EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 14, No. 3, 2021, 1082-1097
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Quantum codes obtained through constacyclic codes
over Z3 + νZ3 + ωZ3 + νωZ3

Jagbir Singh1, Prateek Mor2,∗, Shikha3, Meena4

1 Department of Mathematics, Maharshi Dayanand University, Rohtak-124001, India
2 Department of Mathematics, Government College Israna, Panipat-132103, India
3 Department of Mathematics, S.K Government College Kanwali, Rewari-123411, India
4 Department of Mathematics, Government College Julana, Jind-126101, India

Abstract. This paper is concerned with, structural properties and construction of quantum codes
over Z3 by using constacyclic codes over the finite commutative non-chain ring < = Z3 + νZ3 +
ωZ3 + νωZ3 where ν2 = 1, ω2 = 1, νω = νω and Z3 is field having 3 elements with
characteristic 3. A Gray map is defined between < and Z4

3 . The parameters of quantum codes
over Z3 are obtained by decomposing constacyclic codes into cyclic and negacyclic codes over Z3.
As an application, some examples of quantum codes of arbitrary length, are also obtained.

2020 Mathematics Subject Classifications: 94B05, 94B15, 94B35, 94B60

Key Words and Phrases: Finite ring, Linear codes, Constacyclic codes, Negacyclic, Quantum
codes

1. Introduction

Quantum error correction shows an significant role in quantum computing as it is used to
correct any errors in quantum data due to decoherence and other quantum noise. Firstly,
the existence of quantum error correction code was proven by Shor [14] and individually
by Steane [17]. In 1998, Calderbank et. al [3] published a paper in which they developed
the theory to construct quantum codes using classical error correction codes. In current
years, an essential literature has been established about the quantum error correcting
codes. Some authors constructed quantum codes using the Gray image of cyclic codes
on some finite rings. For example, a new technique of constructing quantum codes from
cyclic codes over finite ring F2 + vF2 where v2 = v given by Qian [13]. Kai and Zhu [8]
created quantum codes from hermitian self orthogonal codes over F4 as Gray images of
linear and cyclic codes over F4 + uF4 where u2 = 0. Yin and Ma [18] gave a condition
for the existence of quantum codes from cyclic codes over F2 + uF2 + u2 + F2 with Lee

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v14i3.4043

Email addresses: prateekmor1992@gmail.com (Prateek Mor)

http://www.ejpam.com 1082 c© 2021 EJPAM All rights reserved.



Prateek Mor et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1082-1097 1083

metric. Some non binary quantum codes are described using classical codes via Gaussian
integers by Ozen et. al [11]. Guenda et. al [5] extended the CSS construction to finite
commutative Frobenius rings. Dertli et. al [4] received quantum codes from cyclic codes
over F2 + uF2 + vF2 + uvF2. Ashraf and Mohammad [1] gave a construction of quantum
codes from cyclic codes overF3 + vF3 where v2 = 1. In 2016, Ozen et al. [12] examined
several ternary quantum codes from the cyclic codes over F3 + uF3 + vF3 + uvF3. Very
recently, several researchers established a number of new quantum codes via Fp from the
classical cyclic and constacyclic codes to which we refer [2, 6, 9–11, 15]. Also, Singh and
Mor [16] constructed quantum codes over the finite non-chain ring < = Zp + νZp where
ν2 = ν in 2021.
The remaining paper is arranged as follows, Section 2 contains preliminaries in which some
fundamental properties and some essential definitions have been given. Section 3 defines
the Gray map from < to Z4

3 and some related properties like self orthogonal, self dual etc.
In Section 4, we presented the development of quantum codes through constacyclic codes
over the ring < which are exemplified in Section 5. Finally, the paper is concluded in the
last Section.

2. Preliminaries

Let Z3 is a finite filed with 3 elements. Now, we first start with a general overview of the
ring < = Z3+νZ3+ωZ3+νωZ3 having characteristic 3 with restrictions ν2 = 1, ω2 = 1
and νω = νω. < is a commutative, principal ideal but non-chain finite ring with 34 = 81
elements. The maximal ideal of < are

< 2 + ν + 2ω >, < 2 + ν + νω >, < ν + ω + 2νω >, < 2ν + 2ω + 2νω > .

Some units of < is 1 + ν + ω + 2νω, 1 + 2ν + 2ω + 2νω, νω, ν, ω for sake of simplicity
we consider ϑ is a unit of < and also we note that ϑ−1 = ϑ for each case.
Let us assume

ξ1 = 1+ν+ω+νω, ξ2 = 1−ν+ω−νω, ξ3 = 1+ν−ω−νω and ξ4 = 1−ν−ω+νω.

It is obvious to obtain that ξ2i = ξi, ξiξj = 0 and
∑4

i=1 ξi = 1 for all i, j = 1, 2, 3, 4 and
i 6= j. Now by chinese remainder theorem, the considered ring can be expressed as

< = ξ1Z3 ⊕ ξ2Z3 ⊕ ξ3Z3 ⊕ ξ4Z3.

