Skew cyclic and skew
(α1 + uα2 + vα3 + uvα4)-constacyclic codes
over Fq + uFq + vFq + uvFq

arXiv:1710.07785v2 [] 8 Jan 2018

Habibul Islam and Om Prakash
Department of Mathematics
Indian Institute of Technology Patna
Patna- 801 106, India
E-mail: habibul.pma17@iitp.ac.in and om@iitp.ac.in
Abstract
In this note, we study skew cyclic and skew constacyclic codes over the ring R =
Fq + uFq + vFq + uvFq where q = pm , p is an odd prime, u2 = u, v 2 = v, uv = vu.
We show that Gray images of a skew cyclic and skew α-constacyclic code of length
n are skew quasi-cyclic code of length 4n over Fq of index 4. Also, it is shown
that skew α-constacyclic codes are either equivalent to α-constacyclic codes or αquasi-twisted codes over R. Further, structural properties, specially, generating
polynomials and idempotent generators for skew cyclic and skew constacyclic codes
are determined by decomposition method.

Key Words : Skew polynomial ring; skew cyclic code; skew quasi-cyclic code; skew constacyclic code; Gray map; generating polynomial; idempotent generator.
2010 MSC : 94B15, 94B05, 94B60.

1. Introduction
Linear codes have been studied in coding theory for last six decades. Initially linear codes
were considered and studied over binary field. In 1994, a remarkable work presented by
Hammons et al. [8], which shows that some good binary non-linear codes can be considered as linear codes over Z4 through Gray map. Consequently, the study of linear codes
over finite rings has got more attention than the study of linear codes over binary field.
One of the most important class of linear codes is cyclic code which is easier to implement
and also playing crucial role in the development of algebraic coding theory. Latter, cyclic
codes over finite rings have got attention of many researchers and hence many new techniques have been discovered to produce cyclic codes over finite commutative ring with
better parameters and properties [[2], [5], [13], [11]]. But all of these works are restricted
1

to the finite commutative rings.
In 2007, Boucher et al. [3] introduced skew cyclic codes which is a generalization of
the class of cyclic codes on a non-commutative ring, namely, skew polynomial ring F [x; θ],
where F is a field and θ is an automorphism on F . Boucher and Ulmer [4] constituted
some skew cyclic codes with Hamming distance larger than the previously known linear
codes with the same parameters. Later on, Abualrub et al. [1] studied skew cyclic codes
over F2 + vF2 , where v 2 = v. In these works, they constructed skew cyclic codes by
taking an automorphism whose order must divide the length of the code. By relaxing the
said restriction, Siap et al. [12] investigated structural properties of skew cyclic codes of
arbitrary length and found that a skew cyclic code of arbitrary length is equivalent to
either a cyclic code or a quasi-cyclic code over Fp + vFp , where v 2 = v. In 2014, Gursoy
et al. [7] constructed a skew cyclic code along with their generators and also found the
idempotent generators of skew cyclic codes over Fp + vFp , where v 2 = v. Jitman et al.
[10] introduced skew constacyclic codes over the skew polynomial ring whose coefficients
are comes from finite chain rings, particularly for the ring Fpm + uFpm where u2 = 0. Very
recently, Gao et al. [6] investigated some structural properties of skew constacyclic codes
over the ring Fq + vFq ; v 2 = v.
Inspired by these study, we consider R = Fq + uFq + vFq + uvFq , where q = pm , p
is an odd prime and u2 = u, v 2 = v, uv = vu and study skew cyclic codes and skew
constacyclic codes over R. Now, for skew polynomial ring, we define an automorphism
θt : R → R
by
t

t

t

t

θt (a + ub + vc + uvd) = ap + ubp + vcp + uvdp ,
where a, b, c, d ∈ Fq and use the same throughout this paper. In this case, the order
of the automorphism is | hθt i |= mt = k (say). Also, the invariant subring under the
automorphism
pt + vFpt + uvFpt . For this automorphism θt on R, the set
 θt is Fpt + uF
2
R[x; θt ] = a0 + a1 x + a2 x + · · · + an xn : ai ∈ R, ∀ i = 0, 1, 2, ..., n forms a ring under
the usual addition of polynomials and the multiplication, denoted by ∗, with respect to
(axi ) ∗ (bxj ) = aθt i (b)xi+j . This is a non-commutative ring, unless θt is identity, known
as a skew polynomial ring.
Presentation of the manuscript is as follows: In section 2, we discuss some preliminaries which have been used later. In section 3, we define Gray map and describe Gray
images of skew cyclic codes over R. Section 4 contains some important results of skew
cyclic codes over R while section 5 gives us the idempotent generators of skew cyclic along
with their duals and includes two examples to elaborate the derived results. In section 6
presents a relation among skew constacyclic, constacyclic and quasi-twisted code. Section
7 discusses the characterization of skew α-constacyclic code along with quasi-twisted shift
operator. Section 8 gives structure of skew constacyclic code by decomposition method
and section 9 concludes the paper.

2

2. Preliminaries
Throughout the paper R represents Fq + uFq + vFq + uvFq , where q = pm , p is an odd
prime and u2 = u, v 2 = v, uv = vu. It can be checked that R is a finite commutative ring
containing q 4 elements and characteristic p.
We recall that a linear code C of length n over R is a R−submodule of Rn and members of
C are called codewords. A linear code C of length n over R is said to be a skew cyclic code
with respect to the automorphism θt if and only if for any codeword a = (a1 , a2 , ..., an ) ∈ C
implies σ(a) = (θt (an ), θt (a0 ), ..., θt (an−1 )) ∈ C, where σ is a skew cyclic shift of C.
The inner product of any a = (a1 , a2 , ..., an ), b = (b1 , b2 , ..., bn ) in C is define as a.b =
P
n
i=1 ai bi and a, b is saidto be orthogonal if a.b = 0. For a code C, its dual denoted by
⊥
C and define as C ⊥ = x ∈ Rn : x.c = 0 ∀c ∈ C . If C = C ⊥ , then C is said to be
a self-dual code. The Hamming weight wH (a) is defined as the number of the non-zero
components in a = (a1 , a2 , ..., an ) ∈ C and Hamming distance between two codewords a
and b of C is defined as dH (a, b) = wH (a − b) while the
 Hamming distance for the code
C is denoted by dH (C) and defined as dH (C) = min dH (a, b) | a 6= b, ∀a, b ∈ C . The
Lee weight of an element r = a + ub + vc + uvd ∈ R is defined by wL (r) = wH (a, a +
b, a + c, aP
+ b + c + d) and Lee weight for the codeword a = (a1 , a2 , ..., an ) ∈ Rn is
n
wL (a) = ni=1 wL (ai ). The
Pn Lee distance between two codewords a, b ∈ R is defined as
dL (a, b) = wL (a − b) = i=1 wL (ai − bi ) and the Lee distance for the code C is defined
by dL (C) = min dL (a, b) | a 6= b, ∀a, b ∈ C .
A code C of length nm over Fq is said to be a skew quasi-cyclic code of index m if
πm (C) = C, where πm is the skew quasi-cyclic shift on (Fqn )m define by
πm (a1 | a2 | ... | am ) = (σ(a1 ) | σ(a2 ) | ... | σ(am )).

