Received December 15, 2019, accepted January 5, 2020, date of publication January 14, 2020, date of current version January 30, 2020.

Digital Object Identifier 10.1109/ACCESS.2020.2966542

On the Structure of Cyclic Codes Over FqRSand Applications in Quantum andLCD Codes ConstructionsHAI Q. DINH 1,2, TUSHAR BAG 3, ASHISH KUMAR UPADHYAY 3, RAMAKRISHNA BANDI 4,AND WARATTAYA CHINNAKUM 51Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh 700000, Vietnam2Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh 700000, Vietnam3Department of Mathematics, IIT Patna, Patna 801103, India4Department of Mathematics, International Institute of Information Technology, Naya Raipur, Atal Nagar 493661, India5Centre of Excellence in Econometrics, Chiang Mai University, Chiang Mai 50200, Thailand

Corresponding author: Hai Q. Dinh (dinhquanghai@tdtu.edu.vn)

ABSTRACT Let p be an odd prime, q = pm, R = Fq + uFq with u2 = 1, and S = Fq + uFq + vFq + uvFqwith u2 = 1, v2 = 1, uv = vu. In this paper, FqRS-cyclic codes over FqRS are studied. As an application,we present a construction of quantum error-correcting codes (QECCs) from the FqRS-cyclic codes overFqRS, which provides new QECCs. We also consider linear complementary dual (LCD) codes from theFqRS-cyclic codes over FqRS. Among others, we construct a Gray map over FqRS and discuss the Grayimages of FqRS-cyclic codes over Fq.

INDEX TERMS Cyclic codes, mixed alphabet codes, QECCs, LCD codes.

I. INTRODUCTIONThe family of cyclic codes is one of the most importantfamilies of codes which was introduced by Prange [41] andSloane and Thompson [47] in 1954. Due to their rich alge-braic structure and ease in implementation, these codes arewidely used. The study of cyclic codes over finite rings hasseen rapid growth after the work of Hammons et al. [27]. Theproperties of cyclic codes and their constructions over variousfinite rings are well explored in the literature.

Since the last two decades, researchers have started study-ing codes over mixed alphabets. The study of linear codesover mixed alphabets was initiated by Brouwer et al. [17]in 1998. In [17], the authors studied mixed alphabet codesas Z2-submodule over Zr2Z

s3. However, thereafter not much

work has been done on mixed alphabet codes. Recently,codes over mixed alphabets have caught attention of theresearchers. In 2010, Borges et al. [15] studiedZ2Z4-additivecodes and the corresponding Z2Z4-linear codes. In this work,they discussed the standard form for generator matrices,parity-checkmatrices ofZ2Z4-additive codes and establishedthe relation between them. They also talked about the auto-morphism groups of these codes. In 2013, Aydogdu and

The associate editor coordinating the review of this manuscript and

approving it for publication was Xueqin Jiang .

Siap [10] studied the structure ofZ2Z2s -additive codes, whichgeneralize Z2Z4-additive codes. In that paper, the authorspresented some fundamental parameters, standard form ofgenerator and parity-check matrices of Z2Z4-additive codes.Furthermore, they also provided two bounds on the min-imum distance of Z2Z4-additive codes. In a similar line,Aydogdu et al. [8] generalized these results of [10], [15] overZprZps . In 2014, Abualrub et al. [1] studied Z2Z4-additivecyclic codes. They showed that dual of a Z2Z4-cyclic codeis also a Z2Z4-cyclic code, and studied infinite family ofMDS codes. Then Borges et al. [16] obtained some impor-tant properties of Z2Z4-additive cyclic codes, and introducedgenerator polynomials of Z2Z4-additive cyclic and dualZ2Z4-additive cyclic codes. They also presented the parame-ters of Z2Z4-additive cyclic codes in terms of the degrees ofthe generator polynomials of these codes.

Then in [5], Aydogdu et al. studied MacWilliams identityover Z2Z2[u]-additive codes, and constructed some optimalbinary codes from this study. Srinivasulu and Bhaintwal [44]studied Z2(Z2 + uZ2)-additive cyclic codes and presentedtheir generators and minimal spanning sets. They also deter-mined the generators of duals Z2(Z2 + uZ2)-additive cycliccodes for odd code-lengths. After that, Aydogdu et al. [6]studied Z2Z2[u]-cyclic and constacyclic codes. They pre-sented the generator polynomials, minimal spanning sets,

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H. Q. Dinh et al.: On the Structure of Cyclic Codes Over FqRS

their sizes and binary images of Z2Z2[u]-cyclic and con-stacyclic codes under a Gray map. Recently, there areseveral papers on mixed alphabets such as Z2Z2[u3] [9];Z2Z2[u] [31].

Using these concepts of mixed alphabet codes,Borges et al. [14] studied double cyclic codes over Z2.Then Gao et al. [21] generalized that to consider doublecyclic codes over Z4. After that, extending this double cycliccode structure, Mostafanasab [39] introduced the triple cycliccodes over Z2. Recently, Wu et al. [47] and Aydogdu andGursoy [7] independently studied Z2Z2Z4-additive cycliccodes and Z2Z4Z8-cyclic codes, respectively.In all of the aforementioned works, researchers mainly

focused on exploring the structural properties of mixed alpha-bet codes such as generator matrices, parity check matrices,generating polynomials, minimal generating sets, generat-ing polynomials for dual codes etc. There are hardly anypapers on the applications of mixed alphabets codes. In thiswork, our main goal is to study mixed alphabet FqRS-cycliccodes as well as finding the tools to apply them in some ofthe recent topics of research like construction of quantumerror-correcting codes (QECC) and linear complementarydual (LCD) codes. To do that, in the first part of our article,we study the properties of FqRS-cyclic codes, and in the laterpart (Sections 6 and 7), we discuss the construction of QECCsand LCD codes from FqRS-cyclic codes.In this article, we first study linear codes and then cyclic

codes over FqRS, where R = Fq + uFq, u2 = 1 and S =Fq+uFq+ vFq+uvFq, u2 = 1, v2 = 1, uv = vu and q = pm

for odd prime p and positive integer m. As an applicationof this work, using the structure of cyclic codes over FqRS,we construct QECCs and also LCD codes over Fq. The paperis organized as follows: In Section 2, we describe the basicterminology to understand cyclic codes over FqRS and theirproperties. In Section 3, a construction of linear codes over Rand S have been discussed. In Section 4, we define a Graymap, through which we discuss some properties of codesover FqRS. In Section 5, cyclic codes over R, S and FqRSare studied. As an application of this study, in Section 6 andSection 7, we construct QECCs and LCD codes, respectively,with worked out examples. All the computations are doneusing Magma Computing Software.

