Negacyclic codes over the local ring $\mathbb{Z}_4[v]/\langle v^2+2v\rangle$ of oddly even length and their Gray images

TL;DR

This paper classifies negacyclic codes over a specific local ring of odd length, analyzes their structure, duals, and Gray images, and constructs new self-dual codes with good parameters.

Contribution

It provides a complete classification of negacyclic codes over the ring, including self-dual codes, and constructs new optimal self-dual codes via Gray images.

Findings

Classification of negacyclic codes over the ring for odd lengths.

Explicit enumeration of self-dual negacyclic codes.

Construction of new self-dual 2-quasi-twisted codes with good parameters.

Abstract

Let $R=\mathbb{Z}_{4}[v]/\langle v^2+2v\rangle=\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ ($v^2=2v$) and $n$ be an odd positive integer. Then $R$ is a local non-principal ideal ring of $16$ elements and there is a $\mathbb{Z}_{4}$-linear Gray map from $R$ onto $\mathbb{Z}_{4}^2$ which preserves Lee distance and orthogonality. First, a canonical form decomposition and the structure for any negacyclic code over $R$ of length $2n$ are presented. From this decomposition, a complete classification of all these codes is obtained. Then the cardinality and the dual code for each of these codes are given, and self-dual negacyclic codes over $R$ of length $2n$ are presented. Moreover, all $23\cdot(4^p+5\cdot 2^p+9)^{\frac{2^{p}-2}{p}}$ negacyclic codes over $R$ of length $2M_p$ and all $3\cdot(4^p+5\cdot 2^p+9)^{\frac{2^{p-1}-1}{p}}$ self-dual codes among them are presented precisely, where $M_p=2^p-1$ is a Mersenne prime. Finally, $36$ new and good self-dual $2$-quasi-twisted linear codes over $\mathbb{Z}_4$ with basic parameters $(28,2^{28}, d_L=8,d_E=12)$ and of type $2^{14}4^7$ and basic parameters $(28,2^{28}, d_L=6,d_E=12)$ and of type $2^{16}4^6$ which are Gray images of self-dual negacyclic codes over $R$ of length $14$ are listed.

Equations

$$(-a_{N-1},a_0,a_1,\dots ,a_{N-2})\in \mathcal{C},\forall (a_0,a_1,\dots ,a_{N-1})\in \mathcal{C}.$$

$$R={\mathbb{Z}}_4[v]/⟨v^2+2v⟩=\{a+bv\mid a,b\in {\mathbb{Z}}_4\}={\mathbb{Z}}_4+v{\mathbb{Z}}_4(v^2=2v)$$

$$\varrho (\alpha )=(a+b,b),\forall \alpha =a+bv\in R\mathrm{where}a,b\in {\mathbb{Z}}_4$$

$$w_L(0)=0,w_L(1)=w_L(3)=1\mathrm{and}{w}_L(2)=2.$$

$$w_L(a+bv)=w_L(a+b)+w_L(b),\forall a,b\in {\mathbb{Z}}_4.$$

$$\phi (a(x))=a(-x)=\sum _{j=0}^{n-1}{a}_j(-x{)}^j(\forall a(x)=\sum _{j=0}^{n-1}{a}_j{x}^j\in R[x]/⟨x^n-1⟩)$$

$$t_0+\pi t_1+{\pi }^2{t}_2+{\pi }^3{t}_3,t_0,t_1,t_2,t_3\in \mathcal{T}.$$

$$G=\left(\begin{array}[]{ccccc}\pi^{0}I_{k_{0}}&M_{0,1}&M_{0,2}&M_{0,3}&M_{0,4}\cr 0&\pi I_{k_{1}}&\pi M_{1,2}&\pi M_{1,3}&\pi M_{1,4}\cr 0&0&\pi^{2}I_{k_{2}}&\pi^{2}M_{2,3}&\pi^{2}M_{2,4}\cr 0&0&0&\pi^{3}I_{k_{3}}&\pi^{3}M_{3,4}\end{array}\right)U$$

$$(0,a+\omega {\pi }^{2}b)\in C,\forall (a,b)\in C$$

$$f(-x^2)=\prod _{k=0}^{m-1}(-x^2-{\zeta }^{2^k})=(-1{)}^m\prod _{k=0}^{m-1}(x^2+{\zeta }^{2^k}).$$

$$f(x{)}^2$$

$$\overline{g}(x)=x\sum _{k=0}^{m-1}{\overline{\zeta }}^{2^k}\prod _{0\le j\ne k\le m-1}(x^2+{\overline{\zeta }}^{2^{j+1}})=x\sum _{k=0}^{m-1}{\overline{\zeta }}^{2^k}\prod _{0\le j\ne k\le m-1}(x-{\overline{\zeta }}^{2^j}{)}^2.$$

$${\mathcal{K}}_f/M=({\mathbb{F}}_2[x]/⟨\overline{f(-x^2)}⟩)/⟨\overline{f}(x)⟩=({\mathbb{F}}_2[x]/⟨\overline{f}(x{)}^2⟩)/⟨\overline{f}(x)⟩={\mathbb{F}}_2[x]/⟨\overline{f}(x)⟩$$

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

$${\mathcal{K}}_i={\mathbb{Z}}_4[x]/⟨f_i(-x^2)⟩,{\mathcal{T}}_i=\{\sum _{j=0}^{m_i-1}{b}_j{x}^j\mid b_0,b_1,\dots ,b_{m_i-1}\in {\mathbb{F}}_2\}\subseteq {\mathcal{K}}_i,$$

$$f_i(x{)}^2=2g_i(x)\mathrm{in}{\mathcal{K}}_i$$

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

$$-(x^{2n}+1)=f_1(-x^2)f_2(-x^2)\dots f_r(-x^2)$$

$${\epsilon }_i(x)\equiv a_i(-x^2)F_i(-x^2)=1-b_i(-x^2)f_i(-x^2)(\mathrm{mod}{x}^{2n}+1).$$

$$\varphi (a_1(x),\dots ,a_r(x))=\sum _{i=1}^r{\epsilon }_i(x)a_i(x)(\mathrm{mod}{x}^{2n}+1),$$

$$\mathcal{A}={\mathbb{Z}}_4[x]/⟨x^{2n}+1⟩=\{\sum _{j=0}^{2n-1}{a}_j{x}^j\mid a_j\in {\mathbb{Z}}_4,j=0,1,\dots ,2n-1\}$$

$$\mathcal{A}+v\mathcal{A}=\mathcal{A}[v]/⟨v^2+2v⟩=\{\alpha +\beta v\mid \alpha ,\beta \in \mathcal{A}\}(v^2=2v)$$

$$\Xi :\alpha +\beta v\mapsto \sum _{i=0}^{2n-1}{\xi }_i{x}^i,\mathrm{where}{\xi }_i=a_i+b_{i}v\in R,i=0,1,\dots ,2n-1.$$

$${\mathcal{K}}_i+v{\mathcal{K}}_i={\mathcal{K}}_i[v]/⟨v^2+2v⟩=\{\alpha +\beta v\mid \alpha ,\beta \in {\mathcal{K}}_i\}$$

$$\mathcal{C}={\epsilon }_1(x)C_1\oplus {\epsilon }_1(x)C_1\oplus \dots \oplus {\epsilon }_r(x)C_r(\mathrm{mod}{x}^{2n}+1)$$

$$\Phi (\xi )$$

$$\Phi (\xi \eta )=\Phi ({\xi }_1{\eta }_1,\dots ,{\xi }_r{\eta }_r)$$

$$\mathcal{C}$$

$$\sigma :{\mathcal{K}}_i^2\to {\mathcal{K}}_i+v{\mathcal{K}}_i\mathrm{via}(\alpha ,\beta )\mapsto \alpha +\beta v(\forall \alpha ,\beta \in {\mathcal{K}}_i).$$

$$(0,\alpha +2\beta )\in S_i,\forall (\alpha ,\beta )\in S_i$$

Full text

††journal: Finite Fields and Their Applications

1 Introduction

The catalyst for the study of codes over rings was the discovery of the connection between the Kerdock and Preparata codes, which are non-linear binary codes, and linear codes over $\mathbb{Z}_{4}$ (see [3] and [4]). Soon after this discovery, codes over many different rings were studied. This led to many new discoveries and concreted the study of codes over rings as an important part of the coding theory discipline. Since $\mathbb{Z}_{4}$ is a chain ring, it was natural to expand the theory to focus on alphabets that are finite commutative chain rings and other special rings (See [1], [2], [5], [7–12], [14], [16–22], for examples).

In 1999, Wood in [23] showed that for certain reasons finite Frobenius rings are the most general class of rings that should be used for alphabets of codes. Then self-dual codes over commutative Frobenius rings were investigated by Dougherty et al. [13]. Especially, in 2014, codes over an extension ring of $\mathbb{Z}_{4}$ were studied in [24] and [25], here the ring was described as $\mathbb{Z}_{4}[u]/\langle u^{2}\rangle=\mathbb{Z}_{4}+u\mathbb{Z}_{4}$ ($u^{2}=0$) which is a local non-principal ring.

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

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

A linear code ${\cal C}$ over $A$ of length $N$ is said to be negacyclic if

[TABLE]

We will use the natural connection of negacyclic codes to polynomial rings, where $c=(c_{0},c_{1},c_{2},\ldots$, $c_{N-1})\in A^{N}$ is viewed as $c(x)=\sum_{j=0}^{N-1}c_{j}x^{j}$ and the negacyclic code ${\cal C}$ is an ideal in the polynomial residue ring $A[x]/\langle x^{N}+1\rangle$.

In this paper, let $n$ be an odd positive integer and denote

[TABLE]

in which the operations are defined by:

$\alpha+\beta=(a+b)+v(c+d)$ and $\alpha\beta=ac+(ad+bc+2bd)v$.

for any $\alpha=a+bv,\beta=c+dv\in\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ with $a,b,c,d\in\mathbb{Z}_{4}$. Then $R$ is a local Frobenius non-principal ideal ring of $16$ elements.

Linear codes over $R$ were studied in [15]. In the paper, a duality preserving Gray map was given and used to present MacWilliams identities and self-dual codes. Some extremal Type II $\mathbb{Z}_{4}$-codes were provided as images of codes over this ring. $\mathbb{Z}_{4}$-codes that are images of linear codes over $R$ were characterised and some well-known families of $\mathbb{Z}_{4}$-codes were proved to be linear over $R$. As in [15] Section 3, we define a map $\varrho:R\rightarrow\mathbb{Z}_{4}^{2}$ by

[TABLE]

and let $\theta:R^{N}\rightarrow\mathbb{Z}_{4}^{2N}$ be such that $\theta(\alpha_{1},\ldots,\alpha_{N})=(\varrho(\alpha_{1}),\ldots,\varrho(\alpha_{N}))$, for all $\alpha_{1},\ldots,\alpha_{N}\in R.$ Let $w_{L}$ denote the Lee weight on $\mathbb{Z}_{4}$ defined by:

[TABLE]

We extend $w_{L}$ on the ring $R$ in a natural way that

[TABLE]

With this distance and Gray map definition, the following conclusions have been verified by Martínez-Moro et al. [15].

Lemma 1.1 ([15] Theorem 3.1) Let $\mathcal{C}$ be a linear code over $R$ of length $N$ and minimum Lee distance $d$. Then $\theta(\mathcal{C})$ is a linear code over $\mathbb{Z}_{4}$ of length $2N$, $|\theta(\mathcal{C})|=|\mathcal{C}|$ and is of minimum Lee distance $d$.

Lemma 1.2 ([15] Proposition 3.3) Let $\mathcal{C}$ be a linear code over $R$ of length $N$. Then $\theta(\mathcal{C}^{\bot})=\theta(\mathcal{C})^{\bot}$. In particular, if $\mathcal{C}$ is self-dual, then $\theta(\mathcal{C})$ is an self-dual code over $\mathbb{Z}_{4}$ of length $2N$ and has the same Lee weight distribution.

Moreover, we have the following properties for Negacyclic codes $R$.

Proposition 1.3 Let $\mathcal{C}$ be a negacyclic code $R$ of length $N$. Then $\theta(\mathcal{C})$ is a $2$-quasi-twisted code over $\mathbb{Z}_{4}$ of length $2N$.

