ISSN 2148-838Xhttp://dx.doi.org/10.13069/jacodesmath.671815

J. Algebra Comb. Discrete Appl.
7(1) • 85–101

Received: 30 June 2019
Accepted: 16 November 2019

Journal of Algebra Combinatorics Discrete Structures and Applications

Zq(Zq + uZq)− linear skew constacyclic codes
Research Article

Ahlem Melakhessou, Nuh Aydin, Zineb Hebbache, Kenza Guenda

Abstract: In this paper, we study skew constacyclic codes over the ring ZqR where R = Zq + uZq, q = ps
for a prime p and u2 = 0. We give the definition of these codes as subsets of the ring Zαq Rβ. Some
structural properties of the skew polynomial ring R[x, Θ] are discussed, where Θ is an automorphism
of R. We describe the generator polynomials of skew constacyclic codes over ZqR, also we determine
their minimal spanning sets and their sizes. Further, by using the Gray images of skew constacyclic
codes over ZqR we obtained some new linear codes over Z4. Finally, we have generalized these codes
to double skew constacyclic codes over ZqR.

2010 MSC: 94B15, 94B60

Keywords: Linear codes, Skew constacyclic codes, ZqZq[u]− linear skew constacyclic codes, Bounds

1. Introduction

Codes over finite rings have been known for several decades, but interest in these codes increased
substantially after the discovery that good non-linear binary codes can be constructed from codes over
rings. Several methods have been introduced to produce certain types of linear codes with good algebraic
structures and parameters. Cyclic codes and their various generalizations such as constacyclic codes and
quasi-cyclic (QC) codes have played a key role in this quest. One particularly useful generalization of
cyclic codes has been the class of quasi-twisted (QT) codes that produced hundreds of new codes with
best known parameters [4, 8, 9, 11, 12, 16, 17] recorded in the database [25]. Yet another generalization
of cyclic codes, called skew cyclic codes, were introduced in [15] and they have been the subject of an
increasing research activity over the past decade. This is due to their algebraic structure and their
applications to DNA codes and quantum codes [14, 19, 20]. Skew constacyclic codes over various rings

Ahlem Melakhessou; Department of Mathematics, Mostefa Ben Boulaïd University (Batna2), Batna, Algeria
(email: a.melakhessou@univ-batna2.dz).
Nuh Aydin; Department of Mathematics and Statistics, Kenyon College, USA (email: aydinn@kenyon.edu).
Zineb Hebbache, Kenza Guenda (Corresponding Author); Faculty of Mathematics, USTHB, Laboratory

of Algebra and Number Theory, BP 32 El Alia, Bab Ezzouar, Algeria (email: zinebhebache@gmail.com,
ken.guenda@gmail.com).

85

https://orcid.org/0000-0002-5618-2427
https://orcid.org/0000-0002-1482-7565


A. Melakhessou et al. / J. Algebra Comb. Discrete Appl. 7(1) (2020) 85–101

have been studied in [1, 2, 5, 13, 21, 23, 26, 30, 32, 33] as a generalization of skew cyclic codes over
finite fields. Recently, P. Li et al. [28] gave the structure of (1 + u)-constacyclic codes over the ring
Z2Z2[u] and Aydogdu et al. [6] studied Z2Z2[u]-cyclic and constacyclic codes. Further, Jitman et al. [27]
considered the structure of skew constacyclic codes over finite chain rings. More recently A. Sharma and
M. Bhaintwal studied skew cyclic codes over ring Z4 + uZ4, where u2 = 0.

The aim of this paper is to introduce and study skew constacyclic codes over the ring Zq(Zq + uZq),
where q is a prime power and u2 = 0. Some structural properties of the skew polynomial ring R[x, Θ] are
discussed, where Θ is an automorphism of R. We describe the generator polynomials of skew constacyclic
codes over R and ZqR. Using Gray images of skew constacyclic codes over ZqR we obtained some new
linear codes over Z4. Further, we generalize these codes to double skew constacyclic codes over ZqR.

The paper is organized as follows. We first give some basic results about the ring R = Zq + uZq,
where q = ps, p is a prime and u2 = 0, and linear codes over ZqR, we construct the non-commutative
ring R[x, Θ], where the structure of this ring depends on the elements of the commutative ring R and an
automorphism Θ of R. We give some results on skew constacyclic codes over the ring R. In Section 3, we
study the algebraic structure of skew constacyclic codes over the ring ZqR, section 4 includes the work
on the generator polynomials of these codes, their minimal spanning sets and their sizes. In section 5,
we determine the Gray images of skew constacyclic codes over R and ZqR. These codes are then further
generalized to double skew constacyclic codes in the next section. Finally. In section 7, we use the Gray
images of skew constacyclic codes over ZqR to obtain some new linear codes over Z4.

2. Preliminaries

Let (α,β) denote n = α + 2β where α and β are positive integers. Consider the ring R = Zq + uZq,
where q = ps, p is a prime and u2 = 0. The ring R is isomorphic to the quotient ring Zq[u]/

〈
u2
〉
. The

ring R is not a chain ring, whereas it is a local ring with the maximal ideal 〈u,p〉. Each element r of R
can be expressed uniquely as

r = a + ub, where a,b ∈ Zq.

2.1. Skew polynomial ring over R

In this subsection we construct the non-commutative ring R[x, Θ]. The structure of this ring depends
on the elements of the commutative ring R and an automorphism Θ of R. Note that an automorphism
Θ in R must fix every element of Zq, hence it satisfies Θ(a + ub) = a + δ(u)b. Therefore, it is determined
by its action on u. Let δ(u) = k + ud, where k is a non-unit in Zq, k2 ≡ 0 mod q and 2kd ≡ 0 mod q.
Then,

Θ(a + ub) = a + δ(u)b = (a + kb) + udb, (1)

for all a + ub ∈ R. Further, let Θ an automorphism of R and let m be its order. The skew polynomial
ring R[x, Θ] is the set of polynomials over R in which the addition is defined as the usual addition of
polynomials and the multiplication is defined by the rule

xa = Θ(a)x.

The multiplication is extended to all elements in R[x, Θ] by associativity and distributivity. The ring
R[x, Θ] is called a skew polynomial ring over R and an element in R[x, Θ] is called a skew polynomial.
Further, an element g(x) ∈ R[x, Θ] is said to be a right divisor (resp. left divisor) of f(x) if there exists
q(x) ∈ R[x, Θ] such that

f(x) = q(x)g(x) ( resp. f(x) = g(x)q(x)).

