Codes over the ring $R=\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2$ were introduced in [1]. The ring $R$ is a characteristic 2 ring of size 16. It turns out to be a commutative, non-chain, and a local Frobenius ring constructed subject to $u^2=v^2=0, uv=vu$. It can be viewed as a natural extension of the ring $\mathbb{F}_2+u\mathbb{F}_2, $ which was studied quite extensively in [2] and [3]. In particalar, the ring $\mathbb F_{2}+u\mathbb F_{2}$ is interesting because it shares some good properties of both $\mathbb Z_{4}$ and Galois field $\mathbb F_{4}$. $(1+u)$-constacyclic codes over $\mathbb F_{2}+u\mathbb F_{2}$ of odd length were first introduced by Qian et al. in [4], where it proved that the Gray image of a linear $(1+u)$-constacyclic code over $\mathbb F_{2}+u\mathbb F_{2}$ of odd length is a binary distance invariant linear cyclic code. Recently, Yildiz and Karadeniz in [5, 6] studied constacyclic codes of odd length over $R$. They found some good binary codes as the Gray images of these cyclic codes. The authors in [7] considered the more general ring $\frac{\mathbb{F}_2[u_1, u_2, \ldots, u_k]}{\langle u_i^2, u_j^2, u_iu_j-u_ju_i\rangle}$, and studied the general properties of cyclic codes over these rings and characterized the nontrivial one-generator cyclic codes. Kemat et al. in [8] extended these studies to cyclic codes over the ring $\frac{\mathbb{Z}_p[u, v]}{\langle u^2, v^2, uv-vu\rangle}=\mathbb{Z}_p+u\mathbb{Z}_p+v\mathbb{Z}_p+uv\mathbb{Z}_p$, where $u^2=0, v^2=0, uv=vu$ and $p$ is a prime number.

In this paper, we study repeated-root $\lambda$-constacyclic codes of length $2^s$ over $R$, where $\lambda$ is unit of $R$. Although repeated-root $\lambda$-constacyclic codes over finite ring $\tilde{R}$ are known to be asymptotically bad, they are optimal in a few cases. They motivated the researchers to further study (see [9-15]).

The paper is organized as follows. In Section 2, we recall some notations and properties about constacyclic codes over finite local rings. In Section 3, we address the cyclic and $(1+uv)$-constacyclic codes of length $2^s$ over $R$. We classify all such cyclic and $(1+uv)$-constacyclic codes by categorizing the ideals of the local ring $R_1=\frac{R[x]}{\langle x^{2^s}-1\rangle }$ and $R_2=\frac{R[x]}{\langle x^{2^s}-(1+uv)\rangle }$ into 13 types. In the last section, we study the $(1+u)$-constacyclic codes of length $2^s$ over $R$. These $(1+u)$-constacyclic codes are the ideals of the ring $R_3=\frac{R[x]}{\langle x^{2^s}-(1+u)\rangle }$, which is a local ring with the maximal ideal $\langle x-1, v\rangle $. We classify all $(1+u)$-constacyclic codes by categorizing the ideals of the local ring $R_3$ into 4 type, and provide a detailed structure of ideal in each type. By using similar discussion of $(1+u)$-constacyclic codes, we obtain the classification and the structure of $(1+v), (1+u+uv), (1+v+uv), (1+u+v), (1+u+v+uv)$-constacyclic codes of length $2^s$ over $R$.

A code of length $n$ over $R$ is a nonempty subset of $R^n$, and a linear code $C$ of length $n$ over $R$ is an $R$-submodule of $R^n$. If $\lambda$ is a unit in $R$, a linear code $C$ is called as $\lambda$-constacyclic if $(\lambda a_{n-1}, a_0, \cdots, a_{n-2})\in C$ for every $(a_0, a_1, \cdots, a_{n-1})\in C$. It is well known that a $\lambda$-constacyclic code of length $n$ over $R$ can be identified as an ideal in the residue ring $\frac{R[x]}{\langle x^n-\lambda\rangle }$ via the $R$-module isomorphism $\varphi:R^n\rightarrow \frac{R[x]}{\langle x^n-\lambda\rangle }$ given by

$ (a_0, a_1, \cdots, a_{n-1})\mapsto a_0+a_1x+\cdots+a_{n-1}x^{n-1}({\rm mod}(x^n-\lambda)). $

If $\lambda=1$, $\lambda$-constacyclic codes are just cyclic codes and if $\lambda=-1$, $\lambda$-constacyclic codes are known as negacyclic codes. A polynomial is said to be regular if it is not a zero divisor. The following version of the Euclidean algorithm holds true for polynomials over finite commutative local rings.

Proposition 2.1 (see [16, Example, Ⅲ.6]) Let $\tilde{R}$ be a finite commutative local ring, and $f, g$ be nonzero polynomials in $\tilde{R}[x]$. If $g$ is regular, then there exist polynomials $q(x), r(x)\in \tilde{R}[x]$ such that $f(x)=q(x)g(x)+r(x)$ and $r(x)=0$ or $\deg(r(x))<\deg(g(x))$.

Cyclic codes of length $2^s$ over $R$ are ideals of the residue ring $R_1=\frac{R[x]}{\langle x^{2^s}-1\rangle }$, and the $(1+uv)$-constacyclic codes of length $2^s$ over $R$ are ideals of the residue ring $R_2=\frac{R[x]}{\langle x^{2^s}-(1+uv)\rangle }$. It is easy to prove the following three lemmas.

Lemma 3.1 The following hold true in $R_1$:

(ⅰ) For any nonegative integer $t$, $(x-1)^{2^t}=x^{2^t}-1$.

(ⅱ) $x-1$ is nilpotent with the nilpotency index $2^s$.

Lemma 3.2 The following hold true in $R_2$.

(ⅰ) For any nonnegative integer $t$, $(x-1)^{2^t}=x^{2^t}-1$. In particular, $(x-1)^{2^s}=uv$.

(ⅱ) $x-1$ is nilpotent with the nilpotency index $2^{s+1}$.

Lemma 3.3 Let $f(x)\in R_1$ or $R_2$. Then $f(x)$ can be uniquely expressed as

$ \begin{aligned} f(x)=&\sum\limits_{j=0}^{2^s-1}a_{0j}(x-1)^j+u\sum\limits_{j=0}^{2^s-1}a_{1j}(x-1)^j+v\sum\limits_{j=0}^{2^s-1}a_{2j}(x-1)^j+uv\sum\limits_{j=0}^{2^s-1}a_{3j}(x-1)^j\\ =&a_{00}+(x-1)\sum\limits_{j=1}^{2^s-1}a_{0j}(x-1)^{j-1}+u\sum\limits_{j=0}^{2^s-1}a_{1j}(x-1)^j+v\sum\limits_{j=0}^{2^s-1}a_{2j}(x-1)^j\\ &+uv\sum\limits_{j=0}^{2^s-1}a_{3j}(x-1)^j, \end{aligned} $

where $a_{0j}, a_{1j}, a_{2j}, a_{3j}\in \mathbb{F}_2$. Furthermore, $f(x)$ is invertible if and only if $a_{00}\neq0$.