Proof. Let $\underline{\alpha}=(\alpha_{0},\alpha_{1},\ldots,\alpha_{N-1})\in\mathcal{C}$, where $\alpha_{i}=a_{i}+b_{i}v$, $a_{i},b_{i}\in\mathbb{Z}_{4}$, for all $i=0,1,\ldots,N-1$. Then $\theta(\underline{\alpha})=(a_{0}+b_{0},b_{0},a_{1}+b_{1},b_{1},\ldots,a_{N-1}+b_{N-1},b_{N-1})\in\theta(\mathcal{C})$. As $\mathcal{C}$ is negacyclic, we have $(-\alpha_{N-1},\alpha_{0},\alpha_{1},\ldots,\alpha_{N-2})$ $\in\mathcal{C}$. This implies $((-1)(a_{N-1}+b_{N-1}),-b_{N-1},a_{0}+b_{0},b_{0},a_{1}+b_{1},b_{1},\ldots,a_{N-2}+b_{N-2},b_{N-2})\in\theta(\mathcal{C})$. So $\theta(\mathcal{C})$ is a $2$-quasi-twisted $\mathbb{Z}_{4}$-code of length $2N$. $\Box$

Since $n$ is odd, the map $\varphi:R[x]/\langle x^{n}-1\rangle\rightarrow R[x]/\langle x^{n}+1\rangle$ defined by

[TABLE]

is an isomorphism of rings preserving Lee distance. Hence $\mathcal{C}$ is a negacyclic code over $R$ of $n$ if and only if there is a unique cyclic code $\mathcal{D}$ over $R$ of length $n$ such that $\varphi(\mathcal{D})=\mathcal{C}$. Moreover, $\mathcal{C}$ and $\mathcal{D}$ has the same Lee weight distribution. A complete classification for cyclic codes over $R$ of odd length and self-dual codes among them had been studied in [5]. In the paper, some good self-dual codes over $\mathbb{Z}_{4}$ of length $30$ and extremal binary self-dual codes with parameters $[60,30,12]$ were obtained from self-dual cyclic codes over $R$ of length $15$. In this paper, we study negacyclic codes over $R$ of length $2n$.

The present paper is organized as follows. In Section 2, we sketch the basic theory of finite rings and linear codes over finite rings needed in this paper. In Section 3, we decompose the ring $\mathbb{Z}_{4}[x]/\langle x^{2n}+1\rangle$ into a direct product of finite chain rings of length $4$. In Section 4, we give a canonical form decomposition for any negacyclic code over $R$ of length $2n$ and present all distinct codes by their generator sets. Using this decomposition, we give the number of codewords for each of these codes and an enumeration for all these codes. In Section 5, we present the dual code and its self-duality for each negacyclic code over $R$ of length $2n$. In Section 6, we focus our attention on negacyclic code over $R$ of length $2M_{p}$, where $p$ is a prime and $M_{p}=2^{p}-1$ is a Mersenne prime. Especially, we present explicitly all $293687$ ngeacyclic code over $R$ of length $14$ and $339$ self-dual codes among them. Finally, we obtain $36$ new and good self-dual $2$-quasi-twisted codes over $\mathbb{Z}_{4}$ of length $28$.

2 Preliminaries

In this section, we sketch the basic theory of finite chain rings and linear codes over finite chain rings needed in this paper.

Lemma 2.1 ([10] Proposition 2.1) Let $\mathcal{K}$ be a finite ring with identity. Then the following conditions are equivalent:

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

(ii) $\mathcal{K}$* is a local principal ideal ring*;

(iii) $\mathcal{K}$* is a chain ring with all ideals given by: $\langle\pi^{i}\rangle=\pi^{i}\mathcal{K}$, $0\leq i\leq s$, where $s$ is the nilpotency of $\pi$*.

Lemma 2.2 ([10] Proposition 2.2) Let $\mathcal{K}$ be a finite chain ring, with maximal ideal $M=\langle\pi\rangle$, and let $s$ be the nilpotency of $\pi$. Then

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

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

Lemma 2.3 ([16] Lemma 2.4)Using the notations in Lemma 2.2, let $\mathcal{T}\subseteq\mathcal{K}$ be a system of representatives for the equivalence classes of $\mathcal{K}$ under congruence modulo $\pi$. (Equivalently, we can define $\mathcal{T}$ to be a maximal subset of $\mathcal{K}$ with the property that $t_{1}-t_{2}\not\in\langle\pi\rangle$ for all $t_{1},t_{2}\in\mathcal{T}$, $t_{1}\neq t_{2}$.) Then

(i) Every element $a$ of $\mathcal{K}$ has a unique $\pi$-expansion: $a=\sum_{j=0}^{s-1}t_{j}\pi^{j}$, $t_{0},t_{1},\ldots,t_{s-1}\in\mathcal{T}$.

(ii) $|\mathcal{K}/\langle\pi\rangle|=|\mathcal{T}|$* and $|\langle\pi^{i}\rangle|=|\mathcal{T}|^{s-i}$ for $0\leq i\leq s$*.

From now on, let $\mathcal{K}$ be an arbitrary finite chain ring with $1\neq 0$, $\pi$ be a fixed generator of the maximal ideal of $\mathcal{K}$ with nilpotency index $4$, and $F=\mathcal{K}/\langle\pi\rangle$. In this case, $\mathcal{K}$ is called a finite chain ring of length $4$. Using the notations of Lemma 2.3, every element $a\in\mathcal{K}$ has a unique $\pi$-adic expansion:

[TABLE]

Hence $|\mathcal{K}|=|F|^{4}$. If $a\neq 0$, the $\pi$-degree of $a$ is defined as the least index $j\in\{0,1,2,3\}$ for which $t_{j}\neq 0$ and written for $\|a\|_{\pi}=j$. If $a=0$ we write $\|a\|_{\pi}=4$. It is clear that $a\in\mathcal{K}^{\times}$ if and only if $t_{0}\neq 0$, i.e. $\|a\|_{\pi}=0$. Hence $|\mathcal{K}^{\times}|=(|F|-1)|F|^{3}$. Moreover, we have $\mathcal{K}/\langle\pi^{0}\rangle=\{0\}$ and $\mathcal{K}/\langle\pi^{l}\rangle=\{\sum_{i=0}^{l-1}\pi^{i}a_{i}\mid a_{0},\ldots,a_{l-1}\in{\cal T}\}$ with $|\mathcal{K}/\langle\pi^{l}\rangle|=|F|^{l}$, $1\leq l\leq 3$.

Let $L$ be a positive integer and $\mathcal{K}^{L}=\{(\alpha_{1},\ldots,\alpha_{L})\mid\alpha_{1},\ldots,\alpha_{L}\in\mathcal{K}\}$ that is a free $\mathcal{K}$-module under componentwise addition and scalar multiplication with elements from $\mathcal{K}$. Then $\mathcal{K}$-submodules of $\mathcal{K}^{L}$ are linear codes over $\mathcal{K}$ of length $L$. Let $C$ be a linear code over $\mathcal{K}$ of length $L$. By [16] Definition 3.1, a matrix $G$ is called a generator matrix for $C$ if the rows of $G$ span $C$ and none of them can be written as a $\mathcal{K}$-linear combination of the other rows of $G$. Furthermore, a generator matrix $G$ is said to be in standard form if there is a suitable permutation matrix $U$ of size $L\times L$ such that

[TABLE]

where the columns are grouped into blocks of sizes $k_{0},k_{1},k_{2},k_{3},k$ with $k_{i}\geq 0$ and $k=L-(k_{0}+k_{1}+k_{2}+k_{3})$. Of course, if $k_{i}=0$, the matrices $\pi^{i}I_{k_{i}}$ and $\pi^{i}M_{i,j}$ ($i<j$) are suppressed in $G$. From [16] Proposition 3.2 and Theorem 3.5, we deduce the following.

Lemma 2.4 Let $C$ be a nonzero linear code of length $L$ over $\mathcal{K}$. Then $C$ has a generator matrix in standard form as in $(1)$. In this case, the number of codewords in $C$ is equal to $|C|=|F|^{4k_{0}+3k_{1}+2k_{2}+k_{3}}=|\mathcal{T}|^{4k_{0}+3k_{1}+2k_{2}+k_{3}}$.

All distinct nontrivial linear codes of length $2$ over $\mathcal{K}$ has been listed (cf. Cao [6] Lemma 2.2 and Example 2.5). In particular, we have

Theorem 2.5 Using the notations above, let $\omega\in\mathcal{K}^{\times}$. Then every linear code $C$ over $\mathcal{K}$ of length $2$ satisfying the following condition

[TABLE]

has one and only one of the following matrices $G$ as its generator matrix in standard form:

(I) $G=(\pi^{2}(a+b\pi),1)$, where $a,b\in\mathcal{T}$.

(II) $G=(0,\pi^{3})$; $G=(\pi^{3}b,\pi^{2})$ where $b\in\mathcal{T}$; $G=(\pi^{3}a,\pi)$* where $a\in\mathcal{T}$*.

(III) $G=\pi^{k}I_{2}$* where $I_{2}$ is the identity matrix of order $2$, $0\leq k\leq 4$*.

(IV) G=\left(\begin{array}[]{cc}0&1\cr\pi&0\end{array}\right); G=\left(\begin{array}[]{cc}\pi^{t-1}z&1\cr\pi^{t}&0\end{array}\right) where $z\in\mathcal{T}$ and $t=2,3$.

(V) G=\left(\begin{array}[]{cc}\pi^{2}z&\pi\cr\pi^{3}&0\end{array}\right)* where $z\in\mathcal{T}$, and G=\left(\begin{array}[]{cc}0&\pi^{t-1}\cr\pi^{t}&0\end{array}\right) where $t=2,3$*.

Therefore, the number of linear codes over $\mathcal{K}$ of length $2$ satisfying Condition $(2)$ is equal to $|\mathcal{T}|^{2}+5|\mathcal{T}|+9$.

Proof. See Appendix. $\Box$

3 A direct sum decomposition of the ring $\mathbb{Z}_{4}[x]/\langle x^{2n}+1\rangle$

From now on, let $n$ be an odd positive integer. In this section, we decompose the ring $\mathbb{Z}_{4}[x]/\langle x^{2n}+1\rangle$ into a direct product of finite chain rings of length $4$. This direct sum decomposition will be needed in the following sections.

It is known that any element $a$ of $\mathbb{Z}_{4}$ is unique expressed as $a=a_{0}+2a_{1}$ where $a_{0},a_{1}\in\mathbb{F}_{2}=\{0,1\}$ in which we regard $\mathbb{F}_{2}$ as a subset of $\mathbb{Z}_{4}$. Denote $\overline{a}=a_{0}\in\mathbb{F}_{2}$. Then ${}^{-}:a\mapsto\overline{a}$ ($\forall a\in\mathbb{Z}_{4}$) is a ring isomorphism from $\mathbb{Z}_{4}$ onto $\mathbb{F}_{2}$, and - can be extended to a ring isomorphism from $\mathbb{Z}_{4}[x]$ onto $\mathbb{F}_{2}[x]$ by: $\overline{f}(x)=\overline{f(x)}=\sum_{i=0}^{m}\overline{b}_{i}x^{i}$ for all $f(x)=\sum_{i=0}^{m}b_{i}x^{i}\in\mathbb{Z}_{4}[x].$

A monic polynomial $f(x)\in\mathbb{Z}_{4}[x]$ is said to be basic irreducible if $\overline{f}(x)$ is an irreducible polynomial in $\mathbb{F}_{2}[x]$. Then we have the following conclusions for monic basic irreducible polynomials in $\mathbb{Z}_{4}[x]$.

Lemma 3.1 Let $f(x)$ be a monic basic irreducible polynomial in $\mathbb{Z}_{4}[x]$ of degree $m$ and denote $\Gamma=\mathbb{Z}_{4}[x]/\langle f(x)\rangle$. Then

(i) ([22] Theorem 6.1]) $\Gamma$* is a Galois ring of characteristic $4$ and cardinality $4^{m}$ and $\Gamma=\mathbb{Z}_{4}[\zeta]$, where $\zeta=x+\langle f(x)\rangle\in\Gamma$ satisfying $\zeta^{2^{m}-1}=1$ in $\Gamma$*.

