Complete classification for simple root cyclic codes over the local ring $\mathbb {Z}_{4}[v]/\langle v^{2}+2v\rangle$

Abstract

Let $R=\mathbb {Z}_{4}[v]/\langle v^{2}+2v\rangle$. Then R is a local non-principal ideal ring of 16 elements. First, we give the structure of every cyclic code of odd length n over R and obtain a complete classification for these codes. Then we determine the cardinality, the type and its dual code for each of these cyclic codes. Moreover, we determine all self-dual cyclic codes of odd length n over R and provide a clear formula to count the number of these self-dual cyclic codes. Finally, we list some optimal 2-quasi-cyclic self-dual linear codes of length 30 over $\mathbb {Z}_{4}$ and obtain 4-quasi-cyclic and formally self-dual binary linear [60,30,12] codes derived from cyclic codes of length 15 over $\mathbb {Z}_{4}[v]/\langle v^{2}+2v\rangle$.

1 Introduction

The catalyst for the study of codes over rings was the discovery of the connection between linear codes over $\mathbb {Z}_{4}$ and the Kerdock and Preparata codes, which are non-linear binary codes [2, 3]. 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 (see [1, 4,5,6, 11, 12, 16, 17, 19] and [20], for examples) and other type of rings [7, 18].

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 in Dougherty et al. [13]. Especially, codes over an extension rings of $\mathbb {Z}_{4}$ were studied in Yildiz et al. [25] and [26] where many good $\mathbb {Z}_{4}$-codes were obtained as images. The ring in the mentioned works was described as $\mathbb {Z}_{4}[u]/\langle u^{2}\rangle =\mathbb {Z}_{4}+u\mathbb {Z}_{4}$ (u² = 0) which is a local non-principal ring. Shi et al. in [18] studied (1 + 2u)-constacyclic codes of odd length n over the ring $\mathbb {Z}_{4}[u]/\langle u^{2}-1\rangle =\mathbb {Z}_{4}+u\mathbb {Z}_{4}$ (u² = 1) which is another extension ring of $\mathbb {Z}_{4}$. Properties of these codes and their $\mathbb {Z}_{4}$ images were investigated.

Let A be an arbitrary finite commutative ring with identity 1 ≠ 0, and A^× be the multiplicative group of invertible elements (units) in A. For any a ∈ A, we denote by 〈a〉_A, or 〈a〉 for simplicity, the ideal of A generated by a, i.e. 〈a〉_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 = {(a₀,a₁,…,a_N− 1)∣a_i ∈ A, i = 0,1,…,N − 1} which is an A-module with componentwise addition and scalar multiplication by elements of A. Then an A-submodule $\mathcal {C}$ of A^N is called a linear code of length N over A. For any vectors a = (a₀,a₁,…,a_N− 1),b = (b₀,b₁,…,b_N− 1) ∈ A^N. The usual Euclidian inner product of a and b is defined by $[a,b]={\sum }_{j=0}^{N-1}a_{j}b_{j}\in A$. Then [−,−] is a symmetric and non-degenerate bilinear form on the A-module A^N. Let $\mathcal {C}$ be a linear code of length N over A. The dual code of $\mathcal {C}$ is defined by $\mathcal {C}^{\bot }=\{a\in A^{N}\mid [a,b]=0, \ \forall b\in \mathcal {C}\}$, and $\mathcal {C}$ is said to be self-dual if $\mathcal {C}=\mathcal {C}^{\bot }$. A linear code $\mathcal {C}$ of length N over A is said to be cyclic if (a_N− 1,a₀,a₁,…, $a_{N-2})\in \mathcal {C}$ for all $(a_{0},a_{1},\ldots ,a_{N-1})\in \mathcal {C}$. Let A be a local ring with residue class field F. Then cyclic codes of length N over R are called simple root cyclic codes if $\gcd (N, \text {char}(F))=1$.

In this paper, every vector c = (c₀,c₁,…,c_N− 1) ∈ A^N is viewed as the polynomial $c(x)={\sum }_{j=0}^{N-1}c_{j}x^{j}$. Then every cyclic code $\mathcal {C}$ is viewed as an ideal in the polynomial residue ring A[x]/〈x^N − 1〉.

In this paper, we adopt the following notation:

• ◇ $\mathbb {Z}_{2}=\{0,1\}$ in which the arithmetic is done modulo 2. Then $\mathbb {Z}_{2}$ is a binary finite field.

• ◇ $\mathbb {Z}_{4}=\{0,1,2,3\}$ in which the arithmetic is done modulo 4. Then $\mathbb {Z}_{4}$ is a finite chain ring with the maximal ideal $2\mathbb {Z}_{4}=\{0,2\}$.

• ◇ $R=\mathbb {Z}_{4}[v]/\langle v^{2}+2v\rangle =\{a+bv\mid a,b\in \mathbb {Z}_{4}\}=\mathbb {Z}_{4}+v\mathbb {Z}_{4} \ (v^{2}=2v)$ in which the operations are defined by:

  $$\alpha+\beta=(a+c)+v(b+d)\ \text{and} \ \alpha\beta=ac+(ad+bc+2bd)v,$$

  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-chain ring of 16 elements.

In 2015, linear codes over $\mathbb {Z}_{4}+v\mathbb {Z}_{4} (v^{2}=2v)$ 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. Recently, we gave a complete classification for negacyclic codes of length 2n over $\mathbb {Z}_{4}+v\mathbb {Z}_{4}$, where n is odd, and obtained some good and new self-dual $\mathbb {Z}_{4}$-codes which are are Gray images of self-dual negacyclic codes over $\mathbb {Z}_{4}+v\mathbb {Z}_{4}$ ([8]).

Now, we follow this line to continue studying cyclic codes of length n over $R=\mathbb {Z}_{4}+v\mathbb {Z}_{4}$, where n is a positive odd integer and n ≥ 3.

The rest of the paper is organized as follows. In Section 2, we give an explicit representation for every cyclic code of length n over R by determining their generator sets as ideals in the ring R[x]/〈xⁿ − 1〉. In Section 3, we determine the dual code for each code, present explicitly all distinct self-dual cyclic codes of length n over R and give a clear formula to count the number of all these self-dual cyclic codes. In Section 4, we present all 583443 cyclic codes of length 15 over R and list all 315 self-dual codes among them. As applications, we obtain 162 good self-dual quasi-cyclic codes of index 2 and length 30 with minimum Lee weight 10 and 12 over $\mathbb {Z}_{4}$. From these $\mathbb {Z}_{4}$-codes, we derive 70 quasi-cyclic type II binary formal self-dual [60,30,12] codes of index 4.

2 Cyclic codes over $\mathbb {Z}_{4}+v\mathbb {Z}_{4}$ of odd length

In this section, we consider cyclic codes of odd length n over the ring $R=\mathbb {Z}_{4}[v]/\langle v^{2}+2v\rangle =\mathbb {Z}_{4}+v\mathbb {Z}_{4} (v^{2}=2v)$, i.e., ideals of the ring

$$ R[x]/\langle x^{n}-1\rangle=\left\{{\sum}_{i=0}^{n-1}r_{i}x^{i}\mid r_{i}\in R, \ i=0,1,\ldots,n-1\right\} $$

in which the arithmetic is done modulo xⁿ − 1. From now on, we denote

• $\mathcal {A}=\mathbb {Z}_{4}[x]/\langle x^{n}-1\rangle =\left \{{\sum }_{i=0}^{n-1}a_{i}x^{i}\mid a_{i}\in \mathbb {Z}_{4}, \ i=0,1,\ldots ,n-1\right \}$

  in which the arithmetic is done modulo xⁿ − 1.

  Then $\mathcal {A}$ is a finite commutative ring containing $\mathbb {Z}_{4}$ as its subring.

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

  and the operations are defined by: for any $\alpha _{1},\alpha _{2},\beta _{1}, \beta _{2}\in \mathcal {A}$, we have

  $$ \begin{array}{@{}rcl@{}} &&(\alpha_{1}+\beta_{1} v)+(\alpha_{2}+\beta_{2} v)=(\alpha_{1}+\alpha_{2})+v(\beta_{1}+\beta_{2});\\ &&(\alpha_{1}+\beta_{1} v)(\alpha_{2}+\beta_{2} v)=\alpha_{1}\alpha_{2}+v(\alpha_{1}\beta_{2}+\beta_{1}\alpha_{2}+2\beta_{1}\beta_{2}). \end{array} $$

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

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

$$ {\varTheta}(\alpha+\beta v)={\sum}_{i=0}^{n-1}\xi_{i}x^{i}, \ \text{where} \ \xi_{i}=a_{i}+b_{i}v\in R, \ i=0,1,\ldots,n-1. $$

Then one can easily verify the following conclusion.

Lemma 1

The mapΘdefinedabove is an isomorphism of rings from$\mathcal {A}+v\mathcal {A}$ontoR[x]/〈xⁿ − 1〉.

In the following, we will identify $\mathcal {A}+v\mathcal {A}$ with R[x]/〈xⁿ − 1〉 under the ring isomorphism Θ. Therefore, in order to determine all cyclic codes of length n over R, we only need to determine all ideals of the ring $\mathcal {A}+v\mathcal {A}$. To do this, we investigate the structure of the ring $\mathcal {A}$ first.

In this paper, we will regard $\mathbb {Z}_{2}$ as a subset of the ring $\mathbb {Z}_{4}$ although $\mathbb {Z}_{2}$ is not a subring of $\mathbb {Z}_{4}$. Then every element a of $\mathbb {Z}_{4}$ has a unique 2-adic expansion: $a=a_{0}+2a_{1}, \ a_{0},a_{1}\in \mathbb {Z}_{2}$. Define $\overline {a}=a_{0}=a$ (mod 2) for all $a\in \mathbb {Z}_{4}$. This map ⁻ is a surjective homomorphism of rings from $\mathbb {Z}_{4}$ onto $\mathbb {Z}_{2}$, which can be extended into a surjective homomorphism of rings from $\mathbb {Z}_{4}[x]$ onto $\mathbb {Z}_{2}[x]$ in the natural way:

$$ \overline{f}(x)=\overline{f(x)}={\sum}_{k}\overline{f}_{k}x^{k}, \ \forall f(x)={\sum}_{k}f_{k}x^{k}\in \mathbb{Z}_{4}[x] \ \text{where} \ a_{k}\in \mathbb{Z}_{4}. $$

Let g(x) be a monic polynomial in $\mathbb {Z}_{4}[x]$ of positive integer. Then g(x) is said to be basic irreducible if $\overline {g}(x)$ is an irreducible polynomial in $\mathbb {Z}_{2}[x]$.

As n is odd, by [22, Theorem 13.8] there are pairwise coprime monic basic irreducible polynomials $f_{0}(x)=x-1,f_{1}(x),\ldots ,f_{r}(x)\in \mathbb {Z}_{4}[x]$ such that

$$ x^{n}-1=f_{0}(x)f_{1}(x){\ldots} f_{r}(x), $$

(1)

where $\overline {f}_{j}(x)$ is irreducible in $\mathbb {Z}_{2}[x]$ and deg(f_j(x)) = m_j for all j = 0,1,…,r. Especially, f₀(x) = x − 1 with degree m₀ = 1.