II. PRELIMINARIESLet Fq denote the finite field of characteristic p with q ele-ments, where q = pm for odd prime p and positive integerm. The set Fnq of all ordered n-tuples over Fq forms a vectorspace with the usual component-wise addition and scalarmultiplication of vectors. A non-empty subset C of Fnq iscalled a code of length n over Fq, and it is called a linear codeifC is a subspace ofFnq. An element ofC is called a codeword.Throughout this article we use the word code to refer a linearcode. By wH (C), we denote the Hamming weight of a codeC , which is defined as the smallest Hamming weight of allof its non-zero codewords. Let x = (x0, x1, . . . , xn−1), y =(y0, y1, . . . , yn−1) ∈ Fnq, then the Hamming distance between

x and y is defined as dH (x, y) = |{i | xi 6= yi}|, i.e., dH (x, y) =wH (x− y). The Hamming distance of a code C is defined asdH (C) = min{dH (x, y)| x, y ∈ C, x 6= y}. The Euclideaninner product of x and y in Fnq is defined as x · y = x0y0 +x1y1 + · · · + xn−1yn−1. The dual code C⊥ of C is defined asC⊥ = {x ∈ Fnq| x · y = 0, ∀ y ∈ C}.A code C of length n over Fq is called a cyclic code

if (c0, c1, . . . , cn−1) ∈ C implies (cn−1, c0, . . . , cn−2)∈ C . In polynomial representation, each codeword c =(c0, c1, . . . , cn−1) ∈ C is identified with a polynomial c(x) =c0 + c1x + · · · + cn−1xn−1 ∈

Fq[x]〈xn−1〉 . By this identification,

it can be easily shown that a linear code C of length n over Fqis a cyclic code if and only if it is an ideal of the ring Fq[x]

〈xn−1〉 .We can extend these concepts to linear code, dual code,

cyclic code over finite commutative rings depending upon thestructure of the rings. Let R be any finite commutative ring.Then we refer a nonempty subset C as a linear code of lengthn over R if it forms an R-submodule of Rn.

Continuing our discussion we extend previous discussionto codes over product of finite commutative rings.

From now onward, we denote R = Fq + uFq, with u2 = 1and S = Fq + uFq + vFq + uvFq, with u2 = 1, v2 = 1,uv = vu, where q = pm for odd prime p and positiveinteger m. Let s = a + ub + vc + uvd be an element of S.Then we define two maps η and τ as follows:

η : S −→ Fq such that η(s) = a,

and

τ : S −→ R such that τ (s) = a+ ub.

It is clear that the maps η and τ are ring homomorphisms.Now for any s ∈ S and (x, y, z) ∈ FqRS we define thefollowing S-scalar multiplication on FqRS as

• : S × FqRS −→ FqRSsuch that s • (x, y, z) = (η(s)x, τ (s)y, sz).

This is a well-defined multiplication. It can be extendedcomponent-wise over Fαq × Rβ × Sγ as follows:

• : S × (Fαq × Rβ× Sγ ) −→ Fαq × R

β× Sγ

where

s • t = (η(s)x0, η(s)x1, . . . , η(s)xα−1, τ (s)y0, τ (s)y1,

. . . , τ (s)yβ−1, sz0, sz1, . . . , szγ−1),

where s ∈ S and t = (x0, x1, . . . , xα−1, y0, y1, . . . , yβ−1, z0,z1, . . . , zγ−1) ∈ Fαq × Rβ × Sγ . By this multiplication,Fαq × Rβ × Sγ forms an S-module.Definition 1: A non-empty subset C of Fαq × Rβ × Sγ is

said to be a FqRS-linear code of length (α, β, γ ) if C is anS-submodule of Fαq × Rβ × Sγ .Let t = (x0, x1, . . . , xα−1, y0, y1, . . . , yβ−1, z0, z1, . . . ,

zγ−1) and t′ = (x ′0, x′

1, . . . , x′

α−1, y′

0, y′

1, . . . , y′

β−1, z′

0,

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H. Q. Dinh et al.: On the Structure of Cyclic Codes Over FqRS

z′1, . . . , z′

γ−1) ∈ Fαq ×Rβ × Sγ . Now we define inner productas,

t · t′ = uvα−1∑i=0

xix ′i + vβ−1∑j=0

yiy′i +γ−1∑k=0

ziz′i.

Let C be a FqRS-linear code of length (α, β, γ ). Then the dualcode of C is defined as

C⊥ = {t′ ∈ Fαq × Rβ× Sγ | t · t′ = 0,∀t ∈ C}.

Definition 2: An S-submodule C of Fαq × Rβ × Sγ is saidto be a FqRS-cyclic code of length (α, β, γ ) if for any c =(x0, x1, . . . , xα−1, y0, y1, . . . , yβ−1, z0, z1, . . . , zγ−1) ∈ C itscyclic shift ℘(c) := (xα−1, x0, x1, . . . , xα−2, yβ−1, y0, y1,. . . , yβ−2, zγ−1, z0, z1, . . . , zγ−2) ∈ C.Proposition 3: Let C be a FqRS-cyclic code of length

(α, β, γ ). Then C⊥ is also a FqRS-cyclic code of length(α, β, γ ).

Proof: Let C be a FqRS-cyclic code of length (α, β, γ )and t′ = (x ′0, x

′

1, . . . , x′

α−1, y′

0, y′

1, . . . , y′

β−1, z′

0, z′

1, . . . ,

z′γ−1) ∈ C⊥. Take lcm(α, β, γ ) = l and t =

(x0, x1, . . . , xα−1, y0, y1, . . . , yβ−1, z0, z1, . . . , zγ−1) ∈ C.We have to show, ℘(t′) = (x ′α−1, x

′

0, x′

1, . . . , x′

α−2, y′

β−1,

y′0, y′

1, . . . , y′

β−2, z′

γ−1, z′

0, z′

1, . . . , z′

γ−2) ∈ C⊥. Now by theabove defined inner product we get,

t · ℘(t′) = uv{x0x ′α−1 + x1x′

0 + · · · + xα−1x′

α−2}

+v{y0y′β−1 + y1y′

0 + · · · + yβ−1y′

β−2}

+{z0z′γ−1 + z1z′

0 + · · · + zγ−1z′

γ−2}.