Denote $\overline{\Gamma}=\mathbb{F}_{2}[x]/\langle\overline{f}(x)\rangle$ and $\overline{\zeta}=x+\langle\overline{f}(x)\rangle\in\overline{\Gamma}$. Then $\overline{\Gamma}=\mathbb{F}_{2}[\overline{\zeta}]$ which is a finite field of cardinality $2^{m}$, $\overline{f}(x)=\prod_{k=0}^{m-1}(x-\overline{\zeta}^{2^{k}})$ and that - can be extended to a ring isomorphism from $\Gamma$ onto $\overline{\Gamma}$ by $\xi\mapsto\overline{\xi}=\sum_{j=0}^{m-1}\overline{a}_{j}\overline{\zeta}^{j}$, for all $\xi=\sum_{j=0}^{m-1}a_{j}\zeta^{j}\in\Gamma$ where $a_{0},a_{1},\ldots,a_{m-1}\in\mathbb{Z}_{4}$.

(ii) ([22] Proposition 6.14 or [7] Lemma 2.3(ii)) $f(x)=\prod_{k=0}^{m-1}(x-\zeta^{2^{k}})$.

Theorem 3.2 Let $f(x)$ be a monic basic irreducible polynomial in $\mathbb{Z}_{4}[x]$ of degree $m$, denote $\mathcal{K}_{f}=\mathbb{Z}_{4}[x]/\langle f(-x^{2})\rangle=\{\sum_{j=0}^{2m-1}a_{j}x^{j}\mid a_{0},a_{1},\ldots,a_{2m-1}\in\mathbb{Z}_{4}\}$ in which the arithmetic is done modulo $f(-x^{2})$, and set $\mathcal{T}_{f}=\{\sum_{j=0}^{m-1}b_{j}x^{j}$ $\mid b_{0},b_{1},\ldots,b_{m-1}\in\mathbb{F}_{2}\}\subseteq\mathcal{K}_{f}$ in which we regard $\mathbb{F}_{2}$ as a subset of $\mathbb{Z}_{4}$. Then

(i) There is an invertible element $g(x)\in\mathcal{K}_{f}$ such that $f(x)^{2}=2g(x)$. Hence $\langle f(x)^{2}\rangle=\langle 2\rangle$ as ideals of $\mathcal{K}_{f}$.

(ii) $\mathcal{K}_{f}$* is a finite chain ring with maximal ideal $\langle f(x)\rangle$ generated by $f(x)$, the nilpotency index of $f(x)$ is equal to $4$ and $\mathcal{K}_{f}/\langle f(x)\rangle$ is a finite field of cardinality $2^{m}$*.

(iii) Every element $\alpha$ of $\mathcal{K}_{f}$ has a unique $f(x)$-adic expansion given by: $\alpha=\sum_{j=0}^{3}b_{j}(x)f(x)^{j},\ b_{0}(x),b_{1}(x),b_{2}(x),b_{3}(x)\in\mathcal{T}_{f}.$

Moreover, we have $\mathcal{T}_{f}=\mathcal{K}_{f}/\langle f(x)\rangle$ as sets and $|\mathcal{T}_{f}|=2^{m}$.

Proof. (i) By Lemma 3.1(ii), we have

[TABLE]

By Lemma 3.1(i), we know that $\zeta^{2^{m}}=\zeta=\zeta^{2^{0}}$. From this, by Lemma 3.1(ii) and $(x-\zeta^{2^{k}})^{2}=(x^{2}+\zeta^{2^{k+1}})-2\zeta^{2^{k}}x$ we deduce that

[TABLE]

where $g(x)=x\sum_{k=0}^{m-1}\zeta^{2^{k}}\prod_{0\leq j\neq k\leq m-1}(x^{2}+\zeta^{2^{j+1}})\in\mathbb{Z}_{4}[x]$, since $g(x)=\frac{1}{2}((-1)^{m}f(-x^{2})-f(x)^{2})\in\mathbb{Z}[x]$, as a polynomial in $\mathbb{F}_{2}[x]$ we have that

[TABLE]

This implies $\overline{g}(\overline{\zeta}^{2^{k}})=\overline{\zeta}^{2^{k+1}}\prod_{0\leq j\neq k\leq m-1}(\overline{\zeta}^{2^{k}}-\overline{\zeta}^{2^{j}})^{2}\neq 0$ for all $k=0,1,2,\ldots$, $m-1$, since $\overline{\zeta},\overline{\zeta}^{2},\ldots,\overline{\zeta}^{2^{m-1}}$ are distinct root of $\overline{f}(x)$ in the finite field $\overline{\Gamma}$ by Lemma 3.1(i). From this and by $\overline{f(-x^{2})}=\overline{f}(x^{2})=(\overline{f}(x))^{2}=\sum_{k=0}^{m-1}(x-\overline{\zeta}^{2^{k}})^{2}$, we deduce that ${\rm gcd}(\overline{f(-x^{2})},\overline{g}(x))=1$ in $\mathbb{F}_{2}[x]$. Therefore, $f(-x^{2})$ and $g(x)$ are coprime in $\mathbb{Z}_{4}[x]$. Hence there exist $a(x),b(x)\in\mathbb{Z}_{4}[x]$ such that $a(x)g(x)+b(x)f(-x^{2})=1.$ This implies that $g(x)$ is an invertible element of the residue class ring $\mathcal{K}_{f}=\mathbb{Z}_{4}[x]/\langle f(-x^{2})\rangle$ and $g(x)^{-1}=a(x)$ (mod $f(-x^{2})$). Then by $f(x)^{2}=(-1)^{d}f(-x^{2})-2g(x)$ in $\mathbb{Z}_{4}[x]$ it follows that $f(x)^{2}=2g(x)$ in ${\cal K}_{f}$. Hence $\langle 2\rangle=\langle f(x)^{2}\rangle$ as ideals of ${\cal R}_{f}$.

(ii) Let $M=\langle 2,f(x)\rangle$ be the ideal of ${\cal K}_{f}$ generated by $2$ and $f(x)$. Then

[TABLE]

up to natural ring isomorphisms, where $\mathbb{F}_{2}[x]/\langle\overline{f}(x)\rangle$ is a finite field of $2^{m}$ elements by Lemma 3.1(i). Hence $M$ is a maximal ideal of ${\cal K}_{f}$.

By $f(x)^{4}=(2g(x))^{2}=0$, we see that both $2$ and $f(x)$ are nilpotent elements of ${\cal K}_{f}$. From this one can verify easily that every element of ${\cal K}_{f}\setminus M$ is invertible. This implies that ${\cal R}$ is a local ring with $M$ as its unique maximal ideal. Furthermore, by $\langle 2\rangle=\langle f(x)^{2}\rangle$ in (i) we conclude that $M=\langle f(x)\rangle$. Hence ${\cal K}_{f}/\langle f(x)\rangle\cong\mathbb{F}_{2}[x]/\langle\overline{f}(x)\rangle$.

As stated above, by Lemma 2.1 we see that ${\cal K}_{f}$ is a finite chain ring. Let $s$ be the nilpotency index of $f(x)$. By Lemma 2.2(i) it follows that $|{\cal K}_{f}|=|\mathbb{F}_{2}[x]/\langle\overline{f}(x)\rangle|^{s}=2^{sm}$. On the other hand, by ${\deg}(f(-x^{2}))=2m$ it follows that $|{\cal K}_{f}|=|\mathbb{Z}_{4}[x]/\langle f(-x^{2})\rangle|=4^{2m}=2^{4m}$. Therefore, $s=4$.

(iii) By ${\cal K}_{f}/\langle f(x)\rangle\cong\mathbb{F}_{2}[x]/\langle\overline{f}(x)\rangle=\{\sum_{j=0}^{m-1}b_{j}x^{j}\mid b_{0},b_{1},\ldots,b_{m-1}\in\mathbb{F}_{2}\}$, we see that $\mathbb{F}_{2}[x]/\langle\overline{f}(x)\rangle={\cal T}_{f}\subseteq{\cal K}_{f}$ as sets. Hence ${\cal T}_{f}$ is a system of representatives for the equivalence classes of ${\cal K}_{f}$ under congruence modulo $f(x)$. Then the conclusion follows from Lemma 2.3(i) immediately. $\Box$

In the rest of this paper, let

[TABLE]

where $f_{1}(x),f_{2}(x),\ldots,f_{r}(x)$ are pairwise coprime monic basic irreducible polynomials in $\mathbb{Z}_{4}[x]$. We assume ${\rm deg}(f_{i}(x))=m_{i}$ and denote

[TABLE]

for each integer $i$, $1\leq i\leq r$. Then by Theorem 3.2, we know that

$\bullet$ There is an invertible element $g_{i}(x)\in\mathcal{K}_{i}$ such that

[TABLE]

where $g_{i}(x)=\frac{1}{2}((-1)^{m_{i}}f_{i}(-x^{2})-f_{i}(x)^{2})$ as a polynomial in $\mathbb{Z}[x]$ (mod $f_{i}(-x^{2})$, mod $2$). Hence $\langle f_{i}(x)^{2}\rangle=\langle 2\rangle$ as ideals of $\mathcal{K}_{i}$.

$\bullet$ ${\cal K}_{i}$ is a finite chain ring with maximal ideal $\langle f_{i}(x)\rangle$, the nilpotency index of $f_{i}(x)$ is equal to $4$ and ${\cal K}_{i}/\langle f_{i}(x)\rangle$ is a finite field of cardinality $2^{m_{i}}$.

$\bullet$ Every element $\alpha$ of ${\cal K}_{i}$ has a unique $f_{i}(x)$-adic expansion: $\alpha=b_{0}(x)+b_{1}(x)f_{i}(x)+b_{2}(x)f_{i}(x)^{2}+b_{3}(x)f_{i}(x)^{3}$, where $b_{j}(x)\in{\cal T}_{i}$ for all $j=0,1,2,3$. Moreover, we have $\mathcal{T}_{i}=\mathcal{K}_{i}/\langle f_{i}(x)\rangle$ as sets, $|\mathcal{T}_{i}|=2^{m_{i}}$ and $|{\cal K}_{i}|=4^{2m_{i}}$.

For each $1\leq i\leq r$, denote $F_{i}(x)=\frac{x^{n}-1}{f_{i}(x)}\in\mathbb{Z}_{4}[x]$. Since $F_{i}(x)$ and $f_{i}(x)$ are coprime, there are polynomials $a_{i}(x),b_{i}(x)\in\mathbb{Z}_{4}[x]$ such that

[TABLE]

Substituting $-x^{2}$ for $x$ in (3) and (5), we obtain

[TABLE]

and $a_{i}(-x^{2})F_{i}(-x^{2})+b_{i}(-x^{2})f_{i}(-x^{2})=1$ in the ring $\mathbb{Z}_{4}[x]$ respectively. In the rest of this paper, we set

[TABLE]

Then from classical ring theory, we deduce the following conclusions.

Theorem 3.3 Denote ${\cal A}=\mathbb{Z}_{4}[x]/\langle x^{2n}+1\rangle$. We have the following:

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

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

(iii) For each $1\leq i\leq r$, define a mapping $\phi_{i}:a(x)\mapsto\varepsilon_{i}(x)a(x)$ $(\forall a(x)\in{\cal K}_{i}=\mathbb{Z}_{4}[x]/\langle f_{i}(-x^{2})\rangle)$. Then $\phi_{i}$ is a ring isomorphism from ${\cal K}_{i}$ onto ${\cal A}_{i}$. Hence $|{\cal A}_{i}|=16^{m_{i}}$.

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

[TABLE]

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

4 Structure of negacyclic codes over $R$ of length $2n$

In this section, we list all distinct negacyclic codes of length $2n$ over the ring $R=\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ ($v^{2}=2v$), i.e. all distinct ideals of the ring $R[x]/\langle x^{2n}+1\rangle$. Using the notation of Theorem 3.3, we denote

[TABLE]

in which the arithmetic is done modulo $x^{2n}+1$, and set

[TABLE]

in which the operations are defined by: for any $\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}\in\mathcal{A}$,

$(\alpha_{1}+\beta_{1}v)+(\alpha_{2}+\beta_{2}v)=(\alpha_{1}+\alpha_{2})+v(\beta_{1}+\beta_{2})$,

$(\alpha_{1}+\beta_{1}v)(\alpha_{2}+\beta_{2}v)=\alpha_{1}\alpha_{2}+v(\alpha_{1}\beta_{2}+\beta_{1}\alpha_{2}+2\beta_{1}\beta_{2})$.

Then $\mathcal{A}+v\mathcal{A}$ is a finite commutative ring containing $\mathcal{A}$ as its subring.