Therefore, an arbitrary element e = e1 + νe2 + ωe3 + νωe4 of < where ei ∈ Z3 can be
uniquely expressed as

e = e1 + νe2 + ωe3 + νωe4 = ξ1k1 + ξ2k2 + ξ3k3 + ξ4k4

where ki ∈ Z3 for all i = 1, 2, 3, 4.
A nonempty subset K of <n is a linear code over < of length n. If K is an <-submodule



Prateek Mor et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1082-1097 1084

of <n and the elements of K are codewords. Let K be a code over < of length n and its
polynomial representation be T (K), that is,

T (K) = {
n−1∑
i=0

χit
i | (χ0, χ1, ..., χn−1) ∈ K}

Let Υ,Λ and f be the maps from <n to <n defined as

Υ(χ0, χ1, ..., χn−1) = (χn−1, χ0, ..., χn−2),

Λ(χ0, χ1, ..., χn−1) = (−χn−1, χ0, ..., χn−2),

f(χ0, χ1, ..., χn−1) = (ϑχn−1, χ0, ..., χn−2),

respectively. Then K is a cyclic, negacyclic, ϑ-constacyclic if Υ(K) = K, Λ(K) = K,
f(K) = K respectively. A code K over < of length n is cyclic, negacyclic and ϑ-constacyclic
if and only if T (K) is an ideal of <[t]/ < tn − 1 >, <[t]/ < tn + 1 > and <[t]/ < tn − ϑ >
respectively.
For the arbitrary elements χ = (χ0, χ1, ..., χn−1) and ψ = (ψ0, ψ1, ..., ψn−1) of <, the inner
product is defined as

χ.ψ =
n−1∑
i=0

χiψi.

If χ.ψ = 0, then χ and ψ are orthogonal. If K is a linear code over < of length n, then
the dual code of K is defined as

K⊥ = { χ ∈ <n : χ.ψ = 0 for all ψ ∈ K},

which is also a linear code over the ring < of length n. A code K is said to be self orthogonal
if K ⊆ K⊥ and said to be self dual if K = K⊥.

3. Gray Map over <

The hamming weight wH(χ) for any codeword χ = (χ0, χ1, ..., χn−1) ∈ <n is defined as
the number of all non-zero components in χ = (χ0, χ1, ..., χn−1). The minimum weight of
a code K, that is, wH(K) is the least weight among all of its non zero codewords. The
Hamming distance between two codes χ = (χ0, χ1, ..., χn−1) and χ̂ = (χ̂0, χ̂1, ..., χ̂n−1) of
<n, denoted by dH(χ, χ̂) = wH(χ− χ̂) and is defined as

dH(χ, ψ) = | {i | χi 6= ψi} | .

Minimum distance of K, denoted by dH and is given by minimum distance between the
different pairs of codewords of the linear code K. For any codeword χ = (χ0, χ1, ..., χn−1)
∈ <n, the lee weight is defined as wL(χ) =

∑n−1
i=0 wL(χi) and lee distance of (χ, χ̂) is given

by dL(χ, χ̂) = wL(χ− χ̂) =
∑n−1

i=0 wL(χi − χ̂i).
Minimum lee distance of K is denoted by dL and is given by minimum lee distance of



Prateek Mor et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1082-1097 1085

different pairs of codewords of the linear code K.
The Gray map ϕ from < to Z4

3, that is, ϕ: < → Z4
3 is defined as

ϕ(k = ξ1k1 + ξ2k2 + ξ3k3 + ξ4k4) = (k1 + k2 + k3 + k4, k1 − k2 + k3 − k4,
k1 + k2 − k3 − k4, k1 − k2 − k3 + k4).

Proposition 1. The Gray map ϕ is linear and distance preserving isometry map from
(<n, dL) to (Z4n

3 , dH), where dL and dH are the lee distance and hamming distance in <n

and Z4n
3 respectively.

Proof. Let k1, k2 ∈ < and α ∈ Z3 then

ϕ(αk1 + k2) = αϕ(k1) + ϕ(k1)

So, ϕ is linear map.
Now we show that ϕ is distance preserving map.
By the above definitions, dL(χ, χ̂) = wL(χ − χ̂) = wH(ϕ(χ − χ̂)) = wH(ϕ(χ) − φ(χ̂)) =
dH(ϕ(χ), ϕ(χ̂)).
Hence ϕ is distance preserving map from (<n, dL) to (Z4n

3 , dH).

Proposition 2. If K is a linear code over the ring < of length n with |K| = 3k, dL(K) = d,
then ϕ(K) is a ternary linear code having parameters [4n, k, d].

Proposition 3. Let K be a linear code over the ring < of length n. If K is self orthogonal,
then ϕ(K) is also self orthogonal.

Proof. Let K be a self orthogonal code and η1, η2 ∈ K such that η1 = ξ1a+ξ2b+ξ3c+ξ4d
and η2 = ξ1a

′
+ ξ2b

′
+ ξ3c

′
+ ξ4d

′
where a, b, c, d, a

′
, b
′
, c
′
, d
′ ∈ Z3 from the definition of

self orthogonality, η1.η2 = 0, that is, ξ1aa
′

+ ξ2bb
′

+ ξ3cc
′

+ ξ4dd
′

= 0, it follows that
aa
′

= bb
′

= cc
′

= dd
′

= 0. Now, applying ϕ on η1, η2, we have ϕ(η1) = (a+b+c+d, a-b+c-
d, a+b-c-d, a-b-c+d) and ϕ(η2) = (a

′
+b
′
+c
′
+d
′
, a
′−b′+c′−d′ , a′+b′−c′−d′ , a′−b′−c′+d′)

and hence ϕ(η1).ϕ(η2) = 4a.a
′
+4bb′+4cc′+4dd′ = 0 that implies ϕ(K) is self orthogonal.

Proposition 4. [7] Let K be a linear code over the ring < of length n. Then ϕ(K⊥) =
(ϕ(K))⊥. Further, K is self dual if and only if ϕ(K) is self dual.

