(1 + u)-Constacyclic codes over Z₄ + uZ₄

Abstract

Constacyclic codes are an important class of linear codes in coding theory. Many optimal linear codes are directly derived from constacyclic codes. In this paper, (1 + u)-constacyclic codes over Z₄ + uZ₄ of any length are studied. A new Gray map between Z₄ + uZ₄ and Z⁴₄ is defined. By means of this map, it is shown that the Z₄ Gray image of a (1 + u)-constacyclic code of length n over Z₄ + uZ₄ is a cyclic code over Z₄ of length 4n. Furthermore, by combining the classical Gray map between Z₄ and F²₂, it is shown that the binary image of a (1 + u)-constacyclic code of length n over Z₄ + uZ₄ is a distance invariant binary quasi-cyclic code of index 4 and length 8n. Examples of good binary codes are constructed to illustrate the application of this class of codes.

Background

Recently, several new classes of rings have been studied in connection with coding theory. Many optimal binary linear codes have been obtained from codes over these rings via some Gray map. In Yildiz and Karadenniz (2010a, b), the authors introduced the ring F₂ + uF₂ + vF₂ + uvF₂ and discussed linear and self-dual codes over F₂ + uF₂ + vF₂ + uvF₂. Later, the structures of cyclic codes and (1 + u)-constacyclic codes over F₂ + uF₂ + vF₂ + uvF₂ were studied and many optimal binary linear codes were constructed from such codes in Yildiz and Karadenniz (2011a, b). More generally, cyclic codes over the ring R_k were investigated in Dougherty et al. (2012). Although the rings mentioned above are not finite chain rings, they have rich algebraic structures and produce binary codes with large automorphism groups and new binary self-dual codes. This demonstrates that linear codes over such non-chain rings have been received increasing attention (see Dougherty et al. 2012; Kai et al. 2012; Shi 2014; Shi et al. 2012; Siap et al. 2012; Zhu and Wang 2011). More recently, linear codes over the non-chain ring Z₄ + uZ₄, where u² = 0, have been explored in Yildiz and Karadenniz (2014). The authors defined a linear Gray map from Z₄ + uZ₄ to Z²₄ and a non-linear Gray map from Z₄ + uZ₄ to (F₂ + uF₂)², and used them to successfully construct formally self-dual codes over Z₄ and good non-linear codes over F₂ + uF₂. In Yildiz and Aydin (2014), the structure of cyclic codes over Z₄ + uZ₄ was determined and many new linear codes over Z₄ were obtained from them. Motivated by the works in Yildiz and Aydin (2014) and Yildiz and Karadenniz (2014), we focus on constacyclic codes over Z₄ + uZ₄ and intend to construct good binary codes from such codes.

The ring Z₄ + uZ₄ is a finite commutative ring with characteristic 4, where u² = 0. The purpose of this paper is to investigate a class of constacyclic codes over this ring, that is, (1 + u)-constacyclic codes over Z₄ + uZ₄. Constacyclic codes over finite commutative rings were first introduced by Wolfmann (1999), where it was proved that the binary image of a linear negacyclic code over Z₄ is a binary cyclic code (not necessarily linear). In Kai et al. (2012), the authors introduced a composite Gray map from F₂ + uF₂ + vF₂ + uvF₂ to F⁴₂ and proved that the image of a (1 + u)-constacyclic code of length n over F₂ + uF₂ + vF₂ + uvF₂ under the Gray map is a distance invariant binary quasi-cyclic code of index 2 and length 4n. It is known that the structure of Z₄ + uZ₄ is similar to that of F₂ + uF₂ + vF₂ + uvF₂. It is natural to ask if there exists a Gray map such that the Gray image of a linear code over Z₄ + uZ₄ has good structure. For this, we introduce a new Gray map from Z₄ + uZ₄ to Z₄, and explore the images of (1 + u)-constacyclic codes over Z₄ + uZ₄ under this Gray map.

