$, in [7] a representation and enumeration formulas for all distinct $(\delta+\alpha u^{2})$-constacyclic codes over $\mathbb{F}_{2^{m}}[u]/\langle u^{4}\rangle$ of length $2n$ were presented explicitly, where $n$ is odd.

$\triangleright$ Mahmoodi and Sobhani [23] gave a complete classification for $(1+\alpha u^{2})$-constacyclic codes of length $p^{k}$ over $R=\mathbb{F}_{p^{m}}[u]/\langle u^{4}\rangle$, where $\alpha\in\mathbb{F}_{p^{m}}^{\times}$. They determined self-dual such codes and enumerate them for the case $p=2$. Moreover, the authors discussed on Gray-maps on $R$ which preserve self-duality, and also discuss on the images of self-dual constacyclic codes under these Gray maps.

Let $e=2\lambda$, where $\lambda$ is an arbitrary integer such that $\lambda\geq 2$.

$\triangleright$ Based on the results of [9], we gave a complete description for repeated-root $(\delta+\alpha u^{2})$-constacyclic codes over $\mathbb{F}_{p^{m}}[u]/\langle u^{2\lambda}\rangle$ for any odd prime $p$ [11]. The expressions and their derivation process for the main results in [11] are heavily depend on that $p$ is an odd prime.

Many methods and techniques used in [11] and [9] can not be directly applied to the case $p=2$.

Motivated by those, we follow the main idea in [11], promote and develop the methods used in [7] to determine all $(\delta+\alpha u^{2})$-constacyclic codes over $R=\mathbb{F}_{2^{m}}[u]/\langle u^{2\lambda}\rangle$ of arbitrary even length. The ideas and methods used in this paper are different to that used in [23] and [26]. Therefore, we can come to clearer and more precise conclusions:

$\bullet$ Provide an explicit representation and enumeration for all distinct $(\delta+\alpha u^{2})$-constacyclic codes over $R$ of length $2^{k}n$ through only one theorem, for any integer $k\geq 2$ and odd positive integer $n$.

Although the proof of this theorem is somewhat complicated, the results expressed by the theorem are very clear and direct.

$\bullet$ Obtain a clear and exact formula to count the number of all $(\delta+\alpha u^{2})$-constacyclic codes over $R$ of length $2^{k}n$, and give a clear formula to count the number of codewords in each code from its generators directly.

The present paper is organized as follows. In Section 2, we provide the notations and review preparation results necessary. Then we give the main result (Theorem 2.5) for representation and enumeration for all distinct $(\delta+\alpha u^{2})$-constacyclic code over $R$ of length $2^{k}n$. In Section 3, we give an explicit representation for a special subclass of $(\delta+\alpha u^{2})$-constacyclic code over $R$ of length $2^{k}n$ including the particular situation of $n=1$. Based on this, we correct a mistake for $(1+u^{2})$-constacyclic codes of length $4$ over $\mathbb{F}_{2}[u]/\langle u^{4}\rangle$ listed by an example of [23]. Moreover, we list precisely all distinct self-dual $(1+\alpha u^{2})$-constacyclic codes of length $4$ over $\mathbb{F}_{2^{m}}[u]/\langle u^{4}\rangle$ for any $\alpha\in\mathbb{F}_{2^{m}}^{\times}$. In Section 4, we give a proof for the main result in Section 2. Section 5 concludes the paper.

2 Main results

In this section, we introduce the necessary notations and review preparation results first. Then we provide the main result on representation and enumeration for $(\delta+\alpha u^{2})$-constacyclic codes over the ring $\mathbb{F}_{2^{m}}[u]/\langle u^{2\lambda}\rangle$ of length $2^{k}n$ where $k\geq 2$.

In this paper, we always assume that $m,n,\lambda,k$ are positive integers such that $n$ is odd and $\lambda,k\geq 2$. As $\delta\in\mathbb{F}_{2^{m}}^{\times}$ and $|\mathbb{F}_{2^{m}}^{\times}|=2^{m}-1$, there exists a unique element $\delta_{0}\in\mathbb{F}_{2^{m}}^{\times}$ such that $\delta=\delta_{0}^{2^{k}}$. This implies $x^{2^{k}n}-\delta=(x^{n}+\delta_{0})^{2^{k}}$ in $\mathbb{F}_{2^{m}}[x]$. In this paper, we adopt the following notation.

$R=\frac{\mathbb{F}_{2^{m}}[u]}{\langle u^{2\lambda}\rangle}=\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}+u^{2}\mathbb{F}_{2^{m}}+\ldots+u^{2\lambda-1}\mathbb{F}_{2^{m}}$ ($u^{2\lambda}=0$). Then $R$ is a finite chain ring with the unique maximal ideal $uR$, and $2\lambda$ is the nilpotent index of $u$.

• 2.

$\mathcal{A}=\frac{\mathbb{F}_{2^{m}}[x]}{\langle(x^{n}+\delta_{0})^{2^{k}\lambda}\rangle}=\{\sum_{i=0}^{2^{k}\lambda n-1}a_{i}x^{i}\mid a_{i}\in\mathbb{F}_{2^{m}},\ i=0,1,\ldots,2^{k}\lambda n-1\}$ in which the arithmetic is done modulo $(x^{n}+\delta_{0})^{2^{k}\lambda}$. Then $\mathcal{A}$ is a finite principal ideal ring and $|\mathcal{A}|=2^{2^{k}\lambda mn}$.

• 3.

$\mathcal{A}+u\mathcal{A}=\frac{\mathcal{A}[u]}{\langle u^{2}-\alpha^{-1}(x^{n}+\delta)^{2^{k}}\rangle}=\{\xi_{0}+u\xi_{1}\mid\xi_{0},\xi_{1}\in\mathcal{A}\}$ in which the operations are defined by

• (a)

$(\xi_{0}+u\xi_{1})+(\eta_{0}+u\eta_{1})=(\xi_{0}+\eta_{0})+u(\xi_{1}+\eta_{1})$,

• (b)

$(\xi_{0}+u\xi_{1})(\eta_{0}+u\eta_{1})=\left(\xi_{0}\eta_{0}+\alpha^{-1}(x^{n}+\delta_{0})^{2^{k}}\xi_{1}\eta_{1}\right)+u(\xi_{0}\eta_{1}+\xi_{1}\eta_{0})$,

for all $\xi_{0},\xi_{1},\eta_{0},\eta_{1}\in\mathcal{A}$. Then $\mathcal{A}$ is a subring of $\mathcal{A}+u\mathcal{A}$.

It is clear that both $\mathcal{A}+u\mathcal{A}$ and $\frac{R[x]}{\langle x^{2^{k}n}-(\delta+\alpha u^{2})\rangle}$ are $\mathbb{F}_{2^{m}}$-spaces of dimension $2^{k+1}\lambda n$. Precisely, we have that

$\diamond$ $\{1,x,\ldots,x^{2^{k}\lambda n-1},u,ux,\ldots,ux^{2^{k}\lambda n-1}\}$ is an $\mathbb{F}_{2^{m}}$-basis of $\mathcal{A}+u\mathcal{A}$;

$\diamond$ $\bigcup_{i=0}^{2\lambda-1}\{u^{i},u^{i}x,\ldots,u^{i}x^{2^{k}n-1}\}$ is an $\mathbb{F}_{2^{m}}$-basis of $\frac{R[x]}{\langle x^{2^{k}n}-(\delta+\alpha u^{2})\rangle}$.

Hence there is a unique isomorphism $\Psi$ of $\mathbb{F}_{2^{m}}$-spaces from $\mathcal{A}+u\mathcal{A}$ onto $\frac{R[x]}{\langle x^{2^{k}n}-(\delta+\alpha u^{2})\rangle}$ such that

[TABLE]

for all integers $i$ and $l$ satisfying $0\leq i\leq 2^{k}n-1$ and $0\leq l\leq\lambda-1$ respectively. Moreover, by $\delta_{0}^{2^{k}}=\delta$ it follows that

$\diamond$ In the ring $\mathcal{A}+u\mathcal{A}$, we have $u^{2}=\alpha^{-1}(x^{n}+\delta_{0})^{2^{k}}$ and $(x^{n}+\delta_{0})^{2^{k}\lambda}=0$. These imply $x^{2^{k}n}=\delta+\alpha u^{2}$ and $u^{2\lambda}=(\alpha^{-1}(x^{n}+\delta_{0})^{2^{k}})^{\lambda}=0$, respectively.

$\diamond$ In the ring $\frac{R[x]}{\langle x^{2^{k}n}-(\delta+\alpha u^{2})\rangle}$, we have $x^{2^{k}n}-(\delta+\alpha u^{2})=0$ and $u^{2\lambda}=0$. These imply $(x^{n}+\delta_{0})^{2^{k}\lambda}=(x^{2^{k}n}-\delta)^{\lambda}=(\alpha u^{2})^{\lambda}=0$.

From these and by an argument similar to the proof of Theorem 2.1 in [11], one can easily verify the following conclusion.

Lemma 2.1 Using the notations above, $\Psi$ is a ring isomorphism from $\mathcal{A}+u\mathcal{A}$ onto $R[x]/\langle x^{2^{k}n}-(\delta+\alpha u^{2})\rangle$.

