On Quantum Codes over Non Local Rings
NOUREDDINE ESSAIDI1, ABDELHAMID TADMORI1, OSSAMA EL ABOUTI2
1Department of Mathematics and Informatics,
University Abdelmalek Essaadi, Faculty of Sciences and Technology Al Hoceima
BP 34. Ajdir 32003 Al Hoceima
M252&&2
2Department of Physics,
University Abdelmalek Essaadi, Faculty of Sciences and Technology Al Hoceima
BP 34. Ajdir 32003 Al Hoceima
M252&&2
Abstract: We study the structural properties of the ring R=Fp+uFp+u2Fp+u3Fp, where p6= 2 is a prime
and u4=u3, as well as the linear codes over R. We investigate the generator’s cyclic codes and their dual codes
over R. An isometric Gray map from Rto F4
pis defined. We offer an equivalence condition that cyclic codes must
satisfy over Rin order to include their dual. Moreover, we establish the existence of quantum error-correcting
codes based on cyclic codes over R. Finally, under various criteria such as cyclic codes length and generator
polynomial degree, we build quantum error-correcting codes over Rwhich their dimensions are divided by pand
24.
Key-Words: - Gray map, cyclic codes, non local finite rings, self-dual codes, Quantum codes
Received: May 16, 2022. Revised: August 18, 2023. Accepted: October 17, 2023. Available online: December 5, 2023.
1 Introduction
Over the past few years, quantum error-correcting
codes have attracted a great deal of curiosity from re-
searchers, especially over finite rings. Without losing
meaning, we will denote them by QECC. A signifi-
cant benefit of these codes is their exceptional adapt-
ability to quantum physical systems of any order. In
addition, finite rings make it less difficult to perform
operations. The conventional error-correcting codes
are necessary to prevent decoherence and other noise
from destroying the classical information, but such in-
formation may also be duplicated. Similarly, the exis-
tence of QECC supplies a powerful manner to avoid
decoherence and other quantum noise during quan-
tum communication and quantum computation. We
suggest the reader to [1], [2] and [3], for further infor-
mation on information theory and coding theory over
finite rings.
Cyclic codes have shown to be an excellent resource
for developing QECC with appropriate parameters.
In [4], the author led the development of the first
QECC. Motivated by this discovery, the researcher,
[5], gives a method of construction QECC over fi-
nite field Fq, with hypothesis that qis a prime num-
ber power. After that, the authors, [6], proposed a
technique for building QECC based on conventional
error-correcting codes. Recently, QECC theory has
advanced quickly. Many QECC were created by var-
ious researchers, [7], [8], [9], [10], [11], using clas-
sical codes with good parameters and properties of
self-orthogonal or dual. Researchers have shown a
strong interest in investigating the built of QECC over
finite rings by using cyclic codes. Across a variety
of finite rings, several QECC constructed from cyclic
codes have been developed, more precisely, over fi-
nite non-chain ring. In [12], the author worked over
Fp[v]/(v2−v)to construct linear codes. Later on, the
paper, [13], examined the linear codes structure over
the finite non-chain ring Fp[u]/(u3−u), where p > 2
is a prime, the building of QECC over Fq[v]/(v4−v)
via cyclic codes, with pis assumed to be an odd prime,
(p−1) is divisible by 3and q=pr, is given in
[14]. Some new QECC based on cyclic codes over
the finite ring F2[u, v]/(u2−u, v2−v, uv −vu)
are presented by the authors of the paper, [15], later
on, they generalized in [16], the creating of QECC
over F2[v1, v2, . . . , vr]/(v2
i−vi, vivj−vjvi)where
1≤i, j≤rfor r≥1, by using cyclic codes. The
building of QECC based on cyclic codes of length not
divisible by two over the chain ring F2[u]/(u2)is pro-
vided in [17]. Further, [18], provided a building of
QECC using cyclic codes of length not divisible by
two over the ring F4[u]/(u2). Again, a building of
QECC over F2[v]/(v2−v)by utilizing cyclic codes,
was established in [19].
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The theory of QECC was further advanced by sev-
eral scientists, whose provided the building of QECC
over the finite non-chain ring F3[u]/(u2−1) bas-
ing on cyclic codes in [20]. Then, over the ring
Fp[v]/(v2−v), they examined QECC derived from
cyclic codes and they established the construction
new non-binary QECC over the ring Fq[u, v]/(u2−
u, v2−v, uv −vu)in [21], [22], respectively. The au-
thors in [23], used cyclic codes that satisfy the condi-
tion of dual-containing, to produce novel QECC over
the ring F2m[u]/(uk+1), where m > 0is an inte-
ger. The creation of QECC over Fpm[u]/(u2), where
pis a prime, from linear codes is presented in [24].
Based on this studies, [25], introduced several new
non-binary QECC over the ring Fq[v]/(v3−v), with
assumption that q=prand p > 2is a prime.