For each integer j, 0 ≤ j ≤ r, denote $F_{j}(x)=\frac {x^{n}-1}{f_{j}(x)}\in \mathbb {Z}_{4}[x]$. Since $\gcd (\overline {F}_{j}(x), \overline {f}_{j}(x))=1$, we see that F_j(x) and f_j(x) are coprime in $\mathbb {Z}_{4}[x]$ (cf. [22, Lemma 13.5]). Hence there are polynomials $c_{j}(x), d_{j}(x)\in \mathbb {Z}_{4}[x]$ such that

$$ c_{j}(x)F_{j}(x)+ d_{j}(x)f_{j}(x)=1. $$

(2)

In this paper, we adopt the following notation:

• Let $e_{j}(x)\in \mathcal {A}$ satisfying

  $$ e_{j}(x)\equiv c_{j}(x)F_{j}(x)=1-d_{j}(x)f_{j}(x) \ (\text{mod} \ x^{n}-1). $$

  (3)

• Denote $\mathcal {K}_{j}=\mathbb {Z}_{4}[x]/\langle f_{j}(x)\rangle =\left \{{\sum }_{i=0}^{m_{j}-1}a_{i}x^{i}\mid a_{i}\in \mathbb {Z}_{4}, \ i=0,1,\ldots ,m_{j}-1\right \}$

  in which the arithmetic is done modulo $f_{j}(x)$. Then $\mathcal {K}_{j}$ is a Galois ring of characteristic 4 and cardinality $4^{m_{j}}$ (cf. [22, Theorem 14.1]).

• Let $\mathcal {K}_{j}+v\mathcal {K}_{j}=\mathcal {K}_{j}[v]/\langle v^{2}+2v\rangle =\{\alpha +\beta v\mid \alpha ,\beta \in \mathcal {K}_{j}\} \ (v^{2}=2v)$

  and the operations are defined by: for any $\alpha _{1},\alpha _{2},\beta _{1}, \beta _{2}\in \mathcal {K}_{j}$, we have

  $$ \begin{array}{@{}rcl@{}} &&(\alpha_{1}+\beta_{1} v)+(\alpha_{2}+\beta_{2} v)=(\alpha_{1}+\alpha_{2})+v(\beta_{1}+\beta_{2});\\ &&(\alpha_{1}+\beta_{1} v)(\alpha_{2}+\beta_{2} v)=\alpha_{1}\alpha_{2}+v(\alpha_{1}\beta_{2}+\beta_{1}\alpha_{2}+2\beta_{1}\beta_{2}). \end{array} $$

  Then $\mathcal {K}_{j}+v\mathcal {K}_{j}$ is a finite commutative local ring containing $\mathcal {K}_{j}$ as its subring.

By [21, Theorem 2.7] and [7, Lemma 3.2], one can easily deduce the following conclusions.

Lemma 2

Using the notation above, we have the following conclusions:

• (i) e₀(x) + e₁(x) + … + e_r(x) = 1,e_j(x)² = e_j(x) ande_j(x)e_l(x) = 0 inthe ring$\mathcal {A}$forall 0 ≤ j≠l ≤ r.

• (ii) For each integer j, 0 ≤ j ≤ r,$\mathcal {A}e_{j}(x)$isa subring of$\mathcal {A}$withe_j(x) as its multiplicative identity.Define

  $$ \varphi_{j}(a(x))=a(x)e_{j}(x) \ (\text{mod} \ x^{n}-1), \ \forall a(x)\in \mathcal{K}_{j}. $$

  Then φ_j is a ring isomorphism from $\mathcal {K}_{j}$ onto $\mathcal {A}e_{j}(x)$ with inverse $\varphi _{j}^{-1}$:

  $$ \varphi_{j}^{-1}(c(x))=c(x) \ (\text{mod} \ f_{j}(x)), \ \forall c(x)\in \mathcal{A}e_{j}(x). $$

• (iii) For any $a_{j}(x)\in \mathcal {K}_{j}$ and j = 0,1,…,r, define

  $$ \varphi(a_{0}(x),a_{1}(x),\ldots,a_{r}(x))=\sum\limits_{j=0}^{r}\varphi_{j}(a_{j}(x)) =\sum\limits_{j=0}^{r}e_{j}(x)a_{j}(x) \ (\text{mod} \ x^{n}-1). $$

  Then φ is a ring isomorphism from $\mathcal {K}_{0}\times \mathcal {K}_{1}\times \ldots \times \mathcal {K}_{r}$ onto $\mathcal {A}$.

Now, we give structural properties for cyclic codes of length n over R.

Lemma 3

Let$\mathcal {C}\subseteq \mathcal {A}+v\mathcal {A}$.Then$\mathcal {C}$isa cyclic code of length n over R if and only if for each integer j,0 ≤ j ≤ r, there is aunique idealC_jofthe ring$\mathcal {K}_{j}+v\mathcal {K}_{j} (v^{2}=2v)$suchthat

$$ \mathcal{C}=e_{0}(x)C_{0}\oplus e_{1}(x)C_{1}\oplus\ldots\oplus e_{r}(x)C_{r} \ (\text{mod} \ x^{n}-1) $$

where

$$ e_{j}(x)C_{j}=\{e_{j}(x)\alpha+v e_{j}(x)\beta\mid \alpha+\beta v\in C_{j}, \ \alpha,\beta \in \mathcal{K}_{j}\} \subseteq \mathcal{A}+v\mathcal{A} $$

for all j = 0,1,…,r. Then the number of codewords in $\mathcal {C}$ equals ${\prod }_{j=0}^{r}|C_{j}|$.

Proof

For any $\xi _{j}=\alpha _{j}+\beta _{j} v\in \mathcal {K}_{j}+v\mathcal {K}_{j}$ with $\alpha _{j},\beta _{j}\in \mathcal {K}_{j}$ for all j = 0,1,…,r, we define

$$ \begin{array}{@{}rcl@{}} {\varPhi}(\xi) &=&\sum\limits_{j=0}^{r}e_{j}(x)\xi_{j} =\sum\limits_{j=0}^{r}e_{j}(x)\left( \alpha_{j}+v\beta_{j} \right) =\sum\limits_{j=0}^{r}e_{j}(x)\alpha_{j}+v\sum\limits_{j=0}^{r}e_{j}(x)\beta_{j}\\ &=&\varphi(\alpha_{0},\alpha_{1},\ldots,\alpha_{r})+v\varphi(\beta_{0},\beta_{1},\ldots,\beta_{r}). \end{array} $$

Then Φ is an isomorphism of rings from the direct product ring $(\mathcal {K}_{0}+v\mathcal {K}_{0})\times (\mathcal {K}_{1}+v\mathcal {K}_{1})\times \ldots \times (\mathcal {K}_{r}+v\mathcal {K}_{r})$ onto $\mathcal {A}+v\mathcal {A}$. This conclusion can be proved by Lemma 2 similarly as that for [8, Theorem 4.2(i)]. Here, we omit it.

From the properties of ring isomorphisms and direct product rings, we conclude that $\mathcal {C}$ is a cyclic code of length n over R, i.e. $\mathcal {C}$ is an ideal of $\mathcal {A}+v\mathcal {A}$, if and only if for each integer j, 0 ≤ j ≤ r, there is a unique ideal C_j of the ring $\mathcal {K}_{j}+v\mathcal {K}_{j}$ such that

$$ \begin{array}{@{}rcl@{}} \mathcal{C}&=&{\varPhi}(C_{0}\times C_{1}\times\ldots\times C_{r}) =\{{\varPhi}(\xi_{0},\xi_{1}, \ldots, \xi_{r})\mid \xi_{j}\in C_{j}, \ j=0,1,\ldots,r\}\\ &=&\left\{{\sum}_{j=0}^{r}e_{j}(x)\xi_{j}\mid \xi_{j}\in C_{j}, \ j=0,1,\ldots,r\right\} ={\sum}_{j=0}^{r}e_{j}(x)\{\xi_{j}\mid \xi_{j}\in C_{j}\}. \end{array} $$

Hence $\mathcal {C}=\bigoplus _{j=0}^{r}e_{j}(x)C_{j}$ (mod xⁿ − 1) and $|\mathcal {C}|=|C_{0}\times C_{1}\times \ldots \times C_{r}|={\prod }_{j=0}^{r}|C_{j}|$. □

By Lemma 3, it is sufficient to determine the ideals of the ring $\mathcal {K}_{j}+v\mathcal {K}_{j}$ for all j, in order to determine cyclic codes of length n over R.

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

$$ \sigma: \mathcal{K}_{j}^{2}\rightarrow \mathcal{K}_{j}+v\mathcal{K}_{j} \ \text{via} \ (\alpha,\beta)\mapsto \alpha+\beta v \ (\forall \alpha,\beta\in \mathcal{K}_{j}). $$

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

Lemma 4

(cf. [10, Lemma 3.4]) Let 0 ≤ j ≤ r.ThenC_jis an ideal of thering$\mathcal {K}_{j}+v\mathcal {K}_{j}$if and only if thereis a unique$\mathcal {K}_{j}$-submoduleS_jof$\mathcal {K}_{j}^{2}$satisfyingthe following condition

$$ (0,\alpha+2\beta)\in S_{j}, \ \forall (\alpha,\beta)\in S_{j} $$

(4)

such that σ(S_j) = C_j.

Therefore, in order to determine all ideals of $\mathcal {K}_{j}+v\mathcal {K}_{j}$ (0 ≤ j ≤ r), it is sufficient to determine all $\mathcal {K}_{j}$-submodule of $\mathcal {K}_{j}^{2}$ satisfying Condition (4). To do this, we sketch some basic theory of linear codes over Galois rings.

In the rest of the paper, for each integer j, 0 ≤ j ≤ r, we denote

• $T_{j}=\{{\sum }_{j=0}^{m-1}a_{j}x^{j}\mid a_{j}\in \mathbb {Z}_{2}\}\subseteq \mathcal {K}_{j}$;

• $F_{j}=\mathbb {Z}_{2}[x]/\langle \overline {f}_{j}(x)\rangle =\{{\sum }_{j=0}^{m-1}a_{j}x^{j}\mid a_{j}\in \mathbb {Z}_{2}\}$ in which the arithmetic is done modulo $\overline {f}_{j}(x)$ in the polynomial ring $\mathbb {Z}_{2}[x]$.

Then F_j is an extension field of the binary field $\mathbb {Z}_{2}$ of $2^{m_{j}}$ elements, and $T_{j}$ is a subset of $\mathcal {K}_{j}$ since we regard $\mathbb {Z}_{2}$ as a subset of $\mathbb {Z}_{4}$.