In this case, f(x) is called a left multiple (resp. right multiple) of g(x).

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A. Melakhessou et al. / J. Algebra Comb. Discrete Appl. 7(1) (2020) 85–101

Lemma 2.1. [31, Lemma 1] Let f(x), g(x) ∈ R[x, Θ] be such that the leading coefficient of g(x) is a
unit. Then there exist q(x), r(x) ∈ R[x, Θ] such that

f(x) = q(x)g(x) + r(x), where r(x) = 0 or deg(r(x)) < deg(g(x)).

Definition 2.2. [31, Definition 3.2] A polynomial f(x) ∈ R[x, Θ] is said to be a central polynomial if

f(x)r(x) = r(x)f(x)

for all r(x) ∈ R[x, Θ].

Theorem 2.3. The center Z(R[x, Θ]) of R[x, Θ] is RΘ[xm], where m is the order of Θ and RΘ is the
subring of R fixed by Θ.

Proof. We know R = Zq + uZq is the fixed ring of Θ. Since order of Θ is m, for any non-negative
integer i, we have

xmia = (Θm)i(a)xmi = axmi

for all a ∈ R. It gives xmi ∈ Z(R[x, Θ]), and hence all polynomials of the form

f = a0 + a1x
m + a2x

2m + · · · + alxlm

with ai ∈ R are in the center.
Conversely, let f = f0 + f1x + f2x2 + · · · + fkxk ∈ Z(R[x, Θ]) we have fx = xf which gives that all

fi are fixed by Θ, so that fi ∈ R. Further, choose a ∈ R such that Θ(a) 6= a. Then it follows from the
relation af = fa that fi = 0 for all indices i not divides m. Thus

f(x) = a0 + a1x
m + a2x

2m + · · · + alxlm ∈ RΘ[xm].

Corollary 2.4. Let f(x) = xβ − 1. Then f(x) ∈ Z(R[x, Θ]) if and only if m | β. Further, xβ − λ ∈
Z(R[x, Θ]) if and only if m | β and λ is fixed by Θ.

2.2. Skew constacyclic codes over R

In this section we generalize the structure and properties from [31] to codes over Zq + uZq. Hence
the proofs of many of the theorems will be omitted.

We start with some structural properties of R[x, Θ]/〈xβ − λ〉. The Corollary 2.4, shows that the
polynomial (xβ −λ) is in the center Z(R[x, Θ]) of the ring R[x, Θ], hence generates a two-sided ideal if
and only if m | β and λ is fixed by Θ. Therefore, in this case R[x, Θ]/〈xβ −λ〉 is a well-defined residue
class ring. If m - β, then the quotient space R[x, Θ]/〈xβ − λ〉 which is not necessarily a ring is a left
R[x, Θ]-module with multiplication defined by

r(x)(f(x) + (xβ −λ)) = r(x)f(x) + (xβ −λ),

for any r(x), f(x) ∈ R[x, Θ].

Next we define the skew λ−constacyclic codes over the ring R. A code of length β over R is a
nonempty subset of Rβ. A code C is said to be linear if it is a submodule of the R−module Rβ. In this
paper, all codes are assumed to be linear unless otherwise stated.

Given an automorphism Θ of R and a unit λ in R, a code C is said to be skew constacyclic, or
specifically, Θ −λ−constacyclic if C is closed under the Θ −λ−constacyclic shift:

ρΘ,λ : R
β → Rβ

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A. Melakhessou et al. / J. Algebra Comb. Discrete Appl. 7(1) (2020) 85–101

defined by

ρΘ,λ((a0,a1, . . . ,aβ−1)) = (λΘ(aβ−1), Θ(a0), . . . , Θ(aβ−2)). (2)

In particular, such codes are called skew cyclic and skew negacyclic codes when λ is 1 and −1, respectively.
When Θ is the identity automorphism, he become classical constacyclic and we denote ρλ the constacyclic
shift.

In the rest of paper, we restrict our study to the case where the length β of codes is a multiple of
the order of Θ and λ is a unit in RΘ, where RΘ denotes the subring of R fixed by Θ.

The proofs of the next theorems are analogous to the proofs of [31] given for the ring Z4 + uZ4,
therefore we omit them.

Theorem 2.5. [31, Theorem 3] A code Cβ of length β in Rβ = R[x, Θ]/〈xβ−λ〉 is a Θ−λ−constacyclic
code if and only if Cβ is a left R[x, Θ]−submodule of the left R[x, Θ]−module Rβ.

Corollary 2.6. [31, Corollary 2] A code C of length β over R is Θ −λ−constacyclic code if and only if
the skew polynomial representation of C is a left ideal in R[x, Θ]/〈xβ −λ〉.

The following theorem is the generalization of the Theorems 4 and 5 of [31].

Theorem 2.7. Let Cβ be a skew contacyclic code of length β over R. Then, Cβ is a free principally
generated skew constayclic code if and only if there exists a minimal degree polynomial gβ(x) ∈ Cβ having
its leading coefficient a unit such that Cβ = 〈gβ(x)〉 and gβ(x) | xβ − λ. Moreover, Cβ has a basis
{gβ(x),xgβ(x), . . . ,xβ−deg(gβ(x))−1} and |Cβ| = |R|β−deg(gβ(x)).

In this next, we study duals of Θ−λ−constacyclic codes over R. Further, the Euclidean inner product
defined by

〈v′,w′〉 =
β−1∑
i=0

v′iw
′
i,

for v′ = (v′0,v
′
1, . . . ,v

′
β−1) and w

′ = (w′0,w
′
1, . . . ,w

′
β−1) in R

β.

Definition 2.8. Let Cβ be a Θ−λ−constacyclic code of length β over R. Then its dual C⊥β is defined as

C⊥β = {v
′ ∈ Rβ; 〈v′,w′〉 = 0 for all w′ ∈ Cβ}

Lemma 2.9. Let Cβ be a code of length β over R, where β is a multiple of the order of the automorphism
Θ and λ is fixed by Θ. Then Cβ is Θ −λ−constacyclic if and only if C⊥β is Θ −λ

−1−constacyclic. In
particular, if λ2 = 1, then Cβ is Θ −λ−constacyclic if and only if C⊥β is Θ −λ−constacyclic.