Let $\alpha,\beta\in\mathcal{A}$. Then $\alpha$ and $\beta$ can be uniquely expressed as $\alpha=\sum_{i=0}^{2n-1}a_{i}x^{i}$ and $\beta=\sum_{i=0}^{2n-1}b_{i}x^{i}$ respectively, where $a_{i},b_{i}\in\mathbb{Z}_{4}$ for all $i=0,1,\ldots,2n-1$. Now, we define a map $\Xi:\mathcal{A}+v\mathcal{A}\rightarrow R[x]/\langle x^{n}-1\rangle$ by

[TABLE]

Then one can easily verify the following conclusion.

Lemma 4.1 The map $\Xi$ defined above is an isomorphism of rings from $\mathcal{A}+v\mathcal{A}$ onto $R[x]/\langle x^{2n}+1\rangle$.

In the following, we will identify $\mathcal{A}+v\mathcal{A}$ with $R[x]/\langle x^{2n}-1\rangle$ under the ring isomorphism $\Xi$. Therefore, in order to determine all negacyclic codes over $R$ of length $2n$, we only need to determine all ideals of the ring $\mathcal{A}+v\mathcal{A}$. To do this, we need to investigate the structure of the ring $\mathcal{A}+v\mathcal{A}$. In the rest of the paper, for each integer $1\leq i\leq r$ we denote

[TABLE]

in which the operations are defined by: $\xi_{1}+\xi_{2}=(\alpha_{1}+\alpha_{2})+v(\beta_{1}+\beta_{2})$ and $\xi_{1}\xi_{2}=\alpha_{1}\alpha_{2}+v(\alpha_{1}\beta_{2}+\beta_{1}\alpha_{2}+2\beta_{1}\beta_{2})$, for any $\xi_{1}=\alpha_{1}+\beta_{1}v$ and $\xi_{2}=\alpha_{2}+\beta_{2}v$ with $\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}\in\mathcal{K}_{i}$.

We give the structure for any negacyclic code over $R$ of length $2n$.

Theorem 4.2 Using the notations above, we have the following conclusions.

(i) Define $\Phi(\xi_{1},\ldots,\xi_{r})=\sum_{i=1}^{r}\varepsilon_{i}(x)\xi_{i}$ $({\rm mod}\ x^{2n}+1)$ $(\forall\xi_{i}\in\mathcal{K}_{i}+v\mathcal{K}_{i},\ i=1,2,\ldots,r)$. Then $\Phi$ is an isomorphism of rings from $(\mathcal{K}_{1}+v\mathcal{K}_{1})\times(\mathcal{K}_{2}+v\mathcal{K}_{2})\times\ldots\times(\mathcal{K}_{r}+v\mathcal{K}_{r})$ onto $\mathcal{A}+v\mathcal{A}$.

(ii) $\mathcal{C}$* is a negacyclic code over $R$ of length $2n$ if and only if for each integer $i$, $1\leq i\leq r$, there is a unique ideal $C_{i}$ of the ring $\mathcal{K}_{i}+v\mathcal{K}_{i}$ such that*

[TABLE]

where $\varepsilon_{i}(x)C_{i}=\{\varepsilon_{i}(x)\alpha+v\varepsilon_{i}(x)\beta\mid\alpha+\beta v\in C_{i},\ \alpha,\beta\in\mathcal{K}_{i}\}\subseteq\mathcal{A}+v\mathcal{A}$ for all $i=1,2,\ldots,r$. Hence the number of codewords in $\mathcal{C}$ is $\prod_{i=1}^{r}|C_{i}|$.

Proof (i) Let $\xi=(\xi_{1},\xi_{2},\ldots,\xi_{r})\in(\mathcal{K}_{1}+v\mathcal{K}_{1})\times(\mathcal{K}_{1}+v\mathcal{K}_{1})\times\ldots\times(\mathcal{K}_{r}+v\mathcal{K}_{r})$ where $\xi_{i}=\alpha_{i}+v\beta_{i}$ and $\alpha_{i},\beta_{i}\in\mathcal{K}_{i}$ for all $1\leq i\leq r$. By the definition $\phi$ defined in Theorem 3.3(iv), we have

[TABLE]

By Theorem 3.3, we know that $\phi$ is a ring isomorphism from $\mathcal{K}_{1}\times\mathcal{K}_{2}\times\ldots\times\mathcal{K}_{r}$ onto $\mathcal{A}$. Then for any $\eta=(\eta_{1},\eta_{2},\ldots,\eta_{r})$, where $\eta_{i}=\gamma_{i}+v\delta_{i}$ with $\gamma_{i},\delta_{i}\in\mathcal{K}_{i}$ for all $i$, by $\Phi(\eta)=\phi(\gamma_{1},\gamma_{2},\ldots,\gamma_{r})+v\phi(\delta_{1},\delta_{2},\ldots,\delta_{r})$, $v^{2}=2v$ and direct calculations one can easily verify that $\Phi(\xi+\eta)=\Phi(\xi)+\Phi(\eta)$ and

[TABLE]

Hence $\Phi$ is a ring isomorphism from $(\mathcal{K}_{1}+v\mathcal{K}_{1})\times\ldots\times(\mathcal{K}_{r}+v\mathcal{K}_{r})$ onto $\mathcal{A}+v\mathcal{A}$.

(ii) From the properties of ring isomorphisms and direct product rings, by (i) we conclude that $\mathcal{C}$ is a negacyclic code over $R$ of length $2n$, i.e. $\mathcal{C}$ is an ideal of $\mathcal{A}+v\mathcal{A}$, if and only if for each integer $i$, $1\leq i\leq r$, there is a unique ideal $C_{i}$ of the ring $\mathcal{K}_{i}+v\mathcal{K}_{i}$ such that

[TABLE]

Hence $\mathcal{C}=\bigoplus_{i=1}^{r}\varepsilon_{i}(x)C_{i}$ and $|\mathcal{C}|=|C_{1}\times C_{2}\times\ldots\times C_{r}|=\prod_{i=1}^{r}|C_{i}|$. $\Box$

Using the notations of Theorem 3.3, ${\cal C}=\bigoplus_{i=1}^{r}\varepsilon_{i}(x)C_{i}$ is called the canonical form decomposition of the negacyclic code ${\cal C}$ over $R$ of length $2n$.

Obviously, $\mathcal{K}_{i}+v\mathcal{K}_{i}$ is a free $\mathcal{K}_{i}$-module with basis $\{1,v\}$. Let $\mathcal{K}_{i}^{2}=\{(\alpha,\beta)\mid\alpha,\beta\in\mathcal{K}_{i}\}$. Then $\mathcal{K}_{i}^{2}$ is a free $\mathcal{K}_{j}$-module of rank $2$ with the componentwise addition and scalar multiplication. Now, define

[TABLE]

Then $\sigma$ is an isomorphism of $\mathcal{K}_{i}$-modules from $\mathcal{K}_{i}^{2}$ onto $\mathcal{K}_{i}+v\mathcal{K}_{i}$. Moreover, we have the following key conclusion.

Lemma 4.3 (cf. [5] Lemma 3.4) Let $1\leq i\leq r$. Then $C_{i}$ is an ideal of the ring $\mathcal{K}_{i}+v\mathcal{K}_{i}$ if and only if there is a unique $\mathcal{K}_{i}$-submodule $S_{i}$ of $\mathcal{K}_{i}^{2}$ satisfying

[TABLE]

such that $\sigma(S_{i})=C_{i}$.

For any ideal $C_{i}$ of $\mathcal{K}_{i}+v\mathcal{K}_{i}$, its annihilating ideal is defined by ${\rm Ann}(C_{i})=\{\beta\in\mathcal{K}_{i}+v\mathcal{K}_{i}\mid\alpha\beta=0,\ \forall\alpha\in C_{i}\}$. Now, we list all distinct ideals and their annihilating ideals of the ring $\mathcal{K}_{i}+v\mathcal{K}_{i}$ by the following theorem.

Theorem 4.4 Let $0\leq i\leq r$. Then all distinct ideals $C_{i}$ and their annihilating ideals ${\rm Ann}(C_{i})$ of the ring $\mathcal{K}_{i}+v\mathcal{K}_{i}$ are given by the following table.

[TABLE]

where $a(x),b(x)\in\mathcal{T}_{i}$ and $N$ is the number of ideals in the same row. Then the number of ideals in $\mathcal{K}_{i}+v\mathcal{K}_{i}$ is equal to $2^{2m_{i}}+5\cdot 2^{m_{i}}+9$.

Proof Let $C_{i}$ be an ideal of $\mathcal{K}_{i}+v\mathcal{K}_{j}$. By Lemma 4.3 there is a unique $\mathcal{K}_{i}$-submodule $S$ of $\mathcal{K}_{i}^{2}$ satisfying condition (7) such that $C_{i}=\sigma(S)$.

By Equation (4) there is an invertible element $g_{i}(x)\in\mathcal{K}_{j}$ such that $f_{i}(x)^{2}=2g_{i}(x)$. Let $\omega_{i}(x)=g_{i}(x)^{-1}\in\mathcal{K}_{j}$. Then we have $2=\omega_{i}(x)f_{i}(x)^{2}$. From this we deduce that $S$ satisfying condition (7) if and only if $S$ satisfies the following condition:

[TABLE]

Recall that $\mathcal{K}_{i}$-submodules $S$ of $\mathcal{K}_{i}^{2}$ are called linear codes over $\mathcal{K}_{i}$ of length $2$. Since ${\cal K}_{i}$ is a finite chain ring with maximal ideal $\langle f_{i}(x)\rangle$, the nilpotency index of $f_{i}(x)$ is equal to $4$ and ${\cal T}_{i}={\cal K}_{i}/\langle f_{i}(x)\rangle$ as sets with cardinality $2^{m_{i}}$, by Theorem 2.5 we conclude that $S$ has one of the following matrix $G$ as its generator matrix in standard form:

(I) $G=(f_{i}(x)^{2}(a(x)+b(x)f_{i}(x)),1)$, where $a(x),b(x)\in\mathcal{T}_{i}$. Since $g_{i}(x)\in\mathcal{K}_{i}^{\times}$, we have $2g_{i}(x)=2\overline{g}_{i}(x)$ where $0\neq\overline{g}_{i}(x)\equiv g_{i}(x)$ (mod $2$, mod $\overline{f}_{i}(x)$). Hence $(a(x),b(x))\mapsto(\overline{g}_{i}(x)a(x),\overline{g}_{i}(x)b(x))$ (mod $\overline{f}_{i}(x)$) is a permutation on the set $\mathcal{T}_{i}\times\mathcal{T}_{i}$. Then by $f_{i}(x)^{2}=2g_{i}(x)$ we have

[TABLE]

where $a^{\prime}(x)=\overline{g}_{i}(x)a(x),b^{\prime}(x)=\overline{g}_{i}(x)b(x)\in\mathcal{T}_{i}$. Moreover, by Lemma 2.4 it follows that $|C_{i}|=|S|=|\mathcal{T}_{i}|^{4\cdot 1}=(2^{m_{i}})^{4}=4^{2m_{i}}$. Obviously, the number of ideals is equal to $|\mathcal{T}_{i}\times\mathcal{T}_{i}|=|\mathcal{T}_{i}|^{2}=2^{2m_{i}}$ in this case.

(II) We have one of the following three subcases:

(II-1) $G=(0,f_{i}(x)^{3})$. Then $C_{i}=\langle\sigma(0,f_{i}(x)^{3})\rangle=\langle vf_{i}(x)^{3}\rangle=\langle 2vf_{i}(x)\rangle$ by Equation (4). Moreover, by Lemma 2.4 we have $|C_{i}|=|S|=|\mathcal{T}_{i}|^{1}=2^{m_{i}}$.

(II-2) $G=(f_{i}(x)^{3}b(x),f_{i}(x)^{2})$ where $b(x)\in\mathcal{T}_{i}$. In this case, we have $C_{i}=\langle\sigma(f_{i}(x)^{3}b(x),f_{i}(x)^{2})\rangle=\langle f_{i}(x)^{3}b(x)+vf_{i}(x)^{2}\rangle=\langle 2(f_{i}(x)b(x)+v)\rangle$ by Equation (4). Moreover, by Lemma 2.4 we have $|C_{i}|=|S|=|\mathcal{T}_{i}|^{2\cdot 1}=2^{2m_{i}}$.