To reduce the number of symbols, in this paper we will identify T_j with F_j. Whether T_j is a subset of the Galois ring $\mathcal {K}_{j}$ or a finite field itself, the reader can easily determine what it means according to the context. In this sense, each element α of $\mathcal {K}_{j}$ has a unique 2-adic expansion:

$$ \alpha=t_{0}(x)+2t_{1}(x), \ \text{where} \ t_{0}(x),t_{1}(x)\in T_{j}. $$

Then $\alpha \in \mathcal {K}_{j}^{\times }$ if and only if t₀(x)≠ 0. Hence $|\mathcal {K}_{j}^{\times }|=(2^{m_{j}}-1)2^{m_{j}}$.

Let 0 ≤ j ≤ r and L ≥ 2 be a positive integer. Assume S is a linear code of length L over the Galois ring $\mathcal {K}_{j}$. By [16, Definition 3.1], a matrix G is called a generator matrix for S if the rows of G span S and none of them can be written as an $\mathcal {K}_{j}$-linear combination of the other rows of G. Moreover, a generator matrix G is said to be in standard form if there is a suitable permutation matrix U of size L × L such that

$$ G=\left( \begin{array}{ccc} I_{k_{0}} & M_{0,1} & M_{0,2} \cr 0 & 2 I_{k_{1}} & 2 M_{1,2} \end{array}\right)U $$

where the columns are grouped into blocks of column sizes k₀,k₁,k with k_i ≥ 0, k = L − (k₀ + k₁), M_0,2 is a matrix over $\mathcal {K}_{j}$, M_0,1 and M_1,2 are matrices over T_j. Of course, if k_i = 0, the matrices $2^{i}I_{k_{i}}$ and 2ⁱM_i,t (i = 0,1) are suppressed in G. In this case, the following map

$$ (\xi,\eta)\mapsto (\xi,\eta)G =\left( \xi(I_{k_{0}}, M_{0,1}, M_{0,2})+\eta(0, 2 I_{k_{1}}, 2 M_{1,2})\right)U $$

(for any $\xi =(\alpha _{1},\ldots ,\alpha _{k_{0}})\in \mathcal {K}_{j}^{k_{0}}$ and $\eta =(b_{1},\ldots ,b_{k_{1}})\in F_{j}^{k_{1}}$) is an isomorphism of groups from $(\mathcal {K}_j^{k_0}\times F_{j}^{k_1},+)$ onto (S,+). Hence S is an abelian group of type $4^{k_{0}m_{j}}2^{k_{1}m_{j}}$ and contain $2^{(2k_{0}+k_{1})m_{j}}$ codewords.

All distinct nontrivial linear codes of length 2 over finite chain rings had been determined (by [9, Example 2.5], for example). Moreover, we have

Lemma 5

All distinct linear code$S_{j}$oflength 2 over the Galois ring$\mathcal {K}_{j}=\mathbb {Z}_{4}[x]/\langle f_{j}(x)\rangle$satisfyingCondition (4) in Lemma 4 are given by thefollowing table, where G is a generator matrix ofS_j:

Case	G	type of Sj	|Sj|
(i)	$I_{2}$	$4^{2m_{j}}2^{0}$	$2^{4m_{j}}$
	$2I_{2}$	$4^{0}2^{2m_{j}}$	$2^{2m_{j}}$
	0	$4^{0}2^{0}$	1
(ii)	(2wj(x),1), where wj(x) ∈ Tj arbitrary	$4^{m_{j}}2^{0}$	$2^{2m_{j}}$
(iii)	(0,2)	$4^{0}2^{m_{j}}$	$2^{m_{j}}$
(iv)	$\left (\begin {array}{cc}0 & 1\cr 2 & 0 \end {array}\right )$	$4^{m_{j}}2^{m_{j}}$	$2^{3m_{j}}$

Then the number of all codes listed above is equalto$2^{m_{j}}+5$.

Proof

By 2² = 0 and [9, Example 2.5] we know that the number of linear codes over the Galois ring $\mathcal {K}_{j}$ of length 2 is equal to ${\sum }_{i=0}^{2}(2i+1)|T_{j}|^{2-i}={\sum }_{i=0}^{2}(2i+1)2^{(2-i)m_{j}}=4^{m_{j}}+3\cdot 2^{m_{j}}+5$. Precisely, every nontrivial linear code S over $\mathcal {K}_{j}$ of length 2 has one and only one of the following matrices G as its generator matrix:

• 1.G = (1,a(x)), $\forall a(x)\in \mathcal {K}_{j}$. 2.G = (2,2b(x)), ∀b(x) ∈ T_j.

• 3.G = (2w(x),1), ∀w(x) ∈ T_j. 4.G = (0,2). 5.G = 2I₂.

• 6.$G=\left (\begin {array}{cc}1 & c(x)\cr 0 & 2 \end {array}\right )$, ∀c(x) ∈ T_j. 7.$G=\left (\begin {array}{cc}0 & 1\cr 2 & 0 \end {array}\right )$.

Then by ordinary careful calculations (cf. [10, Appendix]), we obtain the conclusions. □

As the end of this section, for any integer j, 0 ≤ j ≤ r, we determine the ideals of $\mathcal {K}_{j}+v\mathcal {K}_{j}$ and their annihilating ideals. For any ideal C of $\mathcal {K}_{j}+v\mathcal {K}_{j}$, its annihilating ideal is defined by $\text {Ann}(C)=\{\beta \in \mathcal {K}_{j}+v\mathcal {K}_{j}\mid \alpha \beta =0, \ \forall \alpha \in C\}$.

Theorem 1

Let 0 ≤ j ≤ r.Then all distinct idealsC_jandtheir annihilating ideals of the ring$\mathcal {K}_{j}+v\mathcal {K}_{j} (v^{2}=2v)$aregiven by the following table.

N	$C_{j}$	Type of $C_{j}$	$|C_{j}|$	$\text {Ann}(C_{j})$
1	$\langle 1\rangle$	$4^{2m_{j}}2^{0}$	$2^{4m_{j}}$	$\langle 0\rangle$
1	$\langle 2\rangle$	$4^{0}2^{2m_{j}}$	$2^{2m_{j}}$	$\langle 2\rangle$
1	$\langle 0\rangle$	$4^{0}2^{0}$	1	$\langle 1\rangle$
$2^{m_{j}}$	$\langle 2 w_{j}(x)+v\rangle$	$4^{m_{j}}2^{0}$	$2^{2m_{j}}$	$\langle 2(1+w_{j}(x))+v\rangle$
1	$\langle 2v\rangle$	$4^{0}2^{m_{j}}$	$2^{m_{j}}$	$\langle 2,v\rangle$
1	$\langle 2,v\rangle$	$4^{m_{j}}2^{m_{j}}$	$2^{3m_{j}}$	$\langle 2v\rangle$

where$w_{j}(x)\in T_{j}$arbitraryand N is the number of ideals in the same row.

Therefore, the number of ideals in $\mathcal {K}_{j}+v\mathcal {K}_{j}$ is $2^{m_{j}}+5$ .

Proof

It follows from Lemma 4, Lemma 5 and the definition for the $\mathcal {K}_{j}$-module isomorphism σ from $\mathcal {K}_{j}^{2}$ onto $\mathcal {K}_{j}+v\mathcal {K}_{j}$, immediately. Here, we omit these details. □

As stated above, by Lemma 3 and Theorem 1 we conclude that the number of cyclic codes over R of odd length n is equal to ${\prod }_{j=0}^{r}(2^{m_{j}}+5)$. Precisely, all these cyclic codes can be listed easily by Lemma 3 and Theorem 1.

Using the notation of Lemma 3, $\mathcal {C}=\bigoplus _{j=0}^{r}e_{j}(x)C_{j}$ is called the canonical form decomposition of the cyclic code $\mathcal {C}$ with length n over R.

3 Self-dual cyclic codes over $\mathbb {Z}_{4}+v\mathbb {Z}_{4}$ of odd length

In this section, we consider how to determine all distinct self-dual cyclic codes over $R=\mathbb {Z}_{4}+v\mathbb {Z}_{4}$ of odd length n precisely. To do this, we need to give the dual code for each cyclic code over R of length n first.

As xⁿ = 1 in the ring $\mathcal {A}=\mathbb {Z}_{4}[x]/\langle x^{n}-1\rangle$, we have x^− 1 = x^n− 1. For any $\alpha (x)={\sum }_{i=0}^{n-1}\alpha _{i}x^{i}\in R[x]/\langle x^{n}-1\rangle$, where α_i = a_i + vb_i ∈ R with $a_{i},b_{i}\in \mathbb {Z}_{4}$ for all 0 ≤ i ≤ n − 1, we have that α(x) = a(x) + vb(x) where $a(x),b(x)\in \mathcal {A}$ given by $a(x)={\sum }_{i=0}^{n-1}a_{i}x^{i}$ and $b(x)={\sum }_{i=0}^{n-1}b_{i}x^{i}$. Now, we define

$$\tau(\alpha(x))=\alpha(x^{-1})=\alpha_{0}+\sum\limits_{i=1}^{n-1}\alpha_{i}x^{n-i} =a(x^{-1})+vb(x^{-1}),$$

where $a(x^{-1})=a_{0}+{\sum }_{i=1}^{n-1}a_{i}x^{n-i}, b(x^{-1})=b_{0}+{\sum }_{i=1}^{n-1}b_{i}x^{n-i}\in \mathcal {A}$. It can be verified easily that the map τ is a ring automorphism of R[x]/〈xⁿ − 1〉. Moreover, the following conclusion is well known.

Lemma 6

For any cyclic code$\mathcal {C}$oflength n over R, its dual code is givenby$\mathcal {C}^{\bot }=\tau (\text {Ann}(\mathcal {C})) =\{a(x^{-1})\mid a(x)\in \text {Ann}(\mathcal {C}\},$where

$$ \text{Ann}(\mathcal{C})=\{\eta\in R[x]/\langle x^{n}-1\rangle\mid \xi\eta=0, \ \forall \xi\in \mathcal{C}\} $$

is the annihilating ideal of $\mathcal {C}$ in R[x]/〈xⁿ − 1〉.

Since $R[x]/\langle x^{n}-1\rangle =\mathcal {A}+v\mathcal {A} (v^{2}=2v)$ by Lemma 1, we see that the restriction of τ on its subring $\mathcal {A}$ is a ring automorphism of $\mathcal {A}$. We still denote this ring automorphism by τ. Then τ(a(x)) = a(x^− 1) for all $a(x)\in \mathcal {A}$.