(1 + u)-Constacyclic codes over Z₄ + uZ₄

Throughout this paper, let R denote the ring Z₄ + uZ₄ with u² = 0. Any element in R can be written as a + bu, where a, b ∊ Z₄. The element a + bu is a unit in R if and only if a is a unit in Z₄. The ring R is a local Frobenius ring, but not a finite chain ring. It has a total of 7 ideals given by $I_0=\{0\}\subseteq I_{2u}=2u(Z_4+u{Z}_4)=\{0,2u\}\subseteq I_u,I_2,I_{2+u}\subseteq I_{2,u}\subseteq I_1=Z_4+u{Z}_4$, where

$$\begin{array}{cc}\hfill I_u& =u(Z_4+u{Z}_4)=\{0,u,2u,3u\},\hfill \\ \hfill I_2& =2(Z_4+u{Z}_4)=\{0,2,2u,2+2u\},\hfill \\ \hfill I_{2+u}& =(2+u)(Z_4+u{Z}_4)=\{0,2+u,2u,2+3u\},\hfill \\ \hfill I_{2,u}& =\{0,2,u,2u,3u,2+u,2+2u,2+3u\}.\hfill \\ \hfill \end{array}$$

A code over R of length n is a nonempty subset of R_n, and a code is linear over R of length n if it is an R-submodule of R_n. For some fixed unit λ ∊ R, the λ-constacyclic shift τ on R_n is the shift τ(c₀, c₁, …, c_n−1) = (λc_n−1, c₀, …, c_n−2). A linear code C of length n over R is λ-constacyclic if the code is invariant under the λ-constacyclic shift τ. We identify the code-word c = (c₀, c₁, …, c_n−1) with its polynomial representation c(x) = c₀ + c₁x + ··· + c_n−1xⁿ⁻¹. Then xc(x) corresponds to a λ-constacyclic shift of c(x) in the ring R[x]/(xⁿ − λ). Thus, λ-constacyclic codes of length n over R can be identified as ideals in the ring R[x]/(xⁿ − λ). From the above discuss, we have the following result.

Proposition 1

A subset C of R_nis a linear cyclic code of length n if and only if C is an ideal of$A_n=R[x]/(x^n-1)$. A subset C of R_nis a linear$(1+u)$-constacyclic code of length n over R if and only if C is an ideal of$B_n=R[x]/(x^n-1-u)$.

Now, we determine a set of generators of (1 + u)-constacyclic codes for an arbitrary length over R. We begin by recalling a unique set of generators for cyclic codes over Z₄.

Lemma 2

[cf. Abualrub and Siap (2006), Theorem 6] Let C be a cyclic code of length n over Z₄. Then

1. If n is odd then$C=⟨g(x),2a(x)⟩=⟨g(x)+2a(x)⟩$, where$g(x),a(x)$are binary polynomials with$a(x)\left|g(x)\right.\left|(x^n-1)\right.mod2$.

2. If n is even then

   • 2.1
     If$g(x)=a(x)$, then$C=⟨g(x)+2p(x)⟩$, where$g(x),p(x)$are binary polynomials with$g(x)\left|(x^n-1)\right.mod2$, and$g(x)\left|p(x)\frac{(x^n-1)}{g(x)}\right.$,
   • 2.2
     $C=⟨g(x)+2p(x),2a(x)⟩$, where$g(x),a(x)$and$p(x)$are binary polynomials with$a(x)\left|g(x)\right.\left|(x^n-1)\right.mod2$, $a(x)\left|p(x)\frac{(x^n-1)}{g(x)}\right.$and$degg(x)>dega(x)>degp(x)$.