As C is a FqRS-cyclic code and lcm(α, β, γ ) = l,℘l−1(t) ∈ C, where ℘l−1(t) = (x1, x2, . . . , xα−1, x0, y1,y2, . . . , yβ−1, y0, z1, z2 . . . , zγ−1, z0). Taking the inner prod-uct of ℘l−1(t) and t′, we get ℘l−1(t) · t′ = 0, where

℘l−1(t) · t′ = uv{x1x ′0 + x2x′

1 + · · · + x0x′

α−1}

+v{y1y′0 + y2y′

1 + · · · + y0y′

β−1}

+{z1z′0 + z2z′

1 + · · · + z0z′

γ−1}.

Comparing the coefficients from both sides we get,

x1x ′0 + x2x′

1 + · · · + xα−1x′

α−2 + x0x′

α−1 = 0,

y1y′0 + y2y′

1 + · · · + yβ−1y′

β−2 + y0y′

β−1 = 0,

z1z′0 + z2z′

1 + · · · + zγ−1z′

γ−2 + z0z′

γ−1 = 0.

Therefore, t · ℘(t′) = 0. Hence ℘(t′) ∈ C⊥, implying C⊥ is aFqRS-cyclic code of length (α, β, γ ). �

Let Sα,β,γ =Fq[x]〈xα−1〉 ×

R[x]〈xβ−1〉 ×

S[x]〈xγ−1〉 and f = (a0,

a1, . . . , aα−1, b0, b1, . . . , bβ−1, c0, c1, . . . , cγ−1) ∈ Fαq ×Rβ × Sγ . Then f can be identified by an element in Sα,β,γas

f (x) = (a0 + a1x + · · · + aα−1xα−1,

b0 + b1x + · · · + bβ−1xβ−1,

c0 + c1x + · · · + cγ−1xγ−1)

= (a(x), b(x), c(x)),

which gives the one-to-one identification betweenFαq × Rβ × Sγ and Sα,β,γ .Let s(x) = s0 + s1x + · · · + sδxδ ∈ S[x]

and (a(x), b(x), c(x)) ∈ Sα,β,γ . Then the above definedS-scalar multiplication induces the multiplication ? in Sα,β,γas follows,

s(x) ? (a(x), b(x), c(x))= (η(s(x))a(x), τ (s(x))b(x), s(x)c(x)),

where η(s(x)) = η(s0)+η(s1)x+· · ·+η(sδ)xδ and τ (s(x)) =τ (s0) + τ (s1)x + · · · + τ (sδ)xδ . It is easy to show that, withrespect to the multiplication ?, Sα,β,γ forms an S[x]-module.Proposition 4: A code C is a FqRS-cyclic code of length

(α, β, γ ) if and only if C is an S[x]-submodule of Sα,β,γ .Proof: Suppose C is a FqRS-cyclic code of length

(α, β, γ ). Let f = (a0, a1, . . . , aα−1, r0, r1, . . . , rβ−1,s0, . . . , sγ−1) ∈ C and the corresponding element of f bef (x) = (a(x), r(x), s(x)). Note that

x ? f (x) = (aα−1 + a0x + · · · + aα−2xα−1,

rβ−1 + r0x + · · · + rβ−2xβ−1,

sγ−1 + s0x + · · · + sγ−2xγ−1),

corresponds to the cyclic shift (aα−1, a0, a1, . . . , aα−2,rβ−1, r0, r1, . . . , rβ−2, sγ−1, s0, s1, . . . , sγ−2) of f, thus,x?f (x) ∈ C. Then by the linearity property of C, c(x)?f (x) ∈ Cfor any c(x) ∈ S[x]. Thus, C is an S[x]-submodule of Sα,β,γ .The other side is trivial by the definition. �

III. DECOMPOSITION OF LINEAR CODES OVER R AND SRecall that R = Fq+uFq, where u2 = 1 and S = Fq+uFq+vFq + uvFq, where u2 = 1, v2 = 1, uv = vu. For a, b ∈ Fq,any r ∈ R is of the form r = a + ub = η1a + η2b, wherea, b ∈ Fq such that a = (a− b), b = (a+ b) and

η1 =1− u2

, η2 =1+ u2

.

It is easy to check that η2i = ηi, ηiηj = 0 and η1+η2 = 1, fori, j = 1, 2; i 6= j. Therefore, R = η1R⊕ η2R and any r ∈ Rcan be expressed as r = η1r1 + η2r2, where r1, r2 ∈ Fq.We define a Gray map ψR : R → F2

q given as

ψR(r) = (r1, r2). This map can be extended Rβ → F2βq

component wise as (r0, r1, . . . , rβ−1) 7→ (r0,1, r1,1, . . . ,rβ−1,1, r0,2, r1,2, . . . , rβ−1,2), where r = (r0, r1, . . . , rβ−1) ∈Rβ and ri = η1ri,1 + η2 ri,2 for i = 0, 1, . . . , β − 1. Thenfor ri = η1ri,1 + η2 ri,2 ∈ R, we define the Lee weight ofri as wL(ri) = wH (ψR(ri)), where wH denotes the Hammingweight over Fq. The Lee distance between ri and r ′i ∈ R isdefined as dL(ri, r ′i ) = wL(ri−r ′i ) = wH (ψR(ri−r ′i )). Then itis easy to show that, the Gray mapψR is a distance preservingFq-linear map from Rβ (Lee distance) to F2β

q (Hammingdistance).

Let Bβ be a linear code of length β over R. Then we define

Bβ,1 = {r1 ∈ Fβq | η1r1 + η2r2 ∈ Bβ; r2 ∈ Fβq },Bβ,2 = {r2 ∈ Fβq | η1r1 + η2r2 ∈ Bβ; r1 ∈ Fβq }.

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H. Q. Dinh et al.: On the Structure of Cyclic Codes Over FqRS

Therefore, Bβ,i are linear codes of length β over Fq fori = 1, 2. So a linear code Bβ of length β over R can beuniquely expressed as Bβ = η1Bβ,1 ⊕ η2Bβ,2, then | Bβ |=|Bβ,1 || Bβ,2 |. By definition of the Gray map we haveψR(Bβ ) = Bβ,1 ⊗ Bβ,2, then using the fact ψR is distancepreserving, we get dL(Bβ ) = dH (ψR(Bβ )) = dH (Bβ,1 ⊗Bβ,2) = min{dH (Bβ,i) | i = 1, 2}.Similarly, for x, y,w, z ∈ Fq, any s = x+uy+vw+uvz ∈ S

can be expressed as below

s = x + uy+ vw+ uvz

= ζ1(x + y+ w+ z)+ ζ2(x + y− w− z)

+ζ3(x − y+ w− z)+ ζ4(x − y− w+ z)

= ζ1x + ζ2y+ ζ3w+ ζ4z,

where x, y, w, w ∈ Fq such that x = (x + y + w + z),y = (x+y−w− z), w = (x−y+w− z), z = (x−y−w+ z)and

ζ1 =14(1+ u+ v+ uv), ζ2 =

14(1+ u− v− uv),

ζ3 =14(1− u+ v− uv), ζ4 =

14(1− u− v+ uv).