Proposition 3.4 The ring $R_1$ or $R_2$ is a local ring with the maximal ideal $\langle u, v, x-1\rangle$, but it is not a chain ring.

Proof By Lemma 3.3, the ideal $\langle u, v, x-1\rangle $ is the set of all non-invertible elements of $R_1$ or $R_2$. Hence $R_1$ or $R_2$ is a local ring with maximal ideal $\langle u, v, x-1\rangle $. Suppose $u\in \langle x-1\rangle .$ Then there must exist $f_1(x)$ and $f_2(x)\in R[x]$ such that $u=(x-1)f_1(x)+(x^{2^s}-1)f_2(x)$ or $u=(x-1)f_1(x)+[x^{2^s}-(1+uv)]f_2(x)$. However, this is impossible because plugging in $x=1$ yields $u=0$ or $u=uv$. Hence, $u\notin \langle x-1\rangle $. Similarly, $v\notin \langle x-1\rangle $. Next we suppose $x-1\in \langle u\rangle $ or $\langle v\rangle $. Then $(x-1)^2=0$, which is a contradiction with $s>1$. Therefore, the maximal ideal $\langle u, v, x-1\rangle$ of $R_1$ or $R_2$ is not principal. It means $R_1$ or $R_2$ is not a chain ring.

Theorem 3.5 All of cyclic codes of length $2^s$ over $R$, i.e., ideals of the ring $R_1$ are the following:

● Type 1: $\langle 0\rangle, \langle 1\rangle $;

● Type 2: $I=\langle uv(x-1)^\sigma\rangle $, where $0\leq \sigma \leq 2^s-1$;

● Type 3: $I=\langle v(x-1)^t+uv\sum\limits_{j=0}^{t} m_{1j}(x-1)^j\rangle $, where $0\leq t \leq 2^s-1$;

● Type 4: $I=\langle v(x-1)^t+uv\sum\limits_{j=0}^{t-1} m_{1j}(x-1)^j, uv(x-1)^z\rangle $, where $1\leq t \leq 2^s-1, z<t$;

● Type 5: $I=\langle u(x-1)^l+v\sum\limits_{j=0}^{2^s-1}h_{2j}(x-1)^j+uv\sum\limits_{j=0}^{l}h_{3j}(x-1)^j\rangle $, where $0\leq l \leq 2^s-1$;

● Type 6: $I=\langle u(x-1)^l+v\sum\limits_{j=0}^{2^s-1}h_{2j}(x-1)^j+uv\sum\limits_{j=0}^{l-1}h_{3j}(x-1)^j, uv(x-1)^t\rangle $, where $1\leq l \leq 2^s-1, t<l$;

● Type 7: $I=\langle u(x-1)^l+v\sum\limits_{j=0}^{2^s-1}h_{2j}(x-1)^j+uv\sum\limits_{j=0}^{l}h_{3j}(x-1)^j, v(x-1)^z+uv\sum\limits_{j=0}^{z}b_{3j}(x-1)^j\rangle $, where $0\leq l \leq 2^s-1, 0 \leq z \leq 2^s-1$;

● Type 8: $I=\langle u(x-1)^l+v\sum\limits_{j=0}^{z-1}h_{2j}(x-1)^j+uv\sum\limits_{j=0}^{z-1}h_{3j}(x-1)^j,$ $v(x-1)^z+uv\sum\limits_{j=0}^{z-1}b_{3j}(x-1)^j, uv(x-1)^w\rangle, $where $1\leq l \leq 2^s-1, 1 \leq z \leq 2^s-1, $ and $w< \min \{l, z\}$;

● Type 9: $I=\langle (x-1)^i+u\sum\limits_{j=0}^{i}a_{1j}(x-1)^j+v\sum\limits_{j=0}^{i}a_{2j}(x-1)^j+uv\sum\limits_{j=0}^{i}a_{3j}(x-1)^j\rangle $, where $0\leq i\leq 2^s-1$;

● Type 10: $ I=\langle (x-1)^i+u\sum\limits_{j=0}^{i}a_{1j}(x-1)^j+v\sum\limits_{j=0}^{i}a_{2j}(x-1)^j+uv\sum\limits_{j=0}^{t}a_{3j}(x-1)^j, u(x-1)^l+$$v\sum\limits_{j=0}^{i}h_{2j}(x-1)^j+uv\sum\limits_{j=0}^{t}h_{3j}(x-1)^j\rangle $, where $0\leq i \leq 2^s-1, 0\leq l \leq 2^s-1$ and $t\leq \min\{i, l\}$;

● Type 11: $ I=\langle (x-1)^i+u\sum\limits_{j=0}^{i}a_{1j}(x)^j+v\sum\limits_{j=0}^{i}a_{2j}(x-1)^j+uv\sum\limits_{j=0}^{t}a_{3j}(x-1)^j, u(x-1)^l+$$v\sum\limits_{j=0}^{i}h_{2j}(x-1)^j+uv\sum\limits_{j=0}^{t}h_{3j}(x-1)^j, uv(x-1)^t\rangle $, where $0\leq i \leq 2^s-1, 0\leq l \leq 2^s-1$ and $t\leq \min\{i, l\}$;

● Type 12: $I=\langle (x-1)^i+u\sum\limits_{j=0}^{i}a_{1j}(x-1)^j+v\sum\limits_{j=0}^{i}a_{2j}(x-1)^j+uv\sum\limits_{j=0}^{t}a_{3j}(x-1)^j, u(x-1)^l+$$v\sum\limits_{j=0}^{i}h_{2j}(x-1)^j+uv\sum\limits_{j=0}^{t}h_{3j}(x-1)^j, v(x-1)^z+uv\sum\limits_{j=0}^{t}b_{3j}(x-1)^j\rangle, $ where $0\leq i \leq 2^s-1, 0\leq l \leq 2^s-1, 0\leq z \leq 2^s-1$ and $t\leq \min\{i, l, z\}$;

● Type 13: $ I=\langle (x-1)^i+u\sum\limits_{j=0}^{i}a_{1j}(x-1)^j+v\sum\limits_{j=0}^{i}a_{2j}(x-1)^j+uv\sum\limits_{j=0}^{w}a_{3j}(x-1)^j, u(x-1)^l+$$v\sum\limits_{j=0}^{i}h_{2j}(x-1)^j+uv\sum\limits_{j=0}^{w}h_{3j}(x-1)^j, v(x-1)^z+uv\sum\limits_{j=0}^{w}b_{3j}(x-1)^j, uv(x-1)^w\rangle $, where $0\leq i \leq 2^s-1, 0\leq l \leq 2^s-1, 0\leq z \leq 2^s-1$ and $w\leq \min\{i, l, z\}$.

Proof Ideals of Type 1 are the trivial ideals. Consider an arbitrary nontrivial ideals of $R_1$.