For a linear code C of length n over R, we can denote two linear codes of length n over Z₄ as follows:

1. The torsion code Tor(C) = {x ∊ Zⁿ₄|ux ∊ C},

2. The residue code $\text{Res}(C)=\{x\in Z_4^n\left|\exists y\in Z_4^n:x+uy\in C\right.\}$.

Consider the homomorphism $\mathit{\phi }:R\to Z_4$ defined by $\mathit{\phi }(a+ub)=a$. The map $\mathit{\phi }$ extends naturally to a ring homomorphism $\mathit{\phi }:R_n\to Z_4(n)=\frac{Z_4[x]}{(x^n-1)}$ defined by

$$\mathit{\phi }(c_0+c_{1}x+\cdots ,c_{n-1}{x}^{n-1})=\mathit{\phi }\left(c_0\right)+\mathit{\phi }\left(c_1\right)x+\cdots +\mathit{\phi }\left(c_{n-1}\right)x^{n-1}.$$

Acting $\mathit{\phi }$ on C over R, we define a ring homomorphism

$$\mathit{\phi }:C\to \text{Res}(C),\mathit{\phi }(a+ub)=a\text{where}a,b\in Z_4.$$

We can easily obtain that $Ker\mathit{\phi }$ ≅ Tor(C) and $\mathit{\phi }$(C) = Res(C). By the first isomorphism theorem of finite groups, we have $\left|C\right|=\left|Tor(C)\right|\left|\text{Res}(C)\right|$. It is obvious that the image of C under the map $\mathit{\phi }$ is a cyclic code of length n over Z₄. Combining the above discussion with Lemma 2, we can obtain the set of generators for cyclic codes of length n over R.

Theorem 3

Let C be a$(1+u)$-constacyclic code of length n over R. Then

1. If n is odd then$C=⟨g_1(x)+2a_1(x)+ub(x),u(g_2(x)+2a_2(x))⟩$, where$b(x)$is a polynomial in$Z_4[x]$and$g_i(x),a_i(x)$are binary polynomials with$a_i(x)\left|g_i(x)\right.\left|(x^n\right.-1)mod2fori=1,2$.

2. If n is even then

   • 2.1
     If$g_i(x)=a_i(x)$, then$C=⟨g_1(x)+2p_1(x)+ud(x),u(g_2(x)+2p_2(x))⟩$, where$d(x)$is a polynomial in$Z_4[x]$, $g_i(x),p_i(x)$are binary polynomials with$g_i(x)\left|(x^n\right.-1)mod2$and$g_i(x)\left|p_i(x)\frac{(x^n-1)}{g_i(x)}\right.,fori=1,2$;
   • 2.2
     $C=⟨g_1(x)+2p_1(x)+u{e}_1(x),2a_1(x)+u{e}_2(x),u{g}_2(x)+2u{p}_2(x),2a_2(x)⟩$, where$e_i(x)$is a polynomial in$Z_4[x]$, and$g_i(x),a_i(x),p_i(x)$are binary polynomials with$a_i(x)\left|g_i(x)\right.\left|(x^n\right.-1)mod2$, $a_i(x)\left|p_i(x)\frac{(x^n-1)}{g_i(x)}\right.$and$deg{g}_i(x)>deg{a}_i(x)>deg{p}_i(x)$,$fori=1,2$.

Proof

We only give the proof of the part (1), and the proof of the part (2) is similar.

Assume that n is odd. Let C be a (1 + u)-constacyclic code of length n over R. Then the image of C under the map $\mathit{\phi }$ is Res(C), which is a cyclic code of length n over Z₄. By Lemma 2, we have $\mathit{\phi }$(C) = 〈g₁(x) + 2a₁(x)〉, where g₁(x), a₁(x) are binary polynomials with $a_1(x)\left|g_1(x)\right.\left|(x^n\right.-1)mod2$. Thus, there exists b(x) ∊ Z₄[x] such that g₁(x) + 2a₁(x) + ub(x) ∊ C.