Researchers have recently focused on the structural
characteristics of codes over mixed alphabets (the di-
rect product of finite rings). [26], discovered QECC
and LCD codes using mixed alphabets. QECC over
mixed alphabets was constructed by [27]. The re-
searchers , [28], provide non-binary QECC across
mixed alphabets. Inspired by these studies, we inves-
tigate the creation of QECC employing cyclic codes
over the ring R.
This paper’s remaining sections are structured as be-
low: In sect. 2, fundamental facts about the ring Rare
presented. In sect. 3, an equivalent condition of self-
duality verified by a linear code, is provided, along
with some helpful results on linear codes over this
ring. In sect. 4, the definition of the Gray map is intro-
duced and a method for representing codes that equal
their dual over Fpto be the images of linear codes by
this map over R, is provided. In sect. 5, cyclic codes’
characterisation over the ring Ris covered, where we
also provide a condition that is equivalent to dual con-
taining verified by cyclic codes over R. In sect. 6,
we provide the characteristics of a QECC basing on
cyclic codes over R.
2 Preliminaries
In this work, consider a prime pwhere p6= 2 and Fp
represent a finite field, the ring Fp[u] = Fp+uFp+
u2Fp+u3Fp[u]is denoted by Rwhere uis an inde-
terminate with u4=u3.
The following facts provide some fundamental char-
acteristics of R, which will be employed in the parts
that follow:
1. For any element α∈ R, there exist δ,σ,ρ, λ ∈
Fp,so we can represent αas α=δ+σu +ρu2+
λu3.
2. Recall that Rand Fp[X]/(X4−X3)are iso-
morphic as rings. Besides that, the finiteness and
commutativity are verified by the ring R, further-
more, it has identity and characteristic equals p.
To prove this result, we need to construct a bijec-
tive ring homomorphism between them. In fact,
we consider the map:
ϑ:Fp[X]−→ Fp[u]
P7−→ P(u).
The fact that ϑis a surjective homomorphism is
obvious. It is still necessary to demonstrate that
the kernel of ϑis the ideal (X4−X3).
From the fact that usatisfies u4=u3, it follows
immediately that
(X4−X3)⊆ker(ϑ).
On other hand, Let P∈Fp[X]such that ϑ(P) =
0in R. Then,
0 = ϑ(P) = P(u)which implies that Pis
divisible by (X4−X3). Therefore, ker(ϑ) =
(X4−X3).
Thus, by the isomorphism theorem for rings, we
have
R ' Fp[u],
where u4=u3.
3. For any element α=δ+σu +ρu2+λu3of R,
we have
α is unit ⇐⇒ δ6= 0,
δ+σ+ρ+λ6≡ 0(mod p)
Moreover, from [13], we have | R×|= (p−1)2,
where R×represents the group of units of R.
4. Let α∈ R, then
α∈ R\R×⇐⇒
α=σu +ρu2+λu3
or
α=δ+σu +ρu2−(δ+σ+ρ)u3
where (δ, σ, ρ, λ)∈F4
p.Basing on this fact, Ris
a semi-local ring.
Indeed, consider I1= (σu +ρu2+λu3)and
I2= (δ+σu +ρu2−(δ+σ+ρ)u3)to be two
ideals of R.
Since I16=Rand I26=R, then it is sufficient
to show that I1∪I2is not an ideal.
It is obvious that the elements of I1∪I2are non
invertible elements in R.
Let consider δ, σ, ρ, a, b, c ∈Fp,then we obtain
δ+σu +ρu2−(δ+σ+ρ)u3=au +bu2+cu3
⇒δ+(σ−a)u+(ρ−b)u2−(δ+σ+ρ+c)u3= 0
⇒
δ= 0
σ−a= 0
ρ−b= 0
δ+σ+ρ+c= 0
⇒
δ= 0
σ=a
ρ=b
σ+ρ=−c
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As a result, we have I1∩I2= (au +bu2−cu3).
therefore I1∪I2is not an ideal.
Consequently, I1and I2are two maximal ideals
of R. Then, we obtain the result.
5. According to ring theory, a commutative chain
ring is a ring verifying that its ideals create a
unique chain below the inclusion relation. Based
on the previous result, we can see that the ide-
als of Rdo not create a chain because I1and I2
are incomparable, implying that Ris not a chain
ring.
6. The ring A=Fp[X]/(X4−X3)is isomorphic
to the direct product rings
A ' A1× A2,
where A1=Fpand A2=Fp[v]/(v3).
7. Let consider the following mapping:
π1:A −→ A1=Fp
α=δ+σu +ρu2+λu37−→ δ+σ+ρ+λ
and
π2:A −→ A2
α=δ+σu +ρu2+λu37−→ δ+σv +ρv2,
where v3= 0, then π1and π2are the surjective
morphisms of rings.