From this lemma and by Equation (1), we deduce the following:

[TABLE]

In the rest of this paper, we usually identify $R[x]/\langle x^{2^{k}n}-(\delta+\alpha u^{2})\rangle$ with $\mathcal{A}+u\mathcal{A}$ under the ring isomorphism $\Psi$ determined by Equations (1) and (2), unless otherwise stated. Hence in order to determine all $(\delta+\alpha u^{2})$-constacyclic codes over $R$ of length $2^{k}n$, it is sufficient to give all ideals of the ring $\mathcal{A}+u\mathcal{A}$.

To determine all ideals of the ring $\mathcal{A}+u\mathcal{A}$, we need to study the structure of the ring $\mathcal{A}$ first. As $n$ is odd, there are pairwise coprime monic irreducible polynomials $f_{1}(x),\ldots,f_{r}(x)$ in $\mathbb{F}_{2^{m}}[x]$ such that $x^{n}-\delta_{0}=f_{1}(x)\ldots f_{r}(x)$. This implies

[TABLE]

For any integer $j$, $1\leq j\leq r$, we assume ${\rm deg}(f_{j}(x))=d_{j}$ and denote $F_{j}(x)=\frac{x^{n}-\delta_{0}}{f_{j}(x)}$. Then $F_{j}(x)^{2^{k}\lambda}=\frac{(x^{2^{k}n}-\delta)^{\lambda}}{f_{j}(x)^{2^{k}\lambda}}$ and ${\rm gcd}(F_{j}(x),f_{j}(x))=1$. These imply

[TABLE]

Hence there exist $g_{j}(x),h_{j}(x)\in\mathbb{F}_{2^{m}}[x]$ such that

[TABLE]

As in [11] for odd prime $p$, we adopt the following notation where $j$ be an integer satisfying $1\leq j\leq r$.

Let $\varepsilon_{j}(x)\in\mathcal{A}$ be defined by

[TABLE]

• 2.

$\mathcal{K}_{j}=\frac{\mathbb{F}_{2^{m}}[x]}{\langle f_{j}(x)^{2^{k}\lambda}\rangle}=\{\sum_{i=0}^{2^{k}\lambda d_{j}-1}a_{i}x^{i}\mid a_{i}\in\mathbb{F}_{2^{m}},0\leq i<2^{k}\lambda d_{j}\}$ in which the arithmetics are done modulo $f_{j}(x)^{2^{k}\lambda}$.

• 3.

$\mathcal{F}_{j}=\frac{\mathbb{F}_{2^{m}}[x]}{\langle f_{j}(x)\rangle}=\{\sum_{i=0}^{d_{j}-1}a_{i}x^{i}\mid a_{i}\in\mathbb{F}_{2^{m}},0\leq i<d_{j}\}$ in which the arithmetics are done modulo $f_{j}(x)$. Then $\mathcal{F}_{j}$ is an extension field of $\mathbb{F}_{2^{m}}$ with $2^{md_{j}}$ elements.

Remark $\mathcal{F}_{j}$ is a finite field in which the arithmetic is done modulo $f_{j}(x)$, $\mathcal{K}_{j}$ is a finite ring in which the arithmetic is done modulo $f_{j}(x)^{2^{k}\lambda}$ and $\mathcal{A}$ is a principal ideal ring in which the arithmetic is done modulo $(x^{n}+\delta_{0})^{2^{k}\lambda}$. In this paper, we adopt the following points of view:

[TABLE]

It is worth noting that $\mathcal{F}_{j}$ is not a subfield of $\mathcal{K}_{j}$ and $\mathcal{K}_{j}$ is not a subring of $\mathcal{A}$ when $n\geq 2$.

Then from Chinese remainder theorem for commutative rings, we deduce the following lemma about the structure and properties of the ring $\mathcal{A}$.

Lemma 2.2 Using the notations above, we have the following decomposition via idempotents:

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

(ii) We regard $\mathcal{K}_{j}$ as a subset of $\mathcal{A}$ for all $j$. Then

[TABLE]

where $\varepsilon_{j}(x)\mathcal{K}_{j}=\{\varepsilon_{j}(x)a_{j}(x)\mid a_{j}(x)\in\mathcal{K}_{j}\}$ for all $j=1,\ldots,r$.

For the ring $\mathcal{K}_{j}$, where $1\leq j\leq r$, we know the following conclusion.

Lemma 2.3 (cf. [8] Example 2.1) The ring $\mathcal{K}_{j}$ have the following properties:

(i) $\mathcal{K}_{j}$* is a finite chain ring, $f_{j}(x)$ generates the unique maximal ideal $\langle f_{j}(x)\rangle$ of $\mathcal{K}_{j}$, $2^{k}\lambda$ is the nilpotency index of $f_{j}(x)$ and the residue class field of $\mathcal{K}_{j}$ modulo $\langle f_{j}(x)\rangle$ is $\mathcal{K}_{j}/\langle f_{j}(x)\rangle\cong\mathcal{F}_{j}$*.

(ii) Every element $\xi$ of $\mathcal{K}_{j}$ has a unique $f_{j}(x)$-adic expansion:

[TABLE]

where $\ b_{0}(x),b_{1}(x),\ldots,b_{2^{k}\lambda-1}\in\mathcal{F}_{j}$. Moreover, $\xi$ is invertible in $\mathcal{K}_{j}$ if and only if $b_{0}(x)\neq 0$. Here regard $\mathcal{F}_{j}$ as a subset of $\mathcal{K}_{j}$.

(iii) All distinct $2^{k}\lambda+1$ ideals of $\mathcal{K}_{j}$ are given by:

[TABLE]

Moreover, $|\langle f_{j}(x)^{l}\rangle|=2^{md_{j}(2^{k}\lambda-l)}$ for $l=0,1,\ldots,2^{k}\lambda$.

(iv) Let $1\leq l\leq 2^{k}\lambda$. Then $|\mathcal{K}_{j}/\langle f_{j}(x)^{l}\rangle|=2^{md_{j}l}$. Precisely, we have

[TABLE]

Remark For any integer $l$, $1\leq l\leq 2^{k}\lambda-1$, by Lemma 2.3(iv) we can identify $\mathcal{K}_{j}/\langle f_{j}(x)^{l}\rangle$ with $\frac{\mathbb{F}_{2^{m}}[x]}{\langle f_{j}(x)^{l}\rangle}$ up to a natural ring isomorphism. We will take this view in the rest of this paper. Then for any $0\leq l\leq t\leq 2^{k}\lambda-1$, we stipulate

[TABLE]

Hence $|f_{j}(x)^{l}(\mathcal{K}_{j}/\langle f_{j}(x)^{t}\rangle)|=2^{md_{j}(t-l)}$, where we set $f_{j}(x)^{l}(\mathcal{K}_{j}/\langle f_{j}(x)^{l}\rangle)$ $=\{0\}$ for convenience.

Then we study the structure of the ring $\mathcal{A}+u\mathcal{A}$. To do this, we introduce the following notation.

Let $\alpha_{0}\in\mathbb{F}_{2^{m}}^{\times}$ satisfying $\alpha_{0}^{2}=\alpha^{-1}$.

• 2.

Let $\omega_{j}\in\mathcal{K}_{j}$ be defined by $\omega_{j}=\alpha_{0}F_{j}(x)^{2^{k-1}}\ ({\rm mod}\ f_{j}(x)^{2^{k}\lambda})$.

• 3.

$\mathcal{K}_{j}+u\mathcal{K}_{j}=\frac{\mathcal{K}_{j}[u]}{\langle u^{2}-\omega_{j}^{2}f_{j}(x)^{2^{k}}\rangle}=\{\xi_{0}+u\xi_{1}\mid\xi_{0},\xi_{1}\in\mathcal{K}_{j}\}$ in which the operations are defined by

• (a)

$(\xi_{0}+u\xi_{1})+(\eta_{0}+u\eta_{1})=(\xi_{0}+\eta_{0})+u(\xi_{1}+\eta_{1})$,

• (b)

$(\xi_{0}+u\xi_{1})(\eta_{0}+u\eta_{1})=\left(\xi_{0}\eta_{0}+\omega_{j}^{2}f_{j}(x)^{2^{k}}\xi_{1}\eta_{1}\right)+u(\xi_{0}\eta_{1}+\xi_{1}\eta_{0})$,

for all $\xi_{0},\xi_{1},\eta_{0},\eta_{1}\in\mathcal{K}_{j}$. Then $\mathcal{K}_{j}$ is a subring of $\mathcal{K}_{j}+u\mathcal{K}_{j}$.

Lemma 2.4 Let $1\leq j\leq r$. Then we have the following conclusions.

(i) The element $\omega_{j}$ is invertible in the ring $\mathcal{K}_{j}$ and satisfies

[TABLE]

Hence $\alpha^{-1}(x^{n}+\delta_{0})^{2^{k}}=\omega_{j}^{2}f_{j}(x)^{2^{k}}$ in the ring $\mathcal{K}_{j}$.