4. Quantum codes obtained through ϑ-constacyclic codes

Let S′is be the linear codes for i = 1, 2, 3, 4. we denote

S1 ⊕ S2 ⊕ S3 ⊕ S4 = {s1 + s2 + s3 + s4 | si ∈ Si for i = 1, 2, 3, 4}

and
S1 ⊗ S2 ⊗ S3 ⊗ S4 = {(s1, s2, s3, s4) | si ∈ Si for i = 1, 2, 3, 4}



Prateek Mor et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1082-1097 1086

For a linear code K of length n over <, we define

K∞ = {s1 + s2 + s3 + s4 ∈ Zn
3 such that s1 + s2ν + s3ω + s4νω ∈ K},

K∈ = {s1 − s2 + s3 − s4 ∈ Zn
3 such that s1 + s2ν + s3ω + s4νω ∈ K},

K3 = {s1 + s2 − s3 − s4 ∈ Zn
3 such that s1 + s2ν + s3ω + s4νω ∈ K},

K4 = {s1 − s2 − s3 + s4 ∈ Zn
3 such that s1 + s2ν + s3ω + s4νω ∈ K}.

Clearly, K∞, K∈, K3 and K4 are the linear codes over Z3 of length n.

Theorem 5. [7] Let K be a linear code over the ring < of length n. Then ϕ(K) =
K∞ ⊗ K∈ ⊗ K3 ⊗ K4 and |K|= |K∞||K∈||K3|K4|.

Corollary 6. [7] If ϕ(K) = K∞ ⊗ K∈ ⊗ K3 ⊗ K4 then K= ξ1K∞ ⊕ ξ2K∈ ⊕ ξ3K3 ⊕ ξ4K4

By the help of Theorem 4.1 and Corollary 4.2, we say that the linear code K can be
uniquely expressed as

K = ξ1K∞ ⊕ ξ2K∈ ⊕ ξ3K3 ⊕ ξ4K4

and also
|K| = |K∞||K∈||K3|K4|.

If G1, G2, G3 and G4, are the generator matrices of the linear codes K∞, K∈, K3, and K4
respectively. Then, the generator matrix of K is

G =
[
ξ1G1 ξ2G2 ξ3G3 ξ4G4

]T
,

and that of ϕ(K) is

ϕ(G) =
[
ϕ(ξ1G1) ϕ(ξ2G2) ϕ(ξ3G3) ϕ(ξ4G4)

]T
Note: Now, we consider different case of ϑ.

Case 1. ϑ = 1 + ν + ω + 2νω

Theorem 7. Let K = ξ1K∞ ⊕ ξ2K∈ ⊕ ξ3K3 ⊕ ξ4K4 be a linear code over the ring < of
length n where K∞, K∈, K3, K4 are the linear codes over Z3. Then, K is a ϑ-constacyclic
code over the ring < of length n if and only if K∞, K∈, K3 are negacyclic codes and K4
is cyclic code over Z3 of length n.

Proof.
Let,

ȧ = (ȧ0, ȧ1, ..., ˙an−1) ∈ K∞, ḃ = (ḃ0, ḃ1, ..., ˙bn−1) ∈ K∈,
ċ = (ċ0, ċ1, ..., ˙cn−1) ∈ K3, ḋ = (ḋ0, ḋ1, ..., ˙dn−1) ∈ K4.



Prateek Mor et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1082-1097 1087

For an arbitrary element ζi ∈ K, uniquely expressed as

ζi = ξ1ȧi + ξ2ḃi + ξ3ċi + ξ4ḋi,

where ȧi, ḃi, ċi, ḋi ∈ Z3 for i = 0, 1, ..., n− 1.
Let,

ζ = (ζ0, ζ1, ..., ζn−1) ∈ <n.

First we assume that K is ϑ-constacyclic code over the ring < of length n, then

f(ζ) = ((1 + ν + ω + 2νω)ζn−1, ζ0, ..., ζn−2)

= (−(1 + ν + ω + νω) ˙an−1 − (1− ν + ω − νω) ˙bn−1 − (1 + ν − ω − νω) ˙cn−1

+ (1− ν − ω + νω) ˙dn−1, (1 + ν + ω + νω)ȧ0 + (1− ν + ω − νω)ḃ0 + (1 + ν − ω − νω)ċ0

+ (1− ν − ω + νω)ḋ0, ..., (1 + ν + ω + νω) ˙an−2 + (1− ν + ω − νω) ˙bn−2

+ (1 + ν − ω − νω) ˙cn−2 + (1− ν − ω + νω) ˙dn−2)

= (1 + ν + ω + νω)Λ(ȧ) + (1− ν + ω − νω)Λ(ḃ) + (1 + ν − ω − νω)Λ(ċ)

+ (1− ν − ω + νω)Υ(ḋ)

= ξ1Λ(ȧ) + ξ2Λ(ḃ) + ξ3Λ(ċ) + ξ4Υ(ḋ),

which is an element of the linear code K. Therefore, K∞, K∈, K3, are negacyclic codes
and K4 is a cyclic code over the ring Z3 of length n respectively.
Conversely, for any ζ = (ζ0, ζ1, ..., ζn−1) ∈ K, where ζi = ξ1ȧi + ξ2ḃi + ξ3ċi + ξ4ḋi and
ȧi, ḃi, ċi, ḋi ∈ Z3 for i = 0, 1, ..., n − 1. If K∞, K∈, K3 are negacyclic codes and K4 is a
cyclic code over Z3 of length n, then Λ(ȧ) ∈ K∞, Λ(ḃ) ∈ K∈, Λ(ċ) ∈ K3, Υ(ḋ) ∈ K4.
Hence, we have

(1+ν+ω+νω)Λ(ȧ)+(1−ν+ω−νω)Λ(ḃ)+(1+ν−ω−νω)Λ(ċ)+(1−ν−ω+νω)Υ(ḋ) ∈ K

where it given that

f(ζ) = (1+ν+ω+νω)Λ(ȧ)+(1−ν+ω−νω)Λ(ḃ)+(1+ν−ω−νω)Λ(ċ)+(1−ν−ω+νω)Υ(ḋ),

which implies that f(ζ) ∈ K.
Therefore, K is a ϑ-constacyclic code over the ring < of length n .