Furthermore, note that $Ker\mathit{\phi }$ is a cyclic code of length n over Z₄ + uZ₄, so $Ker\mathit{\phi }$ = u 〈g₂(x) + 2a₂(x)〉, where g₂(x), a₂(x) are binary polynomials with $a_2(x)\left|g_2(x)\right.\left|(x^n\right.-1)mod2$. Hence, 〈g₁(x) + 2a₁(x) + ub(x), u(g₂(x) + 2a₂(x))〉 ⊆ C.

On the other hand, for any f(x) = f₁(x) + uf₂(x) ∊ C, where f_i(x) ∊ Z₄[x], for i = 1, 2, it is obvious that f₁(x) ∊ $\mathit{\phi }$(C). Hence,

$$\begin{array}{cc}\hfill f(x)& =f_1(x)+u{f}_2(x)\hfill \\ \hfill & =m(x)(g_1(x)+2a_1(x))+u{f}_2(x)\hfill \\ \hfill & =m(x)(g_1(x)+2a_1(x)+ub(x))+u(f_2(x)-m(x)b(x))\hfill \\ \hfill \end{array}$$

Since u(f₂(x) − m(x)b(x)) ∊ $Ker\mathit{\phi }$, we have

$$f(x)\in ⟨g_1(x)+2a_1(x)+ub(x),u\left(g_2(x)+2a_2(x)\right)⟩.$$

This shows that C ⊆ 〈g₁(x) + 2a₁(x) + ub(x), u(g₂(x) + 2a₂(x))〉.

Thus, C = 〈g₁(x) + 2a₁(x) + ub(x), u(g₂(x) + 2a₂(x))〉.□

Gray images of (1 + u)-constacyclic codes over R

A new Gray map

Recall that the Gray map ${\mathit{\varphi }}_1$ from Z₄ to F²₂ is defined as ${\mathit{\varphi }}_1$(z) = (q, q + r) where z = r + 2q with r, q ∊ F₂. The map ${\mathit{\varphi }}_1$ can be extended to Zⁿ₄ as follows:

$$\begin{array}{cc}\hfill {\mathit{\varphi }}_1& :Z_4^n\to F_2^{2n}\hfill \\ \hfill & (z_0,z_1,\dots ,z_{n-1})\to (q_0,q_1,\dots ,q_{n-1},q_0+r_0,q_1+r_1,\dots ,q_{n-1}+r_{n-1})\hfill \\ \hfill \end{array}$$

where z_i = r_i + 2q_i with r_i, q_i ∊ F₂for 0 ≤ i ≤ n − 1. It is known that ${\mathit{\varphi }}_1$ is a distance-preserving map from Zⁿ₄ (Lee distance) to F²ⁿ₂ (Hamming distance).

Now, we define a map ${\mathit{\varphi }}_2$ from Rⁿ to Z⁴ⁿ₄. First note that each element c ∊ R can be expressed as c = a + ub, where a, b ∊ Z₄. The map ${\mathit{\varphi }}_2$ is defined as

$${\mathit{\varphi }}_2(c)=\left(b+3a,b+2a,b+a,b\right).$$

Clearly, this map can be also extended to Rⁿ as follows:

$$\begin{array}{cc}\hfill {\mathit{\varphi }}_2& :R^n\to Z_4^{4n}\hfill \\ \hfill & (c_0,c_1,\dots ,c_{n-1})\to (b_0+3a_0,b_1+3a_1,\dots ,b_{n-1}+3a_{n-1},b_0+2a_0,b_1\hfill \\ \hfill & +2a_1,\dots ,b_{n-1}+2a_{n-1},b_0+a_0,b_1+a_1,\dots ,b_{n-1}+a_{n-1},b_0,b_1,\dots ,b_{n-1})\hfill \\ \hfill \end{array}$$

where c_i = a_i + ub_i with a_i, b_i ∊ Z₄for 0 ≤ i ≤ n − 1.

It is well-known that the homogeneous weight has many applications for codes over finite rings and provides a good metric for the underlying ring in constructing superior codes. Next, we define a homogeneous weight on R. We first recall the definition of the homogeneous weight on a finite ring K.