(ii) We regard $\mathcal{K}_{j}+u\mathcal{K}_{j}$ as a subset of $\mathcal{A}+u\mathcal{A}$ for all $j$. Then

[TABLE]

where $\varepsilon_{j}(x)(\mathcal{K}_{j}+u\mathcal{K}_{j})=\{\varepsilon_{j}(x)a_{j}(x)+u\varepsilon_{j}(x)b_{j}(x)\mid a_{j}(x),b_{j}(x)\in\mathcal{K}_{j}\}$.

Proof. (i) Since $\omega_{j}\in\mathcal{K}_{j}$ satisfying $\omega_{j}\equiv\alpha_{0}F_{j}(x)^{2^{k-1}}$ (mod $f_{j}(x)^{2^{k}\lambda}$) and ${\rm gcd}(F_{j}(x),f_{j}(x))=1$, we conclude that ${\rm gcd}(\omega_{j},f_{j}(x)^{2^{k}\lambda})=1$ as polynomials in $\mathbb{F}_{2^{m}}[x]$. This implies that $\omega_{j}$ is an invertible element of the ring $\mathcal{K}_{j}$. Then from $(x^{n}+\delta_{0})^{2^{k}}=f_{1}(x)^{2^{k}}\ldots f_{r}(x)^{2^{k}}$ and $F_{j}(x)^{2^{k}}=\frac{(x^{n}+\delta_{0})^{2^{k}}}{f_{j}(x)^{2^{k}}}$, we deduce that

[TABLE]

This implies $\alpha^{-1}(x^{n}+\delta_{0})^{2^{k}}=\omega_{j}^{2}f_{j}(x)^{2^{k}}$ in $\mathcal{K}_{j}$.

(ii) By Lemma 2.2 and the conclusion of (i), it follows that

[TABLE]

This implies $\mathcal{A}+u\mathcal{A}=\bigoplus_{j=1}^{r}\varepsilon_{j}(x)(\mathcal{K}_{j}+u\mathcal{K}_{j})=\sum_{j=1}^{r}\varepsilon_{j}(x)(\mathcal{K}_{j}+u\mathcal{K}_{j})$. $\Box$

Finally, we list all $(\delta+\alpha u^{2})$-constacyclic codes over $R$ of length $2^{k}n$ by the following theorem. Its proof will be given in Section 4.

Theorem 2.5 For any integers $j,l$: $1\leq j\leq r$ and $0\leq l\leq 2^{k-1}\lambda$, let

[TABLE]

Especially, we have $\frac{\mathbb{F}_{2^{m}}[x]}{\langle f_{j}(x)^{0}\rangle}=\{0\}$ and $\frac{\mathbb{F}_{2^{m}}[x]}{\langle f_{j}(x)\rangle}=\mathcal{F}_{j}$. Then all distinct $(\delta+\alpha u^{2})$-constacyclic codes over $R$ of length $2^{k}n$, as ideals of the ring $\mathcal{A}+u\mathcal{A}$, are given by

[TABLE]

where $C_{j}$ is an ideal of the ring $\mathcal{K}_{j}+u\mathcal{K}_{j}$ listed by the following three cases.

(I) $\sum_{s=0}^{2^{k}\lambda-1}2^{md_{j}(2^{k-1}\lambda-\lceil\frac{s}{2}\rceil)}$ ideals given by the following two subcases:

1. $C_{j}=\left\langle\omega_{j}f_{j}(x)^{2^{k-1}+s}+f_{j}(x)^{2^{k-1}\lambda+\lceil\frac{s}{2}\rceil}h(x)+uf_{j}(x)^{s}\right\rangle$ with $|C|=2^{md_{j}(2^{k}\lambda-s)}$, where $h(x)\in\frac{\mathbb{F}_{2^{m}}[x]}{\langle f_{j}(x)^{2^{k-1}\lambda-\lceil\frac{s}{2}\rceil}\rangle}$ and $0\leq s\leq 2^{k-1}\lambda-1$.

2. $C_{j}=\left\langle f_{j}(x)^{2^{k-1}\lambda+\lceil\frac{s}{2}\rceil}h(x)+uf_{j}(x)^{s}\right\rangle$ with $|C|=2^{md_{j}(2^{k}\lambda-s)}$,

where $h(x)\in\frac{\mathbb{F}_{2^{m}}[x]}{\langle f_{j}(x)^{2^{k-1}\lambda-\lceil\frac{s}{2}\rceil}\rangle}$ and $2^{k-1}\lambda\leq s\leq 2^{k}\lambda-1$.

(II) $2^{k}\lambda+1$ ideals:

3. $C_{j}=\left\langle f_{j}(x)^{s}\right\rangle$* with $|C|=2^{md_{j}(2^{k+1}\lambda-2s)}$, where $0\leq s\leq 2^{k}\lambda$*.

(III) $\sum_{t=1}^{2^{k}\lambda-1}(2^{k}\lambda-t)2^{md_{j}\lfloor\frac{t}{2}\rfloor}$ ideals given by the following three subcases:

4. $C_{j}=\left\langle uf_{j}(x)^{s},f_{j}(x)^{s+1}\right\rangle$* with $|C|=2^{md_{j}(2^{k+1}\lambda-2s-1)}$,

where $0\leq s\leq 2^{k}\lambda-2$*.

5. $C_{j}=\left\langle f_{j}(x)^{s+\lceil\frac{t}{2}\rceil}h(x)+uf_{j}(x)^{s},f_{j}(x)^{s+t}\right\rangle$*

with $|C|=2^{md_{j}(2^{k}\lambda-2s-t)}$, where $h(x)\in\frac{\mathbb{F}_{2^{m}}[x]}{\langle f_{j}(x)^{\lfloor\frac{t}{2}\rfloor}\rangle}$,

$0\leq s\leq 2^{k}\lambda-1-t$ and $2\leq t\leq 2^{k}$.*

6. $C_{j}=\left\langle\omega_{j}f_{j}(x)^{2^{k-1}+s}+f_{j}(x)^{s+\lceil\frac{t}{2}\rceil}h(x)+uf_{j}(x)^{s},f_{j}(x)^{s+t}\right\rangle$*

with $|C|=2^{md_{j}(2^{k+1}\lambda-2s-t)}$, where $h(x)\in\frac{\mathbb{F}_{2^{m}}[x]}{\langle f_{j}(x)^{\lfloor\frac{t}{2}\rfloor}\rangle}$,

$0\leq s\leq 2^{k}\lambda-1-t$ and $2^{k}+1\leq t\leq 2^{k}\lambda-1$.*

In this case, the number of codewords in $\mathcal{C}$ is $|\mathcal{C}|=\prod_{j=1}^{r}|C_{j}|$.

Moreover, the number of all $(\delta+\alpha u^{2})$-constacyclic codes over $R$ of length $2^{k}n$ is equal to $\prod_{j=1}^{r}N_{(2^{m},d_{j},2^{k}\lambda)},$ where

[TABLE]

is the number of ideals in $\mathcal{K}_{j}+u\mathcal{K}_{j}$ for all $j=1,\ldots,r$.

Specifically, one can easily give an explicit representation for all distinct $(\delta+\alpha u^{2})$-constacyclic codes over $R$ of length $2^{k}n$, as ideals of the ring $\frac{R[x]}{\langle x^{2^{k}n}-(\delta+\alpha u^{2})\rangle}$, from Theorem 2.5 by replacing each element in $\mathcal{C}$ with the substitutions determined by Equations (1) and (2).

Similarly, as Corollary 3.8 of [11] one can prove the following conclusion.

Corollary 2.6 Every $(\delta+\alpha u^{2})$-constacyclic code over $R$ of length $2^{k}n$ can be generated by at most $2$ polynomials in the ring $\frac{R[x]}{\langle x^{2^{k}n}-(\delta+\alpha u^{2})\rangle}$.

3 A subclass of $(\delta+\alpha u^{2})$-constacyclic codes over $R$

In this section, let $x^{n}+\delta_{0}$ be an irreducible polynomial in $\mathbb{F}_{2^{m}}[x]$ and $\delta=\delta_{0}^{2^{k}}$. We consider $(\delta+\alpha u^{2})$-constacyclic codes over $R$ of length $2^{k}n$. In this case, we have $r=1$, $f_{1}(x)=x^{n}+\delta_{0}$, $\varepsilon_{1}(x)=1$, $d_{1}={\rm deg}(x^{n}-\delta_{0})=n$, $\omega_{1}=\alpha_{0}$ where $\alpha_{0}^{2}=\alpha^{-1}$, and

$\mathcal{K}_{1}=\mathcal{A}=\frac{\mathbb{F}_{2^{m}}[x]}{\langle(x^{n}+\delta_{0})^{2^{k}\lambda}\rangle}$;

• 2.

$\mathcal{F}_{1}=\frac{\mathbb{F}_{2^{m}}[x]}{\langle x^{n}+\delta_{0}\rangle}=\{\sum_{i=0}^{n-1}a_{i}x^{i}\mid a_{0},a_{1},\ldots,a_{n-1}\in\mathbb{F}_{2^{m}}\}$ and $|\mathcal{F}_{1}|=2^{mn}$.

From these and by Theorem 2.5, we deduce the following conclusion.