8. Let R1and R2be two rings, such that
R1={u3.α |α∈ A}
and
R2={δ+σu+ρu2−(δ+σ+ρ)u3|(δ, σ, ρ)∈F3
p}.
The following mapping:
ξ1:R1−→ A1
u3.α 7−→ δ+σ+ρ+λ,
and
ξ2:R1−→ A2
δ+σu +ρu2−(δ+σ+ρ)u37−→ δ+σv +ρv2,
are the isomorphisms of rings.
9. From the facts above, we can deduce that
R ' Fp×(Fp+vFp+v2Fp),
where v3= 0.
3 Linear codes over R
From coding theory, a code Cthat it is linear and
its length equals nover R, is characterized as an
R-submodule of Rn. A codeword is every element
w∈C.
Let w= (w0,w1, . . . , wn−1)∈C,
• The Hamming weight of wis known to be
wtH(w) =| {i|wi6= 0} |, for 0≤i≤n−1.
• For an element α=δ+σu +ρu2+λu3∈
R, its Lee weight can be given as wtL(α) =
wtH(δ, σ −ρ, σ +ρ−λ, δ −σ+ρ+λ), where
wtH(γ)is the Hamming weight of γ= (δ, σ −
ρ, σ +ρ−λ, δ −σ+ρ+λ)over Fp.
• From previous definition, it can easily define the
Lee weight of a vector z= (z0,z1, . . . , zn−1)∈
Rnto be
wtL(z) =
n−1
X
i=0
wtL(zi),
where wtL(zi) = wtL(δi, σi−ρi, σi+ρi−λi, δi−
σi+ρi+λi)for 0≤i≤n−1.
• The number of places where two codewords x=
(x0,x1, . . . , xn−1)and y= (y0,y1, . . . , yn−1)
are different is called the Hamming distance, i.e
dH(x,y) =| {i|xi6=yi} |.
• The Lee distance is provided by dL(x,y) =
wtL(x−y), where xand yare elements of Rn.
• The minimum Hamming distance of Cis d(C) =
min{wtH(w)|06=w∈C}.
• The smallest dL(x,y)6= 0 present the minimum
Lee distance of a code C, where x,y∈C. The
minimum Lee weight, on the other hand, is the
codeword with the least nonzero Lee weight.
Given that Cis linear, it result that there is equality
between the minimum Lee distance and the minimum
Lee weight.
Recall that
<x,y>Rn=
n
X
i=1
xiyi,
is the definition of the Euclidean inner product of
two components x= (x1,x2, . . . , xn)and y=
(y1,y2, . . . , yn)in Rn.
If <x,y>Rnequals zero, implies that they are or-
thogonal.
The set
C⊥={x∈ Rn|<x,y>Rn= 0,∀y∈C},
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represents the dual of C.
The condition C⊂C⊥implies that Csatisfy the self-
orthogonality and verifying the self-duality if there is
equality.
Now, consider a code Cover the product ring Fp×
(Fp+vFp+v2Fp)with assumption that it is linear and
its length is n. Hence, Cis represented to be (C1,C2),
where the codes C1and C2are considered over the
rings Fpand Fp+vFp+v2Fp, respectively. Moreover,
they are linear and their length is n.
Furthermore, C1and C2are expressed in the manner
that follow:
C1={µ∈Fn
p|(µ, 0) ∈C}
and
C2={ν∈(Fp+vFp+v2Fp)n|(0, ν)∈C}.
As a result, Cis presented by the direct sum of C1and
C2, with denotation C=C1⊕C2, and each codeword
in Cis uniquely written as
w= (r,s+tv+lv2),
where r,s,t,l∈Fn
p.
The generator matrices G1and G2are assumed cor-
responding to C1and C2,respectively. Since Cis a
R-module, hence, the matrix Ggenerating Cis repre-
sented as follows
G=G1
G2.(1)
Lemma 1 Given a code C=C1⊕C2that supposed
linear and its length is nover R. Then,
C⊥=C⊥
1⊕C⊥
2.
In addition, the next propositions are equivalents:
1. Csatisfy the self-orthogonality over R.
2. The self-orthogonality is verified respectively by
C1and C2over the rings Fpand Fp+vFp+v2Fp.
Proof 1 Suppose that C1satisfy the self-
orthogonality over Fpi.e C1⊆C⊥
1and satisfied by
C2over Fp+vFp+v2Fpi.e C2⊆C⊥
2. Since C
is expressed as (C1,C2), moreover, C1and C2are
linear , then C⊆C⊥.
Conversely, let w= (r,s+tv+lv2)∈C, assum-
ing the self-orthogonality of Cover R, we get via
Euclidian inner product:
<w,w>Rn=<(r,s+tv+lv2),(r,s+tv+lv2)>Rn
=<< r,r>Fn
p, < s+tv+lv2,s+tv+lv2>(Fp+vFp+v2Fp)n>
= 0.