It is easy to see that ζ 2i = ζi, ζiζj = 0 and∑4

i=1 ζi = 1,where i, j = 1, 2, 3, 4; i 6= j. Therefore, S = ζ1S ⊕ ζ2S ⊕ζ3S ⊕ ζ4S. Thus, any s ∈ S can be uniquely expressed ass = ζ1s1 + ζ2s2 + ζ3s3 + ζ4s4, where s1, s2, s3, s4 ∈ Fq.

We define a Gray map over S as ψS : S → F4q such that

ψS (s) = (s1, s2, s3, s4). This map can be extended Sγ → F4γq

component wise as

(s0, s1, . . . , sγ−1) 7→ (s0,1, s1,1, . . . , sγ−1,1, s0,2, s1,2,

. . . , sγ−1,2, s0,3, . . . , sγ−1,3, s0,4, s1,4, . . . , sγ−1,4),

where s = (s0, s1, . . . , sγ−1) ∈ Sγ and sj = ζ1sj,1 +ζ2sj,2 + ζ3sj,3 + ζ4sj,4 for j = 0, 1, . . . , γ − 1. As above, forsj = ζ1sj,1 + ζ2sj,2 + ζ3sj,3 + ζ4sj,4 ∈ S, we define wL(sj) =wH (ψS (sj)), and dL(sj, s′j) = wL(sj − s′j) = wH (ψS (sj − s′j))for s′j ∈ S

γ . Then we can easily show that the Gray mapψS isa distance preserving Fq-linear map from Sγ (Lee distance)to F4γ

q (Hamming distance).Let Cγ be a linear code of length γ over S, then we define

Cγ,1 = {s1 ∈ Fγq | ζ1s1 + ζ2s2 + ζ3s3 + ζ4s4 ∈ Cγand s2, s3, s4 ∈ Fγq },

Cγ,2 = {s2 ∈ Fγq | ζ1s1 + ζ2s2 + ζ3s3 + ζ4s4 ∈ Cγand s1, s3, s4 ∈ Fγq },

Cγ,3 = {s3 ∈ Fγq | ζ1s1 + ζ2s2 + ζ3s3 + ζ4s4 ∈ Cγand s1, s2, s4 ∈ Fγq },

Cγ,4 = {s4 ∈ Fγq | ζ1s1 + ζ2s2 + ζ3s3 + ζ4s4 ∈ Cγand s1, s2, s3 ∈ Fγq }.

Arguing as above, any linear code Cγ of length γ over Scan be written as Cγ = ζ1Cγ,1 ⊕ ζ2Cγ,2 ⊕ ζ3 Cγ,3 ⊕ ζ4 Cγ,4then | Cγ |=

∏4j=1 | Cγ,j |. Similarly using the facts ψS is

a distance preserving map and ψS (Cγ ) = ⊗4j=1 Cγ,j, we get

dL(Cγ ) = dH (ψS (Cγ )) = dH (⊗4j=1 Cγ,j) = min{dH (Cγ,j) |

j = 1, 2, 3, 4}.

IV. GRAY MAP OVER FqRSAny arbitrary element of FqRS can be written as (a, r, s) =(a, η1 r1 + η2r2, ζ1 s1 + ζ2s2 + ζ3 s3 + ζ4s4), where a ∈ Fq,r ∈ R and s ∈ S. Define a Gray map 9 : FqRS → F7

q asfollows:

9(a, r, s) = (a, r1, r2, s1, s2, s3, s4).

This is also a Fq-linear map and can be extendedcomponent-wise in the following way:

9 : Fαq × Rβ× Sγ −→ Fα+2β+4γq

defined by

(a0, a1, . . . , aα−1, r0, r1, . . . , rβ−1, s0, s1, . . . , sγ−1)

7→ (a0, a1, . . . , aα−1, r0,1, r1,1, . . . , rβ−1,1,

r0,2, r1,2, . . . , rβ−1,2, s0,1, s1,1, . . . , sγ−1,1,

s02, s1,2, . . . , sγ−1,2, s0,3, s1,3, . . . , sγ−1,3,

s0,4, s1,4, . . . , sγ−1,4),

where (a0, a1, . . . , aα−1) ∈ Fαq , (r0, r1, . . . , rβ−1) ∈ Rβ and(s0, s1, . . . , sγ−1) ∈ Sγ such that each ri = η1 ri,1 + η2ri,2 ∈R and sj = ζ1 sj,1 + ζ2sj,2 + ζ3 sj,3 + ζ4sj,4 ∈ S, fori = 0, 1, . . . , β − 1 and j = 0, 1, . . . , γ − 1.Similar to [47], for any element (a′, r′, s′) ∈ Fαq ×Rβ×Sγ

we define the Lee weight of (a′, r′, s′) as wL(a′, r′, s′) =wH (a′) + wL(r′) + wL(s′), where wH denote the Hammingweight and wL denote the Lee weight. The Lee distancebetween x′, x′′ ∈ Fαq × Rβ × Sγ is defined as dL(x′, x′′) =wL(x′ − x′′) = wH (9(x′ − x′′)).Proposition 5: Let9 be the Gray map defined above. Then,1) 9 is a Fq-linear distance preserving map from FαqRβSγ

(Lee distance) to Fα+2β+4γq (Hamming distance).2) C is a FqRS-linear code of length (α, β, γ ) over FqRS,

then 9(C) is a [α + 2β + 4γ, k, dH ] linear code overFq, where dL = dH .Proof: (1.) Let x′ = (a′, r′, s′), x′′ = (a′′, r′′, s′′) ∈

FαqRβSγ , where

a′ = (a′0, a′

1, . . . , a′

α−1), a′′= (a′′0, a

′′

1, . . . , a′′

α−1) ∈ Fαq ;r′ = η1r′1 + η2r

′

2, r′′= η1r′′1 + η2r

′′

2 ∈ Rβ ,

and

s′ = ζ1s′1 + ζ2s′

2 + ζ3s′

3 + ζ4s′

4,

s′′ = ζ1s′′1 + ζ2s′′

2 + ζ3s′′

3 + ζ4s′′

4 ∈ Sγ ,

such that for i = 1, 2 and j = 1, 2, 3, 4

r′i = (r ′i,0, r′

i,1, . . . , r′

i,β−1),

r′′i = (r ′′i,0, r′′

i,1, . . . , r′′

i,β−1) ∈ Fβq ,s′j = (s′j,0, s

′

j,1, . . . , s′

j,γ−1),

s′′j = (s′′j,0, s′′

j,1, . . . , s′′

j,γ−1) ∈ Fγq .