Case 2. ϑ = 1 + 2ν + 2ω + 2νω

Theorem 8. Let K = ξ1K∞ ⊕ ξ2K∈ ⊕ ξ3K3 ⊕ ξ4K4 be a linear code over the ring < of
length n where K∞, K∈, K3, K4 are the linear codes over Z3. Then, K is a ϑ-constacyclic
code over the ring < of length n if and only if K∞ is cyclic code and K∈, K3, K4 are
negacyclic codes over Z3 of length n.

Proof.
Let,

ȧ = (ȧ0, ȧ1, ..., ˙an−1) ∈ K∞, ḃ = (ḃ0, ḃ1, ..., ˙bn−1) ∈ K∈,



Prateek Mor et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1082-1097 1088

ċ = (ċ0, ċ1, ..., ˙cn−1) ∈ K3, ḋ = (ḋ0, ḋ1, ..., ˙dn−1) ∈ K4.

For an arbitrary element ζi ∈ K, uniquely expressed as

ζi = ξ1ȧi + ξ2ḃi + ξ3ċi + ξ4ḋi,

where ȧi, ḃi, ċi, ḋi ∈ Z3 for i = 0, 1, ..., n− 1.
Let,

ζ = (ζ0, ζ1, ..., ζn−1) ∈ <n.

First we assume that K is ϑ-constacyclic code over the ring < of length n, then

f(ζ) = ((1 + 2ν + 2ω + 2νω)ζn−1, ζ0, ..., ζn−2)

= ((1 + ν + ω + νω) ˙an−1 − (1− ν + ω − νω) ˙bn−1 − (1 + ν − ω − νω) ˙cn−1

− (1− ν − ω + νω) ˙dn−1, (1 + ν + ω + νω)ȧ0 + (1− ν + ω − νω)ḃ0 + (1 + ν − ω − νω)ċ0

+ (1− ν − ω + νω)ḋ0, ..., (1 + ν + ω + νω) ˙an−2 + (1− ν + ω − νω) ˙bn−2

+ (1 + ν − ω − νω) ˙cn−2 + (1− ν − ω + νω) ˙dn−2)

= (1 + ν + ω + νω)Υ(ȧ) + (1− ν + ω − νω)Λ(ḃ) + (1 + ν − ω − νω)Λ(ċ)

+ (1− ν − ω + νω)Λ(ḋ)

= ξ1Υ(ȧ) + ξ2Λ(ḃ) + ξ3Λ(ċ) + ξ4Λ(ḋ),

which is an element of the linear code K. Therefore, K∞ is cyclic code and K∈, K3, K4
are negacyclic codes over Z3 of length n respectively.
Conversely, for any ζ = (ζ0, ζ1, ..., ζn−1) ∈ K, where ζi = ξ1ȧi + ξ2ḃi + ξ3ċi + ξ4ḋi and
ȧi, ḃi, ċi, ḋi ∈ Z3 for i = 0, 1, ..., n−1. If K∞ is cyclic code and K∈, K3, K4 are negacyclic
codes over Z3 of length n., then Υ(ȧ) ∈ K∞, Λ(ḃ) ∈ K∈, Λ(ċ) ∈ K3, Λ(ḋ) ∈ K4.
Hence, we have

(1+ν+ω+νω)Υ(ȧ)+(1−ν+ω−νω)Λ(ḃ)+(1+ν−ω−νω)Λ(ċ)+(1−ν−ω+νω)Λ(ḋ) ∈ K

where it given that

f(ζ) = (1+ν+ω+νω)Υ(ȧ)+(1−ν+ω−νω)Λ(ḃ)+(1+ν−ω−νω)Λ(ċ)+(1−ν−ω+νω)Λ(ḋ),

which implies that f(ζ) ∈ K.
Therefore, K is a ϑ-constacyclic code over the ring < of length n .

Case 3. ϑ = νω

Theorem 9. Let K = ξ1K∞ ⊕ ξ2K∈ ⊕ ξ3K3 ⊕ ξ4K4 be a linear code over the ring
< of length n where K∞, K∈, K3, K4 are the linear codes over Z3. Then, K is a νω-
constacyclic code over the ring < of length n if and only if K∞, K4 are cyclic and K∈, K3
are negacyclic codes over Z3 of length n.



Prateek Mor et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1082-1097 1089

Proof.
Let,

ȧ = (ȧ0, ȧ1, ..., ˙an−1) ∈ K∞, ḃ = (ḃ0, ḃ1, ..., ˙bn−1) ∈ K∈,
ċ = (ċ0, ċ1, ..., ˙cn−1) ∈ K3, ḋ = (ḋ0, ḋ1, ..., ˙dn−1) ∈ K4.

For an arbitrary element ζi ∈ K, uniquely expressed as

ζi = ξ1ȧi + ξ2ḃi + ξ3ċi + ξ4ḋi,

where ȧi, ḃi, ċi, ḋi ∈ Z3 for i = 0, 1, ..., n− 1.
Let,

ζ = (ζ0, ζ1, ..., ζn−1) ∈ <n.