Hence, <r,r>Fn
p= 0 and <s+tv+lv2,s+tv+
lv2>(Fp+vFp+v2Fp)n= 0, which implies that r∈C⊥
1,
and s+tv+lv2∈C⊥
1. Thus, C1⊆C⊥
1and C2⊆C⊥
2.
From literature, we recall the definition of the dimen-
sion of Cover Ras follow:
The dimension of Cis the maximum number of lin-
early independent codewords in C. In other words
|C|=max{|F|;F ⊆ C},
where F={w1,w2, . . . , wj}is a set of linearly in-
dependent codewords of C.
4 Gray map over R
From section 2, every element α∈ R is represented
by the expression α=δ+σu +ρu2+λu3, where
δ, σ, ρ, λ ∈Fp. The following map is known as Gray
map on R:
Γ : R −→ F4
p
α7−→ (δ, σ −ρ, σ +ρ−λ, δ −σ+ρ+λ).
Clearly, Γis linear and an Fp-module isomorphism.
Similarly, the Gray map Γis extended naturally to Rn
by the following manner:
Γ : Rn−→ F4n
p
(α0, α1, . . . , αn−1)7−→ (β0, β1, . . . , βn−1),
where αi=δi+σiu+ρiu2+λiu3and βi=
(δi, σi−ρi, σi+ρi−λi, δi−σi+ρi+λi)for
i= 0,1, . . . , n −1.
In the same way that the matrix generating Cover R
is given in (1). As a R-module isomorphism, the fol-
lowing matrix generate Γ(C)(Gray image of C) as be-
low:
Γ(G) = Γ(G1)
Γ(G2).
According to the definition of Γon Rnthe following
facts are evident:
Lemma 2 [13] Γis an isometry from Rn(Lee dis-
tance) to F4n
p(Hamming distance). Moreover, Γis
Fp-linear.
Lemma 3 [13, 21] Consider a code Cthat supposed
linear and characterized by length n, dimension |C| =
pkand the minimum Lee distance dLover R. It re-
sult that the code Γ(C)is linear and parameterized by
length 4n, dimension kand the minimum Hamming
distance dHover Fp, where dH=dL.
Proof 2 Clearly, Γ(C)is a linear code over Fp, ac-
cording to Fp-linearity of Γfrom lemma 2. Further-
more, the construction of Γimplies that Γ(C)is an
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element of F4n
pthen its length is 4n. It is evident that
Γis a bijective map from Rnto F4n
p, hence, the di-
mension of Γ(C)equals vp(|C|) = k, where vpis the
p-adic valuation.
From the above lemma, the preserving distance of Γ
ensure that the minimum distance of Γ(C)is dH=dL.
Theorem 1 [13] Let a code Cthat supposed linear
over Rand characterized as in Lemma 3. Then
Csatisfy the self −duality over R
⇓
Γ(C)satisfy the self −duality over Fp.
Moreover, the dual of Γ(C)equals Γ(C⊥).
5 Cyclic codes over R
Now, we introduce a few essential structural facts
about cyclic codes over R, that will be useful in build-
ing of the appropriate QECC.
A code Cthat it is linear and its length is nover Ris
known as cyclic if it satisfies the following condition:
for each codeword w= (w0,w1, . . . , wn−1)∈C, the
codeword b
w= (wn−1,w0, . . . , wn−2)∈C.
From literature, the fact that Cis a cyclic code of
length nover Requivalent to Cis viewed as an ideal
in the polynomial ring e
R=R[ε]/(εn−1) by the
following R-module isomorphism:
ϕ:Rn→e
R=R[ε]/(εn−1)
w7→ w0+w1ε+. . . +wn−1εn−1+ (εn−1)
In fact, it is sufficient that we write each w=
(w0,w1, . . . , wn−1)∈Cas polynomial w(ε) =
w0+w1ε+. . . +wn−1εn−1∈ R[ε], which called
the associated polynomial of C. Then, we can write
b
w= (wn−1,w0, . . . , wn−2)as b
w(ε) = wn−1+
w0ε+. . . +wn−2εn−1∈ R[ε]and we obtain b
w(ε) =
w(ε)ε−wn−1(εn−1), it results that
b
w(ε)≡w(ε)ε mod (εn−1).
Furthermore, it is easily seen that
w(ε)∈Cmod(εn−1) ⇐⇒ w(ε)ε∈Cmod(εn−1).
We repeat this procedure, we get
w(ε)ε∈Cmod(εn−1) ⇐⇒ w(ε)ε2∈Cmod(εn−1).
By induction steps, it results that w(ε)εi∈
Cmod (εn−1), for all i∈N.