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Then 9(x′ + x′′)

= (a′ + a′′, r′1 + r′′1, r′

2 + r′′2, s′

1 + s′′1, · · · , s′

4 + s′′4)

= (a′, r′1, r′

2, s′

1, s′

2, s′

3, s′

4)+ (a′′, r′′1, r′′

2, s′′

1, s′′

2, s′′

3, s′′

4)

= 9(x′)+9(x′′),and 9(cx′) = (ca′, cr′1, cr

′

2, cs′

1, cs′

2, cs′

3, cs′

4) = c9(x′),where c ∈ Fq. Therefore, 9 is a Fq-linear map.For the other part, using the fact that 9 is a Fq-linear map

we get, dL(x′, x′′) = wL(x′ − x′′) = wH (9(x′ − x′′)) =dH (9(x′), 9(x′′)). Hence, the result follows.

(2.) It is not difficult to show upon noticing that 9 is aFq-linear distance preserving and bijective map. �Definition 6: Suppose d ∈ Fmnq such that d =

(d1,d2, · · · ,dn−1,dn), where di ∈ Fmq ; i = 1, 2, . . . , n.Let ρ be the cyclic shift from Fmq to Fmq definedby ρ(x0, x1, . . . , xm−1) = (xm−1, x0, . . . , xm−2). Defineχ : Fmnq −→ Fmnq by, χ (d1,d2, . . . ,dn) =

(ρ(d1), ρ(d2), · · · , ρ(dn)). Then a code C is called aquasi-cyclic code of index n if χ (C) = C.Suppose d′ ∈ Fm1

q × Fm2q × · · · × Fmnq such that d′ =

(d′1,d′

2, · · · ,d′

n−1,d′n), where d′i ∈ Fmiq ; i = 1, 2, . . . , n.

Let ρ be the cyclic shift from Fmiq to Fmiq defined byρ(x0, x1, . . . , xmi−1) = (xmi−1, x0, . . . , xmi−2). Define χg :Fm1q × Fm2

q × · · · × Fmnq −→ Fm1q × Fm2

q × · · · × Fmnqby, χg(d′1,d

′

2, · · · ,d′n) = (ρ(d′1), ρ(d

′

2), · · · , ρ(d′n)), then a

code C is called a generalized quasi-cyclic code of index n ifχg(C) = C.Based on Definition 6, we have the following result.Proposition 7: Let χg and 9 be the maps defined above,

and ℘ be the cyclic shift over FqRS. Then 9℘ = χg9.Proof: Let c = (a0, a1, . . . , aα−1, r0, r1, . . . , rβ−1, s0,

s1, . . . , sγ−1) ∈ FαqRβSγ , where each ri = η1 ri,1 + η2 ri,2 ∈R and sj = ζ1 sj,1 + ζ2 sj,2 + ζ3 sj,3 + ζ4 sj,4 ∈ S fori = 0, 1, . . . , β − 1; j = 0, 1, . . . , γ − 1. Then 9℘(c)= 9℘(a0, a1, . . . , aα−1, r0, r1, . . . , rβ−1, s0, s1, . . . , sγ−1)

= 9(aα−1, a0, a1, . . . , aα−2, rβ−1, r0, r1, . . . , rβ−2, sγ−1,

s0, s1, . . . , sγ−2)

= (aα−1, a0, a1, . . . , aα−2, rβ−1,1, r0,1, r1,1, . . . , rβ−2,1,

rβ−1,2, r0,2, r1,2, . . . , rβ−2,2, sγ−1,1, s0,1, s1,1, . . . ,

sγ−2,1, sγ−1,2, s0,2, s1,2, . . . , sγ−2,2, sγ−1,3, s0,3, s1,3,

. . . , sγ−2,3, sγ−1,4, s0,4, s1,4, . . . , sγ−2,4).

On the other hand, χg9(c)= χg9(a0, a1, . . . , aα−1, r0, r1, . . . , rβ−1, s0, . . . , sγ−1)

= χg(a0, a1, . . . , aα−1, r0,1, . . . , rβ−1,1, r0,2, r1,2, . . . ,

rβ−1,2, s0,1, s1,1, . . . , sγ−1,1, s0,2, . . . , sγ−1,2,

s0,3, . . . , sγ−1,3, s0,4, s1,4, . . . , sγ−1,4)

= (aα−1, a0, a1, . . . , aα−2, rβ−1,1, r0,1, r1,1, . . . , rβ−2,1,

rβ−1,2, r0,2, r1,2, . . . , rβ−2,2, sγ−1,1, s0,1, s1,1, . . . ,

sγ−2,1, sγ−1,2, s0,2, s1,2, . . . , sγ−2,2, sγ−1,3, s0,3,

. . . , sγ−2,3, sγ−1,4, s0,4, s1,4, . . . , sγ−2,4).

Therefore, 9℘ = χg9. �

Using Proposition 7, we have the following result.Theorem 8: Let C be a linear code of length (α, β, γ ) over

FqRS. Then the9-Gray image of FqRS-cyclic code of length(α, β, γ ) is a generalized quasi-cyclic code of index 7 overFq. In particular, when α = β = γ , the 9-Gray image of aFqRS-cyclic code of length (α, β, γ ) is a quasi-cyclic code ofindex 7 over Fq.

V. CYCLIC CODES OVER Fq , R, S and FqRSIn this section, we study cyclic codes of length α, β andγ over the rings Fq,R and S, respectively. Then using thestructure of the cyclic codes over Fq,R and S, we discuss thecyclic codes of length (α, β, γ ) over FqRS.

A. CYCLIC CODES OVER Fq

Theorem 9 [28, Theorem 12.9]: Let A be a cyclic code oflength α over Fq. Then there exists a unique monic polyno-mial a(x) ∈ Fq[x]/〈xα − 1〉 such that Aα = 〈a(x)〉 and a(x) |(xα − 1). Moreover, the dimension of Aα is r = α − deg(a)with {a(x), xa(x), · · · , xr−1a(x)} as a basis.