For any polynomial $f(x)={\sum }_{i=0}^{m}a_{i}x^{i}\in \mathbb {Z}_{4}[x]$ of degree m ≥ 0. The reciprocal polynomial of f(x) is defined by $\widetilde {f}(x)=x^{m}f(\frac {1}{x})$, and f(x) is said to be self-reciprocal if $\widetilde {f}(x)=\delta f(x)$ for some $\delta \in \mathbb {Z}_{4}^{\times }=\{1,-1\}$. By (1) in Section 2, we have xⁿ − 1 = f₀(x)f₁(x)…f_r(x). This implies

$$ x^{n}-1=(-1)\widetilde{f}_{0}(x)\widetilde{f}_{1}(x){\ldots} \widetilde{f}_{r}(x). $$

Since f₀(x) = x − 1,f₁(x),…,f_r(x) are pairwise coprime monic basic irreducible polynomials in $\mathbb {Z}_{4}[x]$, $\widetilde {f}_{0}(x), \widetilde {f}_{1}(x), \ldots , \widetilde {f}_{r}(x)$ are pairwise coprime basic irreducible polynomials in $\mathbb {Z}_{4}[x]$ as well. Hence for any integer j, 0 ≤ j ≤ r, there is a unique integer $j^{\prime }$, $0\leq j^{\prime }\leq r$, such that

$$ \widetilde{f}_{j}(x)=\delta_{j}f_{j^{\prime}}(x) \ \text{for} \ \text{some} \ \delta_{j}\in \mathbb{Z}_{4}^{\times}. $$

Especially, we have $0^{\prime }=0$ since $\widetilde {f}_{0}(x)=1-x=(-1)f_{0}(x)$. Then by

$$ x^{n}=1 \ \text{and} \ x^{m_{j}}f_{j}(x^{-1})=\widetilde{f}_{j}(x) \ \text{in} \ \mathcal{A}, $$

from the definition for e_j(x) (see (2) and (3)), we deduce that $e_{j}(x^{-1})=e_{j^{\prime }}(x)$.

We still use τ to denote the map $j\mapsto j^{\prime }$. Then we have

$$ \widetilde{f}_{j}(x)=\delta_{j}f_{\tau(j)}(x) \ \text{and} \ \tau(e_{j}(x))=e_{j}(x^{-1})=e_{\tau(j)}(x). $$

(5)

Whether τ denotes the ring automorphism of $\mathcal {A}$ or this map is determined by the context. The next lemma shows the compatibility of the two uses of τ.

Lemma 7

Using the notation above, we have the following conclusions.

• (i) τisa permutation on the set {0,1,…,r} satisfyingτ^− 1 = τ.

• (ii) After a rearrangement off₁(x),…,f_r(x),there exists a unique pair (λ,ρ) ofnonnegative integers such that

  • λ + 2ρ = r;

  • τ(j) = j,for all 0 ≤ j ≤ λ;

  • τ(λ + l) = λ + l + ρandτ(λ + l + ρ) = λ + l,for all 1 ≤ l ≤ ρ.

• (iii) For any integer j, 0 ≤ j ≤ r,the mapτ_jdefinedby

  $$\tau_{j}(a(x)+vb(x))=a(x^{-1})+vb(x^{-1}) \ (\text{mod} \ f_{\tau(j)}(x)), \forall a(x),b(x)\in \mathcal{K}_{j}$$

  is an isomorphism of rings from $\mathcal {K}_{j}+v\mathcal {K}_{j}$ onto $\mathcal {K}_{\tau (j)}+v\mathcal {K}_{\tau (j)}$, where x^− 1 = x^n− 1(mod f_τ(j)(x)).

Proof

• (i) It follows from the definition of the map τ and that $f_{\tau (\tau (j))}(x)=\delta _{\tau (j)}^{-1}\widetilde {f}_{\tau (j)}(x)= \delta _{\tau (j)}^{-1}\delta _{j}^{-1}\widetilde {\widetilde {f}_{j}}(x)=\delta _{\tau (j)}^{-1}\delta _{j}^{-1}f_{j}(x) =f_{j}(x)$ by (5).

• (ii) It follows from (i) and the properties of permutations on a finite set.

• (iii) The map τ_j is well-defined and makes the following diagram commutative:

  $$ \left.\begin{array}{ccc} \ \ \ \ \mathcal{K}_{j}+v\mathcal{K}_{j} & \stackrel{\tau_{j}}{\longrightarrow} & \mathcal{K}_{\tau(j)}+v\mathcal{K}_{\tau(j)}\cr \varphi_{j}\downarrow & & \ \ \ \ \downarrow \varphi_{j} \cr \ \ \ \ (\mathcal{A}+v\mathcal{A})e_{j}(x) & \stackrel{\tau|_{(\mathcal{A}+v\mathcal{A})e_{j}(x)}}{\longrightarrow} & (\mathcal{A}+v\mathcal{A})e_{\tau(j)}(x) \end{array}\right., $$

  where $\tau |_{(\mathcal {A}+v\mathcal {A})e_{j}(x)}$ is the restriction of the ring automorphism τ to the subring $(\mathcal {A}+v\mathcal {A})e_{j}(x)$ of $\mathcal {A}+v\mathcal {A}$. From this and by Lemma 2(ii), we deduce that τ_j is an isomorphism of rings from $\mathcal {K}_{j}+v\mathcal {K}_{j}$ onto $\mathcal {K}_{\tau (j)}+v\mathcal {K}_{\tau (j)}$.

□

Now, using the notation of Lemma 3 and Theorem 1, we give the dual code for any cyclic code of length n over R.

Theorem 2

Let$\mathcal {C}$bea cyclic code of length n over R with the canonical formdecomposition

$$ \mathcal{C}=\bigoplus_{0\leq j\leq r}e_{j}(x)C_{j}, $$

where C_j is an ideal of $\mathcal {K}_{j}+v\mathcal {K}_{j}$ listed by Theorem 1 for all j. Then

$$ \mathcal{C}^{\bot}=\bigoplus_{0\leq j\leq r}e_{\tau(j)}(x)D_{\tau(j)} \ (\text{mod} \ x^{n}-1), $$

where D_τ(j) = τ_j(Ann(C_j)), 0 ≤ j ≤ r, is an ideal of $\mathcal {K}_{\tau (j)}+v\mathcal {K}_{\tau (j)}$ determined by the following table:

N	$C_{j}$ (mod $f_{j}(x)$)	$D_{\tau (j)}$ (mod $f_{\tau (j)}(x)$)
3	$\langle 2^{k}\rangle$ (k = 0,1,2)	$\langle 2^{2-k}\rangle$
$2^{m_{j}}$	$\langle 2 w_{j}(x)+v\rangle$ ($w_{j}(x)\in T_{j}$)	$\langle 2(1+w_{j}(x^{-1}))+v\rangle$
1	$\langle 2v\rangle$	$\langle 2,v\rangle$
1	$\langle 2,v\rangle$	$\langle 2v\rangle$

where N is the number of the pair$(C_{j},D_{\tau (j)})$of ideals in the same row.

Proof

Let $\alpha ,\beta \in R[x]/\langle x^{n}-1\rangle =\mathcal {A}+v\mathcal {A}$. By Lemma 3 and its proof, we have that

$$ \alpha=\sum\limits_{j=0}^{r}e_{j}(x)\xi_{j}, \ \beta=\sum\limits_{j=0}^{r}e_{j}(x)\eta_{j}, \ \text{where} \ \xi_{j},\eta_{j}\in \mathcal{K}_{j}+v\mathcal{K}_{j}. $$

Since e_j(x)² = e_j(x) and e_j(x)e_l(x) = 0 for all 0 ≤ j≠l ≤ r by Lemma 2(i), it follows that $\alpha \beta ={\sum }_{j=0}^{r}e_{j}(x)(\xi _{j}\eta _{j})$. This implies that

$$ \alpha\beta=0 \ \text{in} \ R[x]/\langle x^{n}-1\rangle \ \Longleftrightarrow \xi_{j}\eta_{j}=0 \ \text{in} \ \mathcal{K}_{j}+v\mathcal{K}_{j}, \ \forall j=0,1,\ldots,r. $$

From this we deduce $\text {Ann}(\mathcal {C})=\bigoplus _{j=0}^{r}e_{j}(x)\text {Ann}(C_{j}) \ (\text {mod} \ x^{n}-1),$ where Ann(C_j) is the annihilating ideal of C_j in $\mathcal {K}_{j}+v\mathcal {K}_{j}$ for any 0 ≤ j ≤ r. As τ is a ring isomorphism, we have

$$ \mathcal{C}^{\bot}=\tau(\text{Ann}(\mathcal{C}))=\sum\limits_{j=0}^{r}\tau(e_{j}(x)\text{Ann}(C_{j})). $$

For any integer j, 0 ≤ j ≤ r, by (5) and the definition of τ we have τ(e_j(x)) = e_j(x^− 1) = e_τ(j)(x) and

$$ \tau(e_{j}(x)c_{j}(x))=e_{j}(x^{-1})c_{j}(x^{-1})=e_{\tau(j)}(x)\tau_{j}(c_{j}(x)), $$

where τ_j(c_j(x)) = c_j(x^− 1) = c(x^n− 1) (mod f_τ(j)(x)) by Lemma 7(iii), for any c_j(x) ∈Ann(C_j). From these, we deduce that

$$ \mathcal{C}^{\bot}=\bigoplus_{j=0}^{r}e_{\tau(j)}(x)\tau_{j}(\text{Ann}(C_{j})). $$

Denote D_τ(j) = τ_j(Ann(C_j)) where 0 ≤ j ≤ r. Since τ_j is a ring isomorphism from $\mathcal {K}_{j}+v\mathcal {K}_{j}$ onto $\mathcal {K}_{\tau (j)}+v\mathcal {K}_{\tau (j)}$ by Lemma 7(iii), we see that D_τ(j) is an ideal of the ring $\mathcal {K}_{\tau (j)}+v\mathcal {K}_{\tau (j)}$.

Let C_j = 〈〈2w_j(x) + v〉〉 where w_j(x) ∈ T_j. By Theorem 1 we have Ann(C_j) = 〈2(1 + w_j(x)) + v〉. Then by the definition of τ_j, it follows that

$$ D_{\tau(j)}=\tau_{j}(\text{Ann}(C_{j}))=\langle \tau_{j}(2(1+w_{j}(x))+v)\rangle=\langle 2(1+w_{j}(x^{-1}))+v\rangle $$

(mod f_τ(j)(x)). The expressions for D_τ(j) in other cases can be calculated easily. We omit these here. □

For any integer j, 1 ≤ j ≤ λ, as f_j(x) is self-reciprocal in $\mathbb {Z}_{4}[x]$ by Lemma 7(ii) and (5), we see that $\overline {f}_{j}(x)$ is self-reciprocal in $\mathbb {Z}_{2}[x]$ and hence its degree m_j must be even. Then it is well known that

$$ x^{-1}=x^{2^{\frac{m_{j}}{2}}} \ \text{in} \ \text{the} \ \text{field} \ F_{j}=\mathbb{Z}_{2}[x]/\langle \overline{f}_{j}(x)\rangle. $$

(6)

In the rest of this paper, we adopt the following notation:

• ${\mathscr{H}}_{j}=\left \{\xi \in F_{j}\mid \xi ^{2^{\frac {m_{j}}{2}}}=\xi \right \}$. Then ${\mathscr{H}}_{j}$ is a subfield of F_j with $2^{\frac {m_{j}}{2}}$ elements.