Proof. Note that, for each unit λ in R,λ ∈ RΘ if and only if λ−1 ∈ RΘ, since λ ∈ RΘ, so is λ−1. Let
v′ = (v′0,v

′
1, . . . ,v

′
β−1) ∈ Cβ and w

′ = (w′0,w
′
1, . . . ,w

′
β−1) ∈ C

⊥
β be two arbitrary elements. Since Cβ is

Θ −λ−constacyclic code,

ρ
β−1
Θ,λ (v

′) =
(
Θβ−1(λv′1), Θ

β−1(λv′2), . . . , Θ
β−1(λv′β−1), Θ

β−1(v′0)
)
∈ Cβ.

Then, we have

0 = 〈ρβ−1Θ,λ (v
′),w′〉

= 〈(Θβ−1(λv′1), Θβ−1(λv′2), . . . , Θβ−1(λv′β−1), Θ
β−1(v′0)), (w

′
0, . . . ,w

′
β−1)〉

= λ〈(Θβ−1(v′1), Θβ−1(v′2), . . . , Θβ−1(v′β−1), Θ
β−1(λ−1v′0)), (w

′
0, . . . ,w

′
β−1)〉

= λ

(
Θβ−1(λ−1v′0)w

′
β−1 +

β−1∑
j=1

Θβ−1(v′j)w
′
j−1

)
.

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A. Melakhessou et al. / J. Algebra Comb. Discrete Appl. 7(1) (2020) 85–101

As β is a multiple of the order of Θ and λ−1 is fixed by Θ, it follows that

0 = Θ(0) = Θ(λΘβ−1(λ−1v′0)w
′
β−1 +

β−1∑
j=1

Θβ−1(v′j)w
′
j−1)

= λ(v′0Θ(λ
−1w′β−1) +

β−1∑
j=1

v′jΘ(w
′
j−1))

= λ〈ρΘ,λ−1 (w′),v′〉.

This implies that, ρΘ,λ−1 (w′) ∈ C⊥β . In addition, assume that λ
2 = 1. Then λ = λ−1. Therefore Cβ is a

Θ −λ−constacyclic code.
The converse follows from the fact that (C⊥β )

⊥ = Cβ.

3. ZqR−linear skew constacyclic codes

In this section, we study skew λ−constacyclic codes over the ring ZqR.
We known that the ring Zq is a subring of the ring R. We construct the ring

ZqR = {(e,r); e ∈ Zq,r ∈ R}.

The ring ZqR is not an R−module under the operation of standard multiplication. To make ZqR an
R−module, we follow the approach in [2] and define the map

η : R → Zq
a + ub 7→ a.

It is clear that the mapping η is a ring homomorphism. Now, for any d ∈ R, we define the multiplication
∗ by

d∗ (e,r) = (η(d)e,dr).

This multiplication can be naturally generalized to the ring Zαq R
β as follows.

For any d ∈ R and v = (e0,e1, . . . ,eα−1,r0,r1, . . . ,rβ−1) ∈ Zαq Rβ define

dv = (η(d)e0,η(d)e1, . . . ,η(d)eα−1,dr0,dr1, . . . ,drβ−1),

where (e0,e1, . . . ,eα−1) ∈ Zαq and (r0,r1, . . . ,rβ−1) ∈ Rβ.
The following results are analogous to the ones obtained in [2, 5] for the ring Z2(Z2 + uZ2).

Lemma 3.1. The ring Zαq R
β is an R-module under the above definition.

The above Lemma allows us to give the next definition.

Definition 3.2. A non-empty subset C of Zαq R
β is called a ZqR-linear code if it is an R−submodule of

Zαq R
β.

We note that the ring R is isomorphic to Zq as an additive group. Hence, for some positive integers
k0, k1 and k2, any ZqR-linear code C is isomorphic to a group of the form

Zk0q ×Z
2k1
q ×Z

k2
q .

Definition 3.3. If C ⊆ Zαq Rβ is a ZqR-linear code, group isomorphic to Zk0q × Z2k1q × Zk2q , then C is
called a ZqR-additive code of type (α,β,k0,k1,k2), where k0, k1, and k2 are as defined above.

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A. Melakhessou et al. / J. Algebra Comb. Discrete Appl. 7(1) (2020) 85–101

The following results and definitions are analogous to the ones obtained in [6].

Let C be a ZqR-linear code and let Cα (respectively Cβ) be the canonical projection of C on the
first α (respectively on the last β) coordinates. Since the canonical projection is a linear map, Cα and
Cβ are linear codes over Zq and over R of length α and β, respectively. A code C is called separable if
C is the direct product of Cα and Cβ, i.e.,

C = Cα ×Cβ.

We introduce an inner product on Zαq R
β. For any two vectors

v = (v0, . . . ,vα−1,v
′
0, . . . ,v

′
β−1),w = (w0, . . . ,wα−1,w

′
0, . . . ,w

′
β−1) ∈ Z

α
q ×R

β

let

〈v,w〉 = u
α−1∑
i=0

viwi +

β−1∑
j=0

v́jẃj.

Let C be a ZqR-linear code. The dual of C is defined by

C⊥ = {w ∈ Zαq ×R
β, 〈v,w〉 = 0,∀v ∈ C}.

If C = Cα ×Cβ is separable, then

C⊥ = C⊥α ×C
⊥
β . (3)

Now we are ready to define the skew constacyclic codes over Zαq R
β. We start by the following Lemma.

Lemma 3.4. Let R = Zq + uZq, where Zq is a subring of R. Then an element λ is unit in R if and only
if η(λ) is unit in Zq.

Proof. Assume that λ is unit in R; where λ = λ1 + uλ2 and λ1,λ2 ∈ Zq, then we have λ.v = v.λ = 1
and since η is a ring homomorphism, then we have η(λ.v) = η(v.λ) = η(1) thus η(λ).v′ = v′.η(λ) = 1
which means that η(λ) is unit in Zq, where v′ = η(v) ∈ Zq.

Conversely, suppose that η(λ) = λ1 is unit in Zq we should prove that λ = λ1 + uλ2 is unit in
R. The fact that λ is unit in R means that λ.λ−1 = 1, therefore λ.λ−1 = (λ1 + uλ2)(λ1 + uλ2)−1 =
(λ1 + uλ2)(λ

−1
1 + uλ3) = λ1λ

−1
1 + u(λ2λ

−1
1 + λ1λ3), then we denote λ3 =

−λ2λ−11
λ1

= −λ2(λ−11 )
2 and since

λ1 is unit in Zq, then λ1λ−11 = 1 which implies that λ.λ
−1 = 1, so λ is unit in R.

Definition 3.5. Let Θ be an automorphism of R. A linear code C over Zαq R
β is called skew constacyclic

code if C satisfies the following two conditions.