Definition 4

[cf. Greferath and O’Sullivan (2004), Definition 1.1] A real-valued function w on the finite ring K is called a (left) homogeneous weight if w(0) = 0 and the following is true:

1. For all x, y ∊ K, Kx = Ky implies that w(x) = w(y) holds.

2. There exists a real number γ such that ∑_y∊Kx w(y) = γ|Kx| for all $x\in K\backslash \{0\}$.

Right homogenous weight is defined accordingly. If a weight is both left homogenous and right homogeneous, we call it simply as a homogeneous weight.

For any element c = a + ub ∊ R, we assign the weight, denoted by w_hom(c), as w_L(b + 3a, b + 2a, b + a, b), i.e., w_hom(c) = w_L(b + 3a, b + 2a, b + a, b). By simple calculation, we can obtain the weight of any element x = a + ub ∊ R as follows:

$$w_{hom}(x)=\left\{\begin{array}{cc}0,\hfill & x=0\\ 8,\hfill & x=2u\\ 4,\hfill & \text{otherwise}\end{array}\right.$$

It is easy to verify that the weight defined above meets the conditions of Definition 4, hence it is actually a homogeneous weight on R. The homogeneous distance of a linear code over R, denoted by d_hom(C), is defined as the minimum homogeneous weight of nonzero codewords of C. It can be checked that the map ${\mathit{\varphi }}_2$ is a distance-preserving map from Rⁿ (homogeneous distance) to Z⁴ⁿ₄ (Lee distance). Using the maps ${\mathit{\varphi }}_1$ and ${\mathit{\varphi }}_2$, we can define a composite map $\mathit{\varphi }:R\to F_2^8\text{as}\mathit{\varphi }={\mathit{\varphi }}_1{\mathit{\varphi }}_2$. Thus, we have obtained three distance-preserving maps as follows:

$$\begin{array}{cc}\hfill {\mathit{\varphi }}_1& :\left(Z_4^n,\text{Lee}\text{distance}\right)\to \left(F_2^{2n},\text{Hamming}\text{distance}\right),\hfill \\ \hfill {\mathit{\varphi }}_2& :\left(R^n,\text{homogeneous}\text{distance}\right)\to \left(Z_4^{4n},\text{Lee}\text{distance}\right),\hfill \\ \hfill \mathit{\varphi }& :\left(R^n,\text{homogeneous}\text{distance}\right)\to \left(F_2^{8n},\text{Hamming}\text{distance}\right).\hfill \\ \hfill \end{array}$$

Gray images of (1 + u)-constacyclic codes

Lemma 5

Let ${\mathit{\varphi }}_2$be defined as above. Let τ be the $(1+u)$-constacyclic shift on Rⁿand σ be the cyclic shift on Z⁴ⁿ₄. Then${\mathit{\varphi }}_2\mathit{\tau }=\mathit{\sigma }{\mathit{\varphi }}_2$.

Proof

Let c = (c₀, c₁, …, c_n−1) ∊ Rⁿ. Let c_i = a_i + ub_i where a_i, b_i ∊ Z₄for 0 ≤ i ≤ n − 1. From definitions, we have

$$\begin{array}{cc}\hfill {\mathit{\varphi }}_2(c)& =({\text{b}}_0+3a_0,{\text{b}}_1+3a_1,\dots ,{\text{b}}_{n-1}+3a_{n-1},{\text{b}}_0+2a_0,{\text{b}}_1+2a_1,\dots ,{\text{b}}_{n-1}\hfill \\ \hfill & +2a_{n-1},{\text{b}}_0+a_0,{\text{b}}_1+a_1,\dots ,{\text{b}}_{n-1}+a_{n-1},{\text{b}}_0,{\text{b}}_1,\dots ,{\text{b}}_{n-1})\hfill \\ \hfill \end{array}$$