• Let Tr_j be the trace function from F_j onto its subfield ${\mathscr{H}}_{j}$ defined by:

  $$ \text{Tr}_{j}(\xi)=\xi+\xi^{2^{\frac{m_{j}}{2}}}, \ \forall \xi\in F_{j}. $$

  Then by [22, Corollary 7.17], we have that $|\text {Tr}_{j}^{-1}(1)|=2^{\frac {m_{i}}{2}}$ where

  $$ \text{Tr}_{j}^{-1}(1)=\{\xi\in F_{j}\mid \text{Tr}_{j}(\xi)=1\}=\{\xi\in F_{j}\mid \xi+\xi^{2^{\frac{m_{j}}{2}}}=1\}. $$

  (7)

Now is the time to list all self-dual cyclic codes over R of length n.

Theorem 3

Using the notation above, all distinct self-dual cyclic codes of length n over$R=\mathbb {Z}_{4}+v\mathbb {Z}_{4} (v^{2}=2v)$are givenby:

$$ {\mathcal{C}}=\bigoplus_{0\leq j\leq\lambda+2\rho} e_{j}(x)C_{j} ({\text{mod}} x^{n}-1), $$

where C_j is an ideal of $\mathcal {K}_{j}+v\mathcal {K}_{j}$ listed as follows.

• (i) When j = 0, there is 1 ideal: $C_{0}=\langle 2\rangle$.

• (ii) When 1 ≤ j ≤ λ, there are $1+2^{\frac {m_{j}}{2}}$ ideals:

  $$C_{j}=\langle 2\rangle, \ \text{and} \ C_{j}=\langle 2w_{j}(x)+v\rangle \ \text{where} \ w_{j}(x)\in \text{Tr}_{j}^{-1}(1) \ \text{arbitrary}.$$

• (iii) When $\lambda +1\leq j\leq \lambda +\rho$, there are $2^{m_{j}}+5$ pairs $(C_{j}, C_{j+\rho })$ of ideals listed by the following table:

N	$C_{j}$ (mod $f_{j}(x)$)	$|C_{j}|$	$C_{j+\rho }$ (mod $f_{j+\rho }(x)$)
3	$\langle 2^{k}\rangle$, $0\leq k\leq 2$	$2^{(4-2k)m_{j}}$	$\langle 2^{2-k}\rangle$
$2^{m_{j}}$	$\langle 2 w_{j}(x)+v\rangle$, $w_{j}(x)\in T_{j}$	$2^{2m_{j}}$	$\langle 2(1+w_{j}(x^{-1}))+v\rangle$
1	$\langle 2v\rangle$	$2^{m_{j}}$	$\langle 2,v\rangle$
1	$\langle 2,v\rangle$	$2^{3m_{j}}$	$\langle 2v\rangle$

where N is the number of pairs in the same row.

Therefore, the number of self-dual cyclic codes of length n over R is

$$ L_{n}=\prod\limits_{1\leq j\leq \lambda}(1+2^{\frac{m_{j}}{2}})\prod\limits_{\lambda+1\leq j\leq \lambda+\rho}(2^{m_{j}}+5). $$

Proof

By Lemma 7(ii) and Theorem 2, we have

$$ \mathcal{C}^{\bot}=\left( \bigoplus_{j=0}^{\lambda} e_{j}(x)D_{j}\right) \oplus\left( \bigoplus_{j=\lambda+1}^{\lambda+\rho}e_{j+\rho}D_{j+\rho}\right) \oplus\left( \bigoplus_{j=\lambda+\rho+1}^{\lambda+2\rho}e_{j-\rho}D_{j-\rho}\right), $$

where D_j+ρ = τ_j(C_j) for all j = λ + 1,…,λ + ρ. Hence the cyclic code $\mathcal {C}$ is self-dual, i.e., $\mathcal {C}=\mathcal {C}^{\bot }$, if and only if the following two conditions are satisfied:

• (‡) C_j = D_j = τ_j(Ann(C_j)), for all 0 ≤ j ≤ λ;

• (†) C_j is an arbitrary ideal of $\mathcal {K}_{j}+v\mathcal {K}_{j}$ and C_j+ρ = D_j+ρ = τ_j+ρ(Ann(C_j)), for all j = λ + 1,…,λ + ρ.

Therefore, the class of self-dual cyclic codes over R of length n is the same as the class of cyclic codes over R of length n: $\mathcal {C}=\bigoplus _{j=0}^{\lambda +2\rho }e_{j}(x)C_{j}$, where C_j is an ideal of $\mathcal {K}_{j}+v\mathcal {K}_{j}$ satisfying Conditions (‡) and (†) for all j.

Every pair (C_j,C_j+ρ) of ideals can be determined by the table in Theorem 2, for all j = λ + 1,…,λ + ρ. Then in order to determine codes in the latter class, we only need to consider ideals of $\mathcal {K}_{j}+v\mathcal {K}_{j}$ satisfying Conditions (‡) for all j = 0,1,…,λ.

Now, let 0 ≤ j ≤ λ and C_j is an ideal of $\mathcal {K}_{j}+v\mathcal {K}_{j}$ listed by Theorem 2. Then C_j satisfies condition (‡) if and only if C_j is given by one of the following two cases:

• (‡-1) C_j = 〈2〉.

• (‡-2) C_j = 〈2w_j(x) + v〉,

• where w_j(x) ∈ T_j satisfying w_j(x) ≡ 1 + w_j(x^− 1) (mod f_j(x), mod 2).

Since w_j(x) is a polynomial in $\mathbb {Z}_{2}[x]$, the condition is equivalent to w_j(x) + w_j(x^− 1) ≡ 1 (mod $\overline {f}_{j}(x)$), i.e.,

$$ w_{j}(x)+w_{j}(x^{-1})=1 \ \text{in} \ F_{j}=\mathbb{Z}_{2}[x]/\langle \overline{f}_{j}(x)\rangle. $$

(8)

Now, we have the following two subcases:

• (†-2-i) When j = 0, we have $\overline {f}_{0}(x)=x-1$ and $F_{0}=\mathbb {Z}_{2}$. In this case, there is no element $w_{j}\in \mathbb {Z}_{2}$ satisfying Condition (8).

• (†-2-ii) Let 1 ≤ j ≤ λ. By (6) and a² = a for all $a\in \mathbb {Z}_{2}$, we obtain

  $$w_{j}(x^{-1})=w_{j}(x^{2^{\frac{m_{j}}{2}}})=(w_{j}(x))^{2^{\frac{m_{j}}{2}}}, \ \forall w_{j}(x)\in F_{j}.$$

  Hence Condition (8) is equivalent to $w_{j}(x)\in \text {Tr}_{j}^{-1}(1)$ by (7).

By Lemma 7(ii) and Theorem 2, we conclude that the class of cyclic codes over R of length n: $\mathcal {C}=\bigoplus _{j=0}^{\lambda +2\rho }e_{j}(x)C_{j}$, where C_j is an ideal of $\mathcal {K}_{j}+v\mathcal {K}_{j}$ satisfying Conditions (‡) and (†) for all j, is exactly the same as the class of cyclic codes over R of length n listed by the three cases (i)–(iii) of this theorem.

As stated above, we proved the theorem. □

As the end of this section, we list the number L_n of self-dual cyclic codes of length n over $\mathbb {Z}_{4}+v\mathbb {Z}_{4}$ (v² = 2v), where n is odd and 3 ≤ n ≤ 49, by the following table.

n	Ln	${\mathscr{M}}_{n}$	n	Ln	${\mathscr{M}}_{n}$
3	3	(2;∅)	27	13851	(2,6,18;∅)
5	5	(4;∅)	29	16385	(28;∅)
7	13	(∅;3)	31	50653	(∅;5,5,5)
9	27	(2,6;∅)	33	107811	(2,10,10,10;∅)
11	33	(10;∅)	35	266565	(4;3,12)
13	65	(12;∅)	37	262145	(36;∅)
15	315	(2,4;4)	39	799695	(2,12;12)
17	289	(8,8;∅)	41	1050625	(20,20;∅)
19	513	(18;∅)	43	2146689	(14,14,14;∅)
21	2691	(2;3,6)	45	11626335	(2,4,6;4,12)
23	2053	(∅;11)	47	8388613	(∅;23)
25	5125	(4,20;∅)	49	27263041	(∅;3,21)

where ${\mathscr{M}}_{n}=(m_{1},\ldots ,m_{\lambda };m_{\lambda +1},\ldots ,m_{\lambda +\rho })$ corresponding to the degrees of monic basic irreducible divisors f₁(x),…,f_λ(x);f_λ+ 1(x),…,f_λ+ρ(x) ofxⁿ − 1 in $\mathbb {Z}_{4}[x]$ (see (1) in Section 1). It is obvious that

$$ 1+m_{1}+\ldots+m_{\lambda}+2(m_{\lambda+1}+\ldots+m_{\lambda+\rho})=n. $$

4 Quasi-cyclic code over $\mathbb {Z}_{4}$ derived from cyclic codes over $\mathbb {Z}_{4}+v\mathbb {Z}_{4}$

In this section, we consider self-dual 2-quasi-cyclic codes of length 2n over $\mathbb {Z}_{4}$ derived from self-dual cyclic codes of length n over the ring $R=\mathbb {Z}_{4}+v\mathbb {Z}_{4}$. As in [15, Section 3], we define $\varrho : R\rightarrow \mathbb {Z}_{4}^{2}$ by

$$ \varrho(\alpha)=(a+b,b), \ \forall \alpha=a+bv\in R \ \text{where} \ a,b\in \mathbb{Z}_{4} $$

and let $\theta : R^{n}\rightarrow \mathbb {Z}_{4}^{2n}$ be such that 𝜃(α₁,…,α_n) = (ϱ(α₁),…,ϱ(α_n)), for all α₁,…,α_n ∈ R. Let w_L denote the Lee weight on $\mathbb {Z}_{4}$ defined by:

$$ w_{L}(0)=0, \ w_{L}(1)=w_{L}(3)=1 \ \text{and} \ w_{L}(2)=2. $$

We extend w_L in a natural way: for a + bv ∈ R with $a,b\in \mathbb {Z}_{4}$, define

$$ w_{L}(a+bv)=w_{L}(a+b)+w_{L}(b). $$

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

Proposition 1

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

Proposition 2

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

Moreover, we have the following properties for cyclic codes over R.

Proposition 3

Let $\mathcal {C}$ be a cyclic codes of length n over R. Then $\theta (\mathcal {C})$ is a 2-quasi-cyclic code of length 2n over $\mathbb {Z}_{4}$.