(II-3) $G=(f_{i}(x)^{3}a(x),f_{i}(x))$ where $a(x)\in\mathcal{T}_{i}$. By $g_{i}(x)\in\mathcal{K}_{i}^{\times}$, an argument similar to (I) shows that the map $a(x)\mapsto\overline{g}_{i}(x)a(x)$ (mod $\overline{f}_{i}(x)$) is a permutation on the set $\mathcal{T}_{i}$. Then by Equation (4) we have

[TABLE]

where $a^{\prime}(x)=\overline{g}_{i}(x)a(x)\in\mathcal{T}_{i}$ (mod $\overline{f}_{i}(x)$). Moreover, by Lemma 2.4 we have $|C_{i}|=|S|=|\mathcal{T}_{i}|^{3\cdot 1}=2^{3m_{i}}$.

(III) $G=f_{i}(x)^{k}I_{2}$ where $0\leq k\leq 4$. In this case, we have $C_{i}=\langle\sigma(f_{i}(x)^{k},0),\sigma(0,f_{i}(x)^{k})\rangle=\langle f_{i}(x)^{k},vf_{i}(x)^{k}\rangle=\langle f_{i}(x)^{k}\rangle$. Moreover, by Lemma 2.4 we have $|C_{i}|=|S|=|\mathcal{T}_{i}|^{(4-k)\cdot 2}=2^{2(4-k)m_{i}}=4^{(4-k)m_{i}}$. Precisely, by Equation (4) it follows that $\langle f_{i}(x)^{2}\rangle=\langle 2\rangle$ and $\langle f_{i}(x)^{3}\rangle=\langle 2f_{i}(x)\rangle$.

(IV) We have one of the following three subcases:

(IV-1) G=\left(\begin{array}[]{cc}0&1\cr f_{i}(x)&0\end{array}\right). Then $C_{i}=\langle\sigma(0,1),\sigma(f_{i}(x),0)\rangle=\langle f_{i}(x),v\rangle$. Moreover, by Lemma 2.4 we have $|C_{i}|=|S|=|\mathcal{T}_{i}|^{4\cdot 1+3\cdot 1}=2^{7m_{i}}$.

(IV-2) G=\left(\begin{array}[]{cc}f_{i}(x)b(x)&1\cr f_{i}(x)^{2}&0\end{array}\right) where $b(x)\in\mathcal{T}_{i}$. In this case, we have $C_{i}=\langle\sigma(f_{i}(x)b(x),1)$, $\sigma(f_{i}(x)^{2},0)\rangle=\langle f_{i}(x)b(x)+v,f_{i}(x)^{2}\rangle=\langle f_{i}(x)b(x)+v,2\rangle$ by Equation (4). Then by Lemma 2.4 we have $|C_{i}|=|S|=|\mathcal{T}_{i}|^{4\cdot 1+2\cdot 1}=2^{6m_{i}}$.

(IV-3) G=\left(\begin{array}[]{cc}f_{i}(x)^{2}a(x)&1\cr f_{i}(x)^{3}&0\end{array}\right) where $a(x)\in\mathcal{T}_{i}$. By Equation (4) we have

[TABLE]

where $a^{\prime}(x)=\overline{g}_{i}(x)a(x)\in\mathcal{T}_{i}$ (mod $\overline{f}_{i}(x)$). Moreover, by Lemma 2.4 we have $|C_{i}|=|S|=|\mathcal{T}_{i}|^{4\cdot 1+1\cdot 1}=2^{5m_{i}}$.

(V) We have one of the following three subcases:

(V-1) G=\left(\begin{array}[]{cc}0&f_{i}(x)\cr f_{i}(x)^{2}&0\end{array}\right). In this case, we have $C_{i}=\langle\sigma(0,f_{i}(x))$, $\sigma(f_{i}(x)^{2},0)\rangle=\langle vf_{i}(x),f_{i}(x)^{2}\rangle=\langle 2,vf_{i}(x)\rangle$ by Equation (4). Moreover, by Lemma 2.4 we have $|C_{i}|=|S|=|\mathcal{T}_{i}|^{3\cdot 1+2\cdot 1}=2^{5m_{i}}$.

(V-2) G=\left(\begin{array}[]{cc}0&f_{i}(x)^{2}\cr f_{i}(x)^{3}&0\end{array}\right). In this case, we have $C_{i}=\langle\sigma(0,f_{i}(x)^{2})$, $\sigma(f_{i}(x)^{3},0)\rangle=\langle vf_{i}(x)^{2},f_{i}(x)^{3}\rangle=\langle 2f_{i}(x),2v\rangle$ by Equation (4). Moreover, by Lemma 2.4 we have $|C_{i}|=|S|=|\mathcal{T}_{i}|^{2\cdot 1+1\cdot 1}=2^{3m_{i}}$.

(V-3) G=\left(\begin{array}[]{cc}f_{i}(x)^{2}b(x)&f_{i}(x)\cr f_{i}(x)^{3}&0\end{array}\right) where $b(x)\in\mathcal{T}_{i}$. Then by Equation (4) we have

[TABLE]

where $b^{\prime}(x)=\overline{g}_{i}(x)b(x)\in\mathcal{T}_{i}$ (mod $\overline{f}_{i}(x)$). Moreover, by Lemma 2.4 we have $|C_{i}|=|S|=|\mathcal{T}_{i}|^{3\cdot 1+1\cdot 1}=2^{4m_{i}}$. By Theorem 2.5, we see that the number of ideals in $\mathcal{K}_{i}+v\mathcal{K}_{i}$ is equal to $|\mathcal{T}_{i}|^{2}+5|\mathcal{T}_{i}|+9=2^{2m_{i}}+5\cdot 2^{m_{i}}+9$.

Since $\mathcal{K}_{i}+v\mathcal{K}_{i}$ is a local ring with maximal ideal $\langle f_{i}(x),v\rangle$, one can deduce the conclusions for annihilating ideals from $v^{2}=2v$, $\omega_{i}(x)f_{i}(x)^{2}=2$ and a direct computation. $\Box$

Finally, from Theorems 4.2 and 4.4 we deduce the following corollary.

Corollary 4.5 Every negacyclic code over $R$ of length $2n$ can be constructed by the following two steps:

(i) For each $i=1,\ldots,r$, choose an ideal $C_{i}$ of ${\cal K}_{i}+v{\cal K}_{i}$ listed in Theorem 4.4.

(ii) Set ${\cal C}=\bigoplus_{i=1}^{r}\varepsilon_{i}(x)C_{i}$ (mod $x^{2n}+1$).

The number of codewords in ${\cal C}$ is equal to $|{\cal C}|=\prod_{i=1}^{r}|C_{i}|$.

Moreover, the number of negacyclic codes over $R$ of length $2n$ is equal to: $\prod_{i=1}^{r}\left(2^{2m_{i}}+5\cdot 2^{m_{i}}+9\right).$

5 Dual codes of negacyclic codes over $R$ of length $2n$

In this section, we give the dual code of each negacyclic code over $R=\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ of length $2n$ and investigate the self-duality of these codes.

Let $\alpha=(\alpha_{0},\alpha_{1},\ldots,\alpha_{2n-1}),\beta=(\beta_{0},\beta_{1},\ldots,\beta_{2n-1})\in R^{2n}$, where $\alpha_{j},\beta_{j}\in R$ for all $j=0,1\ldots,2n-1$. As usual, we will identify the vector $\alpha$ with $\alpha(x)=\sum_{j=0}^{2n-1}\alpha_{j}x^{j}\in R[x]/\langle x^{2n}+1\rangle$ in this paper. Then we define

[TABLE]

It is clear that $\mu$ is a ring automorphism of $R[x]/\langle x^{2n}+1\rangle$ satisfying $\mu^{-1}=\mu$. Now, by a direct calculation we get the following lemma.

Lemma 5.1 Let $\alpha,\beta\in R^{2n}$. Then $[\alpha,\beta]=\sum_{j=0}^{2n-1}\alpha_{j}\beta_{j}=0$ if $\alpha(x)\mu(\beta(x))=0$ in $R[x]/\langle x^{2n}+1\rangle$.

Using the notations of Section 4, we have $R[x]/\langle x^{2n}+1\rangle={\cal A}+v{\cal A}$ where ${\cal A}=\mathbb{Z}_{4}[x]/\langle x^{2n}+1\rangle$ and $v^{2}=2v$. It is obvious that the restriction of $\mu$ to ${\cal A}$ is a ring automorphism of ${\cal A}$. We still denote this automorphism by $\mu$, i.e. $\mu(a(x))=a(x^{-1})$ for any $a(x)\in{\cal A}$. Let $1\leq i\leq r$. By Equation (6) in Section 3, we have

[TABLE]

For any polynomial $f(x)=\sum_{j=0}^{m}c_{j}x^{j}\in\mathbb{Z}_{4}[x]$ of degree $m\geq 1$, recall that the reciprocal polynomial of $f(x)$ is defined as $\widetilde{f}(x)=x^{m}f(\frac{1}{x})=\sum_{j=0}^{m}c_{j}x^{m-j}$, and $f(x)$ is said to be self-reciprocal if $\widetilde{f}(x)=f(x)$ or $-f(x)$. Then by Equation (3) in Section 3, we have

[TABLE]

Since $f_{1}(x),f_{2}(x),\ldots,f_{r}(x)$ are pairwise coprime monic basic irreducible polynomials in $\mathbb{Z}_{4}[x]$, $\widetilde{f}_{1}(x),\widetilde{f}_{2}(x),\ldots,\widetilde{f}_{r}(x)$ are pairwise coprime basic irreducible polynomials in $\mathbb{Z}_{4}[x]$ as well. Hence for each integer $i$, $1\leq i\leq r$, there is a unique integer $i^{\prime}$, $1\leq i^{\prime}\leq r$, such that $\widetilde{f}_{i}(x)=\delta_{i}f_{i^{\prime}}(x)$ where $\delta_{i}\in\{1,-1\}$. We assume that ${\rm deg}(f_{i}(x))=d_{i}$ and $f_{i}(x)=\sum_{j=0}^{m_{i}}c_{j}x^{j}$ where $c_{j}\in\mathbb{Z}_{4}$. Then

[TABLE]

From this, by Equation (6) and $x^{2n}=-1$ in the ring ${\cal A}$, we deduce that

[TABLE]

where $h_{i}(x)=(-1)^{{\rm deg}(b_{i}(x))+m_{i}+1}\delta_{i}x^{2n-2({\rm deg}(b_{i}(x))+m_{i})}\widetilde{b}_{i}(-x^{2})\in{\cal A}$. Similarly, by (6) it follows that $\mu(\varepsilon_{i}(x))=g_{i}(x)F_{i^{\prime}}(-x^{2})$ for some $g_{i}(x)\in{\cal A}$. Then from these and by (6), we deduce that $\mu(\varepsilon_{i}(x))=\varepsilon_{i^{\prime}}(x)$.

As stated above, we see that for each $1\leq i\leq r$ there is a unique integer $i^{\prime}$, $1\leq i^{\prime}\leq r$, such that $\mu(\varepsilon_{i}(x))=\varepsilon_{i^{\prime}}(x)$. We still use $\mu$ to denote this map $i\mapsto i^{\prime}$. Then $\mu(\varepsilon_{i}(x))=\varepsilon_{\mu(i)}(x)$.

Whether $\mu$ denotes the automorphism of ${\cal A}$ or this map on the set $\{1,2$, $\ldots,r\}$ can be determined by the context. The next lemma shows the compatibility of the two uses of $\mu$.

Lemma 5.2 Using the notations above, we have the following conclusions.

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

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

(iii) For each integer $i$, $1\leq i\leq r$, there is a unique element $\delta_{i}$ of $\{1,-1\}$ such that $\widetilde{f}_{i}(x)=\delta_{i}f_{\mu(i)}(x)$.

(iv) For any integer $i$, $1\leq i\leq r$, $\mu(\varepsilon_{i}(x))=\varepsilon_{\mu(i)}(x)$ and $\mu({\cal A}_{i})={\cal A}_{\mu(i)}$ in the ring ${\cal A}$.

(v) For any $c(x)\in{\cal K}_{i}=\mathbb{Z}_{4}[x]/\langle f_{i}(-x^{2})\rangle$, define

[TABLE]

Then $\mu_{i}=\phi_{\mu(i)}^{-1}\mu\phi_{i}$ which is a ring isomorphism from ${\cal K}_{i}$ onto ${\cal K}_{\mu(i)}$.