Hence,

$$\begin{array}{cc}\hfill \mathit{\sigma }{\mathit{\varphi }}_2(c)& =({\text{b}}_{n-1},{\text{b}}_0+3a_0,{\text{b}}_1+3a_1,\dots ,{\text{b}}_{n-1}+3a_{n-1},{\text{b}}_0+2a_0,{\text{b}}_1+2a_1,\dots ,{\text{b}}_{n-1}\hfill \\ \hfill & +2a_{n-1},{\text{b}}_0+a_0,{\text{b}}_1+a_1,\dots ,{\text{b}}_{n-1}+a_{n-1},{\text{b}}_0,{\text{b}}_1,\dots ,{\text{b}}_{n-2})\hfill \\ \hfill \end{array}$$

On the other hand,

$$\begin{array}{cc}\hfill \mathit{\tau }(c)& =((1+u)c_{n-1},c_0,c_1,\dots ,c_{n-2})\hfill \\ \hfill & =(a_{n-1}+u(a_{n-1}+b_{n-1}),a_0+u{b}_0,\dots ,a_{n-2}+u{b}_{n-2})\hfill \\ \hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill {\mathit{\varphi }}_2\mathit{\tau }(c)& =({\text{b}}_{n-1},{\text{b}}_0+3a_0,{\text{b}}_1+3a_1,\dots ,{\text{b}}_{n-1}+3a_{n-1},{\text{b}}_0+2a_0,{\text{b}}_1+2a_1,\dots ,{\text{b}}_{n-1}\hfill \\ \hfill & +2a_{n-1},{\text{b}}_0+a_0,{\text{b}}_1+a_1,\dots ,{\text{b}}_{n-1}+a_{n-1},{\text{b}}_0,{\text{b}}_1,\dots ,{\text{b}}_{n-2})\hfill \\ \hfill \end{array}$$

The result follows.□

Theorem 6

A linear code C of length n over R is a$(1+u)$-constacyclic code if and only if ${\mathit{\varphi }}_2$(C) is a cyclic code of length 4n over Z₄.

Proof

If C is a (1 + u)-constacyclic code, then using Lemma 5 we have

$$\mathit{\sigma }\left({\mathit{\varphi }}_2(C)\right)={\mathit{\varphi }}_2\left(\mathit{\tau }(C)\right)={\mathit{\varphi }}_2(C).$$

Hence, ${\mathit{\varphi }}_2$(C) is a cyclic code of length 4n over Z₄.

Conversely, if ${\mathit{\varphi }}_2$(C) is a cyclic code of length 4n over Z₄, then using Lemma 5 again we get ${\mathit{\varphi }}_2$(τ(C)) = σ(${\mathit{\varphi }}_2$(C)) = ${\mathit{\varphi }}_2$(C).

Note that ${\mathit{\varphi }}_2$ is injection, so τ(C) = C.

Thus, we immediately have the following result.□

Corollary 7

The image of a$(1+u)$-constacyclic code of length n over R under the map ${\mathit{\varphi }}_2$is a distance invariant cyclic code of length 4n over Z₄.

Let σ be the cyclic shift. For any positive integer s, let σ_s be the quasi-shift given by

$${\mathit{\sigma }}_s(a^{(1)}\left|a^{(2)}\right.\left|\cdots \left|a^{(s)}\right.\right.)=(\mathit{\sigma }(a^{(1)})\left|\mathit{\sigma }(a^{(2)})\right.\left|\cdots \left|\mathit{\sigma }(a^{(\text{s})})\right.\right.)$$

where a⁽¹⁾, a⁽²⁾, …, a^(s) ∊ F²ⁿ₂ and “|”denotes the usual vector concatenation. A binary quasi-cyclic code C of index s and length 2ns is a subset of (F²ⁿ₂)^s such that σ_s(C) = C.

Lemma 8

Let $\mathit{\varphi }$be defined as above and let τ be the $(1+u)$-constacyclic shift on Rⁿ. Then  $\mathit{\varphi }\mathit{\tau }={\mathit{\sigma }}_4\mathit{\varphi }$.