First we assume that K is νω-constacyclic code over the ring < of length n, then

f(ζ) = ((νω)ζn−1, ζ0, ..., ζn−2)

= (α(1 + ν + ω + νω) ˙an−1 − (1− ν + ω − νω) ˙bn−1 − (1 + ν − ω − νω) ˙cn−1

+ (1− ν − ω + νω) ˙dn−1, (1 + ν + ω + νω)ȧ0 + (1− ν + ω − νω)ḃ0 + (1 + ν − ω − νω)ċ0

+ (1− ν − ω + νω)ḋ0, ..., (1 + ν + ω + νω) ˙an−2 + (1− ν + ω − νω) ˙bn−2

+ (1 + ν − ω − νω) ˙cn−2 + (1− ν − ω + νω) ˙dn−2)

= (1 + ν + ω + νω)Υ(ȧ) + (1− ν + ω − νω)Λ(ḃ) + (1 + ν − ω − νω)Λ(ċ)

+ (1− ν − ω + νω)Υ(ḋ)

= ξ1Υ(ȧ) + ξ2Λ(ḃ) + ξ3Λ(ċ) + ξ4Υ(ḋ),

which is an element of the linear code K. Therefore, K∞, K4 are cyclic and K∈, K3, are
negacyclic codes over the ring Z3 of length n respectively.
Conversely, for any ζ = (ζ0, ζ1, ..., ζn−1) ∈ K, where ζi = ξ1ȧi + ξ2ḃi + ξ3ċi + ξ4ḋi and
ȧi, ḃi, ċi, ḋi ∈ Z3 for i = 0, 1, ..., n − 1. If K∞, K4 are cyclic and K∈, K3, are negacyclic
codes over the ringZp of length n, then Υ(ȧ) ∈ K∞, Λ(ḃ) ∈ K∈, Λ(ċ) ∈ K3, Υ(ḋ) ∈ K4.
Hence, we have

(1+ν+ω+νω)Υ(ȧ)+(1−ν+ω−νω)Λ(ḃ)+(1+ν−ω−νω)Λ(ċ)+(1−ν−ω+νω)Υ(ḋ) ∈ K

where it given that

f(ζ) = (1+ν+ω+νω)Υ(ȧ)+(1−ν+ω−νω)Λ(ḃ)+(1+ν−ω−νω)Λ(ċ)+(1−ν−ω+νω)Υ(ḋ),

which implies that f(ζ) ∈ K.
Therefore, K is a νω-constacyclic code over the ring < of length n .

Case 4. ϑ = ν



Prateek Mor et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1082-1097 1090

Theorem 10. Let K = ξ1K∞ ⊕ ξ2K∈ ⊕ ξ3K3 ⊕ ξ4K4 be a linear code over the
ring < of length n where K∞, K∈, K3, K4 are the linear codes over Z3. Then, K is
a ν-constacyclic code over the ring < of length n if and only if K∞, K3 are cyclic and
K∈, K4 are negacyclic codes over Z3 of length n.

Proof. The proof of this theorem is similar to proof of Theorem 4.5.

Case 5. ϑ = ω

Theorem 11. Let K = ξ1K∞ ⊕ ξ2K∈ ⊕ ξ3K3 ⊕ ξ4K4 be a linear code over the
ring < of length n where K∞, K∈, K3, K4 are the linear codes over Z3. Then, K is
a ω-constacyclic code over the ring < of length n if and only if K∞, K∈ are cyclic and
K3, K4 are negacyclic codes over Z3 of length n.

Proof. Proof of the theorem is similar to proof of theorem 4.6.

The following Theorem is Similar to Theorem 14 [7].

Theorem 12. Let K be a ϑ-constacyclic code over the ring < of length n. Then

K = < ξ1g1(t), ξ2g2(t), ξ3g3(t), ξ4g4(t) > = < ξ1g1(t) + ξ2g2(t) + ξ3g3(t) + ξ4g4(t) >

where gi(t) are the generator polynomials of K∞, K∈, K3 and K4 for i = 1, 2, 3, 4 respec-

tively. Moreover, |K| = 34n−
∑4

i=0 deg(gi(t))

Theorem 13. Let K be a ϑ-constacyclic code over the ring < of length n. Then K⊥ is
also a ϑ-constacyclic code over the ring < of length n. Moreover,

1. K⊥ = ξ1K⊥∞ ⊕ ξ2K⊥∈ ⊕ ξ3K⊥3 ⊕ ξ4K⊥4
2. K⊥ = < ξ1g

?
1(t), ξ2g

?
2(t), ξ3g

?
3(t), ξ4g

?
4(t) > = < ξ1g

?
1(t) + ξ2g

?
2(t) + ξ3g

?
3(t) + ξ4g

?
4(t) >

3. | K⊥ | = 3
∑4

i=1 deg(gi(t))

where g?i (t) are the reciprocal polynomial of xn+1
g1(t)

, x
n+1
g2(t)

, x
n+1
g3(t)

and xn−1
g4(t)

for i = 1, 2, 3, 4
respectively.

Lemma 1. [4] If K is a cyclic or negacyclic code over the ring Zp with a generator
polynomial g(t). Then, K contains its dual code if and only if

xn − ι ≡ 0 mod(g(t)g?(t))

where ι = ±1.

Case 1. ϑ = 1 + ν + ω + 2νω

Theorem 14. If K = < ξ1g1(t) + ξ2g2(t) + ξ3g3(t) + ξ4g4(t) > is a ϑ-constacyclic code
over the ring < of length n. Then, K⊥ ⊆ K if and only if

xn + 1 ≡ 0 mod(gi(t)g
?
i (t))

and
xn − 1 ≡ 0 mod(gj(t)g

?
j (t)).

for i = 1, 2, 3 and j = 4.