(i) C is an R−submodule of Zαq Rβ,

(ii)

(η(λ)Θ(eα−1), Θ(e0), . . . , Θ(eα−2),λΘ(rβ−1), Θ(r0), . . . , Θ(rβ−2)) ∈ C

whenever

(e0,e1, . . . ,eα−1,r0,r1, . . . ,rβ−1) ∈ C

Remark 3.6. Θ(ei) = ei for 0 ≤ i ≤ α− 1, as ei ∈ Zq (the fixed ring of Θ).

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A. Melakhessou et al. / J. Algebra Comb. Discrete Appl. 7(1) (2020) 85–101

In polynomial representation, each codeword c = (e0,e1, . . . ,eα−1,r0,r1, . . . ,rβ−1) of a skew consta-
cyclic code can be represented by a pair of polynomials

c(x) =
(
e0 + e1x + · · · + eα−1xα−1,r0 + r1x + · · · + rβ−1xβ−1

)
= (e(x),r(x)) ∈ Zq[x]/〈xα −η(λ)〉×R[x, Θ]/〈xβ −λ〉.

Let h(x) = h0 +h1x+· · ·+htxt ∈ R [x, Θ] and let (f(x),g(x)) ∈ Zq[x]/〈xα−η(λ)〉×R[x, Θ]/〈xβ−λ〉.
The multiplication is defined by the basic rule

h(x)(f(x),g(x)) = (η(h(x))f(x),h(x)g(x)),

where η(h(x)) = η(h0) + η(h1)x + · · · + η(ht)xt.

Lemma 3.7. A code C of length (α,β) over ZqR is a Θ −λ−constacyclic code if and only if C is left
R[x, Θ]−submodule of Zq[x]/〈xα −η(λ)〉×R[x, Θ]/〈xβ −λ〉.

Proof. Assume that C is a skew constacyclic code and let c ∈ C. We denote by c(x) = (e(x),r(x)) the
associated polynomial of c. As xc(x) is a skew constacyclic shift of c, xc(x) ∈ C. Then, by linearity of C,
r(x)c(x) ∈ C for any r(x) ∈ R[x, Θ]. Thus C is left R[x, Θ]−submodule of Zq[x]/〈xα−η(λ)〉×R[x, Θ]/〈xβ−
λ〉. Conversely, suppose that C is a left R[x, Θ]−submodule of Zq[x]/〈xα−η(λ)〉×R[x, Θ]/〈xβ −λ〉, then
we have that xc(x) ∈ C. Thus, C is a Θ −λ−constacyclic code.

The converse is straightforward.

Theorem 3.8. Let C be a linear code over ZqR of length (α,β), and let C = Cα×Cβ, where Cα is linear
code over Zq of length α and Cβ is linear code over R of length β. Then C is a skew λ−constacyclic code
if and only if Cα is a η(λ)−constacyclic code over Zq and Cβ is a skew λ−constacyclic code over R.

Proof. Let (e0,e1, . . . ,eα−1) ∈ Cα and let (r0,r1, . . . ,rβ−1) ∈ Cβ. If C = Cα×Cβ is a skew constacyclic
code, then

(η(λ)Θ(eα−1), Θ(e0), . . . , Θ(eα−2),λΘ(rβ−1), Θ(r0), . . . , Θ(rβ−2)) ∈ C,

which implies that

(η(λ)Θ(eα−1), Θ(e0), . . . , Θ(eα−2)) ∈ Cα
as Θ is fixed by Zq, then

(η(λ)eα−1,e0, . . . ,eα−2) ∈ Cα
and

(λΘ(rβ−1), Θ(r0), . . . , Θ(rβ−2)) ∈ Cβ.

Hence, Cα is a constacyclic code over Zq and Cβ is a Θ −λ−constacyclic code over R.
On the other hand, suppose that Cα is a constacyclic code over Zq and Cβ is a Θ −λ−constacyclic

code over R. Note that

(η(λ)eα−1,e0, . . . ,eα−2) ∈ Cα
and

(λΘ(rβ−1), Θ(r0), . . . , Θ(rβ−2)) ∈ Cβ.

Since C = Cα ×Cβ and Θ(ei) = ei, then

(η(λ)Θ(eα−1), Θ(e0), . . . , Θ(eα−2),λΘ(rβ−1), Θ(r0), . . . , Θ(rβ−2)) ∈ C,

so C is a skew constacyclic code over ZqR.

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A. Melakhessou et al. / J. Algebra Comb. Discrete Appl. 7(1) (2020) 85–101

Corollary 3.9. Let C = Cα ×Cβ be a skew λ−constacyclic code over ZqR, where β is a multiple of the
order Θ and λ−1 is fixed by Θ. Then the dual code C⊥ = C⊥α ×C⊥β of C is a skew λ

−1-constacyclic code
over ZqR.

Proof. From Equation (3), we have C⊥ = C⊥α × C⊥β . Clearly, if Cα is a constacyclic code over Zq
then C⊥α is also a constacyclic code over Zq. Moreover, from Lemma (2.9), we have C

⊥
β is a skew

λ−constacyclic code over R. Hence the dual code C⊥ is skew λ−1−constacyclic over ZqR.

4. The generators and the spanning sets for ZqR−skew consta-
cyclic codes

In this section, we find a set of generators for ZqR−skew constacyclic codes as a left
R[x, Θ]−submodules of Zq[x]〈xα −η(λ)〉×R[x, Θ]/〈xβ −λ〉. Let C be a ZqR−skew constacyclic codes,
C and R[x, Θ]/〈xβ −λ〉 are R[x, Θ]−modules and w define the following mapping:

Ψ : C → R[x, Θ]/〈xβ −λ〉

where

Ψ(f1(x),f2(x)) = f2(x).

It is clear that Ψ is a module homomorphism whose image is a R[x, Θ]−submodule of R[x, Θ]/〈xβ −λ〉
and ker(Ψ) is a submodule of C.

Proposition 4.1. Let C be a skew constacyclic code of length n over ZqR. Then

C = 〈(f(x), 0), (l(x),a(x) + ug(x))〉,

where f(x) | (xα −η(λ)) and g(x) | a(x) | (xβ −λ).

Proof. Assume that β is a positive integer coprime to the characteristic of R, by similarly theory of
cyclic codes over Z2Z4 (see. [3]) we have that

Ψ(C) = (a(x) + ug(x)) with a(x),g(x) ∈ R[x, Θ] and g(x) | a(x) | (xβ −λ).