Proof

Let $\underline {\alpha }=(\alpha _{0},\alpha _{1},\ldots ,\alpha _{n-1})\in \mathcal {C}$, where α_i = a_i + b_iv with $a_{i},b_{i}\in \mathbb {Z}_{4}$ for all i = 0,1,…,n − 1. Then $\theta (\underline {\alpha })=(a_{0}+b_{0},b_{0},a_{1}+b_{1},b_{1},\ldots ,a_{n-1}+b_{n-1}, b_{n-1})\in \theta (\mathcal {C})$. Since $\mathcal {C}$ is cyclic, we have $(\alpha _{n-1},\alpha _{0},\alpha _{1},\ldots ,\alpha _{n-2})\in \mathcal {C}$. This implies $(a_{n-1}+b_{n-1}, b_{n-1},a_{0}+b_{0},b_{0},a_{1}+b_{1},b_{1},\ldots ,a_{n-2}+b_{n-2}, b_{n-2})\in \theta (\mathcal {C})$. Hence $\theta (\mathcal {C})$ is a 2-quasi-cyclic code of length 2n over $\mathbb {Z}_{4}$. □

As an application, we consider cyclic codes of length 15 over R. In this case, x¹⁵ − 1 = f₀(x)f₁(x)f₂(x)f₃(x)f₄(x) where f₀(x) = x − 1, f₁(x) = x² + x + 1, f₂(x) = x⁴ + x³ + x² + x + 1, f₃(x) = x⁴ + 2x² + 3x + 1 and $f_{4}(x)=\widetilde {f}_{3}(x)$ are monic basic irreducible polynomials in $\mathbb {Z}_{4}[x]$. Hence m₁ = 2 and m₂ = m₃ = m₄ = 4. Hence the number of cyclic codes of length 15 over $\mathbb {Z}_{4}+v\mathbb {Z}_{4}$ is

$$(2^{1}+5)\cdot(2^{2}+5)\cdot(2^{4}+5)^{3}=583443.$$

First, for each integer j, 0 ≤ j ≤ 4, we denote $F_{j}(x)=\frac {x^{15}-1}{f_{j}(x)}$,

• ◇ $\mathcal {K}_{j}=\mathbb {Z}_{4}[x]/\langle f_{j}(x)\rangle =\left \{{\sum }_{i=0}^{m_{j}-1}a_{i}x^{i}\mid a_{0},a_{1},\ldots ,a_{m_{j}-1}\in \mathbb {Z}_{4}\right \}$.

• ◇ $T_{j}=\left \{{\sum }_{i=0}^{m_{j}-1}b_{i}x^{i}\mid b_{0},b_{1},\ldots ,b_{m_{j}-1}\in \mathbb {Z}_{2}\right \}\subset \mathcal {K}_{j}$.

Then we find polynomials $c_{j}(x),d_{j}(x)\in \mathbb {Z}_{4}[x]$ satisfying c_j(x)F_j(x) + d_j(x)f_j(x) = 1 and set $e_{j}(x)\in \mathbb {Z}_{4}[x]/\langle x^{15}-1\rangle$ such that e_j(x) ≡ c_j(x)F_j(x) (mod x¹⁵ − 1). Precisely, we have

$$ \begin{array}{@{}rcl@{}} e_{0}(x)&=&3{\kern1.7pt}{x}^{14}+3{\kern1.7pt}{x}^{13}+3{\kern1.7pt}{x}^{12}+3{\kern1.7pt}{x}^{11}+3{\kern1.7pt}{x}^{10}+3{\kern1.7pt}{x}^{9} +3{\kern1.7pt}{x}^{8}+3{\kern1.7pt}{x}^{7}+3{\kern1.7pt}{x}^{6}+3{\kern1.7pt}{x}^{5} +3{\kern1.7pt}{x}^{4}\\ &&+3{\kern1.7pt}{x}^{3}+3{\kern1.7pt} {x}^{2}+3{\kern1.7pt}x+3,\\ e_{1}(x)&=&{x}^{14}{+}{\kern1.7pt}{x}^{13}{+}{\kern1.7pt}2{\kern1.7pt}{x}^{12}{+}{\kern1.7pt}{x}^{11}{+}{\kern1.7pt}{x}^{10}{+}{\kern1.7pt}2{\kern1.7pt}{x}^{9}{+}{\kern1.7pt}{x}^{8}{+}{x}^{7}{+}{\kern1.7pt} 2{\kern1.7pt}{x}^{6}{+}{\kern1.7pt}{x}^{5}{+}{\kern1.7pt}{x}^{4}{+}{\kern1.7pt}2{\kern1.7pt}{x}^{3} {+}{\kern1.7pt}{x}^{+}{\kern1.7pt}x+2,\\ e_{2}(x)&=&{x}^{14}+{x}^{13}+{x}^{12}+{x}^{11}+{x}^{9}+{x}^{8}+{x}^{7}+{x}^{6}+{x }^{4}+{x}^{3}+{x}^{2}+x,\\ e_{3}(x)&=&{x}^{12}+2{\kern1.7pt}{x}^{10}+{x}^{9}+3{\kern1.7pt}{x}^{8}+{x}^{6}+2{\kern1.7pt}{x}^{5}+3{\kern1.7pt}{x}^{4}+ {x}^{3}+3{\kern1.7pt}{x}^{2}+3{\kern1.7pt}x,\\ e_{4}(x)&=&3{\kern1.7pt}{x}^{14}+3{\kern1.7pt}{x}^{13}+{x}^{12}+3{\kern1.7pt}{x}^{11}+2{\kern1.7pt}{x}^{10}+{x}^{9}+3{\kern1.7pt}{x }^{7}+{x}^{6}+2{\kern1.7pt}{x}^{5}+{x}^{3}. \end{array} $$

(I) By Lemma 3 and Theorem 1, all distinct 583443 cyclic codes of length 15 over R are given by: $\mathcal {C}=\bigoplus _{j=0}^{4}e_{j}(x)C_{j}$ (mod x¹⁵ − 1), where

• C₀ is one of the following 7 ideals of $\mathbb {Z}_{4}+v\mathbb {Z}_{4}$:

  • C₀ = 〈2^k〉 with |C₀| = 2^{4 − 2k}, where 0 ≤ k ≤ 2;

  • C₀ = 〈2a + v〉 with |C₀| = 4, where $a\in \mathbb {Z}_{2}=\{0,1\}$ arbitrary;

  • C₀ = 〈2v〉 with |C₀| = 2;

  • C₀ = 〈2,v〉 with |C₀| = 8.

• C₁ is one of the following 9 ideals of $\mathcal {K}_{1}+v\mathcal {K}_{1}$:

  • C₁ = 〈2^k〉 with |C₁| = 4^{4 − 2k}, where 0 ≤ k ≤ 2;

  • C₁ = 〈2(a₀ + a₁x) + v〉 with |C₁| = 16, where $a_{0},a_{1}\in \mathbb {Z}_{2}$ arbitrary;

  • C₁ = 〈2v〉 with |C₁| = 4;

  • C₁ = 〈2,v〉 with |C₁| = 64.

• C_j is one of the following 21 ideals of $\mathcal {K}_{j}+v\mathcal {K}_{j}$ for all j = 2,3,4:

  • C_j = 〈2^k〉 with |C_j| = 16^{4 − 2k}, where 0 ≤ k ≤ 2;

  • C_j = 〈2(a₀ + a₁x + a₂x² + a₃x³) + v〉 with |C_j| = 16²,

    where $a_{0},a_{1},a_{2},a_{3}\in \mathbb {Z}_{2}$ arbitrary;

  • C_j = 〈2v〉 with |C_j| = 16;

  • C_j = 〈2,v〉 with |C_j| = 16³.

(II) We have r = 4, λ = 2 and ρ = 1. For j = 1,2, set

• ◇ $F_{j}=\mathbb {Z}_{2}[x]/\langle \overline {f}_{j}(x)\rangle =\{{\sum }_{i=0}^{m_{j}-1}a_{i}x^{i}\mid a_{0},a_{1},\ldots ,a_{m_{j}-1}\in \mathbb {Z}_{2}\}$

  in which the arithmetic is done modulo $\overline {f}_{j}(x)$;

• ◇ $\text {Tr}_{j}^{-1}(1)=\{a(x)\in F_{j}\mid a(x)+a(x^{-1})\equiv 1 \ (\text {mod} \ \overline {f}_{j}(x))\}$.

Then by x^− 1 ≡ x¹⁴ (mod $\overline {f}_{j}(x)$) for j = 1,2, we have the following

$\text {Tr}_{1}^{-1}(1)=\{a+bx\in \mathbb {Z}_{2}[x]/\langle \overline {f}_{1}(x)\rangle \mid (a+bx)+(a+bx^{14})\equiv 1 \ (\text {mod} \ \overline {f}_{1}(x))\}=\{a+bx\mid a+bx+a+b(1+x)+1=0, \ a,b\in \mathbb {Z}_{2}\}=\{x,1+x\}$.

$\text {Tr}_{2}^{-1}(1)=\{a+bx+cx^{2}+dx^{3}\in \mathbb {Z}_{2}[x]/\langle \overline {f}_{2}(x)\rangle \mid (a+bx+cx^{2}+dx^{3})+(a+bx^{14}+cx^{13}+dx^{12})\equiv 1 \ (\text {mod} \ \overline {f}_{2}(x))\}=\{a+bx+cx^{2}+dx^{3}\mid bx+cx^{2}+dx^{3}+b(1+x+x^{2}+x^{3})+cx^{3}+dx^{2}+1=0, \ a,b,c,d\in \mathbb {Z}_{2}\}=\{a+x+cx^{2}+(1+c)x^{3}\mid a,c\in \mathbb {Z}_{2}\}$.

By Theorem 3, all 315 self-dual cyclic codes of length 15 over R are given by: $\mathcal {C}=\bigoplus _{j=0}^{4}e_{j}(x)C_{j}$ (mod x¹⁵ − 1), where

• $C_{0}=\langle 2\rangle =2(\mathbb {Z}_{4}+v\mathbb {Z}_{4})$.

• C₁ = 〈2〉, C₁ = 〈2w₁(x) + v〉 where $w_{1}(x)\in \text {Tr}_{1}^{-1}(1)$.

• C₂ = 〈2〉, C₂ = 〈2w₂(x) + v〉 where $w_{2}(x)\in \text {Tr}_{2}^{-1}(1)$.

• (C₃,C₄) is given by the following table:

N	C3 (mod f3(x))	|C3|	C4 (mod f4(x))
3	〈2k〉, 0 ≤ k ≤ 2	24(4 − 2k)	〈22−k〉
16	〈2w3(x) + v〉	28	〈2(1 + w3(x− 1)) + v〉
1	〈2v〉	24	〈2,v〉
1	〈2,v〉	212	〈2v〉

in which N is the number of pairs (C₃,C₄) in the same row, and

w₃(x) = a + bx + cx² + dx³;

w₃(x^− 1) = a + d + (d + c)x + (c + b)x² + bx³ ≡ w₃(x¹⁴) (mod f₄(x), 2),

for $a,b,c,d\in \mathbb {Z}_{2}$ arbitrary.