Prateek Mor et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1082-1097 1091

Proof.
Let K = < g(t) > = < ξ1g1(t) + ξ2g2(t) + ξ3g3(t) + ξ4g4(t) > be a ϑ-constacyclic code

over < of length n. Then, K = ξ1K∞ ⊕ ξ2K∈ ⊕ ξ3K3 ⊕ ξ4K4 where gi(t) are the
generator polynomial of K∞, K∈, K3 and K4 for i = 1, 2, 3, 4 respectively.
First we consider

xn + 1 ≡ 0 mod(gi(t)g
?
i (t))

and
xn − 1 ≡ 0 mod(gj(t)g

?
j (t)).

for i = 1, 2, 3 and j = 4. Then by above lemma, we have

K⊥∞ ⊆ K∞, K⊥∈ ⊆ K∈,K⊥3 ⊆ K3 and K⊥4 ⊆ K4,

and therefore

ξ1K⊥∞ ⊆ ξ1K∞, ξ2K⊥∈ ⊆ ξ2K∈, ξ3K⊥3 ⊆ ξ3K3 and ξ4K⊥4 ⊆ ξ4K4

which implies that

ξ1K⊥∞ ⊕ ξ2K⊥∈ ⊕ ξ3K⊥3 ⊕ ξ4K⊥4 ⊆ ξ1K∞ ⊕ ξ2K∈ ⊕ ξ3K3 ⊕ ξ4K4

Thus, we have
K⊥ ⊆ K.

Conversely, let us consider
K⊥ ⊆ K,

then

ξ1K⊥∞ ⊕ ξ2K⊥∈ ⊕ ξ3K⊥3 ⊕ ξ4K⊥4 ⊆ ξ1K∞ ⊕ ξ2K∈ ⊕ ξ3K3 ⊕ ξ4K4,

which implies that

ξ1K⊥∞ ⊆ ξ1K∞, ξ2K⊥∈ ⊆ ξ2K∈, ξ3K⊥3 ⊆ ξ3K3 and ξ4K⊥4 ⊆ ξ4K4,

that implies
K⊥∞ ⊆ K∞, K⊥∈ ⊆ K∈, K⊥3 ⊆ K3 and K⊥4 ⊆ K4.

Then by above lemma,
xn + 1 ≡ 0 mod(gi(t)g

?
i (t))

and
xn − 1 ≡ 0 mod(gj(t)g

?
j (t)).

for i = 1, 2, 3 and j = 4.

Case 2. ϑ = 1 + ν + ω + 2νω



Prateek Mor et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1082-1097 1092

Theorem 15. If K = < ξ1g1(t) + ξ2g2(t) + ξ3g3(t) + ξ4g4(t) > is a ϑ-constacyclic code
over the ring < of length n. Then, K⊥ ⊆ K if and only if

xn − 1 ≡ 0 mod(gi(t)g
?
i (t))

and
xn + 1 ≡ 0 mod(gj(t)g

?
j (t)).

for i = 1 and j = 2, 3, 4.

Proof.
Let K = < g(t) > = < ξ1g1(t) + ξ2g2(t) + ξ3g3(t) + ξ4g4(t) > be a ϑ-constacyclic code

over < of length n. Then, K = ξ1K∞ ⊕ ξ2K∈ ⊕ ξ3K3 ⊕ ξ4K4 where gi(t) are the
generator polynomial of K∞, K∈, K3 and K4 for i = 1, 2, 3, 4 respectively.
First we consider

xn − 1 ≡ 0 mod(gi(t)g
?
i (t))

and
xn + 1 ≡ 0 mod(gj(t)g

?
j (t)).

for i = 1 and j = 2, 3, 4. Then by above lemma, we have

K⊥∞ ⊆ K∞, K⊥∈ ⊆ K∈,K⊥3 ⊆ K3 and K⊥4 ⊆ K4,

and therefore

ξ1K⊥∞ ⊆ ξ1K∞, ξ2K⊥∈ ⊆ ξ2K∈, ξ3K⊥3 ⊆ ξ3K3 and ξ4K⊥4 ⊆ ξ4K4

which implies that

ξ1K⊥∞ ⊕ ξ2K⊥∈ ⊕ ξ3K⊥3 ⊕ ξ4K⊥4 ⊆ ξ1K∞ ⊕ ξ2K∈ ⊕ ξ3K3 ⊕ ξ4K4

Thus, we have
K⊥ ⊆ K.

Conversely, let us consider
K⊥ ⊆ K,

then

ξ1K⊥∞ ⊕ ξ2K⊥∈ ⊕ ξ3K⊥3 ⊕ ξ4K⊥4 ⊆ ξ1K∞ ⊕ ξ2K∈ ⊕ ξ3K3 ⊕ ξ4K4,

which implies that

ξ1K⊥∞ ⊆ ξ1K∞, ξ2K⊥∈ ⊆ ξ2K∈, ξ3K⊥3 ⊆ ξ3K3 and ξ4K⊥4 ⊆ ξ4K4,

that implies
K⊥∞ ⊆ K∞, K⊥∈ ⊆ K∈, K⊥3 ⊆ K3 and K⊥4 ⊆ K4.



Prateek Mor et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1082-1097 1093

Then by above lemma,
xn − 1 ≡ 0 mod(gi(t)g

?
i (t))

and
xn + 1 ≡ 0 mod(gj(t)g

?
j (t)).

for i = 1 and j = 2, 3, 4.