Case 1 $I\subseteq \langle v\rangle $. Any element of $I$ must have the form $v\sum\limits_{j=0}^{2^s-1}a_{0j}(x-1)^j+uv\sum\limits_{j=0}^{2^s-1}a_{1j}(x-1)^j$, where $a_{0j}, a_{1j}\in \mathbb{F}_2$.

Let

$ M=\{v\sum\limits_{j=0}^{2^s-1}a_{0j}(x-1)^j+uv\sum\limits_{j=0}^{2^s-1}a_{1j}(x-1)^j\in I|\sum\limits_{j=0}^{2^s-1}a_{0j}(x-1)^j\neq 0\} $

and $N=\{uv\sum\limits_{j=0}^{2^s-1}a_{1j}(x-1)^j\in I\}$. Suppose $M=\Phi$. Then there exists the smallest integer $\sigma$ such that $n(x)=uv(x-1)^\sigma n_1(x)$ for $n(x)\in N$, where $n_1(x)\in R_1$. It is easy to verify that $uv(x-1)^\sigma\in N$. Hence $I=\langle uv(x-1)^\sigma\rangle $, and $I$ is in Type 2.

Suppose $M\neq \Phi$. Setting $\alpha=\min\{\deg (m(x))|m(x)\in M\}$. Then there is an element $m_1(x)=v\sum m_{0j}(x-1)^j+uv\sum m_{1j}(x-1)^j\in M$ with $\deg (m_1(x))=\alpha$. It has the smallest $t$ such that $m_{0t}\neq 0$. Hence we have

$ m_1(x)=v(x-1)^t[m_{0t}+\sum\limits_{j=t+1}^{2^s-1}m_{0j}(x-1)^{j-t}]+uv\sum\limits_{j=0}^{2^s-1} m_{1j}(x-1)^j\in I. $

Let

$ m_2(x)=(x-1)^t[m_{0t}+\sum\limits_{j=t+1}^{2^s-1}m_{0j}(x-1)^{j-t}]+u\sum\limits_{j=0}^{2^s-1} m_{1j}(x-1)^j. $

Then $m_1(t)=vm_2(t).$

Now we have two subcases.

Case 1.1 $N\subseteq \langle m_1(x)\rangle $. For any $f(x)\in M$, obviously, $f(x)$ can be written as $f(x)=vf_1(x)$, where $f_1(x)=\sum\limits_{j=0}^{2^s-1}a_{0j}(x-1)^j+u\sum\limits_{j=0}^{2^s-1}a_{1j}(x-1)^j$. By Proposition 2.1, $f_1(x)$ can be written as $f_1(x)=q(x)m_2(x)+r(x)$, where $q(x), r(x)\in R_1$ and $r(x)=0$ or $\deg(r(x))<\deg(m_2(x))=\deg(m_1(x))$. It implies that $f(x)=q(x)m_1(x)+vr(x)$. Suppose $vr(x)\notin N.$ Then $vr(x)\neq0$. Hence $vr(x)=f(x)-q(x)m_1(x)\in M, $ which contradicts with the assumption of $m_1(x)$. Thus $vr(x)\in N$. Therefore, $I=\langle v(x-1)^t+uv\sum\limits_{j=0}^{t} m_{1j}(x-1)^j\rangle, $ $0\leq t \leq 2^s-1$. Thus $I$ is in Type 3.

Case 1.2 $N\nsubseteq \langle m_1(x)\rangle $. Then there exists the smallest integer $z$ such that $n(x)=uv(x-1)^z n_1(x)$ for every $n(x)\in N$, where $n_1(x)\in R_1$. It is easy to verify that $uv(x-1)^z\in N$, but $uv(x-1)^z\notin \langle m_1(x)\rangle $, and $z<t$. Hence

$ I=\langle v(x-1)^t+uv\sum\limits_{j=0}^{t-1} m_{1j}(x-1)^j, uv(x-1)^z\rangle, 1\leq t \leq 2^s-1, 0\leq z < t. $

Therefore, $I$ is in Type 4.

Case 2 $\langle v\rangle \varsubsetneq I \subseteq \langle u, v\rangle $. Any element of $I$ must have the form

$ u\sum\limits_{j=0}^{2^s-1}a_{1j}(x-1)^j+v\sum\limits_{j=0}^{2^s-1}a_{2j}(x-1)^j+uv\sum\limits_{j=0}^{2^s-1}a_{3j}(x-1)^j, $

and there exists an element $u\sum\limits_{j=0}^{2^s-1}b_{1j}(x-1)^j+v\sum\limits_{j=0}^{2^s-1}b_{2j}(x-1)^j+uv\sum\limits_{j=0}^{2^s-1}b_{3j}(x-1)^j$ in $I$ such that $\sum\limits_{j=0}^{2^s-1}b_{1j}(x-1)^j\neq 0$.

Let

$ \begin{eqnarray*}M&=&\{u\sum\limits_{j=0}^{2^s-1}a_{1j}(x-1)^j+v\sum\limits_{j=0}^{2^s-1}a_{2j}(x-1)^j+uv\sum\limits_{j=0}^{2^s-1}a_{3j}(x-1)^j\in I\\ &&\mid \sum\limits_{j=0}^{2^s-1}a_{1j}(x-1)^j\neq 0, a_{1j}, a_{2j}, a_{3j}\in \mathbb{F}_2\}, \end{eqnarray*} $

and let $N=\{v\sum\limits_{j=0}^{2^s-1}a_{2j}(x-1)^j+uv\sum\limits_{j=0}^{2^s-1}a_{3j}(x-1)^j\in I\mid a_{2j}, a_{3j}\in \mathbb{F}_2\}$. Then there is an element $\tilde{h}_1(x)=u\sum\limits_{j=0}^{2^s-1}\tilde{h}_{1j}(x-1)^j+v\sum\limits_{j=0}^{2^s-1}\tilde{h}_{2j}(x-1)^j+uv\sum\limits_{j=0}^{2^s-1}\tilde{h}_{3j}(x-1)^j$ in $M$ that has the smallest $l$ such that $\tilde{h}_{1l}\neq0$. Hence we have

$ \tilde{h}_1(x)=u(x-1)^l[\tilde{h}_{1l}+\sum\limits_{j=l+1}^{2^s-1}\tilde{h}_{1j}(x-1)^{j-l}]+v\sum\limits_{j=0}^{2^s-1}\tilde{h}_{2j}(x-1)^j+uv\sum\limits_{j=0}^{2^s-1}\tilde{h}_{3j}(x-1)^j\in I. $

Since $\tilde{h}_{1l}\neq0$, $\tilde{h}_{1l}+\sum\limits_{j=l+1}^{2^s-1}\tilde {h}_{1j}(x-1)^{j-l}$ is invertible, and

$ h_2(x)=\tilde{h}_1(x)[\tilde{h}_{1l}+\sum\limits_{j=l+1}^{2^s-1}\tilde {h}_{1j}(x-1)^{j-l}]^{-1}=u(x-1)^l+ v\sum\limits_{j=0}^{2^s-1}h_{2j}(x-1)^j+\\ uv\sum\limits_{j=0}^{2^s-1}h_{3j}(x-1)^j\in I. $