(2.1)

3. Gray map and Fq -images of skew cyclic code over R
A map Ψ : R → Fq4 defined by
Ψ(a + ub + vc + uvd) = (a, a + b, a + c, a + b + c + d)
is called the Gray map [[13]]. It can be checked that Ψ is a linear map and can be
extended to Rn in obvious way by Ψ : Rn → Fq4n such that
Ψ(r1 , r2 , ..., rn ) = (a1 , ..., an , a1 + b1 , ..., an + bn , a1 + c1 , ..., an + cn ,
a1 + b1 + c1 + d1 , ..., an + bn + cn + dn )
where ri = ai + ubi + vci + uvdi ∈ R ∀ i = 1, 2, ..., n.
In light of above definition, the following fact can be easily checked:
Theorem 3.1. The Gray map Ψ is a Fq -linear distance preserving map from Rn (Lee
distance) to Fq4n (Hamming distance).
Theorem 3.2. If C is [n, k, dL ] linear code over R, then Ψ(C) is [4n, k, dH ] linear code
over Fq , where dL = dH .
3

Proof. By Theorem 3.1, Ψ is a Fq -linear map, so Ψ(C) is a linear code of length 4n. Since
Ψ is bijection as well as distance preserving, Ψ(C) has same minimum distance as C i.e,
dL = dH and dimension k.
Proposition 3.1. Let σ be the skew cyclic shift and π4 the skew quasi-cyclic shift as
defined in equation 2.1 and Ψ be the Gray map from Rn to Fq4n . Then Ψσ = π4 Ψ.
Proof. Let ri = ai + ubi + vci + uvdi ∈ R for i = 1, 2, ..., n. Take r = (r1 , r2 , . . . , rn ), then
t
t
t
t
σ(r) = (θt (rn ), θt (r1 ), ..., θt (rn−1 )). Applying Ψ, we have Ψ(σ(r)) = (apn , ap1 , ..., apn−1 , apn +
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
bpn , ..., apn−1 +bpn−1 , apn +cpn , ..., apn−1 +cpn−1 , apn +bpn +cpn +dpn , ..., apn−1 +bpn−1 +cpn−1 +dpn−1 ).
On the other side, Ψ(r) = (a1 , a2 , ..., an , a1 + b1 , ..., an + bn , a1 + c1 , ..., an + cn , ..., a1 + b1 +
t
t
t
t
t
t
c1 + d1 , ..., an + bn + cn + dn ). Therefore, π4 (Ψ(r)) = (apn , ap1 , ..., apn−1 , apn + bpn , ..., apn−1 +
t
t
t
t
t
t
t
t
t
t
t
t
t
bpn−1 , apn + cpn , ..., apn−1 + cpn−1 , apn + bpn + cpn + dpn , ..., apn−1 + bpn−1 + cpn−1 + dpn−1 ). Hence
Ψσ = π4 Ψ.
Theorem 3.3. A linear code C of length n over R is a skew cyclic if and only if Ψ(C) is
a skew quasi-cyclic code of length 4n over Fq of index 4.
Proof. Let C be a skew cyclic code of length n over R. So, σ(C) = C and this implies
Ψ(σ(C)) = Ψ(C). By Proposition 3.1, we have π4 (Ψ(C)) = Ψ(C). This shows that Ψ(C)
is a skew quasi-cyclic code of length 4n over Fq of index 4.
Conversely, suppose Ψ(C) is a skew quasi-cyclic code of length 4n over Fq of index 4.
Then π4 (Ψ(C)) = Ψ(C). Proposition 3.1, we get Ψ(σ(C)) = Ψ(C) and since Ψ is injective,
so σ(C) = C, i.e, C is a skew cyclic code over R.
It is noted that sometimes the use of permuted version of Gray map instead of using
direct Gray map is more convenient. The permuted version of Ψπ defined as Ψπ (r) =
(a0 , a0 + b0 , a0 + c0 , a0 + b0 + c0 + d0 , a1 , a1 + b1 , a1 + c1 , a1 + b1 + c1 + d1 , . . . , an−1 , an−1 +
bn−1 , an−1 + cn−1 , an−1 + bn−1 + cn−1 + dn−1 ). Clearly, the codes obtained by Ψ and Ψπ are
permutation equivalent. Now, for a particular case, | hθt i |= k = 3, we have the following
result:
Proposition 3.2. For any r ∈ Rn , we have Ψπ σ(r) = σ 4 Ψπ (r).
Proof. Let r = (r0 , r1 , . . . , rn−1 ) ∈ Rn where ri = ai +ubi +vci +uvdi for 0 ≤ i ≤ n−1. We
t
t
t
t
t
t
have, Ψπ (σ(r)) = Ψπ (θt (rn−1 ), θt (r0 ), . . . , θt (rn−2 )) = (apn−1 , apn−1 +bpn−1 , apn−1 +cpn−1 , apn−1 +
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
bpn−1 + cpn−1 + dpn−1 , ap0 , ap0 + bp0 , ap0 + cp0 , ap0 + bp0 + cp0 + dp0 , . . . , apn−2 , apn−2 + bpn−2 , apn−2 +
t
t
t
t
t
cpn−2 , apn−2 + bpn−2 + cpn−2 + dpn−2 ).
On the other hand,
σ 4 Ψπ (r) = σ 4 (a0 , a0 + b0 , a0 + c0 , a0 + b0 + c0 + d0 , a1 , a1 + b1 , a1 + c1 , a1 + b1 + c1 +
d1 , . . . , an−1 , an−1 + bn−1 , an−1 + cn−1 , an−1 + bn−1 + cn−1 + dn−1 ) = (θt4 (an−1 ), θt4 (an−1 +
bn−1 ), θt4 (an−1 + cn−1 ), θt4 (an−1 + bn−1 + cn−1 + dn−1 ), θt4 (a0 ), θt4 (a0 + b0 ), θt4 (a0 + c0 ), θt4 (a0 +
b0 + c0 + d0 ), . . . θt4 (an−2 ), θt4 (an−2 + bn−2 ), θt4 (an−2 + cn−2 ), θt4 (an−2 + bn−2 + cn−2 + dn−2 )).
As | hθt i |= 3, θt3 (a) = a, ∀ a ∈ R. Finally,
4

t

t

t

t

t

t

t

t

t

t

t

t

t

σ 4 Ψπ (r) = (apn−1 , apn−1 + bpn−1 , apn−1 + cpn−1 , apn−1 + bpn−1 + cpn−1 + dpn−1 , ap0 , ap0 + bp0 , ap0 +
t
t
t
t
t
t
t
t
t
t
t
t
t
t
cp0 , ap0 + bp0 + cp0 + dp0 , . . . , apn−2 , apn−2 + bpn−2 , apn−2 + cpn−2 , apn−2 + bpn−2 + cpn−2 + dpn−2 ). This
completes the proof.
In the light of the above proposition, we conclude the following:
Theorem 3.4. Let | hθi |= k = 3 and C be a skew cyclic code of length n over R. Then
its Fq -image Ψπ (C) is equivalent to a 4−quasicyclic code of length 4n over Fq .