Case 3. ϑ = νω

Theorem 16. If K = < ξ1g1(t) + ξ2g2(t) + ξ3g3(t) + ξ4g4(t) > is a ϑ-constacyclic code
over the ring < of length n. Then, K⊥ ⊆ K if and only if

xn − 1 ≡ 0 mod(gi(t)g
?
i (t))

and
xn + 1 ≡ 0 mod(gj(t)g

?
j (t)).

for i = 1, 4 and j = 2, 3.

Proof. Proof of the theorem is similar to proof of Theorem 4.12.

Case 4. ϑ = ν

Theorem 17. If K = < ξ1g1(t) + ξ2g2(t) + ξ3g3(t) + ξ4g4(t) > is a ϑ-constacyclic code
over the ring < of length n. Then, K⊥ ⊆ K if and only if

xn − 1 ≡ 0 mod(gi(t)g
?
i (t))

and
xn + 1 ≡ 0 mod(gj(t)g

?
j (t)).

for i = 1, 3 and j = 2, 4.

Proof. Proof of the theorem is similar to proof of theorem 4.13.

Case 5. ϑ = ω

Theorem 18. If K = < ξ1g1(t) + ξ2g2(t) + ξ3g3(t) + ξ4g4(t) > is a ϑ-constacyclic code
over the ring < of length n. Then, K⊥ ⊆ K if and only if

xn − 1 ≡ 0 mod(gi(t)g
?
i (t))

and
xn + 1 ≡ 0 mod(gj(t)g

?
j (t)).

for i = 1, 2 and j = 3, 4.

Proof. Proof of the theorem is similar to proof of theorem 4.14.

By above Theorems, we have the following Corollary for each case of ϑ.



Prateek Mor et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1082-1097 1094

Corollary 19. Let K= ξ1K∞ ⊕ ξ2K∈ ⊕ ξ3K3 ⊕ ξ4K4 be a ϑ-constacyclic code over <
of length n where K∞, K∈, K3, K4 are linear codes of length n over the ring Z3. Then,
K⊥ ⊆ K if and only if K⊥∞ ⊆ K∞, K⊥∈ ⊆ K∈, K⊥3 ⊆ K3 and K⊥4 ⊆ K4.

Lemma 2. [4](CSS Construction) Let K be a linear code over the ring Z3 having pa-
rameters [n, k, d]. Then a quantum code having parameters [[n, 2k − n, ≥ d]]3 can be
obtained if K⊥ ⊆ K.

The following theorem defines the construction of quantum codes by the use of corollary
4.16 and Lemma 4.17.

Theorem 20. If K= ξ1K∞ ⊕ ξ2K∈ ⊕ ξ3K3 ⊕ ξ4K4 = < ξ1g1(t) + ξ2g2(t) + ξ3g3(t) +
ξ4g4(t) > is a ϑ-constacyclic code over the ring < of length n where gi(t) are the generator
polynomials of K∞, K∈, K3 and K4 for i = 1, 2, 3, 4 respectively. If K⊥∞ ⊆ K∞, K⊥∈ ⊆
K∈, K⊥3 ⊆ K3 and K⊥4 ⊆ K4, then K⊥ ⊆ K and there exists a quantum code having
parameters [4n, 2k − 4n, ≥ dL]3 where k is the dimension of linear code ϕ(K) and dL is
minimum lee distance of a linear code K.

5. Examples

In this section some examples are provided to illustrate the main result. Here, the quan-
tum codes through ϑ-constacyclic code over the ring < = Z3 + νZ3 + ωZ3 + νωZ3 where
ν2 = 1, ω2 = 1 and νω = ων are obtains.

Example 1. In Z3[t], t
3 − 1 = (t + 2)3 and t3 + 1 = (t + 1)(t2 − t + 1). Now, let K be

a 1 + ν + ω + 2νω-constacyclic code over the ring < = Z3 + νZ3 + ωZ3 + νωZ3 where
ν2 = 1, ω2 = 1 and νω = ων of length 3. Let g1(t) = g2(t) = g3(t) = t + 1 and
g4(t) = t2+t+1 then g(t) = ξ1(t+1)+ξ2(t+1)+ξ3(t+1)+ξ4(t

2+t+1) be the generator
polynomial of K. Since gi(t)g

∗
i (t)|t3 + 1 for i = 1, 2, 3 respectively and g4(t)g

∗
4(t)|t3 − 1,

then by the use of Theorem 4.11, we get K⊥ ⊆ K Further ϕ(K) is a linear code over the
ring Z3 having parameters [12, 7, 3]. Then, by the application of Theorem 4.18, we obtain
the quantum code having parameters [12, 2, ≥ 3]3.

Example 2. In Z3[t], t
6 − 1 = (t − 1)3(t + 1)3 and t6 + 1 = (t2 + 1)3. Now, let K be a

1+2ν+2ω+2νω-constacyclic code over the ring < = Z3+νZ3+ωZ3+νωZ3 where ν2 = 1,
ω2 = 1 and νω = ων of length 6. Let g1(t) = t+ 1 and g2(t) = g3(t) = g4(t) = t2 + 1
then g(t) = ξ1(t+ 1) + ξ2(t

2 + 1) + ξ3(t
2 + 1) + ξ4(t

2 + 1) be the generator polynomial of
K. Since g1(t)g

∗
1(t)|t6− 1 and gi(t)g

∗
i (t)|t6 + 1 for i = 2, 3, 4 respectively, then by the use

of Theorem 4.12, we get K⊥ ⊆ K Further ϕ(K) is a linear code over the ring Z3 having
parameters [24, 17, 3]. Then, by the application of Theorem 4.18, we obtain the quantum
code having parameters [24, 10, ≥ 3]3.