(vi) The ring isomorphism $\mu_{i}:{\cal K}_{i}\rightarrow{\cal K}_{\mu(i)}$ induces a ring isomorphism from ${\cal K}_{i}+v{\cal K}_{i}$ onto ${\cal K}_{\mu(i)}+v{\cal K}_{\mu(i)}$ $(v^{2}=2v)$ defined by

[TABLE]

We still denote this isomorphism by $\mu_{i}$. Then $\mu_{i}^{-1}=\mu_{\mu(i)}$.

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

(iv) By $\mu(\varepsilon_{i}(x))=\varepsilon_{\mu(i)}(x)$ and ${\cal A}_{i}=\varepsilon_{i}(x){\cal A}$, it follows that $\mu({\cal A}_{i})=\mu(\varepsilon_{i}(x))\mu({\cal A})=\varepsilon_{\mu(i)}(x){\cal A}={\cal A}_{\mu(i)}$.

(v) Since $x^{2n}=-1$ in the ring $\mathcal{A}$, i.e. $x^{2n}\equiv-1$ (mod $x^{2n}+1$), and $f_{i}(-x^{2})$ is a factor of $x^{2n}+1$, it follows that $x^{2n}\equiv-1$ (mod $f_{i}(-x^{2})$). This implies $x^{-1}=-x^{2n-1}$ in the ring $\mathcal{K}_{i}$ for all $i$, $1\leq i\leq r$. Now, let $c(x)\in{\cal K}_{i}$. By Theorem 3.3(iii) and $\varepsilon_{\mu(i)}(x)=\mu(\varepsilon_{i}(x))=1-h_{i}(x)f_{i^{\prime}}(-x^{2})=1-h_{i}(x)f_{\mu(i)}(-x^{2})$, we have

[TABLE]

This implies $\mu_{i}(c(x))=(\phi_{\mu(i)}^{-1}\mu\phi_{i})(c(x))$ for all $c(x)\in{\cal R}_{i}$. Hence $\mu_{i}=\phi_{\mu(i)}^{-1}\mu\phi_{i}$. Since $\mu$ is a ring automorphism of $\mathcal{A}$, by Theorem 3.3(iii) we conclude that $\mu_{i}$ is a ring isomorphism from ${\cal K}_{i}$ onto ${\cal K}_{\mu(i)}$.

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

By Theorem 4.2(i), each element $\xi\in\mathcal{A}+v\mathcal{A}$ can be uniquely expressed as: $\xi=\sum_{i=1}^{r}\phi_{i}(\xi_{i})=\sum_{i=1}^{r}\varepsilon_{i}(x)\xi_{i}$, where $\xi_{i}\in\mathcal{K}_{i}+v\mathcal{K}_{i}$ for all $i=1,\ldots,r$.

Lemma 5.3 Let $\xi=\sum_{i=1}^{r}\varepsilon_{i}(x)\xi_{i},\eta=\sum_{i=1}^{r}\varepsilon_{i}(x)\eta_{i}\in{\cal A}+v{\cal A}$, where $\xi_{i},\eta_{i}\in{\cal K}_{i}+v{\cal K}_{i}$. Then $\xi\cdot\mu(\eta)=\sum_{i=1}^{r}\varepsilon_{i}(x)(\xi_{i}\cdot\mu_{i}^{-1}(\eta_{\mu(i)}))$.

Proof. By Lemma 5.2(v) we have $\mu_{i}^{-1}(\eta_{\mu(i)})\in\mu_{i}^{-1}({\cal K}_{\mu(i)}+v{\cal K}_{\mu(i)})={\cal K}_{i}+v{\cal K}_{i}$. Hence $\xi_{i}\cdot\mu_{i}^{-1}(\eta_{\mu(i)})\in{\cal K}_{i}+v{\cal K}_{i}$ for all $i$. If $j\neq\mu(i)$, then $i\neq\mu(j)$, which implies $\varepsilon_{i}(x)\varepsilon_{\mu(j)}(x)=0$ by Theorem 3.3(i). Therefore,

[TABLE]

This implies $\xi\cdot\mu(\eta)=\sum_{i=1}^{r}\varepsilon_{i}(x)(\xi_{i}\cdot\mu_{i}^{-1}(\eta_{\mu(i)}))$ by $\mu_{\mu(i)}=\mu_{i}^{-1}$. $\Box$

Lemma 5.4 Let $1\leq i\leq r$. Then $\mu_{i}(f_{i}(x))=-\delta_{i}x^{2n-m_{i}}f_{\mu(i)}(x).$

Proof. Since $f_{i}(-x^{2})|(x^{2n}+1)$ in $\mathbb{Z}_{4}[x]$, we have $x^{2n}=-1$ in ${\cal K}_{i}$. By the definition of $\mu_{i}$ and $\widetilde{f}_{i}(x)=\delta_{i}f_{\mu(i)}(x)$ in Lemma 5.2 (v) and (iii), we have that $\mu_{i}(f_{i}(x))=f_{i}(x^{-1})=-x^{2n-m_{i}}(x^{m_{i}}f_{i}(x^{-1}))=-x^{2n-m_{i}}\widetilde{f}_{i}(x)=-\delta_{i}x^{2n-m_{i}}f_{\mu(i)}(x)$. $\Box$

Now, we can give the dual code of each negacyclic code over the ring $R=\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ of length $2n$ by the following theorem, where $h(x^{-1})=h(-x^{2n-1})$ (mod $f_{\mu(i)}(-x^{2})$) for any $h(x)\in\mathcal{K}_{i}$.

Theorem 5.5 Let ${\cal C}$ be a negacyclic code over $R$ of length $2n$ with ${\cal C}=\oplus_{i=1}^{r}\varepsilon_{i}(x)C_{i}$, where $C_{i}$ is an ideal of ${\cal K}_{i}+v{\cal K}_{i}$. Then the dual code ${\cal C}^{\bot}$ is also a negacyclic code over $R$ of length $2n$. Precisely, we have

[TABLE]

where $D_{j}$ is an ideal of ${\cal K}_{j}+v{\cal K}_{j}$ determined by the following table:

[TABLE]

where $a(x),b(x)\in\mathcal{T}_{i}$.

Proof. Let $1\leq i\leq r$. By Theorem 4.4 we see that $C_{i}\cdot{\rm Ann}(C_{i})=\{0\}$ and $|C_{i}||{\rm Ann}(C_{i})|=4^{4m_{i}}$. Since $\mu_{i}$ is a ring isomorphism from $\mathcal{K}_{i}+v\mathcal{K}_{i}$ onto $\mathcal{K}_{\mu(i)}+v\mathcal{K}_{\mu(i)}$, $\mu_{i}({\rm Ann}(C_{i}))$ is an ideal of $\mathcal{K}_{\mu(i)}+v\mathcal{K}_{\mu(i)}$. Denote $D_{\mu(i)}=\mu_{i}({\rm Ann}(C_{i}))$ and set $\mathcal{D}=\sum_{i=1}^{r}\varepsilon_{\mu(i)}(x)D_{\mu(i)}$. Then $\mathcal{D}=\bigoplus_{j=1}^{r}\varepsilon_{j}(x)D_{j}$ by Lemma 5.2(i) and Theorem 3.3(i). From this and by Theorem 4.4(ii), we deduce that $\mathcal{D}$ is a negacyclic code over $R$ of length $2n$.

As $\mu_{i}^{-1}(D_{\mu(i)})={\rm Ann}(C_{i})$ for all $i$, by Lemma 5.3 it follows that

[TABLE]

This implies ${\cal D}\subseteq{\cal C}^{\bot}$ by Lemma 5.1. On the other hand, we have

[TABLE]

by Theorem 4.2(ii). Since $\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ is a Frobenius ring, from the theory of linear codes over Frobenius rings (see [13]) we deduce that ${\cal C}^{\bot}={\cal D}$.

Finally, for any integer $i$, $1\leq i\leq r$, the expression for each ideal

[TABLE]

follows from Theorem 4.4 and Lemma 5.4 immediately. $\Box$

As the end of this section, we list all distinct self-dual negacyclic codes over $R$ by Theorems 4.2 and 5.5.

Theorem 5.6 Using the notations in Theorem 5.5 and Lemma 5.2$({\rm ii})$, denote

[TABLE]

and

[TABLE]

for any $1\leq i\leq\lambda$. Let ${\cal C}$ be a negacyclic code over $R$ of length $2n$ with ${\cal C}=\bigoplus_{i=1}^{r}\varepsilon_{i}(x)C_{i}$, where $C_{i}$ is an ideal of ${\cal K}_{i}+v{\cal K}_{i}$. Then ${\cal C}$ is self-dual if and only if $C_{i}$ satisfies the following conditions:

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

[TABLE]

where $a(x),b(x)\in\mathcal{T}_{i}$.

(ii) If $1\leq i\leq\lambda$, $C_{i}$ is given by one of the following three cases:

(ii-1) $C_{i}=\langle 2\rangle$.

(ii-2) $C_{i}=\langle 2(a(x)+b(x)f_{i}(x))+v\rangle$, where $(a(x),b(x))\in\mathcal{W}_{i}^{(1)}$.

(ii-3) $C_{i}=\langle 2b(x)+vf_{i}(x),2f_{i}(x)\rangle$, where $b(x)\in\mathcal{W}_{i}^{(2)}$.

The number of self-dual negacyclic codes over $R$ of length $2n$ is equal to

[TABLE]

Proof. By Lemma 5.2 (i) and (ii), we have that $\mu(i)=i$ for all $1\leq i\leq\lambda$, $\mu(i)=i+\epsilon$ and $\mu(i+\epsilon)=i$ for all $\lambda+1\leq i\leq\lambda+\epsilon$. From this by Theorem 5.5 and its proof, we deduce that

[TABLE]

Then by Theorem 4.2, we see that $\mathcal{C}=\mathcal{C}^{\bot}$ if and only of $C_{i}=D_{i}$ for all $i=1,\ldots,r$. Now, we have one of the following two cases.

(i) Let $i=\lambda+j$ where $1\leq j\leq\epsilon$. By $\mu(i)=i+\epsilon$, we have $C_{i+\epsilon}=D_{i+\epsilon}=D_{\mu(i)}$. Then the conclusions follow from Theorem 5.5 immediately.

(ii) $1\leq i\leq\lambda$. In this case, $\mu(i)=i$. Then by $|C_{i}|=|D_{i}|$ and Theorem 5.5, we have one of the following three subcases:

(ii-1) $C_{i}=D_{i}=\langle 2\rangle$.

(ii-2) $C_{i}=\langle 2(a(x)+b(x)f_{i}(x))+v\rangle$ and $D_{i}=\langle v+2(1+a(x^{-1})-\delta_{i}x^{2n-m_{i}}b(x^{-1})f_{i}(x))\rangle$ as ideals of $\mathcal{K}_{i}+v\mathcal{K}_{i}$. Then by Theorem 4.4, Theorem 2.5 and Theorem 3.2(iii), we see that $C_{i}=D_{i}$ if and only if $a(x)=1+a(x^{-1})$ and $b(x)=-\delta_{i}x^{2n-m_{i}}b(x^{-1})$ as elements of the finite field $\mathbb{F}_{2}[x]/\langle\overline{f}_{i}(x)\rangle$. As $-\delta_{i}=1$ in $\mathbb{F}_{2}$, these conditions are equivalent to $(a(x),b(x))\in\mathcal{W}_{i}^{(1)}$.

(ii-3) $C_{i}=\langle 2b(x)+vf_{i}(x),2f_{i}(x)\rangle$ and $D_{i}=\langle 2\widehat{b}(x)-vf_{i}(x)\delta_{i}x^{2n-m_{i}}$, $2f_{i}(x)\rangle$. In this case, $C_{i}=D_{i}$ if and only if $b(x)=\frac{1}{-\delta_{i}x^{2n-m_{i}}}b(x^{-1})$ in $\mathbb{F}_{2}[x]/\langle\overline{f}_{i}(x)\rangle$. Obviously, the latter is equivalent to $b(x)\in\mathcal{W}_{i}^{(2)}$. $\Box$

6 Negacyclic codes over $\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ of length $2M_{p}$

Recall that $2^{p}-1$ is called a Mersenne prime, denoted by

[TABLE]

if $p$ is a prime and $2^{p}-1$ is a prime as well. For example, when $p=2,3,5,7,13,17,19,31$, $M_{p}$ is a prime. In this section, we present negacyclic codes over $\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ ($v^{2}=2v$) of length $2M_{p}$.