Because $vh_2(x)=uv(x-1)^l\in I$, we have

$ h_1(x)=u(x-1)^l+ v\sum\limits_{j=0}^{2^s-1}h_{2j}(x-1)^j+uv\sum\limits_{j=0}^{l-1}h_{3j}(x-1)^j\in I. $

For any $m(x)\in M$, obviously, $m(x)$ can be written as

$ m(x)=u(x-1)^l\sum\limits_{j=l}^{2^s-1}a_{1j}(x-1)^{j-l}+v\sum\limits_{j=0}^{2^s-1}a_{2j}(x-1)^j+uv\sum\limits_{j=0}^{2^s-1}a_{3j}(x-1)^j, $

where $a_{1j}, a_{2j}, a_{3j}\in \mathbb{F}_2$. Thus we can assume that

$ m(x)-h_1(x)\sum\limits_{j=l}^{2^s-1}a_{1j}(x-1)^{j-l}=v\sum\limits_{j=0}^{2^s-1}a_{2j}(x-1)^j+uv\sum\limits_{j=0}^{2^s-1}a_{3j}(x-1)^j\in N. $

Now we have two subcases.

Case 2.1 $N=\{0\}$. Then $I=\langle u(x-1)^l+v\sum\limits_{j=0}^{2^s-1}h_{2j}(x-1)^j+uv\sum\limits_{j=0}^{l}h_{3j}(x-1)^j\rangle $, where $0\leq l \leq 2^s-1$. Thus, $I$ is in Type 5.

Case 2.2 $N\neq \{0\}$. We denote

$ N_1=\{v\sum\limits_{j=0}^{2^s-1}a_{2j}(x-1)^j+uv\sum\limits_{j=0}^{2^s-1}a_{3j}(x-1)^j\in N\mid \sum\limits_{j=0}^{2^s-1}a_{2j}(x-1)^j\neq0, a_{2j}, a_{3j}\in \mathbb{F}_2\}, $

and $N_2=\{uv\sum\limits_{j=0}^{2^s-1}a_{3j}(x-1)^j\in N\mid a_{3j}\in \mathbb{F}_2\}$ in the following.

Again we have two subcases.

Case 2.2.1 $N_1=\Phi$. Then $N_2\neq \{0\}$. Therefore, there exists the smallest integer $t$ such that $n_2(x)=uv(x-1)^tq(x)$ for every $n_2(x)\in N_2=N$, where $q(x)\in R_1$. It is easy to verify that $uv(x-1)^t\in N_2=N$. Hence,

$ I=\langle u(x-1)^l+v\sum\limits_{j=0}^{2^s-1}h_{2j}(x-1)^j+uv\sum\limits_{j=0}^{l-1}h_{3j}(x-1)^j, uv(x-1)^t\rangle . $

Suppose that $t\geq l.$ Then

$ uv(x-1)^t=v(x-1)^{t-l}[u(x-1)^l+v\sum\limits_{j=0}^{2^s-1}h_{2j}(x-1)^j+uv\sum\limits_{j=0}^{l-1}h_{3j}(x-1)^j]. $

Thus, $I=\langle u(x-1)^l+v\sum\limits_{j=0}^{2^s-1}h_{2j}(x-1)^j+uv\sum\limits_{j=0}^{l-1}h_{3j}(x-1)^j\rangle $, it is in Type 5. We can assume without loss of generality that $t< l$, and thus

$ I=\langle u(x-1)^l+v\sum\limits_{j=0}^{2^s-1}h_{2j}(x-1)^j+uv\sum\limits_{j=0}^{l-1}h_{3j}(x-1)^j, uv(x-1)^t\rangle, $

where $1\leq l \leq 2^s-1$. Therefore, $I$ is in Type 6.

Case 2.2.2 $N_1\neq\Phi$. Then there is an element

$ \tilde{a}_1(x)=v\sum\limits_{j=0}^{2^s-1}\tilde{a}_{2j}(x-1)^j+uv\sum\limits_{j=0}^{2^s-1}\tilde{a}_{3j}(x-1)^j $

in $N_1$ that has smallest $z$ such that $\tilde{a}_{2z}\neq0$. Hence we have

$ \tilde{b}_1(x)=v(x-1)^z[\tilde{b}_{2z}+\sum\limits_{j=z+1}^{2^s-1}\tilde{b}_{2j}(x-1)^{j-z}]+uv\sum\limits_{j=0}^{2^s-1}\tilde{b}_{3j}(x-1)^j\in N_1. $

Thus $b_1(x)=\tilde{b}_1(x)[\tilde{b}_{2z}+\sum\limits_{j=z+1}^{2^s-1}\tilde{b}_{2j}(x-1)^{j-z}]^{-1}\in I$ and $b_1(x)$ can be expressed as $b_1(x)=v(x-1)^z+uv\sum\limits_{j=0}^{z-1}b_{3j}(x-1)^j$.

For any $n_1(x)\in N_1$, obviously, $n_1(x)$ can be written as

$ n_1(x)=v(x-1)^z\sum\limits_{j=z}^{2^s-1}a_{2j}(x-1)^{j-z}+uv\sum\limits_{j=0}^{2^s-1}a_{3j}(x-1)^j, $

where $a_{2j}, a_{3j}\in \mathbb{F}_2$. Thus, we can assume that

$ n_1(x)-b_1(x)\sum\limits_{j=z}^{2^s-1}a_{2j}(x-1)^{j-z}=uv\sum\limits_{j=0}^{2^s-1}a_{3j}(x-1)^j\in I. $

If $N_2=\{0\}, $ then

$ I=\langle u(x-1)^l+v\sum\limits_{j=0}^{2^s-1}h_{2j}(x-1)^j+uv\sum\limits_{j=0}^{l}h_{3j}(x-1)^j, v(x-1)^z+uv\sum\limits_{j=0}^{l}b_{3j}(x-1)^j\rangle, $

where $0\leq l \leq 2^s-1, 0 \leq z \leq 2^s-1$. Thus, $I$ is in Type 7.

If $N_2\neq \{0\}.$ Then there exists the smallest integer $w$ such that $n_2(x)=uv(x-1)^wq(x)$ for every $n_2(x)\in N_2$, where $q(x)\in R_1$. It is easy to verify that $uv(x-1)^w\in I$. Hence,

$ I=\langle u(x-1)^l+v\sum\limits_{j=0}^{z-1}h_{2j}(x-1)^j+uv\sum\limits_{j=0}^{z-1}h_{3j}(x-1)^j, v(x-1)^z+uv\sum\limits_{j=0}^{z-1}b_{3j}(x-1)^j, \\uv(x-1)^w\rangle , $

where $1\leq l \leq 2^s-1, 1 \leq z \leq 2^s-1, $ and $w<\min \{l, z\}$. Thus, $I$ is in Type 8.