By Proposition 3, we obtain 315 2-quasi-cyclic self-dual codes $\theta (\mathcal {C})$ of length 30 over $\mathbb {Z}_{4}$. Among these codes, there are 70 codes with minimum Lee weight 12. These 70 2-quasi-cyclic self-dual codes over $\mathbb {Z}_{4}$ are given by the following table, with $C_{1}$, $C_{2}$, C₃, C₄ and the type of each $\mathbb {Z}_{4}$-code $\theta (\mathcal {C})$.

C1	C2	C3 (mod f3(x))	C4 (mod f4(x))	Type
〈v + 2x〉	〈2x3 + 2x + v〉	〈2〉	〈2〉	21846
〈v + 2x + 2〉	〈2x3 + 2x + v〉	〈2〉	〈2〉	21846
〈v + 2x〉	〈2x2 + 2x + v〉	〈2〉	〈2〉	21846
〈v + 2x + 2〉	〈2x2 + 2x + v〉	〈2〉	〈2〉	21846
〈v + 2x〉	〈2x3 + 2x + v + 2〉	〈2〉	〈2〉	21846
〈v + 2x + 2〉	〈2x3 + 2x + v + 2〉	〈2〉	〈2〉	21846
〈v + 2x〉	〈2x2 + 2x + v + 2〉	〈2〉	〈2〉	21846
〈v + 2x + 2〉	〈2x2 + 2x + v + 2〉	〈2〉	〈2〉	21846
〈v + 2x〉	〈2〉	〈2v〉	〈2,v〉	21846
〈v + 2x + 2〉	〈2〉	〈2v〉	〈2,v〉	21846
〈v + 2x〉	〈2〉	〈2,v〉	〈2v〉	21846
〈v + 2x + 2〉	〈2〉	〈2,v〉	〈2v〉	21846
〈v + 2x〉	〈2x3 + 2x + v〉	〈2,v〉	〈2v〉	210410
〈v + 2x〉	〈2x2 + 2x + v〉	〈2v〉	〈2,v〉	210410
〈v + 2x〉	〈2x3 + 2x + v + 2〉	〈2,v〉	〈2v〉	210410
〈v + 2x〉	〈2x2 + 2x + v + 2〉	〈2v〉	〈2,v〉	210410
〈v + 2x〉	〈2〉	〈2x3 + 2x2 + v〉	〈2x2 + v〉	210410
〈v + 2x〉	〈2〉	〈2x3 + 2x + v〉	〈2x3 + 2x2 + 2x + v〉	210410
〈v + 2x〉	〈2〉	〈2x3 + 2x2 + 2x + v〉	〈2x3 + v〉	210410
〈v + 2x〉	〈2〉	〈2x3 + v + 2〉	〈v + 2x + 2〉	210410
〈v + 2x〉	〈2〉	〈1〉	〈0〉	210410
〈v + 2x + 2〉	〈2〉	〈1〉	〈0〉	210410
〈v + 2x〉	〈2〉	〈0〉	〈1〉	210410
〈v + 2x + 2〉	〈2〉	〈0〉	〈1〉	210410
〈2〉	〈2x3 + 2x + v〉	〈2x3 + v + 2〉	〈v + 2x + 2〉	26412
〈2〉	〈2x3 + 2x + v〉	〈2x3 + 2x2 + 2x + v + 2〉	〈2x3 + v + 2〉	26412
〈2〉	〈2x3 + 2x + v〉	〈1〉	〈0〉	26412
〈2〉	〈2x2 + 2x + v〉	〈2x3 + 2x + v〉	〈2x3 + 2x2 + 2x + v〉	26412
〈2〉	〈2x2 + 2x + v〉	〈2x3 + 2x2 + v + 2〉	〈2x2 + v + 2〉	26412
〈2〉	〈2x2 + 2x + v〉	〈1〉	〈0〉	26412
〈2〉	〈2x2 + 2x + v〉	〈0〉	〈1〉	26412
〈2〉	〈2x3 + 2x + v + 2〉	〈2x3 + v〉	〈v + 2x〉	26412
〈2〉	〈2x3 + 2x + v + 2〉	〈2x3 + 2x2 + 2x + v〉	〈2x3 + v〉	26412
〈2〉	〈2x3 + 2x + v + 2〉	〈1〉	〈0〉	26412
〈2〉	〈2x2 + 2x + v + 2〉	〈2x3 + 2x2 + v〉	〈2x2 + v〉	26412
〈2〉	〈2x2 + 2x + v + 2〉	〈2x3 + 2x + v + 2〉	〈2x3 + 2x2 + 2x + v + 2〉	26412
〈2〉	〈2x2 + 2x + v + 2〉	〈1〉	〈0〉	26412
〈2〉	〈2x2 + 2x + v + 2〉	〈0〉	〈1〉	26412
〈v + 2x + 2〉	〈2x3 + 2x + v〉	〈v〉	〈v + 2〉	22414
〈v + 2x〉	〈2x3 + 2x + v〉	〈2x3 + 2x2 + v〉	〈2x2 + v〉	22414
〈v + 2x + 2〉	〈2x3 + 2x + v〉	〈2x3 + 2x2 + v〉	〈2x2 + v〉	22414
〈v + 2x〉	〈2x3 + 2x + v〉	〈v + 2〉	〈v〉	22414
〈v + 2x + 2〉	〈2x3 + 2x + v〉	〈2x2 + v + 2〉	〈2x2 + 2x + v〉	22414
〈v + 2x〉	〈2x3 + 2x + v〉	〈2x3 + 2x2 + 2x + v + 2〉	〈2x3 + v + 2〉	22414
〈v + 2x〉	〈2x3 + 2x + v〉	〈1〉	〈0〉	22414
〈v + 2x + 2〉	〈2x3 + 2x + v〉	〈1〉	〈0〉	22414
〈v + 2x〉	〈2x2 + 2x + v〉	〈v〉	〈v + 2〉	22414
〈v + 2x〉	〈2x2 + 2x + v〉	〈v + 2x〉	〈2x3 + 2x2 + v + 2〉	22414
〈v + 2x + 2〉	〈2x2 + 2x + v〉	〈v + 2x〉	〈2x3 + 2x2 + v + 2〉	22414
〈v + 2x〉	〈2x2 + 2x + v〉	〈2x3 + 2x + v〉	〈2x3 + 2x2 + 2x + v〉	22414
〈v + 2x + 2〉	〈2x2 + 2x + v〉	〈v + 2〉	〈v〉	22414
〈v + 2x〉	〈2x2 + 2x + v〉	〈2x3 + v + 2〉	〈v + 2x + 2〉	22414
〈v + 2x + 2〉	〈2x2 + 2x + v〉	〈2x3 + v + 2〉	〈v + 2x + 2〉	22414
〈v + 2x + 2〉	〈2x2 + 2x + v〉	〈1〉	〈0〉	22414
〈v + 2x + 2〉	〈2x3 + 2x + v + 2〉	〈v〉	〈v + 2〉	22414
〈v + 2x〉	〈2x3 + 2x + v + 2〉	〈2x3 + v〉	〈v + 2x〉	22414
〈v + 2x + 2〉	〈2x3 + 2x + v + 2〉	〈2x2 + v〉	〈2x2 + 2x + v + 2〉	22414
〈v + 2x〉	〈2x3 + 2x + v + 2〉	〈2x3 + 2x + v〉	〈2x3 + 2x2 + 2x + v〉	22414
〈v + 2x + 2〉	〈2x3 + 2x + v + 2〉	〈2x3 + 2x + v〉	〈2x3 + 2x2 + 2x + v〉	22414
〈v + 2x〉	〈2x3 + 2x + v + 2〉	〈v + 2〉	〈v〉	22414
〈v + 2x〉	〈2x3 + 2x + v + 2〉	〈1〉	〈0〉	22414
〈v + 2x + 2〉	〈2x3 + 2x + v + 2〉	〈1〉	〈0〉	22414
〈v + 2x〉	〈2x2 + 2x + v + 2〉	〈v〉	〈v + 2〉	22414
〈v + 2x〉	〈2x2 + 2x + v + 2〉	〈2x3 + 2x2 + v〉	〈2x2 + v〉	22414
〈v + 2x〉	〈2x2 + 2x + v + 2〉	〈2x3 + 2x2 + 2x + v〉	〈2x3 + v〉	22414
〈v + 2x + 2〉	〈2x2 + 2x + v + 2〉	〈2x3 + 2x2 + 2x + v〉	〈2x3 + v〉	22414
〈v + 2x + 2〉	〈2x2 + 2x + v + 2〉	〈v + 2〉	〈v〉	22414
〈v + 2x〉	〈2x2 + 2x + v + 2〉	〈v + 2x + 2〉	〈2x3 + 2x2 + v〉	22414
〈v + 2x + 2〉	〈2x2 + 2x + v + 2〉	〈v + 2x + 2〉	〈2x3 + 2x2 + v〉	22414
〈v + 2x〉	〈2x2 + 2x + v + 2〉	〈1〉	〈0〉	22414

There are 92 2-quasi-cyclic self-dual codes $\theta (\mathcal {C})$ of length 30 with minimum Lee weight 10 over $\mathbb {Z}_{4}$ derived from the 315 self-dual codes $\mathcal {C}$ of length 15 over $\mathbb {Z}_{4}+v\mathbb {Z}_{4}$. These 92 codes are given by the following table, withC₁, C₂, C₃, C₄ and the type of each $\mathbb {Z}_{4}$-code $\theta (\mathcal {C})$.