Proof

Let r = (r₀, r₁, …, r_n−1) ∊ Rⁿ. Let r_i = a_i + 2b_i + uc_i + 2ud_i where a_i, b_i, c_i, d_i ∊ F₂for 0 ≤ i ≤ n − 1. Then we have

$$\begin{array}{cc}\hfill \mathit{\varphi }(r)& =(a_0+b_0+d_0,\dots ,a_{n-1}+b_{n-1}+d_{n-1},a_0+d_0,\dots ,a_{n-1}+d_{n-1},b_0+d_0,\dots ,b_{n-1}\hfill \\ \hfill & +d_{n-1},d_0,\dots ,d_{n-1},b_0+c_0+d_0,\dots ,b_{n-1}+c_{n-1}+d_{n-1},a_0+c_0+d_0,\dots ,a_{n-1}+c_{n-1}\hfill \\ \hfill & +d_{n-1},a_0+b_0+c_0+d_0,\dots ,a_{n-1}+b_{n-1}+c_{n-1}+d_{n-1},c_0+d_0,\dots ,c_{n-1}+d_{n-1})\hfill \\ \hfill \end{array}$$

and so

$$\begin{array}{cc}\hfill {\mathit{\sigma }}_4\mathit{\varphi }(r)& =(a_{n-1}+d_{n-1},a_0+b_0+d_0,\dots ,a_{n-1}+b_{n-1}+d_{n-1},a_0+d_0,\dots ,a_{n-2}\hfill \\ \hfill & +d_{n-2},d_{n-1},b_0+d_0,\dots ,b_{n-1}+d_{n-1},d_0,\dots ,d_{n-2},a_{n-1}+c_{n-1}+d_{n-1},b_0+c_0\hfill \\ \hfill & +d_0,\dots ,b_{n-1}+c_{n-1}+d_{n-1},a_0+c_0+d_0,\dots ,a_{n-2}+c_{n-2}+d_{n-2},c_{n-1}\hfill \\ \hfill & +d_{n-1},a_0+b_0+c_0+d_0,\dots ,a_{n-1}+b_{n-1}+c_{n-1}+d_{n-1},c_0+d_0,\dots ,c_{n-2}+d_{n-2})\hfill \\ \hfill \end{array}$$

On the other hand,

$$\begin{array}{cc}\hfill \mathit{\tau }(r)& =((1+u)r_{n-1},r_0,r_1,\dots ,r_{n-2})\hfill \\ \hfill & =((a_{n-1}+2b_{n-1})+u(a_{n-1}+c_{n-1})+2u(b_{n-1}+d_{n-1}),a_0\hfill \\ \hfill & +2b_0+u{c}_0+2u{d}_0,\dots ,a_{n-2}+2b_{n-2}+u{c}_{n-2}+2u{d}_{n-2})\hfill \\ \hfill \end{array}$$

Hence,

$$\begin{array}{cc}\hfill \mathit{\varphi }\mathit{\tau }(r)& =(a_{n-1}+d_{n-1},a_0+b_0+d_0,\dots ,a_{n-1}+b_{n-1}+d_{n-1},a_0+d_0,\dots ,a_{n-2}\hfill \\ \hfill & +d_{n-2},d_{n-1},b_0+d_0,\dots ,b_{n-1}+d_{n-1},d_0,\dots ,d_{n-2},a_{n-1}+c_{n-1}+d_{n-1},b_0+c_0\hfill \\ \hfill & +d_0,\dots ,b_{n-1}+c_{n-1}+d_{n-1},a_0+c_0+d_0,\dots ,a_{n-2}+c_{n-2}+d_{n-2},c_{n-1}\hfill \\ \hfill & +d_{n-1},a_0+b_0+c_0+d_0,\dots ,a_{n-1}+b_{n-1}+c_{n-1}+d_{n-1},c_0+d_0,\dots ,c_{n-2}+d_{n-2})\hfill \\ \hfill \end{array}$$

This completes the proof.□

Theorem 9

A linear code C of length n over R is a $(1+u)$-constacyclic code if and only if $\mathit{\varphi }$(C) is a binary quasi-cyclic code of index 4 and length 8n.