Theorem 6.1 Let $n=M_{p}$ where $p\geq 3$. Then the number of all negacyclic codes over $\mathbb{Z}_{4}+u\mathbb{Z}_{4}$ of length $2n$ and the number of self-dual negacyclic codes over $\mathbb{Z}_{4}+u\mathbb{Z}_{4}$ of length $2n$ are equal to

[TABLE]

Proof. Since $n=M_{p}$ and $M_{p}$ is a Mersenne prime, for any integer $s$, $1\leq s\leq n-1$, the $2$-cyclotomic coset modulo $n$ containing $s$ is $J^{(2)}_{s}=\{s,2s,\ldots,2^{p-1}s\}$ with $|J^{(2)}_{s}|=p$. Therefore, the number of $2$-cyclotomic cosets modulo $n$ is equal to $1+\frac{n-1}{p}=1+\frac{2^{p}-1-1}{p}=1+2\cdot\frac{2^{p-1}-1}{p}$.

Suppose that $2^{j}s\equiv-s$ (mod $n$) for some $1\leq j\leq p-1$. Then $(2^{p}-1)|s(2^{j}+1)$. Since ${\rm gcd}(s,2^{p}-1)=1$, we have $(2^{p}-1)|(2^{j}+1)$. This implies $2^{p}-1\leq 2^{j}+1$. On the other hand, by $p\geq 3$ and $1\leq j<p$, we have $(2^{p}-1)-(2^{j}+1)=2^{j}(2^{p-j}-1)-2>0$. This implies $(2^{p}-1)>(2^{j}+1)$, and we get a contradiction. Therefore, $J^{(2)}_{-s}\neq J^{(2)}_{s}$ for all $1\leq s\leq n-1$.

Using the notations of Lemma 5.2(ii), we have $r=1+2\cdot\frac{2^{p-1}-1}{p}$, $\lambda=1$, $\epsilon=\frac{2^{p-1}-1}{p}$, $m_{1}=|J^{(2)}_{0}|=1$ and $m_{i}=|J^{(2)}_{1}|=p$ for all $i=2,3,\ldots,r$.

By Corollary 4.5, the number of all negacyclic codes over $\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ of length $2n$ is $(2^{2}+5\cdot 2+9)\prod_{i=2}^{r}(2^{2p}+5\cdot 2^{p}+9)=23\cdot(4^{p}+5\cdot 2^{p}+9)^{2\cdot\frac{2^{p-1}-1}{p}}.$

It is obvious that $f_{1}(x)=x-1$ and $\widetilde{f}_{1}(x)=1-x=\delta_{1}f_{1}(x)$ with $\delta_{1}=-1$. Hence ${\cal K}_{1}=\mathbb{Z}_{4}[x]/\langle f_{1}(-x^{2})\rangle=\mathbb{Z}_{4}[x]/\langle x^{2}+1\rangle$. Using the notations in Section 3, by $m_{1}=1$ we have ${\cal T}_{1}=\{0,1\}$ and $\mathbb{F}_{2}[x]/\langle\overline{f}_{1}(x)\rangle=\mathbb{F}_{2}$ where $\overline{f}_{1}(x)=x-1\in\mathbb{F}_{2}[x]$. Then we have

$\mathcal{W}_{1}^{(1)}=\{(a,b)\mid a+a+1=b+b=0,\ a,b\in\mathbb{F}_{2}\}=\emptyset$ with $|\mathcal{W}_{1}^{(1)}|=0$;

$\mathcal{W}_{1}^{(2)}=\{b\in\mathbb{F}_{2}\mid b+b=0\}=\mathbb{F}_{2}$ with $|\mathcal{W}_{1}^{(2)}|=2$.

Therefore, using the notations in the proof of Theorem 4.6(i) we see that there are $3$ ideals of $\mathcal{K}_{1}+v\mathcal{K}_{1}$ satisfying $C_{1}=D_{1}$:

[TABLE]

For any $i=1+j$, where $1\leq j\leq\epsilon=\frac{2^{p-1}-1}{p}$, By Theorem 4.4 and $m_{i}=p$ we know that the number of pairs of $(C_{i},C_{i+\epsilon})$ in Theorem 5.6(ii) is equal to $2^{2p}+5\cdot 2^{p}+9=4^{p}+5\cdot 2^{p}+9$.

As stated above, by Theorem 5.6 we conclude that the number of self-dual negacyclic codes over $\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ of length $2(2^{p}-1)$, where both $p$ and $2^{p}-1$ are prime integers, is equal to $3\cdot(4^{p}+5\cdot 2^{p}+9)^{\frac{2^{p-1}-1}{p}}$. $\Box$

For example, when $p=5$, we have $M_{5}=31$. Hence the number of self-dual negacyclic codes over $\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ of length $62$ is equal to

[TABLE]

All these codes can be listed by Theorem 5.6. From these codes, by Proposition 1.3 we obtain $5093808171$ self-dual $2$-quasi-twisted codes over $\mathbb{Z}_{4}$ of length $124$, and $5093808171$ self-dual $4$-quasi-twisted binary codes of length $248$ by the Gray map from $\mathbb{Z}_{4}$ onto $\mathbb{F}_{2}^{2}$: $0\mapsto 00,1\mapsto 01,2\mapsto 11,3\mapsto 10$.

When $p=7$, we have $M_{7}=127$. Hence the number of self-dual negacyclic codes over $\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ of length $254$ is equal to

[TABLE]

All these codes can be listed by Theorem 5.6.

Now, let $p=3$. Then $n=M_{3}=2^{3}-1=7$. By Theorem 6.1 we know that the number of all negacyclic codes over $\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ of length $14$ is

$23\cdot(4^{3}+5\cdot 2^{3}+9)^{2}=23\cdot 113^{2}=293687$,

and the number of self-dual negacyclic codes over $\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ of length $14$ is equal to $3\cdot(4^{3}+5\cdot 2^{3}+9)=3\cdot 113=339$.

Precisely, we have that $x^{7}-1=f_{1}(x)f_{2}(x)f_{3}(x),$ where $f_{1}(x)=x-1$, $f_{2}(x)=x^{3}+2x^{2}+x+3$ and $f_{3}(x)=x^{3}+3x^{2}+2x+3$ are pairwise coprime basic irreducible polynomials in $\mathbb{Z}_{4}[x]$. Hence $r=3$, $m_{1}=1$ and $m_{2}=m_{3}=3$. Obviously, $\widetilde{f}_{1}(x)=\delta_{1}f_{1}(x)$ and $\widetilde{f}_{2}(x)=\delta_{2}f_{3}(x)$ where $\delta_{1}=\delta_{2}=-1$, which implies that $\mu(1)=1$ and $\mu(2)=3$.

For each integer $i$, $1\leq i\leq 3$, we denote $F_{i}(x)=\frac{x^{7}-1}{f_{i}(x)}$, and find polynomials $a_{i}(x),b_{i}(x)\in\mathbb{Z}_{4}[x]$ satisfying $a_{i}(x)F_{i}(x)+b_{i}(x)f_{i}(x)=1$. Then we set $\varepsilon_{i}(x)\equiv a_{i}(-x^{2})F_{i}(-x^{2})$ (mod $x^{14}+1$). Precisely, we have

$\varepsilon_{1}(x)=3+x^{2}+3x^{4}+x^{6}+3x^{8}+x^{10}+3x^{12}$;

$\varepsilon_{2}(x)=1+x^{2}+3x^{4}+2x^{6}+3x^{8}+2x^{10}+2x^{12}$;

$\varepsilon_{3}(x)=1+2x^{2}+2x^{4}+x^{6}+2x^{8}+x^{10}+3x^{12}$.

Using the notations in Section 3, we have ${\cal K}_{i}=\mathbb{Z}_{4}[x]/\langle f_{i}(-x^{2})\rangle$ for $i=1,2,3$, and ${\cal T}_{2}=\{c_{0}+c_{1}x+c_{2}x^{2}\mid c_{0},c_{1},c_{2}\in\{0,1\}\}$. As $x^{14}=-1$, we have $x^{-1}=-x^{13}$ in ${\cal K}_{2}$. Hence for any $a(x)=a_{0}+a_{1}x+a_{2}x^{2}\in{\cal T}_{2}$, it follows that

[TABLE]

and $\overline{a}(x^{-1})=a_{0}+a_{2}x^{2}+a_{1}x^{3}+a_{2}x^{4}+a_{1}x^{5}$. By Theorem 5.6, all distinct self-dual negacyclic codes over $\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ of length $14$ are given by

[TABLE]

where $C_{i}$ is an ideal of the ring ${\cal K}_{i}+v{\cal K}_{i}$ (${\cal K}_{i}=\mathbb{Z}_{4}[x]/\langle f_{i}(-x^{2})\rangle$) satisfying one of the following conditions:

$\bullet$ $C_{1}$ is one of the following $3$ ideals: $\langle 2\rangle$, $\langle v(x-1),2(x-1)\rangle$, $\langle 2+v(x-1),2(x-1)\rangle$.

$\bullet$ $(C_{2},C_{3})$ is given by one of the following $113$ cases, where $a(x),b(x)\in\mathcal{T}_{2}$:

[TABLE]

Among the $339$ self-dual negacyclic codes over $\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ of length $14$, we obtain $36$ codes $\mathcal{C}$ whose Gray image $\theta(\mathcal{C})$ are new good self-dual $2$-quasi-twisted code over $\mathbb{Z}_{4}$ of length $28$ (these codes does not exist in [21] too). The new codes we obtained have new type, better minimum Lee weight, more algebraic structures and properties.

$\diamondsuit$ When $C_{1}=\langle v(x-1),2(x-1)\rangle$ or $C_{1}=\langle 2+v(x-1),2(x-1)\rangle$, and $(C_{2},C_{3})$ is given by one of the following $12$ cases:

[TABLE]

we obtain $24$ new good self-dual $\mathbb{Z}_{4}$ codes $\theta(\mathcal{C})$ with basic parameters $(28,2^{28}$, $d_{L}=8,d_{E}=12)$ and of type $2^{14}4^{7}$, where $d_{L}$ is the minimum Lee weight and $d_{E}$ is the minimum Euclidean weight of $\mathcal{C}$.

$\diamondsuit$ When $C_{1}=\langle 2\rangle$ and $(C_{2},C_{3})$ is given by one of the above $12$ cases, we obtain $12$ new good self-dual $\mathbb{Z}_{4}$ codes $\theta(\mathcal{C})$ with basic parameters $(28,2^{28}$, $d_{L}=6,d_{E}=12)$ and of type $2^{16}4^{6}$.

Remark The existing self-dual codes over $\mathbb{Z}_{4}$ are listed in the database [26] and the maximal code length is $19$ in the table. In [27], the existing linear codes over $\mathbb{Z}_{4}$ of length $24$ have basic parameters $(24,2^{28},d_{L}=6)$ and $(24,2^{28},d_{L}=5)$ and both of type $4^{14}$.

7 Conclusions and further research

We have developed a theory for negacyclic codes over the ring $\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ ($v^{2}=2v$) of oddly even length, including the enumeration and construction of these codes, the dual code and self-duality for each of these codes. These codes enjoy a rich algebraic structure (which makes the search process much simpler) compared to arbitrary linear codes. Our further interest is to consider negacyclic codes over $\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ of arbitrary even length. On the other hand, obtaining some bounds for minimum distance such as BCH-like of a negacyclic code over the ring $\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ by just looking at the canonical form decomposition would be rather interesting.

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, 11471255).

Appendix: Proof of Theorem 2.5

Using the notations of Section 2, by [6] Example 2.5 we know that the number of linear codes over the finite chain ring $\mathcal{K}$ of length $2$ is equal to $\sum_{i=0}^{4}(2i+1)|\mathcal{T}|^{4-i}$. Moreover, every linear code $C$ over $\mathcal{K}$ of length $2$ has one and only one of the following matrices $G$ as their generator matrices:

(i) $G=(1,a)$, $a\in\mathcal{K}$.       (ii) $G=(\pi^{k},\pi^{k}a)$, $a\in\mathcal{K}/\langle\pi^{4-k}\rangle$, $1\leq k\leq 3$.

(iii) $G=(\pi b,1)$, $b\in\mathcal{K}/\langle\pi^{3}\rangle$.