4. Skew cyclic codes over R

We denote B1 ⊕ B2 ⊕ B3 ⊕ B4 = a1 + a2 + a3 + a4 : ai ∈ Bi for i = 1, 2, 3, 4 .
Also, any a + ub + vc + uvd ∈ R can be written as a + ub + vc + uvd = (1 − u − v +
uv)a + (u − uv)(a + b) + (v − uv)(a + c) + uv(a + b + c + d) where a, b, c, d ∈ Fq and by
[11], a + ub + vc + uvd ∈ R is a unit if and only if a, (a + b), (a + c), (a + b + c + d) are
units in Fq . Let C be a linear code of length n in R and
C1 = a ∈ Fqn | a + ub + vc + uvd ∈ C, for some b, c, d ∈ Fqn ;
C2 = a + b ∈ Fqn | a + ub + vc + uvd ∈ C, for some c, d ∈ Fqn ;

C3 = a + c ∈ Fqn | a + ub + vc + uvd ∈ C, for some b, d ∈ Fqn ;

C4 = a + b + c + d ∈ Fqn | a + ub + vc + uvd ∈ C .
Then C1 , C2 , C3 , C4 are linear codes of length n over Fq and C can be expressed uniquely
as C = (1 − u − v + uv)C1 ⊕ (u − uv)C2 ⊕ (v − uv)C3 ⊕ uvC4 .
Now, we calculate dual code of C, denoted by C ⊥ , as follows:
Theorem 4.1. Let C = (1 − u − v + uv)C1 ⊕ (u − uv)C2 ⊕ (v − uv)C3 ⊕ uvC4 be a linear
code over R where C1 , C2 , C3 and C4 be linear codes over Fq . Then C ⊥ = (1 − u − v +
uv)C1⊥ ⊕ (u − uv)C2⊥ ⊕ (v − uv)C3⊥ ⊕ uvC4⊥ . Further, C is self-dual if and only if C1 , C2 , C3
and C4 are self-dual.
Proof. The proof is similar to the proof of Theorem 5 in [13].
Let D be a linear code of length n over Fq . For any codeword c = (c0 , c1 , ..., cn−1 ) in D,
we identify to a polynomial c(x) in Fq [x; θt ]/hxn − 1i where c(x) = c0 +c1 x+· · ·+cn−1 xn−1 .
By this identification Siap et al. [12] studied skew cyclic codes over the field Fq and
constructed many structural properties like D is a skew cyclic code over Fq if and only
if D is a left Fq [x; θt ]−submodule of Fq [x; θt ]/hxn − 1i and D is generated by a monic
polynomial which is a right divisor of (xn − 1) in Fq [x; θt ]. Analogous to their study we
discuss skew cyclic codes over R in this article.
Theorem 4.2. Let C = (1 − u − v + uv)C1 ⊕ (u − uv)C2 ⊕ (v − uv)C3 ⊕ uvC4 be a linear
code of length n over R where C1 , C2 , C3 and C4 be linear codes of length n over Fq . Then
C is a skew cyclic code over R if and only if C1 , C2 , C3 and C4 are skew cyclic codes over
Fq .

5

Proof. Suppose C1 , C2 , C3 and C4 are skew cyclic codes over Fq . In order to prove C is a skew
cyclic code over R, let r = (r0 , r1 , ..., rn−1 ) ∈ C where ri = (1 − u − v + uv)ai + (u − uv)bi +
(v − uv)ci + uvdi , 0 ≤ i ≤ n − 1. Take a = (a0 , a1 , ..., an−1 ), b = (b0 , b1 , ..., bn−1 ), c =
(c0 , c1 , ..., cn−1 ), d = (d0 , d1 , ..., dn−1). Then a ∈ C1 , b ∈ C2 , c ∈ C3 , d ∈ C4 and since
C1 , C2 , C3 and C4 are skew cyclic, so σ(a) ∈ C1 , σ(b) ∈ C2 , σ(c) ∈ C3 , σ(d) ∈ C4 . Note that
θt (ri ) = (1 − u − v + uv)θt (ai ) + (u − uv)θt(bi ) + (v − uv)θt (ci ) + uvθt(di ) for 0 ≤ i ≤ n − 1.
Then σ(r) = (θt (rn−1 ), θt (r0 ), ...θt (rn−2 ) = (1 − u − v + uv)σ(a) + (u − uv)σ(b) + (v −
uv)σ(c) + uvσ(d) ∈ (1 − u − v + uv)C1 ⊕ (u − uv)C2 ⊕ (v − uv)C3 ⊕ uvC4 = C. This shows
that C is a skew cyclic code over R.
Conversely, suppose C is a skew cyclic code over R. Let a = (a0 , a1 , .., an−1 ) ∈ C1 , b =
(b0 , b1 , .., bn−1 ) ∈ C2 , c = (c0 , c1 , ..., cn−1 ) ∈ C3 , d = (d0 , d1 , .., dn−1 ) ∈ C4 . Consider ri =
(1−u−v+uv)ai +(u−uv)bi +(v−uv)ci +uvdi , 0 ≤ i ≤ n−1. Then r = (r0 , r1 , ..., rn−1 ) ∈ C
and since C is the skew cyclic so σ(r) ∈ C. But σ(r) = (1−u−v +uv)σ(a)+(u−uv)σ(b)+
(v − uv)σ(c) + uvσ(d), it follows that σ(a) ∈ C1 , σ(b) ∈ C2 , σ(c) ∈ C3 , σ(d) ∈ C4 . Hence
C1 , C2 , C3 and C4 are skew cyclic codes over Fq .
Corollary 4.1. The dual code C ⊥ is a skew cyclic code over R, provided C is a skew
cyclic code over R.
Proof. Let C = (1 − u − v + uv)C1 ⊕ (u − uv)C2 ⊕ (v − uv)C3 ⊕ uvC4 be a skew cyclic code
over R. Then by Theorem 4.2, C1 , C2 , C3 and C4 are skew cyclic codes over Fq . Since dual
of skew cyclic code over Fq is also a skew cyclic code over Fq by Corollary 18 of [4], so
C1⊥ , C2⊥ , C3⊥ and C4⊥ are skew cyclic codes over Fq . Thus, by Theorem 4.1 and Theorem
4.2, C ⊥ is a skew cyclic codes over R.
Corollary 4.2. The code C is a self-dual skew cyclic code over R if and only if C1 , C2 , C3
and C4 are self-dual skew cyclic codes over Fq .
Theorem 4.3. Let C = (1 − u − v + uv)C1 ⊕ (u − uv)C2 ⊕ (v − uv)C3 ⊕ uvC4 be a skew
cyclic code of length n over R. Then C has a generating polynomial f (x) which is a right
divisor of (xn − 1) in R[x; θt ].
Proof. Let fi (x) be generator of Ci in Fq [x; θt ] for i = 1, 2, 3, 4. Then (1 − u − v +
uv)f1 (x), (u − uv)f2 (x), (v − uv)f3 (x), uvf4(x) are generators of C. Let f (x) = (1 − u −
v + uv)f1 (x) + (u − uv)f2 (x) + (v − uv)f3 (x) + uvf4(x) and G = hf (x)i. Then G ⊆ C.
Now, (1 − u − v + uv)f (x) = (1 − u − v + uv)f1 (x) ∈ G, (u − uv)f (x) = (u − uv)f2(x) ∈
G, (v − uv)f (x) = (v − uv)f3 (x) ∈ G, uvf (x) = uvf4 (x) ∈ G, it follows that C ⊆ G and
hence C = G = hf (x)i.
Since fi (x) is a right divisor of (xn − 1) in Fq [x; θt ] for i = 1, 2, 3, 4, so there exit hi (x) ∈
Fq [x; θt ] such that (xn − 1) = h1 (x) ∗ f1 (x), (xn − 1) = h2 (x) ∗ f2 (x), (xn − 1) = h3 (x) ∗
f3 (x), (xn −1) = h4 (x)∗f4 (x). Now, [(1−u−v +uv)h1 (x)+(u−uv)h2 (x)+(v −uv)h2 (x)+
uvh4 (x)] ∗ f (x) = (1 − u − v + uv)h1(x) ∗ f1 (x) + (u − uv)h2 (x) ∗ f2 (x) + (v − uv)h3 (x) ∗
f3 (x) + uvh4 (x) ∗ f4 (x) = (xn − 1). Thus, f (x) is a right divisor of (xn − 1).
Corollary 4.3. Each left submodule of R[x; θt ]/hxn − 1i is generated by single element.
Let C be a skew cyclic code over Fq generated by the polynomial f (x) such that
(x − 1) = h(x) ∗ f (x) where f (x) = f0 + f1 x + · · · + fr xr , h(x) = h0 + h1 x + · · · + hn−r xn−r
in Fq [x; θt ]. Then by [4], its dual C ⊥ is a skew cyclic code over Fq generated by the
n