Theorem 3.1 Let $x^{n}+\delta_{0}$ be an irreducible polynomial in $\mathbb{F}_{2^{m}}[x]$ and $\delta=\delta_{0}^{2^{k}}$. Then all distinct $(\delta+\alpha u^{2})$-constacyclic codes over $R$ of length $2^{k}n$ are listed by the following three cases.

(I) $\sum_{s=0}^{2^{k}\lambda-1}2^{mn(2^{k-1}\lambda-\lceil\frac{s}{2}\rceil)}$ codes given by the following two subcases:

1. $\mathcal{C}=\langle\alpha_{0}(x^{n}+\delta_{0})^{2^{k-1}+s}+(x^{n}+\delta_{0})^{2^{k-1}\lambda+\lceil\frac{s}{2}\rceil}h(x)+u(x^{n}+\delta_{0})^{s}\rangle$

with $|C|=2^{mn(2^{k}\lambda-s)}$, *where $h(x)\in\frac{\mathbb{F}_{2^{m}}[x]}{\langle(x^{n}+\delta_{0})^{2^{k-1}\lambda-\lceil\frac{s}{2}\rceil}\rangle}$

and $0\leq s\leq 2^{k-1}\lambda-1$*.

2. $\mathcal{C}=\langle(x^{n}+\delta_{0})^{2^{k-1}\lambda+\lceil\frac{s}{2}\rceil}h(x)+u(x^{n}+\delta_{0})^{s}\rangle$ with $|C|=2^{mn(2^{k}\lambda-s)}$,

where $h(x)\in\frac{\mathbb{F}_{2^{m}}[x]}{\langle(x^{n}+\delta_{0})^{2^{k-1}\lambda-\lceil\frac{s}{2}\rceil}\rangle}$ and $2^{k-1}\lambda\leq s\leq 2^{k}\lambda-1$.

(II) $2^{k}\lambda+1$ codes:

3. $\mathcal{C}=\langle(x^{n}+\delta_{0})^{s}\rangle$* with $|C|=2^{mn(2^{k+1}\lambda-2s)}$, where $0\leq s\leq 2^{k}\lambda$*.

(III) $\sum_{t=1}^{2^{k}\lambda-1}(2^{k}\lambda-t)2^{mn\lfloor\frac{t}{2}\rfloor}$ codes given by the following three subcases:

4. $\mathcal{C}=\langle u(x^{n}+\delta_{0})^{s},(x^{n}+\delta_{0})^{s+1}\rangle$*

with $|C|=2^{mn(2^{k+1}\lambda-2s-1)}$, where $0\leq s\leq 2^{k}\lambda-2$*.

5. $\mathcal{C}=\langle(x^{n}+\delta_{0})^{s+\lceil\frac{t}{2}\rceil}h(x)+u(x^{n}+\delta_{0})^{s},(x^{n}+\delta_{0})^{s+t}\rangle$*

with $|C|=2^{mn(2^{k}\lambda-2s-t)}$, where $h(x)\in\frac{\mathbb{F}_{2^{m}}[x]}{\langle(x^{n}+\delta_{0})^{\lfloor\frac{t}{2}\rfloor}\rangle}$,

$0\leq s\leq 2^{k}\lambda-1-t$ and $2\leq t\leq 2^{k}$.*

6. $\mathcal{C}=\langle\alpha_{0}(x^{n}+\delta_{0})^{2^{k-1}+s}+(x^{n}+\delta_{0})^{s+\lceil\frac{t}{2}\rceil}h(x)+u(x^{n}+\delta_{0})^{s},(x^{n}+\delta_{0})^{s+t}\rangle$*

with $|C|=2^{mn(2^{k+1}\lambda-2s-t)}$, where $h(x)\in\frac{\mathbb{F}_{2^{m}}[x]}{\langle(x^{n}+\delta_{0})^{\lfloor\frac{t}{2}\rfloor}\rangle}$,

$0\leq s\leq 2^{k}\lambda-1-t$ and $2^{k}+1\leq t\leq 2^{k}\lambda-1$.*

In this case, the number of codes of all $(\delta+\alpha u^{2})$-constacyclic codes over $R$ of length $2^{k}n$ is equal to

[TABLE]

Especially, set $n=1$ in Theorem 3.1. We obtain the following corollary.

Corollary 3.2 All distinct $(\delta+\alpha u^{2})$-constacyclic codes over $R$ of length $2^{k}$ are listed by the following three cases.

(I) $\sum_{s=0}^{2^{k}\lambda-1}2^{m(2^{k-1}\lambda-\lceil\frac{s}{2}\rceil)}$ codes given by the following two subcases:

1. $C=\langle\alpha_{0}(x+\delta_{0})^{2^{k-1}+s}+(x+\delta_{0})^{2^{k-1}\lambda+\lceil\frac{s}{2}\rceil}h(x)+u(x+\delta_{0})^{s}\rangle$ with $|C|=2^{m(2^{k}\lambda-s)}$, where $h(x)\in\frac{\mathbb{F}_{2^{m}}[x]}{\langle(x+\delta_{0})^{2^{k-1}\lambda-\lceil\frac{s}{2}\rceil}\rangle}$ and $0\leq s\leq 2^{k-1}\lambda-1$.

2. $C=\langle(x+\delta_{0})^{2^{k-1}\lambda+\lceil\frac{s}{2}\rceil}h(x)+u(x+\delta_{0})^{s}\rangle$ with $|C|=2^{m(2^{k}\lambda-s)}$,

where $h(x)\in\frac{\mathbb{F}_{2^{m}}[x]}{\langle(x+\delta_{0})^{2^{k-1}\lambda-\lceil\frac{s}{2}\rceil}\rangle}$ and $2^{k-1}\lambda\leq s\leq 2^{k}\lambda-1$.

(II) $2^{k}\lambda+1$ codes:

3. $C=\langle(x+\delta_{0})^{s}\rangle$* with $|C|=2^{m(2^{k+1}\lambda-2s)}$, where $0\leq s\leq 2^{k}\lambda$*.

(III) $\sum_{t=1}^{2^{k}\lambda-1}(2^{k}\lambda-t)2^{m\lfloor\frac{t}{2}\rfloor}$ codes given by one of the following three subcases:

4. $C=\langle u(x+\delta_{0})^{s},(x+\delta_{0})^{s+1}\rangle$* with $|C|=2^{m(2^{k+1}\lambda-2s-1)}$,

where $0\leq s\leq 2^{k}\lambda-2$*.

5. $C=\langle(x+\delta_{0})^{s+\lceil\frac{t}{2}\rceil}h(x)+u(x+\delta_{0})^{s},(x+\delta_{0})^{s+t}\rangle$* with $|C|=2^{m(2^{k+1}\lambda-2s-t)}$,

where $h(x)\in\frac{\mathbb{F}_{2^{m}}[x]}{\langle(x+\delta_{0})^{\lfloor\frac{t}{2}\rfloor}\rangle}$, $0\leq s\leq 2^{k}\lambda-1-t$ and $2\leq t\leq 2^{k}$.*

6. $C=\langle\alpha_{0}(x+\delta_{0})^{2^{k-1}+s}+(x+\delta_{0})^{s+\lceil\frac{t}{2}\rceil}h(x)+u(x+\delta_{0})^{s},(x+\delta_{0})^{s+t}\rangle$* with $|C|=2^{m(2^{k+1}\lambda-2s-t)}$, where $h(x)\in\frac{\mathbb{F}_{2^{m}}[x]}{\langle(x+\delta_{0})^{\lfloor\frac{t}{2}\rfloor}\rangle}$, $0\leq s\leq 2^{k}\lambda-1-t$ and $2^{k}+1\leq t\leq 2^{k}\lambda-1$.*

Therefore, the number of ideals of the ring $\mathcal{A}+u\mathcal{A}$ is equal to

[TABLE]

As the end of this section, we list all $(\delta+\alpha u^{2})$-constacyclic code over $R=\frac{\mathbb{F}_{2^{m}}[u]}{\langle u^{4}\rangle}$ of length $4$, where $\delta,\alpha,\delta_{0},\alpha_{0}\in\mathbb{F}_{2^{m}}^{\times}$ satisfying $\delta_{0}^{4}=\delta$ and $\delta_{0}^{2}=\alpha^{-1}$ respectively. In this case, we have have the following:

$\checkmark$ $\Psi((x+\delta_{0})^{4})=\alpha u^{2}=\alpha_{0}^{-2}u^{2}$.

$\checkmark$ $\frac{\mathbb{F}_{2^{m}}[x]}{\langle(x+\delta_{0})^{l}\rangle}=\{\alpha h(x)\mid h(x)\in\frac{\mathbb{F}_{2^{m}}[x]}{\langle(x+\delta_{0})^{l}\rangle}\}$ for all $l=0,1,2,3,4$.

$\checkmark$ $\Psi((x+\delta_{0})^{3}h(x))=(x+\delta_{0})^{3}h_{0}+u^{2}h_{1}$, where $h_{1}=\alpha\widetilde{h}_{1}$, for all $h(x)=h_{0}+\widetilde{h}_{1}(x+\delta_{0})\in\frac{\mathbb{F}_{2^{m}}[x]}{\langle(x+\delta_{0})^{2}\rangle}$ with $h_{0},\widetilde{h}_{1}\in\mathbb{F}_{2^{m}}$.