This leads us to the conclusion that a code Cwith as-
sumption that it is linear and its length is n, is consid-
ered cyclic over Requivalent to ϕ(C)is an ideal of
e
R.
The following results are required for the next part.
Lemma 4 Consider a code C=C1⊕C2with as-
sumption that it is linear and its length is nover R.
Hence, the following equivalence holds:
Cis cyclic over R
m
T he codes C1and C2are cyclic over the
rings Fpand Fp+vFp+v2Fp, resp.
Proof 3 Let (r0,r1, . . . , rn−1)∈C1and (s0+t0v+
l0v2,s1+t1v+l1v2, . . . , sn−1+tn−1v+ln−1v2)∈C2.
Assume that the cyclicity is verified by C1over Fpand
by C2over Fp+vFp+v2Fp. We consider an el-
ement w= (w0,w1, . . . , wn−1)of C, where wi=
(ri,si+tiv+liv2), for i= 0,1, . . . , n −1. We de-
duce that
(wn−1,w0, . . . , wn−2)=(rn−1,r0, . . . , rn−2) +
(sn−1+tn−1v+ln−1v2,s0+t0v+l0v2, . . . , sn−2+
tn−2v+ln−2v2).
Since (rn−1,r0, . . . , rn−2)∈C1and (sn−1+tn−1v+
ln−1v2,s0+t0v+l0v2, . . . , sn−2+tn−2v+ln−2v2)∈
C2, then
(wn−1,w0, . . . , wn−2)∈C1⊕C2=C,
as a result, Cis a cyclic code over R.
On other hand, suppose that wi= (ri,si+tiv+liv2),
for i= 0,1, . . . , n −1. Then (w0,w1, . . . , wn−1)is
an element of C. According to the hypothesis, Cis a
cyclic code over R, hence (wn−1,w0, . . . , wn−2)∈
C. Furthermore,
(wn−1,w0, . . . , wn−2)=(rn−1,r0, . . . , rn−2) +
(sn−1+tn−1v+ln−1v2,s0+t0v+l0v2, . . . , sn−2+
tn−2v+ln−2v2).
It results that (rn−1,r0, . . . , rn−2)∈C1and (sn−1+
tn−1v+ln−1v2,s0+t0v+l0v2, . . . , sn−2+tn−2v+
ln−2v2)∈C2, which ensure the result.
In the following parts, we set a code C=C1⊕C2over
Rthat supposed cyclic and its length equals n. The
result below gives the polynomial generating Cover
R.
Theorem 2 A unique polynomial χ(ε)∈ R[ε]exists
and generating Cas:
C= (χ(ε))
= (χ1(ε), χ2(ε) + vχ3(ε) + v2χ4(ε)),
where the polynomial χ1(ε)generate C1over Fpand
the polynomial χ2(ε)+vχ3(ε)+v2χ4(ε)generate C2
over Fp+vFp+v2Fp. Additionally, εn−1is divided
by χ(ε)over R.
Proof 4 The existence and uniqueness of χ(ε)are en-
sured by that of χ1(ε), χ2(ε), χ3(ε)and χ4(ε).
In fact, we know there are unique polynomials χ1(ε),
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χ2(ε), χ3(ε)and χ4(ε)in Fp[ε]such that C1=
(χ1(ε)) and C2= (χ2(ε) + vχ3(ε) + v2χ4(ε)) where
χi(ε)divides the polynomial εn−1in Fp[ε]for i=
1,2,3,4and χ2(ε) + vχ3(ε) + v2χ4(ε)divides the
polynomial εn−1in (Fp+vFp+v2Fp)[ε]. Hence
C= (C1,C2)
= (χ1(ε), χ2(ε) + vχ3(ε) + v2χ4(ε))
= (χ(ε)),
In conclusion, χ(ε)divides εn−1over R.
Remark 1 The fact that C=C1⊕C2and C1=
(χ1(ε)) and C2= (χ2(ε)+vχ3(ε)+v2χ4(ε)), allows
us to note that:
1.
|C|=|C1|| C2|
=pn−deg(χ1(ε))p3n−deg(χ2(ε))−deg(χ3(ε))−deg(χ4(ε))
=p4n−deg(χ1(ε))−deg(χ2(ε))−deg(χ3(ε))−deg(χ4(ε))
2. From the existence and uniqueness of χ(ε), we
can deduce that every ideal of e
Ris principal, e
R
is principal.
Recall that, the polynomial:
φ∗(ε) = εdeg(φ(ε))φ(ε−1),
present the reciprocal of the polynomial φ(ε) = e0+
e1ε+. . . +emεm,where ei∈R for 0≤i≤m.
Moreover, φ(ε)is called self-reciprocal if φ(ε) =
φ∗(ε).