6

polynomial b
h(x) = hn−r + θt (hn−r−1 )x + · · · + θt (h0 )xn−r .
Corollary 4.4. Let C = (1 − u − v + uv)C1 ⊕ (u − uv)C2 ⊕ (v − uv)C3 ⊕ uvC4 be a skew
cyclic code over R and fi the generator of Ci such that (xn − 1) = hi (x) ∗ fi (x), for
i = 1, 2, 3, 4. Then
C ⊥ = h(1 − u − v + uv)b
h1(x) + (u − uv)b
h2(x) + (v − uv)b
h3(x) + uvb
h4 (x)i
P4
⊥
f
(x)
i
i=1
and | C |= q
.
Proof. Since b
hi (x) is generator of Ci⊥ for i = 1, 2, 3, 4, then by Theorem 4.1 and Theorem
4.3, C ⊥ = h(1 − u − v + uv)b
h1 (x) + (uP
− uv)b
h2(x) + (v − uv)b
h3 (x) + uvb
h4(x)i.
4
⊥
⊥
⊥
⊥
⊥
fi (x)
i=1
Also, | C |=| C1 || C2 || C3 || C4 |= q
.

5. Idempotent generators of skew cyclic codes and their dual codes over R
Since underlying ring is the skew polynomial ring which is non-commutative, so polynomials exhibit here more factorizations. Therefore, it is not too easy to find exact number
of skew cyclic codes over R[x; θt ] or the number of idempotent generators of skew cyclic
codes over R. But, when we impose conditions gcd(n, k) = 1 and gcd(n, q) = 1 [ as given
by [7]], where k is the order of the automorphism and n is the length of the code, we can
find idempotent generators. Towards this, we have the following:
Theorem 5.1. ([7]) Let f (x) ∈ Fq [x; θt ] be a monic right divisor of (xn − 1) and
C = hf (x)i. If gcd(n, k) = 1 and gcd(n, q) = 1, then there exits an idempotent polynomial
e(x) ∈ Fq [x; θt ]/hxn − 1i such that C = he(x)i.
Theorem 5.2. Let C = (1−u−v +uv)C1 ⊕(u−uv)C2 ⊕(v −uv)C3 ⊕uvC4 be a skew cyclic
code of length n over R with gcd(n, k) = 1 and gcd(n, q) = 1. Then C has an idempotent
generator e(x) in R[x; θt ].
Proof. By Theorem 5.1, there exists idempotent generator ei (x) of Ci for i = 1, 2, 3, 4 in
Fq [x; θt ]. Then by Theorem 4.3, e(x) = (1 − u − v + uv)e1 (x) + (u − uv)e2 (x) + (v −
uv)e3 (x) + uve4 (x) is a generator of C which is also idempotent.
Theorem 5.3. Let C = (1 − u − v + uv)C1 ⊕ (u − uv)C2 ⊕ (v − uv)C3 ⊕ uvC4 be a skew
cyclic code of length n over R and gcd(n, k) = 1 and gcd(n, q) = 1. Then C ⊥ has an
idempotent generator in R[x; θt ].
Proof. Suppose e(x) = (1 − u − v + uv)e1 (x) + (u − uv)e2 (x) + (v − uv)e3(x) + uve4(x) is
an idempotent generator of C where ei (x) is idempotent generator of Ci for i = 1, 2, 3, 4.
Then Ci⊥ has idempotent generator 1 − ei (x−1 ), for i = 1, 2, 3, 4 [See Lemma 12.3.23 (i) in
[9]]. Hence, by Theorem 4.3, C ⊥ has idempotent generator (1 − u − v + uv)(1 − e1 (x−1 )) +
(u − uv)(1 − e2 (x−1 )) + (v − uv)(1 − e3 (x−1 )) + uv(1 − e4 (x−1 )) = 1 − e(x−1 ).
Example 5.1. Consider the field F25 = F5 [α], where α2 + α + 1 = 0. Take n = 4
and Frobenius automorphism θt : F25 → F25 defined by θt (α) = α5 . Now, x4 − 1 =
(x + 2)(x + 3)(x + α)(x + α + 1) = (x + 2)(x + 3)(x + α + 1)(x + α). Take f1 (x) = f2 (x) =
f3 (x) = f4 (x) = (x + α + 1). Then C1 = hf1 (x)i, C2 = hf2 (x)i, C3 = hf3 (x)i, C4 = hf4 (x)i
are skew cyclic codes of length 4 with dimension 3 over F25 . Let f (x) = (1 − u − v +
7