$\checkmark$ $\Psi((x+\delta_{0})^{3}h(x))=(x+\delta_{0})^{3}h_{0}+u^{2}h_{1}+u^{2}(x+\delta_{0})h_{2}$, where $h_{1}=\alpha\widetilde{h}_{1}$ and $h_{2}=\alpha\widetilde{h}_{2}$, for all $h(x)=h_{0}+\widetilde{h}_{1}(x+\delta_{0})+\widetilde{h}_{2}(x+\delta_{0})^{2}\in\frac{\mathbb{F}_{2^{m}}[x]}{\langle(x+\delta_{0})^{3}\rangle}$ with $h_{0},\widetilde{h}_{1},\widetilde{h}_{2}\in\mathbb{F}_{2^{m}}$.

First, by Corollary 3.2 the number of $(\delta+\alpha u^{2})$-constacyclic code over $R=\frac{\mathbb{F}_{2^{m}}[u]}{\langle u^{4}\rangle}$ of length $4$ is equal to

[TABLE]

Then we list all these codes by the following corollary.

Corollary 3.3 Denote $y=x+\delta_{0}$ and $\mathcal{F}_{l}=\frac{\mathbb{F}_{2^{m}}[x]}{\langle(x+\delta_{0})^{l}\rangle}$ for $l=2,3,4$. Then all distinct $(\delta+\alpha u^{2})$-constacyclic codes over $\frac{\mathbb{F}_{2^{m}}[u]}{\langle u^{4}\rangle}$ of length $4$ are given by the following tables:

[TABLE]

where $\mathcal{L}$ is the number of cyclic codes in the same row.

Remark When $m=1$, the number of $(\delta+\alpha u^{2})$-constacyclic codes of length $4$ over $\frac{\mathbb{F}_{2}[u]}{\langle u^{4}\rangle}$ is equal to $N_{(2,1,2^{3})}=135$. But in Example 1 of [23], the authors list all distinct $(1+u^{2})$-constacyclic codes of length $4$ over $\frac{\mathbb{F}_{2}[u]}{\langle u^{4}\rangle}$ as $131$ distinct $(1+u^{2})$-constacyclic codes in Pages 3119–3120 of [23].

Finally, let $\delta=1$. Then one can easily verify the following conclusion for $(1+\alpha u^{2})$-constacyclic codes of length $4$. Here we omit the process of proof.

Theorem 3.4 All distinct self-dual $(1+\alpha u^{2})$-constacyclic codes over $\frac{\mathbb{F}_{2^{m}}[u]}{\langle u^{4}\rangle}$ of length $4$ are given by the following four cases:

(i) $C=\langle u^{2}\rangle$.

(ii) $C=\langle u^{2}b_{0}+u(x+1)^{3},u^{2}(x+1)\rangle$, where $b_{0}\in\mathbb{F}_{2^{m}}$.

(iii) $C=\langle u^{2}h(x)+u(x+1)^{2},u^{2}(x+1)^{2}\rangle$, where $h(x)=b_{1}+b_{2}(x+1)$ and $b_{1},b_{2}\in\mathbb{F}_{2^{m}}$.

(iii) $C=\langle\alpha_{0}(x+1)^{3}+u^{2}h(x)+u(x+1),u^{2}(x+1)^{3}\rangle$, where $h(x)=\alpha_{0}+b_{3}(x+1)+b_{4}(x+1)^{2}$ and $b_{3},b_{4}\in\mathbb{F}_{2^{m}}$.

Therefore, the number of self-dual $(1+\alpha u^{2})$-constacyclic codes over $\frac{\mathbb{F}_{2^{m}}[u]}{\langle u^{4}\rangle}$ of length $4$ is $1+2^{m}+2(2^{m})^{2}$.

Finally, let $m=1$. By Theorem 3.4 we deduce that there are $11$ self-dual $(1+u^{2})$-constacyclic codes of length $4$ over $\mathbb{F}_{2}[u]/\langle u^{4}\rangle$ given by:

(i) $\langle u^{2}\rangle$.

(ii) $\langle u(x+1)^{3},u^{2}(x+1)\rangle$, $\langle u^{2}+u(x+1)^{3},u^{2}(x+1)\rangle$.

(iii) $\langle u^{2}+u(x+1)^{2},u^{2}(x+1)^{2}\rangle$, $\langle u^{2}+u(x+1)^{2},u^{2}(x+1)^{2}\rangle$,

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

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

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

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

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

4 Proof of Theorem 2.5

In this section, we give a detailed proof of Theorem 2.5.

Let $\mathcal{C}$ be a $(\delta+\alpha u^{2})$-constacyclic codes over $R$ of length $2^{k}n$, here we regard $\mathcal{C}$ as an ideal of the ring $\mathcal{A}+\mathcal{A}$ under the ring isomorphism $\Psi$ determined by Equations (1) and (2) in Section 2. Then by Lemma 2.2(i) and Lemma 2.4(ii), for each integer $j$, $1\leq j\leq r$, there is a unique ideal $C_{j}$ of the ring $\mathcal{K}_{j}+u\mathcal{K}_{j}$ such that

[TABLE]

Moreover, the number of codewords in $\mathcal{C}$ equals $|\mathcal{C}|=\prod_{j=1}^{r}|C_{j}|$.

Now, let $1\leq j\leq r$ and denote $\pi_{j}=f_{j}(x)\in\mathcal{K}_{j}$. We consider how to determine all ideals of the ring $\mathcal{K}_{j}+u\mathcal{K}_{j}$ ($u^{2}=\omega_{j}^{2}\pi_{j}^{2^{k}}$). Since $\mathcal{K}_{j}$ is a subring of $\mathcal{K}_{j}+u\mathcal{K}_{j}$, we see that $\mathcal{K}_{j}+u\mathcal{K}_{j}$ is a free $\mathcal{K}_{j}$-module of rank $2$ with the basis $\{1,u\}$. Now, we define

[TABLE]

One can easily verify that $\theta$ is an $\mathcal{K}_{j}$-module isomorphism from $\mathcal{K}_{j}^{2}$ onto $\mathcal{K}_{j}+u\mathcal{K}_{j}$. The following lemma can be verified by an argument similar to the proof of Lemma 3.7 of [7]. Here, we omit its proof.

Lemma 4.1 Using the notations above, $C$ is an ideal of the ring $\mathcal{K}_{j}+u\mathcal{K}_{j}$ $(u^{2}=\omega_{j}^{2}\pi_{j}^{2^{k}})$ if and only if there is a unique $\mathcal{K}_{j}$-submodule $S$ of $\mathcal{K}_{j}^{2}$ satisfying

[TABLE]

such that $C=\theta(S)$.

Recall that every $\mathcal{K}_{j}$-submodule of $\mathcal{K}_{j}^{2}$ is called a linear code over the finite chain ring $\mathcal{K}_{j}$ of length $2$. A general discussion and description for linear codes over arbitrary finite chain ring can be found in [24]. Let $S$ be a linear code over $\mathcal{K}_{j}$ of length $2$. A matrix $G$ is called a generator matrix for $S$ if every codeword in $S$ is a $\mathcal{K}_{j}$-linear combination of the row vectors of $G$ and any row vector of $G$ can not be written as a $\mathcal{K}_{j}$-linear combination of the other row vectors of $G$.

In the following lemma, we use lowercase letters to denote the elements of $\mathcal{K}_{j}$ and $\mathcal{K}_{j}/\langle\pi_{j}^{l}\rangle$ ($1\leq l\leq 2^{k}\lambda-1$) in order to simplify the expressions.

Lemma 4.2 (cf. [8] Lemma 2.2 and Example 2.5) Using the notations above, $\sum_{i=0}^{2^{k}\lambda}(2i+1)2^{md_{j}(2^{k}\lambda-i)}$ is the number of linear codes over $\mathcal{K}_{j}$ of length $2$.

Moreover, every linear code over $\mathcal{K}_{j}$ of length $2$ has one and only one of the following matrices $G$ as their generator matrices:

(i) $G=(1,a)$, $a\in\mathcal{K}_{j}$.

(ii) $G=(\pi_{j}^{s},\pi_{j}^{s}a)$, $a\in\mathcal{K}_{j}/\langle\pi_{j}^{2^{k}\lambda-s}\rangle$, $1\leq s\leq 2^{k}\lambda-1$.

(iii) $G=(\pi_{j}b,1)$, $b\in\mathcal{K}_{j}/\langle\pi_{j}^{2^{k}\lambda-1}\rangle$.

(iv) $G=(\pi_{j}^{s+1}b,\pi_{j}^{s})$, $b\in\mathcal{K}_{j}/\langle\pi_{j}^{2^{k}\lambda-1-s}\rangle$, $1\leq s\leq 2^{k}\lambda-1$.

(v) G=\left(\begin{array}[]{cc}\pi_{j}^{s}&0\cr 0&\pi_{j}^{s}\end{array}\right), $0\leq s\leq 2^{k}\lambda$.

(vi) G=\left(\begin{array}[]{cc}1&c\cr 0&\pi_{j}^{t}\end{array}\right), $c\in\mathcal{K}_{j}/\langle\pi_{j}^{t}\rangle$, $1\leq t\leq 2^{k}\lambda-1$.