Case 3 $I\nsubseteq \langle u, v\rangle $. Let $I_{u, v}=\{f(x)\in \frac{\mathbb{F}_2[x]}{\langle x^{2^s}-1\rangle }\mid $ there are $g(x), h(x), m(x)\in \mathbb{F}_2[x]\diagup \langle x^{2^s}-1\rangle $ such that $f(x)+ug(x)+vh(x)+uvm(x)\in R_1\}$. Then $I_{u, v}$ is a nonzero ideal of the ring $\frac{\mathbb{F}_2[x]}{\langle x^{2^s}-1\rangle }$. It is a chain ring with ideals $\langle (x-1)^j\rangle $, where $0\leq j \leq 2^s$. Hence there is an integer $i\in \{0, 1, \cdots, 2^s-1\}$ such that $I_{u, v}=\langle (x-1)^i\rangle \subseteq \frac{\mathbb{F}_2[x]}{\langle x^{2^s}-1\rangle }.$ Therefore, there are three elements

$ c_i(x)=\sum\limits_{j=0}^{2^s-1}c^{(i)}_{0j}(x-1)^j+u\sum\limits_{j=0}^{2^s-1}c^{(i)}_{1j}(x-1)^j+v\sum\limits_{j=0}^{2^s-1}c^{(i)}_{2j}(x-1)^j+uv\sum\limits_{j=0}^{2^s-1}c^{(i)}_{3j}(x-1)^j\in R_1 $

for $i=1, 2, 3$ such that $(x-1)^i+uc_1(x)+vc_2(x)+uvc_3(x)\in I$, where $c^{(i)}_{0j}, c^{(i)}_{1j}, c^{(i)}_{2j}, c^{(i)}_{3j}\in \mathbb{F}_2$. Because

$ \begin{aligned} &(x-1)^i+uc_1(x)+vc_2(x)+uvc_3(x)\\ =&(x-1)^i+u\sum\limits_{j=0}^{2^s-1}c^{(1)}_{0j}(x-1)^j +v\sum\limits_{j=0}^{2^s-1}c^{(2)}_{0j}(x-1)^j +uv\sum\limits_{j=0}^{2^s-1}(c^{(1)}_{2j}+c^{(2)}_{1j}+c^{(3)}_{0j})(x-1)^j\\ &=(x-1)^i+u\sum\limits_{j=0}^{2^s-1}a_{1j}(x-1)^j+v\sum\limits_{j=0}^{2^s-1}a_{2j}(x-1)^j+uv\sum\limits_{j=0}^{2^s-1}a_{3j}(x-1)^j, \end{aligned} $

where $a_{1j}=c^{(1)}_{0j}, a_{2j}=c^{(2)}_{0j}, a_{3j}=c^{(1)}_{2j}+c^{(2)}_{1j}+c^{(3)}_{0j}, $ and for all $l$ with $i\leq l \leq 2^s-1$,

$ uv(x-1)^l=uv[(x-1)^i+u\sum\limits_{j=0}^{2^s-1}a_{1j}(x-1)^j+v\sum\limits_{j=0}^{2^s-1}a_{2j}(x-1)^j+\\ uv\sum\limits_{j=0}^{2^s-1}a_{3j}(x-1)^j](x-1)^{l-i}\in I. $

It follows that

$ (x-1)^i+u\sum\limits_{j=0}^{2^s-1}a_{1j}(x-1)^j+v\sum\limits_{j=0}^{2^s-1}a_{2j}(x-1)^j+uv\sum\limits_{j=0}^{l-1}a_{3j}(x-1)^j\in I. $

Hence it can be assumed without loss of generality that

$ a(x)=(x-1)^i+u\sum\limits_{j=0}^{i}a_{1j}(x-1)^j+v\sum\limits_{j=0}^{i}a_{2j}(x-1)^j+uv\sum\limits_{j=0}^{i}a_{3j}(x-1)^j\in I. $

Now we have two subcases.

Case 3.1 $I=\langle (x-1)^i+u\sum\limits_{j=0}^{i}a_{1j}(x-1)^j+v\sum\limits_{j=0}^{i}a_{2j}(x-1)^j+uv\sum\limits_{j=0}^{i}a_{3j}(x-1)^j\rangle $, where $0\leq i \leq 2^s-1$. Then $I$ is in Type 9.

Case 3.2 $\langle (x-1)^i+u\sum\limits_{j=0}^{i}a_{1j}(x-1)^j+$$v\sum\limits_{j=0}^{i}a_{2j}(x-1)^j+uv\sum\limits_{j=0}^{i}a_{3j}(x-1)^j\rangle\varsubsetneq I $. For every $f(x)\in I \backslash \langle (x-1)^i+u\sum\limits_{j=0}^{i}a_{1j}(x-1)^j+$$v\sum\limits_{j=0}^{i}a_{2j}(x-1)^j+uv\sum\limits_{j=0}^{i}a_{3j}(x-1)^j\rangle $, there is an element $g(x)\in R_1$ such that

$ 0\neq b_f(x)=f(x)-g(x)[(x-1)^i+u\sum\limits_{j=0}^{i}a_{1j}(x-1)^j+v\sum\limits_{j=0}^{i}a_{2j}(x-1)^j+\\ uv\sum\limits_{j=0}^{i}a_{3j}(x-1)^j]\in I, $

and $b_f(x)$ can be expressed as

$ b_f(x)=\sum\limits_{j=0}^{i}b_{0j}(x-1)^j+u\sum\limits_{j=0}^{i}b_{1j}(x-1)^j+v\sum\limits_{j=0}^{i}b_{2j}(x-1)^j+uv\sum\limits_{j=0}^{i}b_{3j}(x-1)^j\in I, $

where $b_{0j}, b_{1j}, b_{2j}, b_{3j}\in \mathbb{F}_2.$ Now, by the definition of $I_{u, v}$, we have

$ \sum\limits_{j=0}^{i}b_{0j}(x-1)^j\in I_{u, v}=\langle (x-1)^i\rangle. $

It implies that $b_{0j}$ for all $0\leq j \leq i$, i.e.,

$ b_f(x)=u\sum\limits_{j=0}^{i}b_{1j}(x-1)^j+v\sum\limits_{j=0}^{i}b_{2j}(x-1)^j+uv\sum\limits_{j=0}^{i}b_{3j}(x-1)^j\in I. $

Similarly with Case 2, we have

$ \begin{aligned} I=&\langle (x-1)^i+u\sum\limits_{j=0}^{i}a_{1j}(x-1)^j+v\sum\limits_{j=0}^{i}a_{2j}(x-1)^j+uv\sum\limits_{j=0}^{t}a_{3j}(x-1)^j, \\ &u(x-1)^l+v\sum\limits_{j=0}^{i}h_{2j}(x-1)^j+uv\sum\limits_{j=0}^{t}h_{3j}(x-1)^j\rangle, \end{aligned} $

where $0\leq i \leq 2^s-1, 0\leq l \leq 2^s-1$ and $t\leq \min\{i, l\}$. Therefore, $I$ is in Type 10.