uv)f1 (x)+(u−uv)f2 (x)+(v −uv)f3 (x)+uvf4 (x) = (x+α+1). Then C = hf (x)i is a skew
cyclic code of length 4 over R = F25 + uF25 + vF25 + uvF25 , where u2 = u, v 2 = v, uv = vu.
Here, the Gray image Ψ(C) is a skew quasi-cyclic code of index 4 over F25 with parameters
[16, 12, 2], which is an optimal code.
Example 5.2. Consider the field F9 = F3 [2α + 1]; where α2 + 1 = 0. Take n = 6
and Frobenius automorphism θt : F9 → F9 defined by θt (α) = α3 . Now, x6 − 1 =
(2 + x + (1 + 2α)x2 + x3 )(1 + x + (2α + 2)x2 + x3 ) = (2 + αx + 2αx3 + x4 )(1 + αx + x2 ).
Take f1 (x) = f2 (x) = f3 (x) = (2+αx+2αx3+x4 ) and f4 (x) = (2+x+(1+2α)x2+x3 ), then
C1 = hf1 (x)i, C2 = hf2 (x)i, C3 = hf3 (x)i, C4 = hf4 (x)i are skew cyclic codes of length 6 over
F9 with dimension 2, 2, 2 and 3 respectively. If we take f (x) = (1 −u −v + uv)f1 (x) + (u −
uv)f2 (x)+(v−uv)f3 (x)+uvf4 (x) = (1−uv)(2+αx+2αx3 +x4 )+uv(2+x+(1+2α)x2 +x3 ),
then C = hf (x)i is a skew cyclic code of length 6 over R = F9 + uF9 + vF9 + uvF9, where
u2 = u, v 2 = v, uv = vu. The Gray image Ψ(C) is a skew quasi-cyclic code of index 4 over
F9 with parameters [24, 9, 4].

6. Skew Constacyclic codes over R
Definition 6.1. Let α = (α1 + uα2 + vα3 + uvα4) be a unit in R where αi ∈ Fpt \{0} and
θt be the automorphism on R. A linear code C of length n is said to be skew α-constacyclic
code over R if and only if C is invariant under the skew α-constacyclic shift operation τα
where τα : Rn → Rn defined by τα (c0 , c1 , .., cn−1) = (αθt (cn−1 ), θt (c0 ), .., θt (cn−2 )), i.e, C is
skew α-constacyclic code if and only if τα (C) = C.
Clearly, C is said to be skew cyclic code for α = 1 and skew negacyclic code for α = −1.
In this section our study restricted to the condition α ∈ R such that α2 = 1. By
identifying each codeword c = (c0 , c1 , . . . cn−1 ) ∈ Rn to a polynomial c(x) = c0 + c1 x +
· · · + cn−1 xn−1 in the left R-module Rn = R[θt ; x]/hxn − αi, we call a linear code C is a
skew α-constacyclic code if and only if it is a left R-submodule of Rn = R[θt ; x]/hxn −αi.
Theorem 6.1. Define a map ρ : Rn = R[θt ; x]/hxn − 1i 7→ Rn , α = R[θt ; x]/hxn − αi by
ρ(f (x)) = f (αx). If n is odd, then ρ is a left R-module isomorphism.
Proof. Justification is straightforward. Only need to observe that if
f (x) = g(x) mod (xn − 1)
⇔ f (x) − g(x) = h(x) ∗ (xn − 1) f or some h(x) ∈ R[x; θt ]
⇔ f (αx) − g(αx) = h(αx) ∗ (αn xn − 1)
= h(αx) ∗ (αxn − α2 )(as αn = α f or odd n)
= αh(αx) ∗ (xn − α)
⇔ f (αx) = g(αx) mod (xn − α).
Corollary 6.1. There is a one-to-one correspondence between the skew cyclic codes and
skew α-constacyclic codes over R of odd length.
Corollary 6.2. Let n be an odd positive integer. Define a permutation µ on Rn as
µ(c0 , c1 , . . . , cn−1 ) = (c0 , αc1 , . . . , αn−1 cn−1 ). Then C is a skew cyclic code of length n if
and only if µ(C) is skew α-constacyclic code of length n over R.
8

6.1. Relations
Definition 6.2. Let C be a linear code of length n over R and n = ml. Then R is said
to be an α-quasi-twisted code if for any
(c0 0 , c0 1 , ..., c0

l−1 , ..., cm−1 0 , cm−1 1 , ..., cm−1 l−1 )

∈C

implies
(αcm−1 0 , αcm−1 1 , ..., αcm−1

l−1 , ..., cm−2 0 , cm−2 1 , ..., cm−2 l−1 )

∈ C.

If l be the least positive integer satisfying n = ml, then C is known as α-quasi-twisted
code of length n over R.
There is a nice relationship among skew α-constacyclic codes, constacyclic codes and
α-quasi-twisted code over R which are obtain from following results.
Theorem 6.2. Let C be a skew α-constacyclic code of length n over R and gcd(n, k) = 1.
Then C is a α-constacyclic code of length n over R.
Proof. Since gcd(n, k) = 1, by elementary concept of number theory, there exists an
integer L > 0 such that T k = 1 + Ln. Let c(x) = c0 + c1 x + · · · + cn−1 xn−1 ∈ C. As
C is a skew α-constacyclic code and x ∗ c(x) represents skew α-constacyclic shift of the
codeword c(x), so x ∗ c(x), x2 ∗ c(x), ..., xT k ∗ c(x) are belong to C, where
xT k ∗ c(x) = xT k ∗ (c0 + c1 x + · · · + cn−1 xn−1 )
= θtT k (c0 )xT k + θtT k (c1 )xT k+1 + · · · + θtT k (cn−1 )xT k+n−1
= c0 x1+Ln + c1 x2+Ln + · · · + cn−1 xLn+n
= αL (c0 x + c1 x2 + · · · + cn−2 xn−1 + αcn−1 ) (as in Rn , xn = α)
=⇒ αL xT k ∗ c(x) = c0 x + c1 x2 + · · · + cn−2 xn−1 + αcn−1 ∈ C (as α2 = 1).
This proves that C is a α-constacyclic code of length n over R.
Corollary 6.3. Let gcd(n, k) = 1. If f (x) is a right divisor of xn − α in the skew
polynomial ring R[x; θt ], then f (x) is a factor of xn − α in the polynomial ring R[x].
Theorem 6.3. Let C be a skew α-constacyclic code of length n and gcd(n, k) = l. Then
C is a α-quasi-twisted code of index l over R.
Proof. Since gcd(n, k) = l, there exit two integers T and D such that T k = l +Dn; D > 0.
Let r = (c0 0 , c0 1 , ..., c0 l−1 , ..., cm−1 0 , cm−1 1 , ..., cm−1 l−1 ) ∈ C. Since C is a skew αconstacyclic code, τα (r), τα2 (r), ..., ταl (r) belong to C, where
ταl (r) = (θtl (αcm−1 0 ), ..., θtl (αcm−1
θtl (cm−2 0 ), ..., θtl (cm−2
=⇒

ταl+Dn (r)

=

l
l
l−1 ), θt (c0 0 ), ...θt (c0 l−1 ), ...,

l−1 )).