B. CYCLIC CODES OVER R AND STheorem 10: Let Bβ = η1Bβ,1 ⊕ η2Bβ,2 be a linear code oflength β over R. Then Bβ is a cyclic code of length β overR if and only if Bβ,i are cyclic codes of length β over Fq, fori = 1, 2.

Proof: Let (x0, x1, . . . , xβ−1) ∈ Bβ,1 and (y0, y1, . . . ,yβ−1) ∈ Bβ,2. Then z = (z0, z1, . . . , zβ−1) ∈ Bβ , where zi =η1xi + η2yi, and xi, yi ∈ Fq for i = 0, 1, . . . , β − 1. SupposeBβ is a cyclic code of length β over R, then ρ(z) ∈ Bβ , where

ρ(z) = (zβ−1, z0, . . . , zβ−2)

= (η1 xβ−1 + η2 yβ−1, η1x0 + η2y0,

. . . , η1xβ−2 + η2yβ−2)

= η1(xβ−1, x0, . . . , xβ−2)+ η2(yβ−1, y0, . . . , yβ−2).

Therefore, by the direct sum decomposition of the linearcode Bβ we get (xβ−1, x0, . . . , xβ−2) ∈ Bβ,1 and(yβ−1, y0, . . . , yβ−2) ∈ Bβ,2. Hence, Bβ,i are cyclic codes oflength β over Fq, for i = 1, 2.Conversely, let z′ = (z′0, z

′

1, . . . , z′

β−1) ∈ Bβ such thateach z′i = η1x ′i + η2y

′i, where x

′i , y′i ∈ Fq for i = 0, 1,

. . . , β − 1. Then x′ = (x ′0, x′

1, . . . , x′

β−1) ∈ Bβ,1 andy′ = (y′0, y

′

1, . . . , y′

β−1) ∈ Bβ,2. Suppose that Bβ,i are cycliccodes of length β over Fq, for i = 1, 2. Then ρ(x′) ∈ B1 andρ(y′) ∈ B2. Note that η1ρ(x′)+ η2ρ(y′)

= η1(x ′β−1, x′

0, . . . , x′

β−2)+ η2(y′

β−1, y′

0, . . . , y′

β−2)

= (η1 x ′β−1 + η2 y′

β−1, η1x′

0 + η2y′

0, . . . , η1x′

β−2 + η2y′

β−2)

= (z′β−1, z′

0, . . . , z′

β−2)

= ρ(z′) ∈ Bβ .

Therefore, Bβ is a cyclic code of length β over R. �Using the direct sum decomposition of dual linear code B⊥β

over R, and arguing as above we can show the direct sumdecomposition of the dual cyclic code over R as follows.

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Corollary 11: Let Bβ be a cyclic code of length β over R.Then its dual B⊥β = η1B

⊥

β,1 ⊕ η2 B⊥

β,2 is also a cyclic code oflength β over R if and only if B⊥β,i are cyclic codes of lengthβ over Fq, for i = 1, 2.

In Theorem 10, we have presented the direct sum decom-position of a cyclic code Bβ over R. Here we give the gener-ator polynomial for such codes.Theorem 12: Let Bβ = η1Bβ,1⊕η2Bβ,2 be a cyclic code of

length β over R and ri(x) be the generator monic polynomialof the cyclic code Bβ,i, for i = 1, 2. Then

1) Bβ = 〈η1r1(x), η2r2(x)〉 and |Bβ | = q2β−∑2

i=1 deg(ri),

2) Bβ = 〈r(x)〉, where r(x) = η1r1(x) +η2r2(x) such thatr(x) | (xβ − 1).

Proof: (1.) Suppose Bβ is a cyclic code of length β overR. Then by Theorem 10, we get that Bβ,i are cyclic codesof length β over Fq, implying Bβ,i = 〈ri(x)〉 ⊆

Fq[x]〈xβ−1〉 for

i = 1, 2. As Bβ = η1Bβ,1 ⊕ η2Bβ,2, we can write Bβ ={r(x) | r(x) = η1r1(x) + η2r2(x); ri(x) ∈ Bβ,i, i = 1, 2},which implies Bβ ⊆ 〈η1r1(x), η2r2(x)〉 ⊆

R[x]〈xβ−1〉 .

On the other hand, let η1r1(x) h1(x) + η2r2(x) h2(x) ∈〈η1r1(x), η2r2(x)〉, where h1(x), h2(x) ∈

R[x]〈xβ−1〉 . Suppose that

the coefficients of hi(x), i = 1, 2 are hij = gj+ug′j ∈ R. Usingthe decomposition of elements inR, we can write hij = η1gj+η2gj

′, where gj, gj′∈ Fq. Thus, we get η1 h1(x) = η1 g1(x)

and η2 h2(x) = η2 g2(x), where g1(x) = g0 + g1x + · · · +gβ−1xβ−1 and g2(x) = g0

′+ g1

′x + · · · + ˆgβ−1′xβ−1 ∈

Fq[x]〈xβ−1〉 . Therefore, 〈η1r1(x), η2r2(x)〉 ⊆ Bβ . Hence,Bβ = 〈η1r1(x), η2r2(x)〉.For the other part, note that |Bβ | = |Bβ,1||Bβ,2|. Therefore,|Bβ | = qβ−deg(r1) × qβ−deg(r2) = q2β−(deg(r1)+deg(r2)).

(2.) Let ri(x) be the generator monic polynomial ofthe cyclic code Bβ,i, for i = 1, 2. By the first part,Bβ = 〈η1r1(x), η2r2(x)〉. Suppose r(x) = η1r1(x) + η2r2(x)such that B = 〈r(x)〉. Then r(x) ∈ Bβ , implying B ⊆ Bβ .Also as ηi(η1r1(x) + η2r2(x)) = ηiri(x) for i = 1, 2,implies Bβ ⊆ B. Therefore Bβ = B = 〈r(x)〉, wherer(x) = η1r1(x)+ η2r2(x).