$ \begin{aligned} I=&\langle (x-1)^i+u\sum\limits_{j=0}^{i}a_{1j}(x-1)^j+v\sum\limits_{j=0}^{i}a_{2j}(x-1)^j+uv\sum\limits_{j=0}^{t}a_{3j}(x-1)^j, u(x-1)^l\\ &+v\sum\limits_{j=0}^{i}h_{2j}(x-1)^j+uv\sum\limits_{j=0}^{t}h_{3j}(x-1)^j, uv(x-1)^t\rangle, \end{aligned} $

where $0\leq i \leq 2^s-1, 0\leq l \leq 2^s-1$ and $t\leq \min\{i, l\}$. Thus, $I$ is in Type 11.

$ \begin{aligned} I=&\langle (x-1)^i+u\sum\limits_{j=0}^{i}a_{1j}(x-1)^j+v\sum\limits_{j=0}^{i}a_{2j}(x-1)^j+uv\sum\limits_{j=0}^{t}a_{3j}(x-1)^j, \\ &u(x-1)^l+v\sum\limits_{j=0}^{i}h_{2j}(x-1)^j+uv\sum\limits_{j=0}^{t}h_{3j}(x-1)^j, v(x-1)^z+uv\sum\limits_{j=0}^{t}b_{3j}(x-1)^j\rangle, \end{aligned} $

where $0\leq i \leq 2^s-1, 0\leq l \leq 2^s-1, 0\leq z \leq 2^s-1$ and $t\leq \min \{i, l, z\}$. Thus, $I$ is in Type 12.

$ \begin{aligned} I=&\langle (x-1)^i+u\sum\limits_{j=0}^{i}a_{1j}(x-1)^j+v\sum\limits_{j=0}^{i}a_{2j}(x-1)^j+uv\sum\limits_{j=0}^{w}a_{3j}(x-1)^j, u(x-1)^l\\ &+v\sum\limits_{j=0}^{i}h_{2j}(x-1)^j+uv\sum\limits_{j=0}^{w}h_{3j}(x-1)^j, v(x-1)^z+uv\sum\limits_{j=0}^{w}b_{3j}(x-1)^j, uv(x-1)^w\rangle, \end{aligned} $

where $0\leq i \leq 2^s-1, 0\leq l \leq 2^s-1, 0\leq z \leq 2^s-1$ and $w\leq \min\{i, l, z\}$. Thus, $I$ is in Type 13.

Similar to discussion in Theorem 3.5, and note that $(x-1)^{2^s}=uv$, we have following theorem.

Theorem 3.6 $(1+uv)$-constacyclic codes of length $2^s$ over $R$, i.e., ideals of the ring $R_2$ are

● Type 1: $\langle 0\rangle, \langle 1\rangle $;

● Type 2: $I=\langle uv(x-1)^\sigma\rangle $, where $0\leq \sigma \leq 2^{s}-1$;

● Type 3: $I=\langle v(x-1)^t+uv\sum\limits_{j=0}^{t} m_{1j}(x-1)^j\rangle $, where $0\leq t \leq 2^{s}-1$;

● Type 4: $I=\langle v(x-1)^t+uv\sum\limits_{j=0}^{t-1} m_{1j}(x-1)^j, uv(x-1)^z\rangle $, where $1\leq t \leq 2^{s}-1, z<t$;

● Type 5: $I=\langle u(x-1)^l+v\sum\limits_{j=0}^{2^{s}-1}h_{2j}(x-1)^j+uv\sum\limits_{j=0}^{l}h_{3j}(x-1)^j\rangle $, where $0\leq l \leq 2^{s}-1$;

● Type 6: $I=\langle u(x-1)^l+v\sum\limits_{j=0}^{2^{s}-1}h_{2j}(x-1)^j+uv\sum\limits_{j=0}^{l-1}h_{3j}(x-1)^j, uv(x-1)^t\rangle $, where $1\leq l \leq 2^{s}-1, t<l$;

● Type 7: $I=\langle u(x-1)^l+v\sum\limits_{j=0}^{2^{s}-1}h_{2j}(x-1)^j+uv\sum\limits_{j=0}^{l}h_{3j}(x-1)^j, v(x-1)^z+uv\sum\limits_{j=0}^{z}b_{3j}(x-1)^j\rangle $, where $0\leq l \leq 2^{s}-1, 0 \leq z \leq 2^{s}-1$;

● Type 8: $I=\langle u(x-1)^l+v\sum\limits_{j=0}^{z-1}h_{2j}(x-1)^j+uv\sum\limits_{j=0}^{z-1}h_{3j}(x-1)^j ,$$v(x-1)^z+uv\sum\limits_{j=0}^{z-1}b_{3j}(x-1)^j, uv(x-1)^w\rangle, $ where $1\leq l \leq 2^{s}-1, 1 \leq z \leq 2^{s}-1, $ and $w< \min \{l, z\}$;

● Type 9: $I=\langle (x-1)^i+u\sum\limits_{j=0}^{i}a_{1j}(x-1)^j+v\sum\limits_{j=0}^{i}a_{2j}(x-1)^j+uv\sum\limits_{j=0}^{i}a_{3j}(x-1)^j\rangle $, where $0\leq i \leq 2^{s}-1$;

● Type 10: $ I=\langle (x-1)^i+u\sum\limits_{j=0}^{i}a_{1j}(x-1)^j+v\sum\limits_{j=0}^{i}a_{2j}(x-1)^j+uv\sum\limits_{j=0}^{t}a_{3j}(x-1)^j, u(x-1)^l+$$ v\sum\limits_{j=0}^{i}h_{2j}(x-1)^j+uv\sum\limits_{j=0}^{t}h_{3j}(x-1)^j\rangle $, where $0\leq i \leq 2^{s}-1, 0\leq l \leq 2^{s}-1$ and $t\leq \min\{i, l\}$;

● Type 11: $ I=\langle (x-1)^i+u\sum\limits_{j=0}^{i}a_{1j}(x-1)^j+v\sum\limits_{j=0}^{i}a_{2j}(x-1)^j+uv\sum\limits_{j=0}^{t}a_{3j}(x-1)^j, u(x-1)^l+$$ v\sum\limits_{j=0}^{i}h_{2j}(x-1)^j+uv\sum\limits_{j=0}^{t}h_{3j}(x-1)^j, uv(x-1)^t\rangle $, where $0\leq i \leq 2^s-1, 0\leq l \leq 2^{s}-1$ and $t\leq \min\{i, l\}$;

● Type 12: $I=\langle (x-1)^i+u\sum\limits_{j=0}^{i}a_{1j}(x-1)^j+v\sum\limits_{j=0}^{i}a_{2j}(x-1)^j+uv\sum\limits_{j=0}^{t}a_{3j}(x-1)^j, u(x-1)^l+$$v\sum\limits_{j=0}^{i}h_{2j}(x-1)^j+uv\sum\limits_{j=0}^{t}h_{3j}(x-1)^j, v(x-1)^z+uv\sum\limits_{j=0}^{t}b_{3j}(x-1)^j\rangle $, where $0\leq i \leq 2^{s}-1, 0\leq l \leq 2^{s}-1, 0\leq z \leq 2^{s}-1$ and $t\leq \min \{i, l, z\}$;