(θtl+Dn (cm−1 0 ), ..., θtl+Dn (cm−1 l−1 ), ...,
θtl+Dn (αcm−2 0 ), ..., θtl+Dn (αcm−2 l−1 ))

= (θtT k (cm−1 0 ), ..., θtT k (cm−1

l−1 ), .

.., θtT k (αcm−2 0 ), ..., θtT k (αcm−2 l−1 ))
= (cm−1 0 , ..., cm−1
=⇒

αταl+Dn (r)

l−1 , ..., αcm−2 0 , ..., αcm−2 l−1 ).

= (αcm−1 0 , ..., αcm−1

l−1 , ..., cm−2 0 , ..., cm−2 l−1 )

This proves that C is an α-quasi-twisted code of index l.
9

∈ C (as α2 = 1)

7. Constacyclic code with other shift constant
In this section we characterize the Gray images of skew α-constacyclic codes over R as a
skew quasi-twisted code over Fq . We define the quasi-twisted shift operator ωl on (Fqn )l by
ωl ((c1 ) | (c2 ) | (c3 ) | · · · | (cl )) = (τα (c1 ) | τα (c2 ) | τα (c3 ) | · · · | τα (cl ))
where ci ∈ Fqn and τα is skew α-constacyclic shift operators as define in last section 6.
A linear code C of length nl over Fq is said to be a skew quasi-twisted code of index l if
ωl (C) = C.
With the help of above definition, we get the following results:
Proposition 7.1. Let Ψ be the Gray map as define earlier, then Ψτα = ω4 Ψ.
Proof. Let r = (r0 , r1 , . . . , rn−1 ) ∈ R. We have, Ψτα (r) = Ψ(αθt (rn−1 ), θt (r0 ),
t
t
t
t
t
t
t
t
t
t
. . . , θt (rn−2 )) = (αapn−1 , ap0 , . . . , apn−2 , αapn−1 +αbpn−1 , . . . , apn−2 +bpn−2 , αapn−1+αcpn−1 , . . . , apn−2 +
t
t
t
t
t
t
t
t
t
cpn−2 , αapn−1 + αbpn−1 + αcpn−1 + αdpn−1 , . . . , apn−2 + bpn−2 + cpn−2 + dpn−2 ).
On the other hand,
ω4 Ψ(r) = ω4 (a0 , a1 , . . . , an−1 , a0 + b0 , a1 + b1 , . . . , an−1 + bn−1 , a0 + c0 , a1 + c1 , . . . , an−1 +
t
t
t
t
cn−1 , a0 +b0 +c0 +d0 , a1 +b1 +c1 +d1 , . . . , an−1 +bn−1 +cn−1 +dn−1 ) = (αapn−1 , ap0 , . . . , apn−2 , αapn−1 +
t
t
t
t
t
t
t
t
t
t
t
t
αbpn−1 , . . . , apn−2 +bpn−2 , αapn−1+αcpn−1 , . . . , apn−2 +cpn−2 , αapn−1 +αbpn−1 +αcpn−1+αdpn−1 , . . . , apn−2 +
t
t
t
bpn−2 + cpn−2 + dpn−2 ). Therefore, Ψτα = ω4 Ψ.
As a consequence of the Proposition 7.1, we have the following:
Theorem 7.1. If C is a skew α-constacyclic codes of length n over R, then its Fq -image
Ψ(C) is a skew quasi-twisted code of length 4n over Fq of index 4 and vice versa.

8. Decomposition of skew (α1 + uα2 + vα3 + uvα4 )-constacyclic codes over
R
In this section, we discuss skew (α1 + uα2 + vα3 + uvα4 )-constacyclic codes of arbitrary
length n over R by decomposition method.
Gao et al. [6] have shown that a skew α-constacyclic code C of length n over Fq is a
left Fq [x; θt ]-submodule of Fq [x; θt ]/hxn − αi generated by a monic polynomial f (x) with
minimal degree in C and f (x) is a right divisor of (xn − α). Motivated by these study
we find some structural properties of skew α-constacyclic codes over R which are listed
below.
Theorem 8.1. Let C = (1 − u − v + uv)C1 ⊕ (u − uv)C2 ⊕ (v − uv)C3 ⊕ uvC4 be a linear
code of length n over R. Then C is (α1 + uα2 + vα3 + uvα4 )-constacyclic code over R
if and only if C1 , C2 , C3 and C4 are skew α1 -constacyclic code, skew (α1 + α2 )-constacyclic
code, skew (α1 + α3 )-constacyclic code and skew (α1 + α2 + α3 + α4 )-constacyclic code of
length n over Fq respectively, where (α1 + uα2 + vα3 + uvα4) is a unit in R.
10