(vii) G=\left(\begin{array}[]{cc}\pi_{j}^{s}&\pi_{j}^{s}c\cr 0&\pi_{j}^{s+t}\end{array}\right), $c\in\mathcal{K}_{j}/\langle\pi_{j}^{t}\rangle$, $1\leq t\leq 2^{k}\lambda-1-s$, $1\leq s\leq 2^{k}\lambda-2$.

(viii) G=\left(\begin{array}[]{cc}c&1\cr\pi_{j}^{t}&0\end{array}\right), $c\in\pi_{j}(\mathcal{K}_{j}/\langle\pi_{j}^{t}\rangle)$, $1\leq t\leq 2^{k}\lambda-1$.

(ix) G=\left(\begin{array}[]{cc}\pi_{j}^{s}c&\pi_{j}^{s}\cr\pi_{j}^{s+t}&0\end{array}\right), $c\in\pi_{j}(\mathcal{K}_{j}/\langle\pi_{j}^{t}\rangle)$, $1\leq t\leq 2^{k}\lambda-1-s$, $1\leq s\leq 2^{k}\lambda-2$.

Let $\beta\in\mathcal{K}_{j}$ and $\beta\neq 0$. By Lemma 2.3(ii), there is a unique integer $t$, $0\leq t\leq 2^{k}\lambda-1$, such that $\beta=\pi_{j}^{t}w$ for some $w\in\mathcal{K}_{j}^{\times}$. We call $t$ the $\pi_{j}$-degree of $\beta$ and denote it by $\|\beta\|_{\pi_{j}}=t$. If $\beta=0$, we write $\|\beta\|_{\pi_{j}}=2^{k}\lambda$. For any vector $(\beta_{1},\beta_{2})\in\mathcal{K}_{j}^{2}$, we define the $\pi_{j}$-degree of $(\beta_{1},\beta_{2})$ by

[TABLE]

Now, as a special case of [24] Proposition 3.2 and Theorem 3.5, we deduce the following lemma.

Lemma 4.3 (cf. [8] Lemma 2.3) Let $S$ be a nonzero linear code over $\mathcal{K}_{j}$ of length $2$, and $G$ be a generator matrix of $S$ with row vectors $G_{1},\ldots,G_{\rho}\in\mathcal{K}_{j}^{2}\setminus\{0\}$ satisfying

[TABLE]

Then the number of codewords in $S$ is equal to $|S|=|\mathcal{F}_{j}|^{\sum_{i=1}^{\rho}(2^{k}\lambda-t_{i})}=2^{md_{j}\sum_{i=1}^{\rho}(2^{k}\lambda-t_{i})}$.

For any positive integer $i$, let $\lceil\frac{i}{2}\rceil={\rm min}\{l\in\mathbb{Z}^{+}\mid l\geq\frac{i}{2}\}$ and $\lfloor\frac{i}{2}\rfloor={\rm max}\{l\in\mathbb{Z}^{+}\cup\{0\}\mid l\leq\frac{i}{2}\}$. It is well known that $\lceil\frac{i}{2}\rceil+\lfloor\frac{i}{2}\rfloor=i$. Using these notations, we list all distinct $\mathcal{K}_{j}$-submodules of $\mathcal{K}_{j}^{2}$ satisfying Condition (4) in Lemma 4.1 by the following lemma.

Lemma 4.4 Using the notations in Section 2, every linear code $S$ over $\mathcal{K}_{j}$ of length $2$ satisfying Condition $(4)$ in Lemma 4.1 has one and only one of the following matrices $G$ as its generator matrix:

(I) $\sum_{s=0}^{2^{k}\lambda-1}2^{md_{j}(2^{k-1}\lambda-\lceil\frac{s}{2}\rceil)}$ matrices given by the following two cases:

1. (I-1)

$G=(\omega_{j}f_{j}(x)^{2^{k-1}+s}+f_{j}(x)^{2^{k-1}\lambda+\lceil\frac{s}{2}\rceil}h(x),f_{j}(x)^{s})$*

where $h(x)\in\frac{\mathbb{F}_{2^{m}}[x]}{\langle f_{j}(x)^{2^{k-1}\lambda-\lceil\frac{s}{2}\rceil}\rangle}$, if $0\leq s\leq 2^{k-1}\lambda-1$*. 2. (I-2)

$G=(f_{j}(x)^{2^{k-1}\lambda+\lceil\frac{s}{2}\rceil}h(x),f_{j}(x)^{s})$*

where $h(x)\in\frac{\mathbb{F}_{2^{m}}[x]}{\langle f_{j}(x)^{2^{k-1}\lambda-\lceil\frac{s}{2}\rceil}\rangle}$, if $2^{k-1}\lambda\leq s\leq 2^{k}\lambda-1$*.

(II) $2^{k}\lambda+1$ matrices:

1. G=\left(\begin{array}[]{cc}f_{j}(x)^{s}&0\cr 0&f_{j}(x)^{s}\end{array}\right), $0\leq s\leq 2^{k}\lambda$.

(III) $\sum_{t=1}^{2^{k}\lambda-1}(2^{k}\lambda-t)2^{md_{j}\lfloor\frac{t}{2}\rfloor}$ matrices given by the following three cases:

1. (III-1)

G=\left(\begin{array}[]{cc}0&f_{j}(x)^{s}\cr f_{j}(x)^{s+1}&0\end{array}\right),\ 0\leq s\leq 2^{k}\lambda-2. 2. (III-2)

G=\left(\begin{array}[]{cc}f_{j}(x)^{s+\lceil\frac{t}{2}\rceil}h(x)&f_{j}(x)^{s}\cr f_{j}(x)^{s+t}&0\end{array}\right) where $h(x)\in\frac{\mathbb{F}_{2^{m}}[x]}{\langle f_{j}(x)^{\lfloor\frac{t}{2}\rfloor}\rangle}$ and $0\leq s\leq 2^{k}\lambda-1-t$, if $2\leq t\leq 2^{k}$. 3. (III-3)

G=\left(\begin{array}[]{cc}\omega_{j}f_{j}(x)^{2^{k-1}+s}+f_{j}(x)^{s+\lceil\frac{t}{2}\rceil}h(x)&f_{j}(x)^{s}\cr f_{j}(x)^{s+t}&0\end{array}\right) where $h(x)\in\frac{\mathbb{F}_{2^{m}}[x]}{\langle f_{j}(x)^{\lfloor\frac{t}{2}\rfloor}\rangle}$ and $0\leq s\leq 2^{k}\lambda-1-t$, if $2^{k}+1\leq t\leq 2^{k}\lambda-1$.

Proof. In order to simplify the symbol, we write $\mathcal{K}_{j}$, $\mathcal{F}_{j}$, $\pi_{j}=f_{j}(x)$ and $\omega_{j}$ simply as $\mathcal{K}$, $\mathcal{F}$, $\pi=f(x)$ and $\omega$ respectively, in the following. Let $S$ be the $\mathcal{K}$-submodule of $\mathcal{K}^{2}$ with generator matrix $G$, where $G$ is a matrix given by Lemma 4.2. Then we only need to consider the following five cases.

Case 1. Let $G$ be given by Lemma 4.2 (i) and (ii), i.e., $G=(\pi^{s},\pi^{s}a)$ where $a\in\mathcal{K}/\langle\pi^{2^{k}\lambda-s}\rangle$ and $0\leq s\leq 2^{k}\lambda-1$.

Suppose that $S$ satisfies Condition $(4)$ in Lemma 4.1. Then

[TABLE]

which is equivalent to that $\omega^{2}\pi^{2^{k}+s}a=\pi^{s}b$ and $\pi^{s}=\pi^{s}ab$. The latter implies

[TABLE]

From this and by $0\leq s<{\rm min}\{2^{k}\lambda,2^{k}+s\}$, we obtain a contradiction. Hence $S$ does not satisfy Condition (4) in Lemma 4.1.

Case 2. Let $G$ be given by Lemma 4.2 (iii) and (iv), i.e., $G=(\pi^{s+1}b,\pi^{s})$ where $b\in\mathcal{K}/\langle\pi^{2^{k}\lambda-1-s}\rangle$ and $0\leq s\leq 2^{k}\lambda-1$.