Proof

If C is (1 + u)-constacyclic, then using Lemma 8 we have

$${\mathit{\sigma }}_4\left(\mathit{\varphi }(C)\right)=\mathit{\varphi }\left(\mathit{\tau }(C)\right)=\mathit{\varphi }(C).$$

Hence, $\mathit{\varphi }$(C) is a binary quasi-cyclic code of index 4 and length 8n. Conversely, if $\mathit{\varphi }$(C) is a binary quasi-cyclic code of index 4 and length 8n, then using Lemma 8 again we get $\mathit{\varphi }$(τ(C)) = σ₄($\mathit{\varphi }$(C)) = $\mathit{\varphi }$(C).

Also, $\mathit{\varphi }$ is injection, hence τ(C) = C.□

From Theorem 9 and the definition of the map $\mathit{\varphi }$, we immediately have the following result.

Corollary 10

The image of a (1 + u)-constacyclic code of length n over R under the map $\mathit{\varphi }$is a distance invariant binary quasi-cyclic code of index 4 and length 8n.

Now, we can construct some binary codes with good parameters based on the new Gray map.

Example 11

Consider (1 + u)-constacyclic codes over Z₄ + uZ₄ of length 3. In F₂[x], x³ − 1=(x − 1)(x² + x + 1).

1. In Theorem 3, we take g₁(x) = x−1, a₁(x) = 1, b(x) = 3x, and g₂(x) = a₂(x) = x³ − 1. Then, we obtain the (1 + u)-constacyclic code C₁ over R of length 3 with generator polynomial (1 + u)x + 1. That is 〈(1 − u)x + 1〉 = 〈x + (1 + u)〉, It is easy to see that both Res(C₁) and Tor(C₁) have size 16. Moreover, d_hom(C₁) = 8. By Corollary 7, $\mathit{\varphi }$₂(C₁) is a Z₄-linear code of length 12 with size 256 and Lee distance 8. By Theorem 9, $\mathit{\varphi }$(C₁) is a binary quasi-cyclic code of index 4 and length 24. We find that $\mathit{\varphi }$(C₁) is a non-linear binary code with parameters (24, 256, 8). The code $\mathit{\varphi }$(C₁) attains the performance of the best-known binary linear code with the same parameters based on Grassl’s codetables (Grassl 2007).

2. In Theorem 3, $g_1(x)=x^3+1,a_1(x)=1,b(x)=3,g_2(x)=x+1,\text{and}{a}_2(x)=x+1$. Then, we obtain the code C₂ = 〈3u(x + 1)〉 = 〈u(x + 1)〉. Obviously, Res(C₂) = {0} and Tor(C₂) has size 4. By Corollary 7, $\mathit{\varphi }$(C₂) is a Z₄-linear code of length 12 with size 4 and Lee distance 16. By Theorem 9, $\mathit{\varphi }$(C₂) is a binary quasi-cyclic code of index 4 and length 24. The code $\mathit{\varphi }$(C₂) is a linear binary code with parameters [24, 2, 16], which is optimal based on Grassl’s codetables (Grassl 2007).

Conclusion

We study the structure of (1 + u)-constacyclic codes over Z₄ + uZ₄ of an arbitrary length, and obtain the examples of good binary codes from them. Our results show that a (1 + u)-constacyclic code of length n over Z₄ + uZ₄ under certain map is equivalent to a cyclic code of length 4n over Z₄. Furthermore, we discuss the relation between (1 + u)-constacyclic codes of length n over Z₄ + uZ₄ and their binary Gray images. It would be interesting to study other constacyclic codes over Z₄ + uZ₄ and use them to construct more good codes over Z₄ or F₂.