Corollary 1 [13] There exist polynomials φi(ε)di-
vides εn−1, i.e. φi(ε)χi(ε) = εn−1in Fp[ε], for
i= 1,2,3,4, such that
C⊥= (φ(ε))
= (φ∗
1(ε), φ∗
2(ε) + vφ∗
3(ε) + v2φ∗
4(ε)),
and |C⊥|=pdeg(χ1(ε))+deg(χ2(ε))+deg(χ3(ε))+deg(χ4(ε)),
where, φ∗
i(ε)is the reciprocal polynomials of φi(ε),
for i= 1,2,3,4.
6 QECC from cyclic codes over R
We start this part by presenting the CSS construction,
which is a fundamental structure of QECC and was
presented by Calderbank, Shor and Steane. Next, we
give our contribution regarding QECC over R.
The following arguments explain why creating QECC
from cyclic codes over Ris preferable: The ring R
has some similar characteristics as the finite field Fp.
Furthermore, the ring Rmay be used to generate op-
timal cyclic codes. Since every ideal over ˜
Ris princi-
pal, QECC of any length may be simply created. The
number of cyclic codes over Rfor a particular length
nis substantially more than those over finite field Fp.
In addition, cyclic codes over Rcan result in good
QECC. We anticipate that cyclic codes over Rwill
be an excellent source for creating excellent QECC.
Lemma 5 [3] Consider two codes C1and C2,with
assumption that they are linear over Fp, where p
is a prime and parameterized by [n, k1, d1]pand
[n, k2, d2]p, respectively, and satisfying the condition
C⊥
2⊆C1. Moreover, let d=min{wt(x)|x∈
(C1\C⊥
2)∪(C2\C⊥
1)}with d≥min{d1, d2}.
It result, the existence of a QECC over Fpis guar-
anteed and parameterized by length n, dimension
k1+k2−nand minimum distance d.
Furthermore, if C⊥
1⊆C1, hence there exist a QECC
over Fqcharacterized by length n, dimension n−2k1
and minimum distance d1,where d1=min{wt(x)|
x∈(C⊥
1\C1)}.
Calderbank and colleagues have provided an es-
sential conclusion that establishes the equivalence
condition verified by cyclic codes over finite fields
of dual containing as follows:
Lemma 6 [6] Consider a cyclic code C1over the fi-
nite field Fpand generated by the polynomial χ1(ε).
Then,
C⊥
1⊆C1⇐⇒ εn−1≡0 (mod χ1(ε)χ∗
1(ε)),
where χ∗
1(ε)is the reciprocal polynomial of χ1(ε).
In a similar manner, we can derive the following con-
clusion:
Lemma 7 Consider a cyclic code C2over Fp+
vFp+v2Fpand generated by the polynomial χ2(ε) +
vχ3(ε) + v2χ4(ε). Then,
C⊥
2⊆C2⇐⇒ εn−1≡0 (mod χi(ε)χ∗
i(ε)),
where χ∗
i(ε)is the reciprocal polynomial of χi(ε)for
i= 2,3,4.
Proof 5 The proof can be given by using the method
in [6].
Using the fact that C=C1⊕C2, the next theorem
provides an equivalence condition satisfied by cyclic
code of dual containing over R.
Theorem 3 Let Cgenerated by the polynomial χ(ε).
Then the following equivalence is established,
C⊥⊆C⇐⇒
εn−1≡0 (mod χ1(ε)χ∗
1(ε))
and
εn−1≡0 (mod χ2(ε)χ∗
2(ε))
and
εn−1≡0 (mod χ3(ε)χ∗
3(ε))
and
εn−1≡0 (mod χ4(ε)χ∗
4(ε)),
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where χ∗
i(ε)is the reciprocal polynomial of χi(ε)for
i= 1,2,3,4.
Proof 6 Via using Theorem 2, Lemma 6 and Lemma
7, the proof can be easily obtained.
Definition 1 A QECC Qover Rof length nis a sub-
space of the tensor product R⊗n. The encoding pro-
cess is represented by an encoding map:
E:R⊗k−→ Q,
which encodes klogical qubits into nphysical qubits.
Based on lemma 5 and theorem 3, the next QECC con-
struction can be derived.
Theorem 4 Let Γ(C)the Gray image of Ccharacter-
ized by length 4n, dimension kand minimum distance
dL.
Suppose that C⊥⊂C, then the existence of a QECC
over Ris ensured and parameterized by length 4n,
dimension 2k−4nand minimum distance dL, where
dLdenotes the minimum Lee distance of C.
We can denote the parameters of this QECC by
[4n, 2k−4n, dL]over R.