Note that:

ker(Ψ) = {(f(x), 0) ∈ C : f(x) ∈ Zq[x]/〈xα −η(λ)〉}.

Define the set I to be

I = {f(x) ∈ Zq[x]/〈xα −η(λ)〉 : (f(x), 0) ∈ ker(Ψ)}.

Clearly, I is an ideal of Zq[x]/〈xα −η(λ)〉. Therefore, there exist a polynomial f(x) ∈ Zq[x]/〈xα −η(λ)〉,
such that I = 〈f(x)〉. Now, for any element (c1(x), 0) ∈ ker(Ψ), we have c1(x) ∈ I = 〈f(x)〉 and
there exists some polynomials m(x) ∈ Zq[x]/〈xα −η(λ)〉 such that c1(x) = m(x)f(x). Thus (c1(x), 0) =
m(x)∗(f(x), 0), which implies that ker(Ψ) is a left submodule of C generated by one element of the form
(f(x), 0) where f(x) | (xα −η(λ)). Thus, by the first isomorphism theorem, we have

C/ker(Ψ) ∼= 〈a(x) + ug(x)〉.

Let (l(x),a(x) + ug(x)) ∈ C, with

Ψ(l(x),a(x) + ug(x)) = 〈a(x) + ug(x))〉.

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Then any ZqR−skew constacyclic code of length (α,β) can be generated as left R[x, Θ]−submodule of
Zq[x]/〈xα −η(λ)〉×R[x, Θ]/〈xβ −λ〉 by two elements of the form (f(x), 0) and (l(x),a(x) + ug(x)), in
other word, any element in the code C can be described as

d1(x) ∗ (f(x), 0) + d2(x) ∗ (l(x),a(x) + ug(x)),

where d1(x) and d2(x) are polynomials in the ring R[x, Θ]. In fact, the element d1(x) can be restricted
to be an element in the ring Zq[x]. We will write this as:

C = 〈(f(x), 0), (l(x),a(x) + ug(x))〉,

where, f(x) | (xα −η(λ)) and g(x) | a(x) | (xβ −λ).

Lemma 4.2. If C = 〈(f(x), 0), (l(x),a(x) + ug(x))〉 is a ZqR−skew constacyclic code, then we may
assume that deg (l(x)) ≤ deg (f(x)).

Proof. Suppose that deg (l(x)) ≥ deg (f(x)) with deg (l(x)) = i. Consider an other ZqR−skew con-
stacyclic code of length (α,β) with generators of the form

D = 〈(f(x), 0), (l(x),a(x) + ug(x)) + xi ∗ (f(x), 0)〉
= 〈(f(x), 0), (l(x) + xif(x),a(x) + ug(x))〉.

Clearly, D ⊆ C. However, we also have that:

(l(x),a(x) + ug(x)) = (l(x) + xif(x),a(x) + ug(x)) −xi ∗ (f(x), 0),

which implies that (l(x),a(x) + ug(x)) ∈ C. Therefore, C ⊆ D implying C = D.

Lemma 4.3. If C = 〈(f(x), 0), (l(x),a(x) + ug(x))〉 is a ZqR−skew constacyclic code, then we may
assume that f(x) | x

β−λ
g(x)

l(x).

Proof. Since x
β−λ
g(x)
∗(l(x),a(x)+ug(x)) = (x

β−λ
g(x)

l(x), 0), it follow that Ψ(x
β−λ
g(x)
∗(l(x),a(x)+ug(x))) = 0.

Therefore, (x
β−λ
g(x)

l(x), 0) ∈ ker(Ψ) ⊆ C and f(x) | (x
β−λ
g(x)

)l(x).

The above Lemma shows that if the ZqR−skew constacyclic code C has only one generator of the
form C = 〈l(x),a(x) + ug(x)〉 then, (xα − η(λ)) | x

β−λ
g(x)

l(x) with g(x) | a(x) | (xβ − λ). Thus from this
discussion and Lemma 4.2 and 4.3, we have the following results.

Theorem 4.4. Let C be a skew constacyclic code of length n over ZqR. Then C can be identified uniquely
as

C = 〈(f(x), 0), (l(x),a(x) + ug(x))〉,

where f(x) | (xα −η(λ)) and g(x) | a(x) | (xβ −λ). and l(x) is a skew polynomial satisfying deg (l(x)) ≤
deg (f(x)) and f(x) | x

β−λ
g(x)

l(x).

Proof. Following from Proposition 4.1, Lemma 4.2 and 4.3, we can easily see that C = 〈(f(x), 0), (l(x),
a(x) + ug(x))〉,where the polynomials f(x), l(x),a(x) and g(x) are stated in the theorem. Now, we will
prove the uniqueness of the generators. Since 〈f(x)〉 and 〈a(x) + ug(x)〉 are skew constacyclic codes over
Zq and R respectively, then, the skew polynomials f(x),a(x) and g(x) are unique. Now, suppose that

C = 〈(f(x), 0), (l1(x),a(x) + ug(x))〉

= 〈(f(x), 0), (l2(x),a(x) + ug(x))〉,

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then, we have

((l1(x) − l2(x)), 0) ∈ ker (Ψ) = 〈f(x), 0〉,

which implies that

l1(x) − l2(x) = f(x)j(x),

for some skew polynomial j(x), and since deg (l1(x) − l2(x)) ≤ deg (l1(x)) ≤ deg (f(x)) then j(x) = 0
and l1(x) = l2(x).

Definition 4.5. Let A be an R−module. A linearly independent subset B of A that spans A is called a
basis of A. If an R−module has a basis, then it is called a free R−module.

Note that if C is a ZqR−skew constacyclic code of the form

C = 〈(f(x), 0), (l(x),a(x) + ug(x))〉,

with g(x) 6= 0, then C is a free R−module. If C is not of this form then it is not a free R−module. But
we still present a minimal spanning set for the code. The following theorem gives us a spanning minimal
set for ZqR−skew constacyclic codes.

Theorem 4.6. Let C be a skew constacyclic code of length n over ZqR, where f(x), l(x),a(x) and g(x)
are as in Theorem 4.4 and f(x)hf (x) = xα −η(λ),a(x)ha(x) = xβ −λ,a(x) = g(x)m(x). Let

S1 =

deg (hf )−1⋃
i=0

{xi ∗ (f(x), 0)},

S2 =

deg (ha)−1⋃
i=0

{xi ∗ (l(x),a(x) + ug(x))},

and

S3 =

deg (m)−1⋃
i=0

{xi ∗ (η(ha(x))l(x),uha(x)g(x))}.