As ri(x) is the monic generator polynomial of the cycliccode Bβ,i, ri(x) divides xβ−1 such that there is fi(x) ∈

Fq[x]〈xβ−1〉

with xβ − 1 = ri(x)fi(x), for i = 1, 2. Thus, xβ − 1 = (η1 +η2)(xβ − 1) = (η1r1(x)f1(x) + η2r2(x)f2(x)) = (η1f1(x) +η2f2(x))(η1r1(x) +η2r2(x)), for i = 1, 2. Therefore, xβ−1 =(η1f1(x) + η2f2(x))r(x). Hence, r(x) | (xβ − 1). �Using Corollary 11 and Theorem 12, we present the fol-

lowing result.Corollary 13: Let Bβ = η1Bβ,1⊕η2Bβ,2 be a cyclic code of

length β over R. Suppose ri(x) are the monic generator poly-nomials of the cyclic codes Bβ,i and f ∗i (x) are the reciprocalpolynomials of fi(x) such that xβ−1 = fi(x)ri(x), for i = 1, 2.Then

1) B⊥β = 〈η1f∗

1 (x), η2f∗

2 (x)〉 and |B⊥β | = q

∑2i=1 deg(ri),

2) B⊥β = 〈f′(x)〉, where f ′(x) = η1f ∗1 (x) + η2f

∗

2 (x).

We demonstrate our results, presenting the followingexamples.Example 14: Let β = 5 and R = F49 + uF49, where

u2 = 1. Take F49 =F7[x]

〈x2+6x+3〉and w is a zero of the

polynomial x2 + 6x + 3 in F49, then

x5 − 1 = (x + 6)(x2 + w27x + 1)(x2 + w45x + 1) ∈ F49[x].

Let r1(x) = x2 +w27x + 1 and r2(x) = x2 +w45x + 1. ThenBβ,1 = 〈r1(x)〉 and Bβ,2 = 〈r2(x)〉 are cyclic codes of length5 over F49. Therefore, Bβ = η1Bβ,1⊕η2Bβ,2 is a cyclic codeof length 5 over R. Thus, Bβ = 〈η1r1(x), η2r2(x)〉 = 〈r(x)〉,where r(x) = η1r1(x) + η2r2(x). Then η1r1(x) + η2r2(x)

=12(1− u)(x2 + w27x + 1)+

12(1+ u)(x2 + w45x + 1)

=12{2x2 + (w27

+ w45)x + u(w45− w27)x + 2}

=12{2x2 + x + uw44x + 2}

= x2 + 4x + 4uw44x + 1

= x2 + (4+ uw28)x + 1.

By direct computation, we get

x5 − 1 = (x3 + (3+ uw4)x2 + (4+ uw28)x + 6)

×(x2 + (4+ uw28)x + 1).

Thus, r(x) | (x5 − 1). Also | Bβ |= 4910−4 = 496. For thedual code of Bβ , note that

f1(x) = (x + 6)(x2 + w45x + 1),

f2(x) = (x + 6)(x2 + w27x + 1).

Then, we get

f ∗1 (x) = 6x3 + w27x2 + w3x + 1,

f ∗2 (x) = 6x3 + w45x2 + w21x + 1.

Thus, B⊥β,i = 〈f∗i (x)〉 are dual cyclic codes of length 5

over F49, for i = 1, 2. Therefore, B⊥β = η1B⊥β,1 ⊕η2 B⊥β,2 is a dual cyclic code of length 5 over R and B⊥β =〈η1f ∗1 (x), η2f

∗

2 (x)〉 = 〈f′(x)〉, where

f ′(x) = η1f ∗1 (x) + η2f∗

2 (x)

= 6x3 + (4+ uw28)x2 + (3+ uw4)x + 1.

Also | B⊥β |= 494. �Using similar arguments as in the case of cyclic codes over

R, we obtain the following results on cyclic codes over S.Theorem 15: Let Cγ = ζ1Cγ,1⊕ζ2Cγ,2⊕ζ3 Cγ,3⊕ζ4 Cγ,4

be a linear code of length γ over S. Then Cγ is a cyclic codeof length γ over S if and only if Cγ,j are cyclic codes of lengthγ over Fq, where j = 1, 2, 3, 4.Corollary 16: Let Cγ = ζ1Cγ,1 ⊕ ζ2Cγ,2 ⊕ ζ3 Cγ,3 ⊕

ζ4 Cγ,4 be a cyclic code of length γ over S. Then the dualC⊥γ = ζ1C

⊥

γ,1⊕ζ2C⊥

γ,2⊕ζ3 C⊥

γ,3⊕ζ4 C⊥

γ,4 is a cyclic code oflength γ over S if and only if C⊥γ,j are cyclic codes of lengthγ over Fq, for j = 1, 2, 3, 4.

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H. Q. Dinh et al.: On the Structure of Cyclic Codes Over FqRS

Theorem 17: Let Cγ = ζ1Cγ,1⊕ζ2Cγ,2⊕ζ3 Cγ,3⊕ζ4 Cγ,4be a cyclic code of length γ over S and sj(x) be generatormonic polynomial of Cγ,j, for j = 1, 2, 3, 4. Then1) Cγ = 〈ζ1s1(x), ζ2s2(x), ζ3s3(x), ζ4s4(x)〉 and| Cγ |= q4γ−(deg(s1)+deg(s2)+deg(s3)+deg(s4)).

2) Cγ = 〈s(x)〉, where s(x) = ζ1s1(x) + ζ2s2(x) +ζ3s3(x) + ζ4s4(x) such that s(x) | (xγ − 1).

Corollary 18: Let Cγ = ζ1Cγ,1 ⊕ ζ2Cγ,2 ⊕ ζ3 Cγ,3 ⊕ζ4 Cγ,4 be a cyclic code of length γ over S. Suppose sj(x)are the generator monic polynomials of Cγ,j and g∗j (x) are thereciprocal polynomials of gj(x) such that xγ −1 = gj(x)sj(x),for j = 1, 2, 3, 4. Then

1) C⊥γ = 〈ζ1g∗1(x), ζ2g∗

2(x), ζ3g∗

3(x), ζ4g∗

4(x)〉 and |C⊥γ |= q(deg(s1)+deg(s2)+deg(s3)+deg(s4)).

2) C⊥γ = 〈g′(x)〉, where g′(x) = ζ1g∗1(x) + ζ2g

∗

2(x) +ζ3g∗3(x) + ζ4g

∗

4(x).Similar to Example 14, here we present an example to

illustrate the above results of cyclic codes over S.Example 19: Let γ = 4 and S = F9 + uF9 + vF9 +

uvF9, where u2 = 1, v2 = 1, uv = vu. Clearly x2 + 2x +2 is irreducible in F3, fix w to be a zero of the polynomialx2 + 2x + 2 in F9. We view F9 =

F3[x]〈x2+2x+2〉

, then

x4 − 1 = (x + 1)(x + 2)(x + w2)(x + w6) ∈ F9[x].