Example 1 Let R=F3×(F3+vF3+v2F5),where
v3= 0 and n= 24. Then, ε24−1=(ε+ 1)(ε+
2)(ε2+1)(ε2+ε+2)(ε2+ 2ε+ 2)(ε4+ε2+ 2)(ε4+
2ε2+ 2) over F3[ε]. Let χ(ε)=(χ1(ε), χ2(ε) +
vχ3(ε) + v2χ4(ε)) where χ1(ε) = (ε2+ 1) and
χ2(ε) = χ3(ε) = χ4(ε) = (ε2+ε+ 2) and let a
code Cthat it is linear over R, where
C= (χ(ε))
= ((χ1(ε), χ2(ε) + vχ3(ε) + v2χ4(ε))).
Clearly, the code Cis cyclic, generated by χ(ε)over
Rand characterized by length 24, dimension 323.3
and dL= 2. Therefore, the code Γ(C)satisfy the
linearity and parameterized by [26,23.3,2] over F3.
From Corollary 1, we can represent the dual code C⊥
by
C⊥= ((φ∗
1(ε), φ∗
2(ε) + vφ∗
3(ε) + v2φ∗
4(ε))).
It result that C⊥⊆C. Then, by Theorem 4, it
is possible to construct a QECC parameterized by
[26,24.3,2] over F3×(F3+vF3+v2F3).
Example 2 Let R=F5×(F5+vF5+v2F5),where
v3= 0 and n= 26. Then, ε26−1=(ε+ 1)(ε+
2)(ε+3)(ε+ 4)(ε2+2)(ε2+ 3)(ε4+2)(ε4+ 3)(ε8+
2)(ε8+ 3)(ε16 + 2)(ε16 + 3) over F5[ε]. Let χ(ε) =
(χ1(ε), χ2(ε) + vχ3(ε) + v2χ4(ε)) where χ1(ε) =
(ε2+ 2) and χ2(ε) = χ3(ε) = χ4(ε) = (ε2+ 3) and
let a code Cthat it is linear over R, where
C= (χ(ε))
= ((χ1(ε), χ2(ε) + vχ3(ε) + v2χ4(ε))).
Clearly, the code Cis cyclic, generated by χ(ε)over
Rand characterized by length 26, dimension 323.31
and dL= 2. Therefore, the code Γ(C)satisfy the
linearity and parameterized by [28,23.31,2] over F5.
From Corollary 1, we can represent the dual code C⊥
by
C⊥= ((φ∗
1(ε), φ∗
2(ε) + vφ∗
3(ε) + v2φ∗
4(ε))).
It result that C⊥⊆C. Then, by Theorem 4, it
is possible to construct a QECC parameterized by
[28,24.3.5,2] over F5×(F5+vF5+v2F5).
7 Discussion
We used PARI GP and Magma to create some new
QECC compared with pre-existing research. Table
1 (Appendix 8), Table 2 (Appendix 8), Table 3 (Ap-
pendix 8), Table 4 (Appendix 8), Table 5 (Appendix
8), Table 6 (Appendix 8), and Table 7 (Appendix 8),
indicate our results of constructing QECC from cyclic
codes over Fp×(Fp+vFp+v2Fp)for 3≤p≤19,
respectively. We varied ifrom 4to 40 and we took
deg(χj) = 2 for j= 1,2,3,4. As a result, we ob-
tained QECC over Fp×(Fp+vFp+v2Fp)which its
dimensions 2k–4nis divided by pand 24. The first
column 2irepresents the length of cyclic code Cover
Fp×(Fp+vFp+v2Fp), The parameters of the Γ(C)
are denoted in the second column and the third col-
umn stands for the characteristics of the QECC over
Fp×(Fp+vFp+v2Fp).
8 Conclusion
In this study, we use the Calderbank, Shor, and Steane
(CSS) building to create QECC from cyclic codes
over the finite nonlocal ring R. In addition, we or-
ganize and provide several interesting examples. The
results ensure that cyclic codes over finite nonlocal
rings are an excellent resource for building QECC.
This study is extremely important for quantum com-
munication. We will build better QECC in the fu-
ture using cyclic codes over-generalized finite non-
local rings.
Acknowledgment:
The authors show their appreciation to the
anonymous reviewers and the editor for sharing their
informative comments.
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The authors equally contributed in the present re-
search, at all stages from the formulation of the prob-
lem to the final findings and solution.
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Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflicts of Interest
The authors have no conflicts of interest to
declare that are relevant to the content of this
article.