Then

S = S1 ∪S2 ∪S3,

forms a minimal spanning set for C and C has qdeg (hf )q2deg (ha)qdeg (m) codewords.

Proof. Let C(x) = η(d(x))(f(x), 0) + e(x)(l(x),a(x) + ug(x)) ∈ Zq[x]/〈xα −η(λ)〉×R[x, Θ]/〈xβ −λ〉
be a codeword in C where d(x) and e(x) are skew polynomials in R[x, Θ]. Now, if deg (η(d(x))) ≤
deg (hf (x)) − 1, then η(d(x))(f(x), 0) ∈ Span(S1). Otherwise, by using right division algorithm we have

η(d(x)) = hf (x)η(q1(x)) + η(r1(x)),

where q1(x),r1(x) ∈ R[x, Θ] and η(r1(x)) = 0 or deg (η(r1(x))) ≤ deg (hf (x)) − 1. Therefore,

η(d(x))(f(x), 0) = (hf (x)η(q1(x)) + η(r1(x)))(f(x), 0)

= η(r1(x))(f(x), 0).

Hence, we can assume that η(d(x))(f(x), 0) ∈ Span(S1).

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Now, if deg (η(e(x))) ≤ deg (ha(x)) − 1, then η(e(x))(l(x),a(x) + ug(x)) ∈ Span(S2). Otherwise,
again by the right division algorithm, we get polynomials q2(x) and r2(x) such that:

e(x) = q2(x)ha(x) + r2(x),

where r2(x) = 0 or deg (r2(x)) ≤ deg (ha(x)) − 1. So, we have

e(x)(l(x),a(x) + ug(x)) = (q2(x)ha(x) + r2(x))(l(x),a(x) + ug(x))

= q2(x)(η(ha(x))l(x),uha(x)g(x)) + r2(x)(l(x),a(x) + ug(x)).

Since r2(x) = 0 or deg (r2(x)) ≤ deg (ha(x)) − 1, then r2(x)(l(x),a(x) + ug(x)) ∈ Span(S2). Let us
consider

q2(x)(η(ha(x))l(x),uha(x)g(x)) ∈ Span(S),

we know that xβ − λ = a(x)ha(x) = g(x)m(x)ha(x) and also we have f(x) | x
β−λ
g(x)

l(x). Therefore,
xβ−λ
g(x)

l(x) = f(x)k(x). Again, if deg (q2(x)) ≤ deg (m(x)) − 1 then q2(x)(η(ha(x))l(x),uha(x)g(x)) ∈
Span(S3). Otherwise, q2(x) = x

β−λ
ha(x)g(x)

q3(x) + r3(x) with r3(x) = 0 or deg (r3(x)) ≤ deg (m(x)) − 1. So,

q2(x)(η(ha(x))l(x),uha(x)g(x))

=

(
xβ −λ

ha(x)g(x)
q3(x)η(ha(x))l(x),

xβ −λ
ha(x)g(x)

q3(x)uha(x)g(x)

)
+r3(x)(η(ha(x))l(x),uha(x)g(x))

=

(
xβ −λ

ha(x)g(x)
q3(x)η(ha(x))l(x), 0

)
+ r3(x)(η(ha(x))l(x),uha(x)g(x)).

Since x
β−λ
g(x)

l(x) = f(x)k(x), then
(

xβ−λ
ha(x)g(x)

q3(x)η(ha(x))l(x), 0
)
∈ Span(S1) and hence

r3(x)(η(ha(x))l(x),uha(x)g(x)) ∈ Span(S3).
Consequently, S = S1 ∪S2 ∪S3 forms a minimal spanning set for C.

5. Gray images of skew constacyclic codes over ZqR

In this section, we define a Gray map on ZqR, and then extend it to Zαq R
β. We discuss the Gray

images of ZqR−skew constacyclic codes where λ is fixed by Θ. We start by recall some results which we
will need its in the next.

From [24, Definition 2] we have the following definition

Definition 5.1. Let Cβ be a linear code over R of length β = N` and let λ be unit in R. If for any
codeword (

c0,0,c0,1, . . . ,c0,`−1,c1,0,c1,1, . . . ,c1,`−1, . . . ,
cN−1,0,cN−1,1, . . . ,cN−1,`−1

)
∈ Cβ,

then (
λΘ(cN−1,0),λΘ(cN−1,1), . . . ,λΘ(cN−1,`−1), Θ(c0,0), Θ(c0,1), . . . , Θ(c0,`−1), . . . ,

Θ(cN−2,0), Θ(cN−2,1), . . . , Θ(cN−2,`−1)

)
∈ Cβ.

Then we say that Cβ is a Θ −λ−quasi-twisted code of length β. If ` is the least positive integer satisfies
that β = N`, then Cβ is said to be a Θ − λ−quasi-twisted code with index `. Furthermore, if Θ is the
identity map, we call Cβ a quasi-twisted code of index l over R.

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According to [31], we define a Gray map φ over R by

φ : Rβ → Z2βq
φ(a + ub) = (b,a + b),

where a,b ∈ Zβq
Furthermore, for r = a + ub ∈ R, we define a map

Φ : ZqR 7→ Z3q

by

Φ(e,r) = (e,φ(r)) = (e,b,a + b)

and it can be extended componentwise Zαq R
β to Znq as

Φ(e0,e1, . . . ,eα−1,r0,r1, . . . ,rβ−1) = (e0,e1, . . . ,eα−1,φ(r0),φ(r1), . . . ,φ(rβ−1)),

for all (e0,e1, . . . ,eα−1) ∈ Zαq and (r0,r1, . . . ,rβ−1) ∈ Rβ, where n = α + 2β. Φ is known as the Gray
map on Zαq R

β.

Let a ∈ Z2βq with a = (a0,a1) = (a(0) | a(1)), a(i) ∈ Zβq , for i = 0, 1. Let σ⊗2 be a map from Z2βq to
Z2βq given by

σ⊗2(a) = (σλ(a
(0)) | σλ(a(1))),

where σλ is a constacyclic shift from Zβq to Z
β
q given by

σλ(a
(i)) = (λai,β−1,a(i,0), . . . ,a(i,β−2)),

for every a(i) = (a(i,0),a(i,1), . . . ,a(i,β−1)) where a(i,j) ∈ Zq, for j = 0, 1, . . . ,β − 1. A linear code Cβ of
length 2β over Zq is said to be a quasi-twisted of index 2 if σ⊗2(Cβ) = Cβ.