C1	C2	C3 (mod f3(x))	C4 (mod f4(x))	Type
〈2〉	〈2x3 + 2x + v〉	〈2v〉	〈2,v〉	21448
〈2〉	〈2x3 + 2x + v〉	〈2,v〉	〈2v〉	21448
〈2〉	〈2x2 + 2x + v〉	〈2v〉	〈2,v〉	21448
〈2〉	〈2x2 + 2x + v〉	〈2,v〉	〈2v〉	21448
〈2〉	〈2x3 + 2x + v + 2〉	〈2v〉	〈2,v〉	21448
〈2〉	〈2x3 + 2x + v + 2〉	〈2,v〉	〈2v〉	21448
〈2〉	〈2x2 + 2x + v + 2〉	〈2v〉	〈2,v〉	21448
〈2〉	〈2x2 + 2x + v + 2〉	〈2,v〉	〈2v〉	21448
〈v + 2x〉	〈2x3 + 2x + v〉	〈2v〉	〈2,v〉	210410
〈v + 2x + 2〉	〈2x3 + 2x + v〉	〈2v〉	〈2,v〉	210410
〈v + 2x + 2〉	〈2x3 + 2x + v〉	〈2,v〉	〈2v〉	210410
〈v + 2x + 2〉	〈2x2 + 2x + v〉	〈2v〉	〈2,v〉	210410
〈v + 2x〉	〈2x2 + 2x + v〉	〈2,v〉	〈2v〉	210410
〈v + 2x + 2〉	〈2x2 + 2x + v〉	〈2,v〉	〈2v〉	210410
〈v + 2x〉	〈2x3 + 2x + v + 2〉	〈2v〉	〈2,v〉	210410
〈v + 2x + 2〉	〈2x3 + 2x + v + 2〉	〈2v〉	〈2,v〉	210410
〈v + 2x + 2〉	〈2x3 + 2x + v + 2〉	〈2,v〉	〈2v〉	210410
〈v + 2x + 2〉	〈2x2 + 2x + v + 2〉	〈2v〉	〈2,v〉	210410
〈v + 2x〉	〈2x2 + 2x + v + 2〉	〈2,v〉	〈2v〉	210410
〈v + 2x + 2〉	〈2x2 + 2x + v + 2〉	〈2,v〉	〈2v〉	210410
〈v + 2x〉	〈2〉	〈2x2 + v〉	〈2x2 + 2x + v + 2〉	210410
〈v + 2x + 2〉	〈2〉	〈2x2 + v〉	〈2x2 + 2x + v + 2〉	210410
〈v + 2x〉	〈2〉	〈v + 2x〉	〈2x3 + 2x2 + v + 2〉	210410
〈v + 2x + 2〉	〈2〉	〈v + 2x〉	〈2x3 + 2x2 + v + 2〉	210410
〈v + 2x〉	〈2〉	〈2x2 + v + 2〉	〈2x2 + 2x + v〉	210410
〈v + 2x + 2〉	〈2〉	〈2x2 + v + 2〉	〈2x2 + 2x + v〉	210410
〈v + 2x〉	〈2〉	〈v + 2x + 2〉	〈2x3 + 2x2 + v〉	210410
〈v + 2x + 2〉	〈2〉	〈v + 2x + 2〉	〈2x3 + 2x2 + v〉	210410
〈2〉	〈2x3 + 2x + v〉	〈v〉	〈v + 2〉	26412
〈2〉	〈2x3 + 2x + v〉	〈2x3 + 2x2 + v〉	〈2x2 + v〉	26412
〈2〉	〈2x3 + 2x + v〉	〈2x3 + 2x + v〉	〈2x3 + 2x2 + 2x + v〉	26412
〈2〉	〈2x3 + 2x + v〉	〈v + 2〉	〈v〉	26412
〈2〉	〈2x3 + 2x + v〉	〈2x3 + 2x2 + v + 2〉	〈2x2 + v + 2〉	26412
〈2〉	〈2x3 + 2x + v〉	〈2x3 + 2x + v + 2〉	〈2x3 + 2x2 + 2x + v + 2〉	26412
〈2〉	〈2x3 + 2x + v〉	〈0〉	〈1〉	26412
〈2〉	〈2x2 + 2x + v〉	〈v〉	〈v + 2〉	26412
〈2〉	〈2x2 + 2x + v〉	〈2x3 + v〉	〈v + 2x〉	26412
〈2〉	〈2x2 + 2x + v〉	〈2x3 + 2x2 + 2x + v〉	〈2x3 + v〉	26412
〈2〉	〈2x2 + 2x + v〉	〈v + 2〉	〈v〉	26412
〈2〉	〈2x2 + 2x + v〉	〈2x3 + v + 2〉	〈v + 2x + 2〉	26412
〈2〉	〈2x2 + 2x + v〉	〈2x3 + 2x2 + 2x + v + 2〉	〈2x3 + v + 2〉	26412
〈2〉	〈2x3 + 2x + v + 2〉	〈v〉	〈v + 2〉	26412
〈2〉	〈2x3 + 2x + v + 2〉	〈2x3 + 2x2 + v〉	〈2x2 + v〉	26412
〈2〉	〈2x3 + 2x + v + 2〉	〈2x3 + 2x + v〉	〈2x3 + 2x2 + 2x + v〉	26412
〈2〉	〈2x3 + 2x + v + 2〉	〈v + 2〉	〈v〉	26412
〈2〉	〈2x3 + 2x + v + 2〉	〈2x3 + 2x2 + v + 2〉	〈2x2 + v + 2〉	26412
〈2〉	〈2x3 + 2x + v + 2〉	〈2x3 + 2x + v + 2〉	〈2x3 + 2x2 + 2x + v + 2〉	26412
〈2〉	〈2x3 + 2x + v + 2〉	〈0〉	〈1〉	26412
〈2〉	〈2x2 + 2x + v + 2〉	〈v〉	〈v + 2〉	26412
〈2〉	〈2x2 + 2x + v + 2〉	〈2x3 + v〉	〈v + 2x〉	26412
〈2〉	〈2x2 + 2x + v + 2〉	〈2x3 + 2x2 + 2x + v〉	〈2x3 + v〉	26412
〈2〉	〈2x2 + 2x + v + 2〉	〈v + 2〉	〈v〉	26412
〈2〉	〈2x2 + 2x + v + 2〉	〈2x3 + v + 2〉	〈v + 2x + 2〉	26412
〈2〉	〈2x2 + 2x + v + 2〉	〈2x3 + 2x2 + 2x + v + 2〉	〈2x3 + v + 2〉	26412
〈v + 2x〉	〈2x3 + 2x + v〉	〈v〉	〈v + 2〉	22414
〈v + 2x〉	〈2x3 + 2x + v〉	〈2x3 + 2x + v〉	〈2x3 + 2x2 + 2x + v〉	22414
〈v + 2x + 2〉	〈2x3 + 2x + v〉	〈2x3 + 2x + v〉	〈2x3 + 2x2 + 2x + v〉	22414
〈v + 2x + 2〉	〈2x3 + 2x + v〉	〈v + 2〉	〈v〉	22414
〈v + 2x〉	〈2x3 + 2x + v〉	〈2x2 + v + 2〉	〈2x2 + 2x + v〉	22414
〈v + 2x〉	〈2x3 + 2x + v〉	〈v + 2x + 2〉	〈2x3 + 2x2 + v〉	22414
〈v + 2x + 2〉	〈2x3 + 2x + v〉	〈v + 2x + 2〉	〈2x3 + 2x2 + v〉	22414
〈v + 2x〉	〈2x3 + 2x + v〉	〈0〉	〈1〉	22414
〈v + 2x + 2〉	〈2x3 + 2x + v〉	〈0〉	〈1〉	22414
〈v + 2x + 2〉	〈2x2 + 2x + v〉	〈v〉	〈v + 2〉	22414
〈v + 2x〉	〈2x2 + 2x + v〉	〈2x3 + 2x2 + 2x + v〉	〈2x3 + v〉	22414
〈v + 2x + 2〉	〈2x2 + 2x + v〉	〈2x3 + 2x2 + 2x + v〉	〈2x3 + v〉	22414
〈v + 2x〉	〈2x2 + 2x + v〉	〈v + 2〉	〈v〉	22414
〈v + 2x〉	〈2x2 + 2x + v〉	〈2x2 + v + 2〉	〈2x2 + 2x + v〉	22414
〈v + 2x + 2〉	〈2x2 + 2x + v〉	〈2x2 + v + 2〉	〈2x2 + 2x + v〉	22414
〈v + 2x〉	〈2x2 + 2x + v〉	〈2x3 + 2x2 + 2x + v + 2〉	〈2x3 + v + 2〉	22414
〈v + 2x〉	〈2x2 + 2x + v〉	〈1〉	〈0〉	22414
〈v + 2x〉	〈2x2 + 2x + v〉	〈0〉	〈1〉	22414
〈v + 2x + 2〉	〈2x2 + 2x + v〉	〈0〉	〈1〉	22414
〈v + 2x〉	〈2x3 + 2x + v + 2〉	〈v〉	〈v + 2〉	22414
〈v + 2x〉	〈2x3 + 2x + v + 2〉	〈2x2 + v〉	〈2x2 + 2x + v + 2〉	22414
〈v + 2x〉	〈2x3 + 2x + v + 2〉	〈2x3 + 2x2 + v〉	〈2x2 + v〉	22414
〈v + 2x + 2〉	〈2x3 + 2x + v + 2〉	〈2x3 + 2x2 + v〉	〈2x2 + v〉	22414
〈v + 2x〉	〈2x3 + 2x + v + 2〉	〈v + 2x〉	〈2x3 + 2x2 + v + 2〉	22414
〈v + 2x + 2〉	〈2x3 + 2x + v + 2〉	〈v + 2x〉	〈2x3 + 2x2 + v + 2〉	22414
〈v + 2x + 2〉	〈2x3 + 2x + v + 2〉	〈v + 2〉	〈v〉	22414
〈v + 2x〉	〈2x3 + 2x + v + 2〉	〈0〉	〈1〉	22414
〈v + 2x + 2〉	〈2x3 + 2x + v + 2〉	〈0〉	〈1〉	22414
〈v + 2x + 2〉	〈2x2 + 2x + v + 2〉	〈v〉	〈v + 2〉	22414
〈v + 2x〉	〈2x2 + 2x + v + 2〉	〈2x3 + v〉	〈v + 2x〉	22414
〈v + 2x〉	〈2x2 + 2x + v + 2〉	〈2x2 + v〉	〈2x2 + 2x + v + 2〉	22414
〈v + 2x + 2〉	〈2x2 + 2x + v + 2〉	〈2x2 + v〉	〈2x2 + 2x + v + 2〉	22414
〈v + 2x〉	〈2x2 + 2x + v + 2〉	〈v + 2〉	〈v〉	22414
〈v + 2x〉	〈2x2 + 2x + v + 2〉	〈2x3 + v + 2〉	〈v + 2x + 2〉	22414
〈v + 2x + 2〉	〈2x2 + 2x + v + 2〉	〈2x3 + v + 2〉	〈v + 2x + 2〉	22414
〈v + 2x + 2〉	〈2x2 + 2x + v + 2〉	〈1〉	〈0〉	22414
〈v + 2x〉	〈2x2 + 2x + v + 2〉	〈0〉	〈1〉	22414
〈v + 2x + 2〉	〈2x2 + 2x + v + 2〉	〈0〉	〈1〉	22414

Let ϕ be the Gray map from $\mathbb {Z}_{4}^{30}$ onto $\mathbb {F}_{2}^{60}$ extended by $0\rightarrow 00, 1\rightarrow 01, 2\rightarrow 11, 3\rightarrow 10$ in the natural way. Then ϕ is a distance and orthogonality preserving bijection from $(\mathbb {Z}_{4}^{30}, \text {Lee} \ \text {distance})$ onto $(\mathbb {F}_{2}^{60}, \text {Hamming} \ \text {distance})$. From the 70 2-quasi-cyclic self-dual codes with minimal Lee weight 12 and 92 2-quasi-cyclic self-dual codes with minimum Lee weight 10 over $\mathbb {Z}_{4}$ above and by the Gray map ϕ, we derive 70 4-quasi-cyclic type II binary formally self-dual [60,30,12] codes and 92 4-quasi-cyclic type II binary formally self-dual [60,30,10] codes. It is well known that binary self-dual [60,30,12] codes are extremal (cf. [14] and [24]).