Then $S$ satisfies Condition (4) in Lemma 4.1 if and only if there exists $a\in\mathcal{K}$ such that $(\omega^{2}\pi^{2^{k}}\cdot\pi^{s},\pi^{s+1}b)=a(\pi^{s+1}b,\pi^{s})=(\pi^{s+1}ab,\pi^{s}a)$, i.e., $\omega^{2}\pi^{2^{k}+s}=\pi^{s+1}ab$ and $\pi^{s+1}b=\pi^{s}a$. These conditions are simplified to that $b\in\mathcal{K}/\langle\pi^{2^{k}\lambda-1-s}\rangle$ satisfying $\omega^{2}\pi^{2^{k}+s}=(\pi^{s}a)\pi b=\pi^{s+2}b^{2}$. As $\omega$ is an invertible element of $\mathcal{K}$ by Lemma 2.4(i), we can set $z=\omega^{-1}b\in\mathcal{K}/\langle\pi^{2^{k}\lambda-1-s}\rangle$. Then the Condition (4) is equivalent to

[TABLE]

Then we have one of the following subcases:

Subcase 2.1. If $0\leq s\leq 2^{k}(\lambda-1)-1$, we have $2^{k}\leq 2^{k}+s\leq 2^{k}\lambda-1$. Now, by Lemma 2.3(ii) we may assume $z=\sum_{i=0}^{2^{k}\lambda-2-s}z_{i}\pi^{i}\in\mathcal{K}/\langle\pi^{2^{k}\lambda-1-s}\rangle$ where $z_{i}\in\mathcal{F}$ for all $i=0,1,\ldots,2^{k}\lambda-2-s$. In the the ring $\mathcal{K}$ we have

[TABLE]

Hence $b=\omega z$ satisfies Condition (5) if and only if the elements $z_{i}\in\mathcal{F}$ satisfying:

1. $\diamond$

$z_{0}^{2}+z_{1}^{2}\pi^{2}+\ldots+z_{2^{k-1}-2}^{2}\pi^{2^{k}-4}\equiv 0$ (mod $\pi^{2^{k}-2}$), i.e. $(z_{0}+z_{1}\pi+\ldots+z_{2^{k-1}-1}\pi^{2^{k-1}-2})^{2}\equiv 0$ (mod $\pi^{2^{k}-2}$) in $\mathcal{K}$. The letter is equivalent to that

$z_{i}=0$ for all integer $i$: $0\leq i\leq 2^{k-1}-2$. 2. $\diamond$

$z_{2^{k-1}-1}=1$. 3. $\diamond$

$z_{i}=0$, if $2^{k}+s<s+2+2i<2^{k}\lambda$, i.e., $2^{k-1}\leq i\leq 2^{k-1}\lambda-2-\lfloor\frac{s}{2}\rfloor$.

As stated above, we conclude that

[TABLE]

By $\lceil\frac{s}{2}\rceil+\lfloor\frac{s}{2}\rfloor=s$, we have $(s+1)+(2^{k-1}\lambda-1-\lfloor\frac{s}{2}\rfloor)=2^{k-1}\lambda+\lceil\frac{s}{2}\rceil$ and

[TABLE]

From these, we deduce

[TABLE]

where $h(x)=\sum_{i=0}^{2^{k-1}\lambda-\lceil\frac{s}{2}\rceil-1}h_{i}y^{i}=\sum_{i=0}^{2^{k-1}\lambda-\lceil\frac{s}{2}\rceil-1}h_{i}f(x)^{i}\in\frac{\mathbb{F}_{2^{m}}[x]}{\langle f(x)^{2^{k-1}\lambda-\lceil\frac{s}{2}\rceil}\rangle}$ with $h_{i}\in\mathcal{F}$ for all $i$.

Subcase 2.2. If $2^{k}(\lambda-1)\leq s\leq 2^{k}\lambda-1$, we have $\pi^{2^{k}+s}=0$ and that the condition (5) is simplified to $\pi^{s+2}b^{2}=0$. The latter condition is equivalent to $s+2+2\|b\|_{\pi}\geq 2^{k}\lambda$, i.e., $\|b\|_{\pi}\geq 2^{k-1}\lambda-1-\lfloor\frac{s}{2}\rfloor$, and hence

[TABLE]

In this case, by $(2^{k}\lambda-1-s)-(2^{k-1}\lambda-1-\lfloor\frac{s}{2}\rfloor-1)=2^{k-1}\lambda-\lceil\frac{s}{2}\rceil$ we have

[TABLE]

Now, the conclusions in (I) follows from $|\frac{\mathbb{F}_{2^{m}}[x]}{\langle f(x)^{2^{k-1}\lambda-\lceil\frac{s}{2}\rceil}\rangle}|=2^{md_{j}(2^{k-1}\lambda-\lceil\frac{s}{2}\rceil)}$.

Case 3. Let $G$ be given by Lemma 4.2(v). Then it is obvious that every submodule $S$ satisfies Condition (4) in Lemma 4.1.

Case 4. Let $S$ be generated by matrix $G$ where $G$ is given by Lemma 4.2 (vi) and (vii), i.e., G=\left(\begin{array}[]{cc}\pi^{s}&\pi^{s}c\cr 0&\pi^{s+t}\end{array}\right), $c\in\mathcal{K}/\langle\pi^{t}\rangle$, $1\leq t\leq 2^{k}\lambda-1-s$ and $0\leq s\leq 2^{k}\lambda-2$. Suppose that $S$ satisfies Condition (4) in Lemma 4.1. Then $(\omega^{2}\pi^{2^{k}}\cdot\pi^{s}c,\pi^{s})\in S$. Hence there exist $a,b\in\mathcal{K}$ such that

[TABLE]

which is equivalent to $\omega^{2}\pi^{2^{k}+s}c=\pi^{s}a$ and $\pi^{s}=\pi^{s}ac+\pi^{s+t}b$. These imply

[TABLE]

and we get a contradiction.

Case 5. Let $G$ be given by Lemma 4.2 (viii) and (ix). Then we have G=\left(\begin{array}[]{cc}\pi^{s}c&\pi^{s}\cr\pi^{s+t}&0\end{array}\right), $c\in\pi(\mathcal{K}/\langle\pi^{t}\rangle)$, $1\leq t\leq 2^{k}\lambda-1-s$ and $0\leq s\leq 2^{k}\lambda-2$. It is clear that the conditions $1\leq t\leq 2^{k}\lambda-1-s$ and $0\leq s\leq 2^{k}\lambda-2$ is equivalent to

[TABLE]

Obviously, we have $(\omega^{2}\pi^{2^{k}}\cdot 0,\pi^{s+t})=\pi^{t}(\pi^{s}c,\pi^{s})-c(\pi^{s+t},0)\in S$. Hence $S$ satisfies Condition (4) if and only if there exist $a,b\in\mathcal{K}$ such that

[TABLE]

which is equivalent to $\omega^{2}\pi^{2^{k}+s}=\pi^{s}ac+\pi^{s+t}b$ and $\pi^{s}c=\pi^{s}a$. These conditions are simplified to

[TABLE]

Then we have one of the following four subcases:

Subcase 5.1. If $t=1$, we have $\pi(\mathcal{A}/\langle\pi^{t}\rangle)=\{0\}$. In this case, it is obvious that $c=0$ satisfies $\omega^{2}\pi^{2^{k}+s}=\pi^{s}\cdot 0^{2}+\pi^{s+t}b$ where $b=\omega^{2}\pi^{2^{k}-t}$. Then we have $2^{k+1}-1$ matrices satisfing Condition (1) in Lemma 2.2:

[TABLE]

Subcase 5.2. If $t=2$, we have $\pi(\mathcal{K}/\langle\pi^{t}\rangle)=\{c_{1}\pi\mid c_{1}\in\mathcal{F}\}$. In this case,

[TABLE]

Then we have $|\mathcal{F}|=2^{md_{j}}$ matrices: G=\left(\begin{array}[]{cc}\pi^{s+1}c_{1}&\pi^{s}\cr\pi^{s+2}&0\end{array}\right), where $c_{1}\in\mathcal{F}$ arbitrary.

In the following we assume $t\geq 3$. Set $g=\omega^{-1}c\in\pi(\mathcal{K}/\langle\pi^{t}\rangle)$. Then $c=\omega g$ and Equation (7) is equivalent to

[TABLE]

Now, let $g=\sum_{i=1}^{t-1}g_{i}\pi^{i}\in\pi(\mathcal{K}/\langle\pi^{t}\rangle)$ where $g_{i}\in\mathcal{F}$ for all $i=1,\ldots,t-1$, and $b=\sum_{j=0}^{2^{k}\lambda-1}b_{j}\pi^{j}$ where $b_{j}\in\mathcal{F}$ for all $j$. Then in the the ring $\mathcal{K}$, we have

[TABLE]

Subcase 5.3. Let $t=2l+1\leq 2^{k}\lambda-1$ where $1\leq l\leq 2^{k-1}\lambda-1$. Then $l=\lfloor\frac{t}{2}\rfloor$ and $l+1=\lceil\frac{t}{2}\rceil$. We have two situations:

1. (${\dagger}$)

When $1\leq l\leq 2^{k-1}-1$, we have $3\leq t\leq 2^{k}-1$. In this case, the element $g\in\pi(\mathcal{K}/\langle\pi^{t}\rangle)$ satisfies Condition (8) if and only if

[TABLE]

i.e. $g=\sum_{i=l+1}^{t-1}g_{i}\pi^{i}$, where $g_{l+1},\ldots,g_{t-1}\in\mathcal{F}$ arbitrary. In fact, we have $\pi^{2^{k}+s}=\pi^{s}g^{2}+\pi^{s+t}b\ {\rm with}\ b=\pi^{2^{k}-t}+\sum_{i=l+1}^{t-1}g_{i}^{2}\pi^{2i-t}\in\mathcal{K}.$

As stated above, we have $(t-1)-(l+1)=t-\lceil\frac{t}{2}\rceil-1=\lfloor\frac{t}{2}\rfloor-1$ and