● Type 13: $ I=\langle (x-1)^i+u\sum\limits_{j=0}^{i}a_{1j}(x-1)^j+v\sum\limits_{j=0}^{i}a_{2j}(x-1)^j+uv\sum\limits_{j=0}^{w}a_{3j}(x-1)^j, u(x-1)^l+$$v\sum\limits_{j=0}^{i}h_{2j}(x-1)^j+uv\sum\limits_{j=0}^{w}h_{3j}(x-1)^j, v(x-1)^z+uv\sum\limits_{j=0}^{w}b_{3j}(x-1)^j, uv(x-1)^w\rangle $, where $0\leq i \leq 2^{s}-1, 0\leq l \leq 2^{s}-1, 0\leq z \leq 2^{s}-1$ and $w\leq \min\{i, l, z\}$.

In this section, we discuss the $\lambda$-constacyclic codes,

$ \lambda=1+u, 1+v, 1+u+uv, 1+v+uv, 1+u+v, 1+u+v+uv. $

First, we study the structure of the $(1+u)$-constacyclic codes of length $2^s$ over $R$. Obviously, $(1+u)$-constacyclic codes of length $2^s$ over $R$ are ideals of the residue ring $R_3=\frac{R[x]}{\langle x^{2^s}-(1+u)\rangle }$. It is easy to verify the following lemmas.

Lemma 4.1 The following hold true in $R_3$.

(ⅰ) For any nonnegative integer $t$, $(x-1)^{2^t}=x^{2^t}-1$. In particular, $(x-1)^{2^s}=u$.

(ⅱ) $x-1$ is nilpotent with the nilpotency index $2^{s+1}$.

Lemma 4.2 Let $f(x)\in R_3$. Then $f(x)$ can be uniquely expressed as

$ f(x)=\sum\limits_{j=0}^{2^{s+1}-1}a_{0j}(x-1)^j+v\sum\limits_{j=0}^{2^{s+1}-1}a_{1j}(x-1)^j, $

where $a_{0j}, a_{1j}\in \mathbb{F}_2$. Furthermore, $f(x)$ is invertible if and only if $a_{00}\neq0$

Proof Since each element $r\in R$ has a unique presentation $r=r_1+ur_2+vr_3+uvr_4$, where $r_1, r_2, r_3, r_4\in \mathbb{F}_2, $ each element $f(x)$ over $R_3$ can be uniquely represented as

$ f(x)=\sum\limits_{j=0}^{2^{s+1}-1}b_{0j}(x-1)^j+u\sum\limits_{j=0}^{2^{s+1}-1}b_{1j}(x-1)^j+v\sum\limits_{j=0}^{2^{s+1}-1}b_{2j}(x-1)^j+uv\sum\limits_{j=0}^{2^{s+1}-1}b_{3j}(x-1)^j. $

Because $(x-1)^{2^s}=u$ in $R_3$, it can be uniquely represented without loss of generality that $f(x)=\sum\limits_{j=0}^{2^{s+1}-1}a_{0j}(x-1)^j+v\sum\limits_{j=0}^{2^{s+1}-1}a_{1j}(x-1)^j$, where $a_{0j}, a_{1j}\in \mathbb{F}_2$. The last assertion follows from the fact that $v$ and $x-1$ are both nilpotent in $R_3$.

Similar to the discussions in Proposition 3.3, we have the following proposition.

Proposition 4.3 The ring $R_{3}$ is a local ring with the maximal ideals $\langle v, x-1\rangle$, but it is not a chain ring.

Following the proposition and lemmas above, we can list all $(1+u)$-constacyclic codes of length $2^s$ over $R_3$ as follows.

Theorem 4.4 $(1+u)$-constacyclic codes of length $2^s$ over $R$, i.e., ideals of the ring $R_3$ are

● Type 1: $\langle 0\rangle, \langle 1\rangle $;

● Type 2: $I=\langle v(x-1)^k\rangle $, where $0\leq k \leq 2^{s}-1$;

● Type 3: $I=\langle (x-1)^l+v\sum\limits_{j=0}^{l-1}c_{1j}(x-1)^j\rangle $, where $1\leq l \leq 2^{s+1}-1$;

● Type 4: $I=\langle (x-1)^l+v\sum\limits_{j=0}^{w-1}c_{1j}(x-1)^j, v(x-1)^w\rangle $, where $1\leq l \leq 2^{s+1}-1, ~w\leq l$.

Proof Ideals of Type 1 are the trivial ideals. Consider an arbitrary nontrivial ideal of $R_3$.

Case 1 $I\subseteq \langle v\rangle $. Any element of $I$ must have the form $v\sum\limits_{j=0}^{2^{s+1}-1}a_{0j}(x-1)^j$, where $a_{0j}\in \mathbb{F}_2$. Let $b(x)\in I$ be an element that has the smallest $k$ such that $a_{0k}\neq 0$. Hence all element $a(x)\in I$ have the form

$ a(x)=v(x-1)^k\sum\limits_{j=k}^{2^{s+1}-1}a_{0j}(x-1)^{j-k}, $

which implies $I\subseteq \langle u(x-1)^k\rangle $.

On the other hand, we have $b(x)\in I$ with

$ b(x)=v(x-1)^k[a_{0k}+\sum\limits_{j=k+1}^{2^{s+1}-1}a_{0j}(x-1)^{j-k}]. $

As $a_{0k}\neq 0$, $a_{0k}+\sum\limits_{j=k+1}^{2^{s+1}-1}a_{0j}(x-1)^{j-k}$ is invertible, and $v(x-1)^k\in I.$ That is to say, the ideal of $R_3$ contained in $\langle v\rangle $ are $\langle v(x-1)^k\rangle, 0 \leq k \leq 2^{s+1}-1.$ It means that $I$ is in Type 2.