Proof. Let r = (r0 , r1 , .., rn−1 ) ∈ C, where ri = (1 − u − v + uv)ai + (u − uv)bi + (v −
uv)ci + uvdi , 0 ≤ i ≤ n − 1. Take a = (a0 , .., an−1 ), b = (b0 , .., bn−1 ), c = (c0 , .., cn−1), d =
(d0 , .., dn−1), then a ∈ C1 , b ∈ C2 , c ∈ C3 and d ∈ C4 . Suppose C1 , C2 , C3 and C4 are
skew α1 -constacyclic, skew (α1 + α2 )-constacyclic, skew (α1 + α3 )-constacyclic, skew
(α1 + α2 + α3 + α4 )-constacyclic code over Fq respectively. So τα1 (a) ∈ C1 , τα1 +α2 (b) ∈
C2 , τα1 +α3 (c) ∈ C3 , and τα1 +α2 +α3 +α4 (d) ∈ C4 . Now, τα1 +uα2 +vα3 +uvα4 (r)= ((α1 + uα2 +
vα3 + uvα4 )θt (rn−1 ), θt (r0 ), .., θt (rn−2 )) = (1 − u − v + uv)τα1 (a) + (u − uv)τα1 +α2 (b) + (v −
uv)τα1 +α3 (c)+uvτα1 +α2 +α3 +α4 (d) ∈ (1−u−v +uv)C1 ⊕(u−uv)C2 ⊕(v −uv)C3 ⊕uvC4 = C.
This shows that C is (α1 + uα2 + vα3 + uvα4 )-constacyclic code over R.
Conversely, let a = (a0 , .., an−1 ) ∈ C1 , b = (b0 , .., bn−1 ) ∈ C2 , c = (c0 , .., cn−1 ) ∈ C3 , d =
(d0 , .., dn−1) ∈ C4 . Take ri = (1 − u − v + uv)ai + (u − uv)bi + (v − uv)ci + uvdi , 0 ≤
i ≤ n − 1. Then r = (r0 , r1 , .., rn−1) ∈ C. Suppose C is (α1 + uα2 + vα3 + uvα4 )constacyclic code over R. So, τα1 +uα2 +vα3 +uvα4 (r) ∈ C, where τα1 +uα2 +vα3 +uvα4 (r) =
(1 − u − v + uv)τα1 (a) + (u − uv)τα1 +α2 (b) + (v − uv)τα1 +α3 (c) + uvτα1 +α2 +α3 +α4 (d). It
follows that τα1 (a) ∈ C1 , τα1 +α2 (b) ∈ C2 , τα1 +α3 (c) ∈ C3 and τα1 +α2 +α3 +α4 (d) ∈ C4 . Hence
C1 , C2 , C3 and C4 are skew α1 -constacyclic code, skew (α1 + α2 )-constacyclic code, skew
(α1 + α3 )-constacyclic code and skew (α1 + α2 + α3 + α4 )-constacyclic code of length n
over Fq respectively.
Corollary 8.1. Let C = (1 − u − v + uv)C1 ⊕ (u − uv)C2 ⊕ (v − uv)C3 ⊕ uvC4 be a
skew (α1 + uα2 + vα3 + uvα4)-constacyclic code of length n over R. Then the dual code
C ⊥ = (1 − u − v + uv)C1⊥ ⊕ (u − uv)C2⊥ ⊕ (v − uv)C3⊥ ⊕ uvC4⊥ is skew (α1 + uα2 +
vα3 + uvα4 )−1 -constacyclic code over R, where C1⊥ , C2⊥ , C3⊥ , C4⊥ are skew α1−1 -constacyclic
code, skew (α1 + α2 )−1 -constacyclic code, skew (α1 + α3 )−1 -constacyclic code and skew
(α1 + α2 + α3 + α4 )−1 -constacyclic code over Fq respectively, provided n is a multiple of
k(order of the automorphism).
Proof. As (α1 +uα2 +vα3 +uvα4) is fixed by θt and n is a multiple of k, then by Lemma 3.1
of [10], C ⊥ is (α1 +uα2 +vα3 +uvα4 )−1 -constacyclic over R. Since (α1 +uα2 +vα3 +uvα4 ) =
(1 − u − v + uv)α1 + (u − uv)(α1 + α2 ) + (v − uv)(α1 + α3 ) + uv(α1 + α2 + α3 + α4 ), it
follows (α1 + uα2 + vα3 + uvα4 )−1 = (1 − u − v + uv)α1−1 + (u − uv)(α1 + α2 )−1 + (v −
uv)(α1 + α3 )−1 + uv(α1 + α2 + α3 + α4 )−1 . Hence by Theorem 8.1, C1⊥ , C2⊥ , C3⊥ , C4⊥ are skew
α1−1 -constacyclic code, skew (α1 + α2 )−1 -constacyclic code, skew (α1 + α3 )−1 -constacyclic
code and skew (α1 + α2 + α3 + α4 )−1 -constacyclic code over Fq respectively.
Corollary 8.2. Let C = (1 − u − v + uv)C1 ⊕ (u − uv)C2 ⊕ (v − uv)C3 ⊕ uvC4 be a skew
(α1 + uα2 + vα3 + uvα4 )-constacyclic code of length n over R. Then C is self-dual if and
only if α1 + uα2 + vα3 + uvα4 = 1, −1, 1 − 2u, 1 − 2v, 1 − 2uv, −1 + 2u, −1 + 2v, −1 +
2uv, 1 − 2u + 2uv, 1 − 2v + 2uv, −1 + 2u − 2uv, −1 + 2v − 2uv, 1 − 2u − 2v + 2uv, −1 +
2u + 2v − 2uv, 1 − 2u − 2v + 4uv, −1 + 2u + 2v − 4uv.
Proof. It can easily be proved that C is self-dual if and only if α1 = ±1, α1 + α2 =
±1, α1 + α2 + α3 = ±1 and α1 + α2 + α3 + α4 = ±1.
Theorem 8.2. Let C = (1 − u − v + uv)C1 ⊕ (u − uv)C2 ⊕ (v − uv)C3 ⊕ uvC4 be a skew
(α1 + uα2 + vα3 + uvα4 )-constacyclic code over R. Then there exits a polynomial f (x)
in R[x; θt ] which is a right divisor of xn − (α1 + uα2 + vα3 + uvα4 ) and C = hf (x)i.