[TABLE]

where $h(x)=\sum_{i=l+1}^{t-1}c_{i}\pi^{i-(l+1)}=\sum_{j=0}^{\lfloor\frac{t}{2}\rfloor-1}c_{j+\lceil\frac{t}{2}\rceil}f(x)^{j}$. Hence there are $2^{md_{j}\lfloor\frac{t}{2}\rfloor}$ matrices satisfying Condition (4) in Lemma 4.1:

[TABLE] 2. (${\ddagger}$)

When $2^{k-1}\leq l\leq 2^{k-1}\lambda-1$, we have $2^{k}+1\leq t=2l+1\leq 2^{k}\lambda-1$. This implies $s+t\geq 2^{k}+s+1$. In this case, the element $c\in\pi(\mathcal{K}/\langle\pi^{t}\rangle)$ satisfies Condition (7) if and only if

[TABLE]

In fact, we have $\omega^{2}\pi^{2^{k}+s}=\pi^{s}c^{2}+\pi^{s+t}b$ where $b=\sum_{i=l+1}^{t-1}c_{i}^{2}\pi^{2i-t}\in\mathcal{K}$. Hence there are $2^{md_{j}\lfloor\frac{t}{2}\rfloor}$ matrices satisfying Condition (4) in Lemma 4.1:

[TABLE]

Subcase 5.4. Let $t=2l\leq 2^{k}\lambda-1$ where $2\leq l\leq 2^{k-1}\lambda-1$. Then $l=\lfloor\frac{t}{2}\rfloor=\lceil\frac{t}{2}\rceil$. We have two situations:

1. (${\dagger}$)

When $2\leq l\leq 2^{k-1}$, we have $4\leq t\leq 2^{k}$. In this case, the element $c\in y(\mathcal{A}/\langle y^{t}\rangle)$ satisfies Condition (7) if and only if

[TABLE]

In fact, for any $c_{l},\ldots,c_{t-1}\in\mathcal{F}$ we have

[TABLE]

As stated above, we have $(t-1)-l=l-1=\lfloor\frac{t}{2}\rfloor-1$ and

[TABLE]

Hence there are $2^{md_{j}\lfloor\frac{t}{2}\rfloor}$ matrices satisfying Condition (4) in Lemma 4.1 given by Equation (9). 2. (${\ddagger}$)

When $2^{k-1}+1\leq l\leq 2^{k-1}\lambda-1$, we have $2^{k}+2\leq t=2l\leq 2^{k}\lambda-2$. This implies $s+t\geq 2^{k}+s+2$. In this case, the element $c\in\pi(\mathcal{K}/\langle\pi^{t}\rangle)$ satisfies Condition (7) if and only if

[TABLE]

In fact, we have $\omega^{2}\pi^{2^{k}+s}=\pi^{s}c^{2}+\pi^{s+t}b$ where $b=\sum_{i=l}^{t-1}c_{i}^{2}\pi^{2i-t}\in\mathcal{K}$. Hence there are $2^{md_{j}\lfloor\frac{t}{2}\rfloor}$ matrices satisfying Condition (4) in Lemma 4.1 given by Equation (10).

As stated above, the number of matrices satisfying Condition (4) in Case (viii) and (ix) of Lemma 4.2 is equal to

[TABLE]

by Equation (6). $\Box$

Finally, we give a proof for Theorem 2.5 as follows.

Let $\mathcal{C}$ be a $(\delta+\alpha u^{2})$-constacyclic code over $R$ of length $2^{k}n$. Here we regard $\mathcal{C}$ as an ideal of the ring $\mathcal{A}+u\mathcal{A}$. Then by Lemma 2.4(ii) for each integer $j$, $1\leq j\leq r$, there is a unique ideal $C_{j}$ of the ring $\mathcal{K}_{j}+u\mathcal{K}_{j}$ such that

[TABLE]

This implies $|\mathcal{C}|=\prod_{j=1}^{r}|C_{j}|$. Furthermore, since $C_{j}$ is an ideal of the ring $\mathcal{K}_{j}+u\mathcal{K}_{j}$ $(u^{2}=\omega_{j}^{2}f_{j}(x)^{2^{k}})$, by Lemma 4.1 there is a unique $\mathcal{K}_{j}$-submodule $S$ of $\mathcal{K}_{j}^{2}$ satisfying Condition (4) such that $C_{j}=\theta(S)$. From this and by Lemma 4.4, we deduce that $S$ has one and only one of the following matrices $G$ as its generator matrix:

1. $G$ is given by Lemma 4.4(I-1). In this case, by the definition of $\theta$ above Lemma 4.1 we have

[TABLE]

This implies $C_{j}=\theta(S)=\langle\omega_{j}y^{2^{k-1}+s}+f_{j}(x)^{2^{k-1}\lambda+\lceil\frac{s}{2}\rceil}h(x)+uf_{j}(x)^{s}\rangle.$

As $\|(\omega_{j}f_{j}(x)^{2^{k-1}+s}+f_{j}(x)^{2^{k-1}\lambda+\lceil\frac{s}{2}\rceil}h(x),f_{j}(x)^{s})\|_{f_{j}(x)}=s$, by Lemma 4.3 it follows that $|C_{j}|=|S|=2^{md_{j}(2^{k}\lambda-s)}$.

2. $G$ is given by Lemma 4.4(I-2). In this case, an argument similar to (i-1) shows that $C_{j}=\theta(S)=\langle f_{j}(x)^{2^{k-1}\lambda+\lceil\frac{s}{2}\rceil}h(x)+uf_{j}(x)^{s}\rangle$ and $|C_{j}|=|S|=2^{md_{j}(2^{k}\lambda-s)}$.

3. $G$ is given by Lemma 4.4(II). In this case, we have

[TABLE]

Then by Lemma 4.3 and $\|(f_{j}(x)^{s},0)\|_{f_{j}(x)}=\|(0,f_{j}(x)^{s})\|_{f_{j}(x)}=s$, we obtain $|C_{j}|=|S|=2^{md_{j}((2^{k}\lambda-s)+(2^{k}\lambda-s))}=2^{md_{j}(2^{k+1}\lambda-2s)}$.

4. $G$ is given by Lemma 4.4(III-1). In this case, we have

[TABLE]

Then by Lemma 2.4, $\|(0,f_{j}(x)^{s})\|_{f_{j}(x)}=s$ and $\|(f_{j}(x)^{s+1},0)\|_{f_{j}(x)}=s+1$, we obtain $|C_{j}|=|S|=2^{md_{j}((2^{k}\lambda-s)+(2^{k}\lambda-s-1))}=2^{md_{j}(2^{k+1}\lambda-2s-1)}$.

5. $G$ is given by Lemma 4.4(III-2). In this case, an argument similar to (i) shows that

[TABLE]

Then by Lemma 4.3, we obtain $|C_{j}|=|S|=2^{md_{j}((2^{k}\lambda-s)+(2^{k}\lambda-s-t))}=2^{md_{j}(2^{k+1}\lambda-2s-t)}$, since$\|(f_{j}(x)^{s+\lceil\frac{t}{2}\rceil}h(x),f_{j}(x)^{s})\|_{f_{j}(x)}=s$ and $\|(f_{j}(x)^{s+t}$, $0)\|_{f_{j}(x)}=s+t$.

6. $G$ is given by Lemma 4.4(III-3). Then an argument similar to (v) shows that

[TABLE]

and $|C_{j}|=|S|=2^{md_{j}(2^{k+1}\lambda-2s-t)}$.

Finally, by Lemmas 4.1 and 4.4 we see that the number of ideals in the ring $\mathcal{K}_{j}+u\mathcal{K}_{j}$

[TABLE]

This implies $N_{(2^{m},d_{j},2^{k}\lambda)}=\sum_{i=0}^{2^{k-1}\lambda}(1+4i)2^{md_{j}(2^{k-1}\lambda-i)}$. Therefore, Theorem 2.5 was proved.

5 Conclusions and further research

We give an explicit representation and enumeration for all distinct $(\delta+\alpha u^{2})$-constacyclic codes over $\frac{\mathbb{F}_{2^{m}}[u]}{\langle u^{2\lambda}\rangle}$ of length $2^{k}n$ for any integers $k,\lambda\geq 2$ and odd positive integer $n$, provide a clear and exact formula to count the number of this class of constacyclic codes and obtain a clear formula to count the number of codewords in each code from its generators directly.

Our further interest is to consider the construction of self-dual $(1+\alpha u^{2})$-constacyclic codes over $\frac{\mathbb{F}_{2^{m}}[u]}{\langle u^{4}\rangle}$ of arbitrary even length and study the properties of self-dual codes over $\mathbb{F}_{2^{m}}$ derived from these codes.

Acknowledgments Part of this work was done when Yonglin Cao was visiting Chern 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, 11801324), the Shandong Provincial Natural Science Foundation, China (Grant No. ZR2018BA007), the Scientific Research Fund of Hubei Provincial Key Laboratory of Applied Mathematics (Hubei University) (Grant No. AM201804) and the Scientific Research Fund of Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (No. 2018MMAEZD09). H.Q. Dinh is grateful for the Centre of Excellence in Econometrics, Chiang Mai University, Thailand, for partial financial support.