Example 3. In Z3[t], t
9 − 1 = (t − 1)9 and t9 + 1 = (t + 1)9. Now, let K be a νω-

constacyclic code over the ring < = Z3 + νZ3 + ωZ3 + νωZ3 where ν2 = 1, ω2 = 1 and



Prateek Mor et al. / Eur. J. Pure Appl. Math, 14 (3) (2021), 1082-1097 1095

νω = ων of length 9. Let g1(t) = g4(t) = t − 1 and g2(t) = g3(t) = t + 1 then
g(t) = ξ1(t− 1) + ξ2(t+ 1) + ξ3(t+ 1) + ξ4(t− 1) be the generator polynomial of K. Since
gi(t)g

∗
i (t)|t9 − 1 for i = 1, 4 respectively and gj(t)g

∗
j (t)|t9 + 1 for j = 2, 3 respectively,

then by the use of Theorem 4.13, we get K⊥ ⊆ K Further ϕ(K) is a linear code over the
ring Z3 having parameters [36, 32, 2]. Then, by the application of Theorem 4.18, we
obtain the quantum code having parameters [36, 28, ≥ 2]3.

Example 4. In Z3[t], t
12−1 = (t+1)3(t+2)3(t2+1)3 and t12+1 = (t2+2t+2)3(t2+t+2)3.

Now, let K be a ν-constacyclic code over < = Z3 + νZ3 + ωZ3 + νωZ3 where ν2 = 1,
ω2 = 1 and νω = ων and ν2ω2 = νω of length 12. Let g1(t) = g3(t) = t + 1 and
g2(t) = g4(t) = t2 + t+ 2, g(t) = ξ1(t+ 1) + ξ2(t

2 + t+ 2) + ξ3(t+ 1) + ξ4(t
2 + t+ 2)

be the generator polynomials of K. Since gi(t)g
∗
i (t)|t12 − 1 for i = 1, 3 respectively

and gj(t)g
∗
j (t)|t12 + 1 for j = 2, 4 respectively, then by the use of Theorem 4.14, we

get K⊥ ⊆ K. Further ϕ(K) is a linear code over Z3 having parameters [48, 43, 3].
Then, by the application of Theorem 4.18, we obtain the quantum code having parameters
[48, 38, ≥ 3]3.

Example 5. In Z3[t], t
15−1 = (t+2)3(t4+t3+t2+t+1)3 and t15+1 = (t+1)3(t4+2t3+

t2 + 2t+ 1)3. Now, let K be a ω-constacyclic code over < = Z3 +νZ3 +ωZ3 +νωZ3 where
ν2 = 1, ω2 = 1 and νω = ων and ν2ω2 = νω of length 15. Let g1(t) = g2(t) = t+ 2 and
g3(t) = g4(t) = t+ 1, g(t) = ξ1(t+ 2) + ξ2(t+ 2) + ξ3(t+ 1) + ξ4(t+ 1) be the generator
polynomials of K.Since gi(t)g∗i (t)|t15−1 for i = 1, 2 respectively and gj(t)g

∗
j (t)|t15 +1 for

j = 3, 4 respectively, then by the use of Theorem 4.15, we get K⊥ ⊆ K. Further ϕ(K) is
a linear code over Z3 having parameters [60, 56, 2]. Then, by the application of Theorem
4.18, we obtain the quantum code having parameters [60, 52, ≥ 2]3.

Example 6. In Z3[t], t
20 − 1 = (t+ 1)(t+ 2)(t2 + 1)(t4 + t3 + 2t+ 1)(t4 + t3 + t2 + t+

1)(t4 + 2t3 + t+ 1)(t4 + 2t3 + t2 + 2t+ 1) and t20 + 1 = (t2 + t+ 2)(t2 + 2t+ 2)(t4 + t2 +
t+ 1)(t4 + t2 + 2t+ 1)(t4 + t3 + t2 + 1)(t4 + 2t3 + t2 + 1). Now, let K be a 1 + ν +ω+ 2νω-
constacyclic code over < = Z3 + νZ3 + ωZ3 + νωZ3 where ν2 = 1, ω2 = 1 and νω = ων
and ν2ω2 = νω of length 20. Let g1(t) = g2(t) = g3(t) = t2 + t+ 2 and g4(t) = t+ 2,
g(t) = ξ1(t

2 + t+2)+ξ2(t
2 + t+2)+ξ3(t

2 + t+2)+ξ4(t+2) be the generator polynomials
of K. Since gi(t)g

∗
i (t)|t20 + 1 for i = 1, 2, 3 respectively and g4(t)g

∗
4(t)|t20 − 1, then by

the use of Theorem 4.11, we get K⊥ ⊆ K. Further ϕ(K) is a linear code over Z3 having
parameters [80, 73, 3]. Then, by the application of Theorem 4.18, we obtain the quantum
code having parameters [80, 66, ≥ 3]3.

6. Conclusion

In this work, we have given a construction of quantum code through ϑ-constacyclic code
over the finite non-chain ring < = Z3 + νZ3 + ωZ3 + νωZ3 where ν2 = 1, ω2 = 1 and
νω = ων and ν2ω2 = νω. We have derived self-orthogonal code over the ring Z3 as Gray
images of linear code over the ring Z3 + νZ3 +ωZ3 + νωZ3. In particular, the parameters
of quantum code over the ring Z3 are obtained by decomposing ϑ-constacyclic code into



REFERENCES 1096

cyclic and negacyclic codes over the ring Z3. For the future scope, one can look at other
classes of constacyclic codes over Z3 + νZ3 + ωZ3 + νωZ3 and Zp + νZp + ωZp + νωZp.

Acknowledgements

The authors are grateful to the anonymous referees for useful comments and sugges-
tions.

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