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Appendix
Table 1: QECC over F3×(F3+vF3+v2F3)
2iΓ(C) : [4n, k, dL]QECC: [n, 2k−4n, dL]
24[26,23.7,2] [26,24.3,2]
26[26,23.31,2] [28,24.3.5,2]
28[210,23.127,2] [210 ,24.32.7,2]
210 [212,23.7.73,3] [212 ,24.3.5.17,3]
212 [214,23.23.89,2] [212 ,24.3.11.31,2]
214 [216,23.8191,4] [216 ,24.32.5.7.13,4]
216 [218,23.7.31.151,2] [218 ,24.3.43.127,2]
218 [220,23.131071,2] [220 ,24.3.5.17.257,2]
220 [222,23.524287,2] [222 ,24.33.7.19.73,2]
222 [224,23.72.127.137,2] [224 ,24.3.52.11.31.41,2]
224 [226,23.47.178481,2] [226 ,24.3.23.89.683,2]
226 [228,23.31.601.1801,2] [228 ,24.32.5.7.13.17.241,2]
228 [230,23.7.73.262657,2] [230 ,24.3.2731.8191,2]
230 [232,23.233.1103.2089,2] [232 ,24.3.5.29.43.113.127,2]
232 [234,23.2147483647,2] [234 ,24.32.7.11.31.151.331,2]
234 [236,23.7.23.89.599479,2] [236 ,24.3.5.17.257.65537,2]
236 [238,23.31.71.127.122921,2] [238 ,24.3.43691.131071,2]
238 [240,23.223.616318177,2] [240 ,24.3.174763.524287,2]
240 [242,23.7.79.8191.121369,2] [242 ,24.3.174763.524287,2]
Table 2: QECC over F5×(F5+vF5+v2F5)
2iΓ(C) : [4n, k, dL]QECC: [n, 2k−4n, dL]
26[28,23.31,2] [28,24.3.5,2]
210 [212,23.7.73,2] [212 ,24.3.5.17,2]
214 [216,23.8191,2] [216 ,24.32.5.7.13,2]
218 [220,23.131071,3] [220 ,24.3.5.17.257,3]
222 [224,23.72.127.337,2] [224 ,24.3.52.11.31.41,2]
226 [228,23.31.601.1801,3] [228 ,24.32.5.7.13.17.241,3]
230 [232,23.233.1103.2089,2] [232 ,24.3.5.29.43.113.127,2]
234 [236,23.7.23.89.599479,2] [236 ,24.3.5.17.257.65537,2]
238 [240,23.223.616318177,2] [240 ,24.33.5.7.13.19.37.73.109,2]
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Table 3: QECC over F7×(F7+vF7+v2F7)
2iΓ(C) : [4n, k, dL]QECC: [n, 2k−4n, dL]
25[27,23.3.5,2] [27,24.7,2]
28[210,23.127,2] [210 ,24.32.7,2]
211 [213,23.3.11.31,2] [213 ,24.7.73,2]
214 [216,23.8191,2] [216 ,24.32.5.7.13,2]
217 [219,23.3.5.17.257,2] [219 ,24.7.31.151,2]
220 [222,23.524287,2] [222 ,24.33.7.19.73,2]
223 [225,23.3.23.89.683,2] [225 ,24.72.127.337,2]
226 [228,23.31.601.1801,2] [228 ,24.32.5.7.13.17.241,2]
229 [231,23.3.5.29.43.113.127,2] [231 ,24.33.7.73.262657,2]
232 [234,23.2147483647,2] [234 ,24.32.7.11.31.151.331,2]
235 [237,23.3.43691.131071,2] [237 ,24.7.23.89.599479,2]
238 [240,23.223.616318177,2] [240 ,24.33.5.7.13.19.37.73.109,2]
Table 4: QECC over F11 ×(F11 +vF11 +v2F11)
2iΓ(C) : [4n, k, dL]QECC: [n, 2k−4n, dL]
212 [214,23.23.89,3] [214 ,24.3.11.31,3]
222 [224,23.72.127.337,3] [224 ,24.3.52.11.31.41,3]
232 [234,23.2147483647,3] [234 ,24.32.7.11.31.151.331,3]
Table 5: QECC over F13 ×(F13 +vF13 +v2F13)
2iΓ(C) : [4n, k, dL]QECC: [n, 2k−4n, dL]
214 [216,23.8191,3] [216 ,24.32.5.7.13,3]
226 [228,23.31.601.1801,3] [228 ,24.32.5.7.13.17.241,3]
238 [240,23.223.616318177,3] [240 ,24.33.5.7.13.19.37.73.109,3]
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Table 6: QECC over F17 ×(F17 +vF17 +v2F17)
2iΓ(C) : [4n, k, dL]QECC: [n, 2k−4n, dL]
210 [212,23.7.73,2] [212 ,24.3.5.17,2]
218 [220,23.131071,2] [220 ,24.3.5.17.257,2]
226 [228,23.31.601.1801,2] [228 ,24.32.5.7.13.17.241,2]
234 [236,23.7.23.89.599479,3] [236 ,24.3.5.17.257.65537,3]
Table 7: QECC over F19 ×(F19 +vF19 +v2F19)
2iΓ(C) : [4n, k, dL]QECC: [n, 2k−4n, dL]
220 [222,23.524287,3] [222 ,24.33.7.19.73,3]
238 [240,23.223.616318177,2] [240 ,24.33.5.7.13.19.37.73.109,2]
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