11

Proof. Let fi (x) be generator of Ci for i = 1, 2, 3, 4. Then (1 − u − v + uv)f1 (x), (u −
uv)f2 (x), (v − uv)f3(x) and uvf4 (x) are generators of C. Take f (x) = (1 − u − v +
uv)f1 (x) + (u − uv)f2(x) + (v − uv)f3 (x) + uvf4 (x) and G = hf (x)i. Then G ⊆ C. On
the other hand, (1 − u − v + uv)f (x) = (1 − u − v + uv)f1(x) ∈ G, (u − uv)f (x) =
(u − uv)f2 (x) ∈ G, (v − uv)f (x) = (v − uv)f3 (x) ∈ G and uvf (x) = uvf4 (x) ∈ G. This
implies that C ⊆ G and hence C = G = hf (x)i.
Since f1 (x), f2 (x), f3 (x) and f4 (x) are right divisors of xn −α1 , xn −(α1 +α2 ), xn −(α1 +α3 ),
and xn − (α1 + α2 + α3 + α4 ) respectively, so there exist h1 (x), h2 (x), h3 (x), h4 (x) such
that xn − α1 = h1 (x) ∗ f1 (x), xn − (α1 + α2 ) = h2 (x) ∗ f2 (x), xn − (α1 + α3 ) = h3 (x) ∗ f3 (x)
and xn − (α1 + α2 + α3 + α4 ) = h4 (x) ∗ f4 (x). Also, [(1 − u − v + uv)h1 + (u − uv)h2 +
(v − uv)h3 + uvh4] ∗ f (x) = (1 − u − v + uv)h1 ∗ f1 + (u − uv)h2 ∗ f2 + (v − uv)h3 ∗ f3 +
uvh4 ∗ f4 = xn − (α1 + uα2 + vα3 + uvα4 ). This proves that f (x) is a right divisor of
xn − (α1 + uα2 + vα3 + uvα4 ).
Corollary 8.3. Each left submodule of R[x; θt ]/hxn −(α1 +uα2 +vα3 +uvα4)i is generated
by single element where α1 + uα2 + vα3 + uvα4 is a unit in R.
Corollary 8.4. Let C = (1 − u − v + uv)C1 ⊕ (u − uv)C2 ⊕ (v − uv)C3 ⊕ uvC4 be a
skew (α1 + uα2 + vα3 + uvα4 )-constacyclic code of length n over R and f1 (x), f2 (x), f3 (x)
and f4 (x) be generators of C1 , C2 , C3 and C4 in Fq [x; θt ] respectively such that xn − α1 =
h1 (x) ∗ f1 (x), xn − (α1 + α2 ) = h2 (x) ∗ f2 (x), xn − (α1 + α3 ) = h3 (x) ∗ f3 (x) and xn − (α1 +
α2 + α3 + α4 ) = h4 (x) ∗ f4 (x). Then
C ⊥ = h(1 − u − v + uv)b
h1 (x) + (u − uv)b
h2 (x) + (v −
P4
⊥
f
(x)
uv)b
h3 (x) + uvb
h4 (x)i and | C |= q i=1 i .
Proof. Same as Corollary 4.4.
Theorem 8.3. Let C = (1 − u − v + uv)C1 ⊕ (u − uv)C2 ⊕ (v − uv)C3 ⊕ uvC4 be a skew
(α1 + uα2 + vα3 + uvα4 )-constacyclic code over R. Let gcd(n, k) = 1 and gcd(n, q) = 1.
Then there exists an idempotent generator e(x) = (1 − u − v + uv)e1 (x) + (u − uv)e2 (x) +
(v − uv)e3 (x) + uve4(x) in R[x; θt ]/hxn − (α1 + uα2 + vα3 + uvα4 )i such that C = he(x)i,
where e1 (x) ∈ Fq [x; θt ]/hxn −α1 i, e2 (x) ∈ Fq [x; θt ]/hxn −(α1 +α2 )i, e3 (x) ∈ Fq [x; θt ]/hxn −
(α1 +α3 )i, e4 (x) ∈ Fq [x; θt ]/hxn −(α1 +α2 +α3 +α4 )i are idempotent generators of C1 , C2 , C3
and C4 respectively.
Proof. By using same argument of the proof of Theorem 16 of [12], we conclude that there
exist idempotent generators e1 (x), e2 (x), e3 (x) and e4 (x) of C1 , C2 , C3 and C4 respectively
in Fq [x; θt ]. Then by Theorem 8.2, e(x) = (1 − u − v + uv)e1 (x) + (u − uv)e2 (x) + (v −
uv)e3 (x) + uve4 (x) is an idempotent generator of C.
Example 8.1. Consider the field F49 = F7 [α]; where α2 − α + 3 = 0. Take n = 4
and Frobenius automorphism θt : F49 → F49 defined by θt (α) = α7 . Now, x4 − 1 =
(x+1)(x−1)(x2 +1); x4 +1 = (x2 +3x+1)(x2 +4x+1). Take f1 (x) = f2 (x) = f3 (x) = (x+1)
and f4 (x) = x2 + 4x+ 1. Then C = h(1 −u −v + uv)f1 (x) + (u −uv)f2 (x) + (v −uv)f3 (x) +
uvf4 (x)i = h(1 − uv)(x + 1) + uv(x2 + 4x + 1)i is a self-dual skew (1 − 2uv)-constacyclic
code of length 4 over R = F49 + uF49 + vF49 + uvF49 , where u2 = u, v 2 = v, uv = vu. Also,
| hθt i |= k = 2 and gcd(n, k) = 2. So, by Theorem 6.3, C is a (1 − 2uv)-quasi-twisted
code of length 4 over R of index 2.

12

Example 8.2. Consider the field F9 = F3 [2α + 1]; where α2 + 1 = 0. Take Frobenius
automorphism θt : F9 → F9 defined by θt (α) = α3 and R = F9 + uF9 + vF9 + uvF9 ,
where u2 = u, v 2 = v, uv = vu. The polynomial f (x) = x6 + (1 − 2v)x5 + x4 + (1 −
2v)x3 + x2 + (1 − 2v)x + 1 is a right divisor of x7 − (1 − 2v − 2uv) in R[θt ; x] and also
gcd(n, k) = gcd(7, 2) = 1. Therefore, by Theorem 6.2, C = hf (x)i is a (1 − 2v − 2uv)constacyclic code of length 7 over R.

9. Conclusion
In this paper, we considered skew cyclic and skew constacyclic codes over R = Fpm +
uFpm + vFpm + uvFpm , where p is an odd prime and u2 = u, v 2 = v, uv = vu. It is shown
that skew cyclic codes and skew constacyclic codes over R are principally generated.
Also, we have given the necessary and sufficient conditions for codes being self-dual over
R.

Acknowledgement
The authors are thankful to University Grant Commission (UGC), Government of India
for financial support under Ref. No. 20/12/2015(ii)EU-V dated 31/08/2016 and Indian
Institute of Technology Patna for providing the research facilities. The authors would
like to thank the anonymous referees for their useful comments and suggestions.

References
[1] T. Abualrub, N. Aydin and P. Seneviratne, On θ-cyclic codes over F2 + vF2 , Australasian J. Combinatorics 54 (2012), 115-126.
[2] T. Abualrub and R. Oehmke, On the generators of Z4 cyclic codes of length 2e , IEEE
Trans. Inf. Theory 49(9) (2003), 2126-2133.
[3] D. Boucher, W. Geiselmann and F. Ulmer, Skew cyclic codes, Appl. Algebra Eng.
Comm. 18 (2007), 379-389.
[4] D. Boucher and F. Ulmer, Coding with skew polynomial rings, J. Symb. Comput.
44(12) (2009), 1644-1656.
[5] H. Q. Dinh, Constacyclic codes of length ps over Fpm + uFpm , J. Algebra 324(5)
(2010), 940-950.
[6] J. Gao, F. Ma and F. Fu, Skew constacyclic codes over the ring Fq + vFq , Appl.
Comput. Math. 6(3) (2017) 286–295.
[7] F. Gursoy, I. Siap and B. Yildiz, Construction of skew cyclic codes over Fq + vFq ,
Adv. Math. Commun. 8(3) (2014), 313-322.

13

[8] A. R. Hammons Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solè,
The Z4 - linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans.
Inform. Theory 40(2) (1994), 301-319.
[9] W. C. Huffman and V. Pless, Fundamentals of Error Correcting Codes. Cambridge
University Press, Cambridge, 2003.
[10] S. Jitman, S. Ling and P. Udomkavanich, Skew constacyclic codes over finite chain
rings, Adv. Math. Commun. 6(1) (2012), 29-63.
[11] J. Kaborè and M. E. Charkani, constacyclic codes over Fp + uFp + vFp + uvFp ,
arXiv:1507.03084 [] (2015).
[12] I. Siap, T. Abualrub, N. Aydin and P. Seneviratne, Skew cyclic codes of arbitrary
length, Int. J. Inform. Coding Theory 2(1) (2011), 10-20.
[13] X. Zheng and B. Kong, Cyclic codes and λ1 + λ2 u + λ3 v + λ4 uv -constacyclic codes
over Fp + uFp + vFp + uvFp , Appl. Math. Comput. 306 (2017), 86-91.

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