Case 2 $I\nsubseteq \langle v\rangle $. Any element of $I$ must have the form

$ \sum\limits_{j=0}^{2^{s+1}-1}a_{0j}(x-1)^j+v\sum\limits_{j=0}^{2^{s+1}-1}a_{1j}(x-1)^j, $

and there exists a polynomial

$ \sum\limits_{j=0}^{2^{s+1}-1}b_{0j}(x-1)^j+v\sum\limits_{j=0}^{2^{s+1}-1}b_{1j}(x-1)^j $

in $I$ such that $\sum\limits_{j=0}^{2^{s+1}-1}b_{0j}(x-1)^j\neq 0$. Let

$ M=\{\sum\limits_{j=0}^{2^{s+1}-1}a_{0j}(x-1)^j+v\sum\limits_{j=0}^{2^{s+1}-1}a_{1j}(x-1)^j\in I\mid \sum\limits_{j=0}^{2^{s+1}-1}a_{0j}(x-1)^j\neq0, a_{0j}, a_{1j}\in \mathbb{F}_2\} $

and $N=\{v\sum\limits_{j=0}^{2^{s+1}-1}a_{1j}(x-1)^j\in I\mid a_{1j}\in \mathbb{F}_2\}.$ Setting $\delta= \min\{\deg (m(x))\mid m(x)\in M\}$. Suppose that $H=\{h(x)\in M\mid \deg (h(x))=\delta\}$. Then there is an element

$ h_1(x)=\sum\limits_{j=0}^{2^{s+1}-1}h_{0j}(x-1)^j+v\sum\limits_{j=0}^{2^{s+1}-1}h_{1j}(x-1)^j $

in $H$ that has the smallest $l$ such that $h_{0l}\neq 0$. Hence we have

$ h_1(x)=(x-1)^l[h_{0l}+\sum\limits_{j=l+1}^{2^{s+1}-1}h_{0j}(x-1)^{j-l}]+v\sum\limits_{j=0}^{2^{s+1}-1}h_{1j}(x-1)^j\in M \subset I. $

Now, we have two subcases.

Case 2.1 $N\subseteq \langle h_1(x)\rangle $. For any $f(x)\in M$, by Proposition 2.1, $f(x)$ can be written as $f(x)=q(x)h_1(x)+r(x), $ where $q(x), r(x)\in R_3, $ and $r(x)=0$ or $\deg (r(x))< \deg(h_1(x))$. Suppose $r(x)\notin N.$ Then $r(x)\neq 0$. Hence $r(x)=f(x)-q(x)h_1(x)\in M, $ which contradicts with the assumption of $h_1(x)$. Thus $r(x)\in N.$ Therefore, $I=\langle h_1(x)\rangle $. Because $vh_1(x)=v(x-1)^l[h_{0l}+\sum\limits_{j=l+1}^{2^{s+1}-1}h_{1j}(x-1)^{j-t}]\in I$ and $h_{0l}+\sum\limits_{j=l+1}^{2^{s+1}-1}h_{0j}(x-1)^{j-t}$ is an invertible element in $R_3$, $v(x-1)^l\in I, $ it follows that

$ \tilde{h}_1(x)=(x-1)^l[h_{0l}+\sum\limits_{j=l+1}^{2^{s+1}-1}h_{0j}(x-1)^{j-l}]+v\sum\limits_{j=0}^{l-1}h_{1j}(x-1)^j\in I. $

Thus

$ c(x)=\tilde{h}_1(x)[h_{0l}+\sum\limits_{j=l+1}^{2^{s+1}-1}h_{0j}(x-1)^{j-l}]^{-1}\in I $

and $c(x)$ can be expressed as $c(x)=(x-1)^l+v\sum\limits_{j=0}^{l-1}c_{1j}(x-1)^j$, where $c_{1j} \in \mathbb{F}_2.$ Therefore,

$ I=\langle (x-1)^l+v\sum\limits_{j=0}^{l-1}c_{1j}(x-1)^j\rangle, $

where $1\leq l \leq 2^{s+1}-1.$ Hence $I$ is in Type 3.

Case 2.2 $N\nsubseteq \langle h_1(x)\rangle =\langle c(x)\rangle $. Then there exists the smallest integer $w$ such that $n(x)=v(x-1)^wn_1(x)$ for every $n(x)\in N, $ where $n_1(x)\in R_3.$ Obviously, $v(x-1)^w\in N, $ but $v(x-1)^w\notin \langle h_1(x)\rangle =\langle c(x)\rangle $. Hence $I=\langle (x-1)^l+v\sum\limits_{j=0}^{l-1}c_{1j}(x-1)^j, v(x-1)^w\rangle .$

Suppose that $w> l.$ Then

$ v(x-1)^w=v(x-1)^{w-l}[(x-1)^l+v\sum\limits_{j=0}^{l-1}c_{1j}(x-1)^j]\in \langle c(x)\rangle . $

It is impossible. Thus

$ I=\langle (x-1)^l+v\sum\limits_{j=0}^{w-1}c_{1j}(x-1)^j, v(x-1)^w\rangle, $

where $1\leq l \leq 2^{s+1}-1, ~w\leq l$. Therefore, $I$ is in Type 4.

Next we study the structure of $(1+v), (1+u+uv), (1+v+uv), (1+u+v), (1+u+v+uv)$-constacyclic codes of length $2^s$ over $R$. Similar to the discussion in Theorem 4.4, we have the following theorems.

Theorem 4.5 $(1+u+uv), (1+u+v), (1+u+v+uv)$-constacyclic codes over $R$ are

● Type 1: $\langle 0\rangle, \langle 1\rangle $;

● Type 2: $I=\langle v(x-1)^k\rangle $, where $0\leq k \leq 2^{s+1}-1$;

● Type 3: $I=\langle (x-1)^l+v\sum\limits_{j=0}^{l-1}c_{1j}(x-1)^j\rangle $, where $1\leq l \leq 2^{s+1}-1$;

● Type 4: $I=\langle (x-1)^l+v\sum\limits_{j=0}^{l-1}c_{1j}(x-1)^j, v(x-1)^w\rangle $, where $1\leq l \leq 2^{s+1}-1, ~w\leq l$.

Proof

$ f(x)=\sum\limits_{j=0}^{2^{s+1}-1}a_{0j}(x-1)^j+v\sum\limits_{j=0}^{2^{s+1}-1}a_{1j}(x-1)^j, $

where $a_{0j}, a_{1j}\in \mathbb{F}_2$. Furthermore, $f(x)$ is invertible if and only if $a_{00}\neq0$.

Now, Similar to the discussion in Theorem 4.4, we can complete the proof of statement.

Theorem 4.6 $(1+v), (1+v+uv)$-constacyclic codes over $R$ are

● Type 1: $\langle 0\rangle, \langle 1\rangle $;

● Type 2: $I=\langle u(x-1)^k\rangle $, where $0\leq k \leq 2^{s+1}-1$;

● Type 3: $I=\langle (x-1)^l+u\sum\limits_{j=0}^{l-1}c_{1j}(x-1)^j\rangle $, where $1\leq l \leq 2^{s+1}-1$;

● Type 4: $I=\langle (x-1)^l+u\sum\limits_{j=0}^{l-1}c_{1j}(x-1)^j, u(x-1)^w\rangle $, where $1\leq l \leq 2^{s+1}-1, w\leq l$.

Proof $f(x)=\sum\limits_{j=0}^{2^{s+1}-1}a_{0j}(x-1)^j+u\sum\limits_{j=0}^{2^{s+1}-1}a_{1j}(x-1)^j$, where $a_{0j}, a_{1j}\in \mathbb{F}_2$. Furthermore, $f(x)$ is invertible if and only if $a_{00}\neq0$.

Now, in the proof of Theorem 4.4, we replace each $v$ by $u$ and get our statement.