(iv) $G=(\pi^{k+1}b,\pi^{k})$, $b\in\mathcal{K}/\langle\pi^{4-k-1}\rangle$, $1\leq k\leq 3$.

(v) $G=\pi^{k}I_{2}$, $0\leq k\leq 4$.   (vi) G=\left(\begin{array}[]{cc}1&c\cr 0&\pi^{t}\end{array}\right), $c\in\mathcal{K}/\langle\pi^{t}\rangle$, $1\leq t\leq 3$.

(vii) G=\left(\begin{array}[]{cc}\pi^{k}&\pi^{k}c\cr 0&\pi^{k+t}\end{array}\right), $c\in\mathcal{K}/\langle\pi^{t}\rangle$, $1\leq t\leq 4-k-1$, $1\leq k\leq 2$.

(viii) G=\left(\begin{array}[]{cc}c&1\cr\pi^{t}&0\end{array}\right), where $c\in\pi(\mathcal{K}/\langle\pi^{t}\rangle)$ and $1\leq t\leq 3$.

(ix) G=\left(\begin{array}[]{cc}\pi^{k}c&\pi^{k}\cr\pi^{k+t}&0\end{array}\right), $c\in\pi(\mathcal{K}/\langle\pi^{t}\rangle)$, $1\leq t\leq 4-k-1$, $1\leq k\leq 2$.

Therefore, we only need to consider the nine cases listed above:

(i) Suppose that $C$ satisfies Condition (2). By $(1,a)\in C$, we have $(0,1+\omega\pi^{2}a)\in C$. Since $G$ is the generator matrix of $C$, there exists $b\in\mathcal{K}$ such that $(0,1+\omega\pi^{2}a)=b(1,a)=(b,ba)$, i.e. $0=b$ and $1+\omega\pi^{2}a=ba$. This implies $1+\omega\pi^{2}a=0$. So we get a contradiction, since $1+\omega\pi^{2}a$ is an invertible element of $\mathcal{K}$. Hence $C$ does not satisfy Condition (2) in this case.

(ii) Suppose that $C$ satisfies Condition (2). By $(\pi^{k},\pi^{k}a)\in C$, we have $(0,\pi^{k}+\omega\pi^{k+2})=(0,\pi^{k}+\omega\pi^{2}\cdot\pi^{k}a)\in C$. Then there exists $b\in\mathcal{K}$ such that $(0,\pi^{k}+\omega\pi^{k+2}a)=b(\pi^{k},\pi^{k}a)=(\pi^{k}b,\pi^{k}ba)$. This implies $0=\pi^{k}b$ and $\pi^{k}(1+\omega\pi^{2}a)=\pi^{k}ba$. Hence $\pi^{k}(1+\omega\pi^{2}a)=0$. Since $1+\omega\pi^{2}a$ is an invertible element of $\mathcal{K}$, we deduce $\pi^{k}=0$, which contradict that $k\leq 3$.

(iii) In this case, $C$ satisfies Condition (2) if and only if there exists $a\in\mathcal{K}$ such that $(0,\pi b+\omega\pi^{2})=(0,\pi b+\omega\pi^{2}\cdot 1)=a(\pi b,1)=(\pi ab,a)$, i.e. $0=\pi ab$ and $\pi(b+\omega\pi)=\pi b+\omega\pi^{2}=a$. These conditions are equivalent to that $b$ satisfies $\pi^{2}(b+\omega\pi)b=0$, i.e. $(b+\omega\pi)b\in\pi^{2}\mathcal{K}$. The latter condition is equivalent to $\|(b+\omega\pi)b\|_{\pi}\geq 2$. As $b\in\mathcal{K}/\langle\pi^{3}\rangle$, $b$ has a unique $\pi$-expansion: $b=r_{0}+r_{1}\pi+r_{2}\pi^{2}$ with $r_{0},r_{1},r_{2}\in\mathcal{T}$. This implies $b+\omega\pi=r_{0}+(\omega+r_{1})\pi+r_{2}\pi^{2}$. Suppose that $r_{0}\neq 0$. Then $\|b\|_{\pi}=\|b+\omega\pi\|_{\pi}=0$. This implies $\|(b+\omega\pi)b\|_{\pi}=0$ and we get a contradiction. Now, let $r_{0}=0$. Then by $(b+\omega\pi)b=\pi^{2}(\omega+r_{1}+r_{2}\pi)(r_{1}+r_{2}\pi)$ it follow that $\|(b+\omega\pi)b\|_{\pi}\geq 2$. Hence $b=\pi(r_{1}+r_{2}\pi)$ for all $r_{1},r_{2}\in\mathcal{T}$ in this case.

(iv) In this case, $C$ satisfies Condition (2) if and only if there exists $a\in\mathcal{K}$ such that $(0,\pi^{k+1}(b+\omega\pi))=(0,\pi^{k+1}b+\omega\pi^{2}\cdot\pi^{k})=a(\pi^{k+1}b,\pi^{k})=(\pi^{k+1}ab,\pi^{k}a)$, i.e. $0=\pi^{k+1}ab=\pi b\cdot\pi^{k}a$ and $\pi^{k+1}(b+\omega\pi)=\pi^{k}a$. These conditions are equivalent to that $b$ satisfies $\pi^{k+2}(b+\omega\pi)b=0$, i.e. $\|(b+\omega\pi)b\|_{\pi}\geq 4-k-2$ where $1\leq k\leq 3$. Then we have one of the following three cases:

(iv-1) Let $k=3$. Then $\|(b+\omega\pi)b\|_{\pi}\geq 4-k-2=-1$ for any $b\in\mathcal{K}/\langle\pi^{4-3-1}\rangle=\mathcal{K}/\langle 1\rangle=\{0\}$. Hence $G=(0,\pi^{3})$.

(iv-2) Let $k=2$. $\|(b+\omega\pi)b\|_{\pi}\geq 4-k-2=0$ for any $b\in\mathcal{K}/\langle\pi^{4-2-1}\rangle=\mathcal{K}/\langle\pi\rangle=\mathcal{T}$. In this case, we have $G=(\pi^{3}b,\pi^{2})$.

(iv-3) Let $k=1$. $\|(b+\omega\pi)b\|_{\pi}\geq 4-k-2=1$ if and only if $b\in\pi(\mathcal{K}/\langle\pi^{4-1-1}\rangle)=\pi(\mathcal{K}/\langle\pi^{2}\rangle)$. In this case, we have $G=(\pi^{2}b,\pi)=(\pi^{3}a,\pi)$ for all $b=\pi a$ with $a\in\mathcal{T}$.

(v) In this case, $C$ satisfies Condition (2) for all $0\leq k\leq 4$.

(vi) Suppose that $C$ satisfies Condition (2). Then there exist $a,b\in\mathcal{K}$ such that $(0,1+\omega\pi^{2}c)=a(1,c)+b(0,\pi^{t})=(a,ac+\pi^{t}b)$, i.e. $0=a$ and $1+\omega\pi^{2}c=ac+\pi^{t}b$. This implies $1=\pi(\pi^{t-1}b-\omega\pi c)$ and we get a contradiction. Hence $C$ does not satisfy Condition (2) in this case.

(vii) Suppose that $C$ satisfies Condition (2). Then there exist $a,b\in\mathcal{K}$ such that $(0,\pi^{k}+\omega\pi^{2}\cdot\pi^{k}c)=a(\pi^{k},\pi^{k}c)+b(0,\pi^{k+t})=(\pi^{k}a,\pi^{k}ac+\pi^{k+t}b)$, i.e. $0=\pi^{k}a$ and $\pi^{k}+\omega\pi^{k+2}c=\pi^{k}a\cdot c+\pi^{k+t}b$. This implies $\pi^{k}=\pi^{k+1}(\pi^{t-1}b-c\omega\pi)$, and we get a contradiction as $1\leq k\leq 3$.

(viii) It is clear that $(0,\pi^{t})=\pi^{t}(c,1)-c(\pi^{t},0)\in C$. Hence $C$ satisfies Condition (2) if and only if there exist $a,b\in\mathcal{K}$ such that $(0,c+\omega\pi^{2}\cdot 1)=a(c,1)+b(\pi^{t},0)=(ac+\pi^{t}b,a)$, i.e. $0=ac+\pi^{t}b$ and $c+\omega\pi^{2}=a$, which are equivalent to that $(c+\omega\pi^{2})c=-\pi^{t}b\in\pi^{t}\mathcal{K}$ for some $b\in\mathcal{K}$, i.e. $\|(c+\omega\pi^{2})c\|_{\pi}\geq t$, where $c\in\pi(\mathcal{K}/\langle\pi^{t}\rangle)$ and $1\leq t\leq 3$. Then we have one of the following three subcases:

(viii-1) When $t=1$, then $c\in\pi(\mathcal{K}/\langle\pi\rangle)=\{0\}$, i.e. $c=0$. In this case, $\|(c+\omega\pi^{2})c\|_{\pi}=\|0\|_{\pi}=4>t$. Hence G=\left(\begin{array}[]{cc}0&1\cr\pi&0\end{array}\right).

(viii-2) When $t=2$, then $c\in\pi(\mathcal{K}/\langle\pi^{2}\rangle)=\pi\mathcal{T}$, i.e. $c=\pi z$ where $z\in\mathcal{T}$. In this case, $\|(c+\omega\pi^{2})c\|_{\pi}=\|(\pi h+\omega\pi^{2})\cdot\pi h\|_{\pi}\geq 2=t$. Hence G=\left(\begin{array}[]{cc}\pi z&1\cr\pi^{2}&0\end{array}\right).

(viii-3) When $t=3$, then $c\in\pi(\mathcal{K}/\langle\pi^{3}\rangle)$, i.e. $c=r_{1}\pi+r_{2}\pi^{2}$ where $r_{1},r_{2}\in\mathcal{T}$. Suppose $r_{1}\neq 0$. Then $\|c\|_{\pi}=\|c+\omega\pi^{2}\|_{\pi}=1$. This implies $\|(c+\omega\pi^{2})c\|_{\pi}$ $=2<3=t$. Now, let $r_{1}=0$. Then it is obvious that $\|(c+\omega\pi^{2})c\|_{\pi}=4>t$ for all $c=\pi^{2}z$ where $z=r_{2}\in\mathcal{T}$. Hence G=\left(\begin{array}[]{cc}\pi^{2}z&1\cr\pi^{3}&0\end{array}\right).

(xi) It is clear that $(0,\pi^{k+t})=\pi^{t}(\pi^{k}c,\pi^{k})-c(\pi^{k+t},0)\in C$. Hence $C$ satisfies Condition (2) if and only if there exist $a,b\in\mathcal{K}$ such that $(0,\pi^{k}c+\omega\pi^{2}\cdot\pi^{k})=a(\pi^{k}c,\pi^{k})+b(\pi^{k+t},0)=(\pi^{k}ac+\pi^{k+t}b,\pi^{k}a)$, i.e. $0=\pi^{k}ac+\pi^{k+t}b$ and $\pi^{k}(c+\omega\pi^{2})=\pi^{k}a$, which are equivalent to that $\pi^{k}(c+\omega\pi^{2})c=-\pi^{k+t}b\in\pi^{k+t}\mathcal{K}$, i.e. $\|(c+\omega\pi^{2})c\|_{\pi}\geq t$, where $c\in\pi(\mathcal{K}/\langle\pi^{t}\rangle)$, $1\leq t\leq 3-k$ and $1\leq k\leq 2$. Then we have one of the following two subcases:

(xi-1) Let $k=1$. Then $1\leq t\leq 2$. If $t=1$, then $c\in\pi(\mathcal{K}/\langle\pi\rangle)=\{0\}$, i.e. $c=0$. Obviously, $\|(c+\omega\pi^{2})c\|_{\pi}=4>t$. Hence G=\left(\begin{array}[]{cc}0&\pi\cr\pi^{2}&0\end{array}\right).

If $t=2$, then $c\in\pi(\mathcal{K}/\langle\pi^{2}\rangle)=\pi\mathcal{T}$, i.e. $c=\pi z$ where $z\in\mathcal{T}$. Obviously, $\|(c+\omega\pi^{2})c\|_{\pi}=2\geq t$. Hence G=\left(\begin{array}[]{cc}\pi^{2}z&\pi\cr\pi^{3}&0\end{array}\right).

(xi-2) Let $k=2$. Then $t=1$ and $c=0$. Hence G=\left(\begin{array}[]{cc}0&\pi^{2}\cr\pi^{3}&0\end{array}\right).