In addition, for each Θ ∈ Aut(R), let TΘ : Rβ 7→ Rβ be a linear transformation given by

TΘ(a0,a1 . . . ,aβ−1) = (Θ(a0), Θ(a1) . . . , Θ(aβ−1)).

Remark 5.2. Cβ is a skew constacyclic code if and only if TΘ ◦ρλ(Cβ) = Cβ.

Proposition 5.3. With the previous notation, we have TΘ ◦φ◦ρλ = σ⊗2 ◦φ.

Proof. Let ri = ai + ubi be the elements of R for i = 0, 1, . . . ,β − 1, we have ρλ(r0,r1, . . . ,rβ−1) =
(λrβ−1,r0,r1, . . . ,rβ−2). If we apply φ, we have

φ(ρλ(r)) = φ(λrβ−1,r0, . . . ,rβ−2)

= (λbβ−1,b0, . . . ,bβ−2,λ(aβ−1 + bβ−1),a0 + b0, . . . ,aβ−2 + bβ−2).

where φi(r) = (bi,ai + ubi), now we apply TΘ in the above equation we get,

TΘ ◦φ(ρλ(r)) = TΘ(λbβ−1,b0, . . . ,bβ−2,λ(aβ−1 + bβ−1),a0 + b0, . . . ,aβ−2 + bβ−2)

=

(
Θ(λbβ−1), Θ(b0), . . . , Θ(bβ−2),λΘ(aβ−1 + bβ−1),

Θ(a0 + b0), . . . , Θ(aβ−2 + bβ−2)

)
,

since λ is fixed by Θ and by (1), for any a ∈ Zq, we have Θ(a) = a. So, we have

TΘ ◦φ◦ρλ(r) =
(
λbβ−1,b0, . . . ,bβ−2,λ(aβ−1 + bβ−1),

(a0 + b0), . . . , (aβ−2 + bβ−2)

)
.

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For the other direction,

σ⊗2(φ(r)) = σ⊗2(b0,b1, . . . ,bβ−1,a0 + b0,a1 + b1, . . . ,aβ−1 + bβ−1)

=

(
λbβ−1,b0, . . . ,bβ−2,λ(aβ−1 + bβ−1),

(a0 + b0), . . . , (aβ−2 + bβ−2)

)
,

and the result follows.

As a consequence of the above Proposition, we have the following theorem.

Theorem 5.4. Let Cβ be a code of length β over R. Then Cβ is a skew λ−constacyclic code of length
β over R if and only if φ(Cβ) is a quasi-twisted code of length 2β over Zq of index 2.

Proof. The necessary part follows from Proposition 5.3, i.e.,

σ⊗2 ◦φ(Cβ) = TΘ ◦φ◦ρλ(Cβ) = φ(Cβ).

For the sufficient part, assume that φ(Cβ) is a quasi-twisted code of index 2, then

φ(Cβ) = σ
⊗2 ◦φ(Cβ) = TΘ ◦φ◦ρλ(Cβ).

The injectivity of φ implies that TΘ (ρλ(Cβ)) = Cβ, i.e., Cβ is a skew constacyclic code over R.

Theorem 5.5. Let C = Cα × Cβ be Θ − λ−constacyclic code of length n = α + 2β over Zq[x]/〈xα −
η(λ)〉×R[x, Θ]/〈xβ −λ〉.

(i) If α = β, then Φ(C) is a quasi-twisted code of index 3 and length 3α.

(ii) If α 6= β and λ = 1, then Φ(C) is a generalized quasi cyclic code of index 3.

Proof. Assume that C = Cα ×Cβ is a skew λ−constacyclic code over ZqR then by Theorem 3.8, we
have that Cα is a constacyclic codes over Zq and Cβ is skew constacyclic codes over R. Further, from
Theorem 5.4, we have that, if Cβ is skew constacyclic code over R then φ(Cβ) is a quasi twisted code of
length 2β over Zq of index 2. Which implies that

Φ(e,r) =

(
λeα−1,e0, . . . ,eα−2,λbβ−1,b0, . . . ,bβ−2,

λ(aβ−1 + bβ−1), (a0 + b0), . . . , (aβ−2 + bβ−2)

)
,

for any (e0,e1, . . . ,eα−1,r0,r1, . . . ,rβ−1) ∈ C. Therefore,

1. if α = β, then Φ(C) is a quasi-twisted code of length 3α over Zq of index 3.

2. if α 6= β and λ = 1, then according to [22], Φ(C) is a generalized quasi-cyclic code of index 3.

6. Double skew constacyclic codes over ZqR

In this subsection, we study double skew constacyclic codes over ZqR. Let ń = ά+2β́ and ´́n = ´́α+2
´́
β

be integers such that n = ń + ´́n. We consider a partition of the set of the n coordinates into two subsets
of ń and ´́n coordinates, respectively, so that C is a subset of Zάq R

β́ ×Z´́αq R
´́
β.

Definition 6.1. A linear code C of length n over ZqR is called a double skew constacyclic code if C
satisfies the following conditions.

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(i) C is an R−submodule of Zά+ ´́αq Rβ́+
´́
β.

(ii) (η(λ)Θ(éά−1), Θ(é0), . . . , Θ(éά−2),λΘ(ŕβ́−1), Θ(ŕ0), . . . , Θ(ŕβ́−2) |
η(λ)Θ(´́e´́α−1), Θ(

´́e0), . . . , Θ(´́e´́α−2),λΘ(
´́r ´́
β−1

), Θ(´́r0), . . . , Θ(´́r ´́
β−2

)) ∈ C.

whenever

(é0, . . . , éά−1, ŕ0, . . . , ŕβ́−1 | ´́e0, . . . , ´́e´́α−1, ´́r0, . . . , ´́r ´́β−1) ∈ C.

Remark 6.2. Θ(éi) = éi and Θ(´́ei) = ´́ei for 0 ≤ i ≤ α− 1, as éi, ´́ei ∈ Zq (the fixed ring of Θ).

Denote by R
ά,β́,´́α,

´́
β
the ring:

Zq[x]/〈xά −η(λ)〉×R[x, Θ]/〈xβ́ −λ〉×Zq[x]/〈x
´́α −η(λ)〉×R[x, Θ]/〈x

´́
β −λ〉.

In polynomial representation, each codeword

c = (é0, é1, . . . , éά−1, ŕ0, . . . , ŕβ́−1 | ´́e0, ´́e1, . . . , ´́e´́α−1, ´́r0, . . . , ´́r ´́β−1)

of a skew constacyclic code can be represented by four polynomials

c(x) =



é0 + é1x + · · · + éά−1xά−1,
ŕ0 + ŕ1x + · · · + ŕβ́−1x

β́−1,
´́e0 + ´́e1x + · · · + ´́e´́α−1x

´́α−1,

´́r0 + ´́r1x + · · · + ´́r ´́
β−1

x
´́
β−1


 = (é(x), ŕ(x) | ´́e(x), ´́r(x)) ∈ Rά,β́,´́α, ´́β.

Let

h(x) = h0 + h1x + · · · + htxt ∈ R [x, Θ]

and let

(f́(x), ǵ(x) | ´́f(x), ´́g(x)) ∈ R
ά,β́,´́α,

´́
β
.

We define the multiplication of h(x) and (f́(x), ǵ(x) | ´́f(x), ´́g(x)) by

h(x)(f́(x), ǵ(x) | ´́f(x), ´́g(x)) = (η(h(x))f́(x),h(x)ǵ(x) | η(h(x)) ´́f(x),h(x)´́g(x)),

where η(h(x)) = η(h0) + η(h1)x + · · · + η(ht)xt. This gives us the following Theorem. But before that,
we need to give the following Remark.

Remark 6.3. If c(x) = (é(x), ŕ(x) | ´́e(x), ´́r(x)) represents the code word c, then xc(x) represents the
ń´́n−skew constacyclic shift of c.

Theorem 6.4. A linear code C is a double skew constacyclic code if and only if it is a left R[x, Θ]−sub
-module of the left-module R

ά,β́,´́α,
´́
β
.

Proof. Assume that C is a double skew constacyclic code. Let c ∈ C, and let the associated polynomial
of c be c(x) = (é(x), ŕ(x) | ´́e(x), ´́r(x)). Since xc(x) is an ń´́n−skew constacyclic shift of c. (See Remark 6.3),
then xc(x) ∈ C. Further, by the linearity of C, it follows that h(x)c(x) ∈ C, for any h(x) ∈ R[x, Θ].
Therefore C is a left R[x, Θ]−submodule of R

ά,β́,´́α,
´́
β
. Converse is straightforward.

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7. New linear codes over Z4

Codes over Z4, sometimes called quaternary codes as well, have a special place in coding theory.
Due to their importance, a database of quaternary codes was introduced in [7] and it is available online
[18]. Hence we consider the case q = 4 to possibly obtain quaternary codes with good parameters. We
conducted a computer search using Magma software [29] to find skew cyclic codes over Z4(Z4 + uZ4)
whose Gray images are quaternary linear codes with better parameters than the currently best known
codes. We have found ten such codes which are listed in the table below.

The automorphism of R = Z4 + uZ4 that we used is Θ(a + bu) = a + 3bu = a− bu. In addition to
the Gray map given in Section 4.1, there are many other possible linear maps from Z4 + uZ4 to Z`4 for
various values of `. For example, the following map was used in [10] a + bu → (b, 2a + 3b,a + 3b) which
triples the length of the code. We used both of these Gray maps in our computations, and obtained new
codes from each map.

We first chose a cyclic code Cα over Z4 generated by gα(x). The coefficients of this polynomial is
given in ascending order of the terms in the table. Therefore, the entry 31212201, for example, represents
the polynomial 3 + x + 2x2 + x3 + 2x4 + 2x5 + x7. Then we searched for divisors of xβ − 1 in the skew
polynomial ring R[x, Θ] where R = Z4 + uZ4 and Θ(a + bu) = a − bu. For each such divisor gβ(x) we
constructed the skew cyclic code over Z4R generated by (gα(x),gβ(x)) and its Z4-images under each
Gray map described above. As a result of the search, we obtained ten new linear codes over Z4. They
are now added to the database ([18]) of quaternary codes. In the table below, which Gray map is used
to obtain each new code is not explicitly stated, but it can be inferred from the values of α,β and n, the
length of the Z4 image. If n = α + 2β, then it is the map given in section 4.1 and if n = α + 3β it is the
map described in this section. For example, the second code in the table has length 57 = 15 + 3 · 14.
This means that the Gray map that triples the length of a code over R is used to obtain this code.

When xβ − 1 = g(x)h(x) we can use either the generator polynomial g(x) or the parity check
polynomial h(x) to define the skew cyclic code over R. For the codes given in the table below we used
the parity check polynomial because it has smaller degree. In general a linear code C over Z4 has
parameters [n, 4k1 2k2 ], and when k2 = 0, C is a free code. In this case C has a basis with k vectors just
like a linear code over a field. All of the codes in the table below are free codes, hence we will simply
denote their parameters by [n,k,d] where d is the Lee weight over Z4.

Our computational results suggest that considering skew cyclic and skew constacyclic codes over
Zq(Zq + uZq) is promising to obtain codes with good parameters over Zq.

Table 1. New quaternary codes

α β gα gβ Z4 Parameters
15 14 31212201 x4 + (u + 1)x3 + x2 + (3u + 2)x + 3u + 3 [43, 8, 26]

15 14 31212201 x4 + (u + 1)x3 + x2 + (3u + 2)x + 3u + 3 [57, 8, 38]

15 14 3021310231 x3 + 2ux2 + (3u + 3)x + 2u + 3 [43, 6, 30]

15 14 3021310231 x3 + 3x2 + (3u + 2)x + 1 [57, 6, 42]

7 14 3121 x4 + (3u + 3)x3 + 3x2 + (u + 2)x + 3u + 3 [35, 8, 20]

7 14 3121 x4 + (u + 3)x3 + (u + 1)x2 + (u + 2)x + 3u + 3 [49, 8, 32]

7 14 12311 x3 + (2u + 1)x2 + 3ux + 3u + 3 [35, 6, 22]

7 14 12311 x3 + (2u + 1)x2 + ux + u + 1 [35, 6, 24]

7 14 12311 x3 + ux2 + (3u + 3)x + 1 [49, 6, 35]

7 14 12311 x3 + (u + 2)x2 + x + 1 [49, 6, 36]

99



A. Melakhessou et al. / J. Algebra Comb. Discrete Appl. 7(1) (2020) 85–101

Acknowledgment: The authors wish to express their thanks to the anonymous reviewers for their
careful checking and valuable remarks that improved the presentation and the content of the paper.

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	Introduction
	Preliminaries
	ZqR-linear skew constacyclic codes
	The generators and the spanning sets for ZqR-skew constacyclic codes
	Gray images of skew constacyclic codes over ZqR 
	Double skew constacyclic codes over ZqR 
	New linear codes over Z4
	References