Let

s1(x) = (x + 1)(x + 2), s2(x) = (x + 1)(x + w2),

s3(x) = (x + 2)(x + w2), s4(x) = (x + w2)(x + w6).

Then Cγ,j(x) = 〈sj(x)〉 are cyclic codes of length 4 over F9,for j = 1, 2, 3, 4. Therefore, Cγ = ζ1Cγ,1 ⊕ ζ2Cγ,2 ⊕ζ3 Cγ,3 ⊕ ζ4 Cγ,4 is a cyclic code of length 4 over S, whereζj; j = 1, 2, 3, 4 are as above.

Thus, Cγ = 〈s(x)〉, where

s(x) = ζ1s1(x) + ζ2s2(x)+ ζ3s3(x) + ζ4s4(x)

=14(1+ u+ v+ uv)(x + 1)(x + 2)

+14(1+ u− v− uv)(x + 1)(x + w2)

+14(1− u+ v− uv)(x + 2)(x + w2)

+14(1− u− v+ uv)(x + w2)(x + w6)

= x2 + (w6+ 2u+ v+ uvw2)x + (uw5

+ vw7).

By direct computation we get,

x4 − 1 = (x2 + (w2+ u+ 2v+ uvw6)x + (uw3

+ vw))

×(x2 + (w6+ 2u+ v+ uvw2)x + (uw5

+ vw7)).

Thus, s(x) | (x4−1). Also | Cγ |= 932−8 = 924. For the dualcode of Cγ , note that

g1(x) = x2 + 1, g2(x) = x2 + w3x + w2,

g3(x) = x2 + w5x + w6, g4(x) = x2 + 2.

Then the reciprocal polynomials are as follows,

g∗1(x) = x2 + 1, g∗2(x) = w2x2 + w3x + 1,

g∗3(x) = w6x2 + w5x + 1, g∗4(x) = 2x2 + 1.

Then C⊥γ,j(x) = 〈gj(x)〉 are dual cyclic codes of length 4over F9, for j = 1, 2, 3, 4 and C⊥γ = ζ1C⊥γ,1 ⊕ ζ2C

⊥

γ,2 ⊕

ζ3 C⊥γ,3 ⊕ ζ4 C⊥

γ,4 is a dual cyclic code of length 4 over S.Therefore, C⊥γ = 〈g

′(x)〉, where

g′(x) = ζ1g∗1(x) + ζ2g∗

2(x)+ ζ3g∗

3(x) + ζ4g∗

4(x)

= (2+ uw6+ vw2

+ 2uv)x2

+(w2+ u+ 2v+ w6uv)x + (2+ 2u+ 2v+ uv).

Also | C⊥γ |= 98. �

C. FqRS-CYCLIC CODESIn Theorem 9, Theorem 12 and Theorem 17, we have dis-cussed the generator polynomials for cyclic codes over Fq,Rand S respectively. Now we present the generator polynomi-als for FqRS-cyclic codes as follows.Theorem 20: Let C be a FqRS-cyclic code. Then

C = 〈(a(x)|0|0), (0|r(x)|0), (l1(x)|l2(x)|s(x))〉, wherea(x)|(xα − 1), r(x)|(xβ − 1), s(x)|(xγ − 1) and l1(x) ∈Fq[x]/〈xα − 1〉, l2(x) ∈ R[x]/〈xβ − 1〉.

Proof: From Theorems 9, 12 and 17, we have Aα =〈a(x)〉, Bβ = 〈r(x)〉 and Cγ = 〈s(x)〉 such that a(x)|(xα − 1),r(x)|(xβ − 1), s(x)|(xγ − 1). Then the proof follows fromTheorem 3.1 of [47]. �A FqRS-linear code C of length (α, β, γ ) is called a sepa-

rable code if C = A′α ⊗ B′β ⊗ C

′γ , while considering A

′α,B′β ,

and C ′γ as punctured codes of C by deleting the coordinatesoutside the α, β and γ components, respectively.Lemma 21: Let C = 〈(a(x)|0|0), (0|r(x)|0), (l1(x)|l2(x)|

s(x))〉 be a FqRS-cyclic code. Then1) deg(l1(x)) ≤ deg(a(x)), deg(l2(x)) ≤ deg(r(x)) and

a(x)|r2(x)l1(x), r(x)|s4(x)l2(x);2) A′α = 〈gcd(a(x), l1(x))〉, B

′β = 〈gcd(r(x), l2(x))〉, and

C ′γ = 〈s(x)〉.Proof: Proof is parallel to that of Lemmas 3.2, 3.3

and 3.4 of [47]. �Lemma 22: Let C = 〈(a(x)|0|0), (0|r(x)|0), (l1(x)|l2(x)|

s(x))〉 be a FqRS-cyclic code. Then1) a(x)|l1(x) if and only if l1(x) = 0,2) r(x)|l2(x) if and only if l2(x) = 0.Proof: Similar to Lemma 5.8 and Lemma 5.9 of [47]. �

The following Lemma is a direct consequence ofLemma 22.Lemma 23: Let C = 〈(a(x)|0|0), (0|r(x)|0), (l1(x)|l2(x)|

s(x))〉 be a FqRS-cyclic code. Then the following areequivalent:1) C is separable;2) a(x)|l1(x), r(x)|l2(x);3) C = 〈(a(x)|0|0), (0|r(x)|0), (0|0|s(x))〉.Thus, for a separable code, we get

A′α = 〈gcd(a(x), 0)〉 = 〈a(x)〉 = Aα,B′β = 〈gcd(r(x), 0)〉 = 〈r(x)〉 = Bβ ,C ′γ = 〈s(x)〉 = Cγ .

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Theorem 24: Suppose C = Aα⊗Bβ ⊗Cγ is a FqRS-linearcode of length (α, β, γ ), where Aα,Bβ and Cγ are linearcodes of length α, β and γ over Fq,R and S respectively.Then C is a FqRS-cyclic code of length (α, β, γ ) if and onlyif Aα,Bβ and Cγ are cyclic codes of length α, β and γ overFq,R and S, respectively.

Proof: Let C be a FqRS-cyclic code of length (α, β, γ )and (a0, a1, . . . , aα−1, b0, b1, . . . , bβ−1, c0, c1, . . . , cγ−1) ∈C, where (a0, a1, . . . , aα−1) ∈ Aα , (b0, b1, . . . , bβ−1) ∈ Bβand (c0, c1, . . . , cγ−1) ∈ Cγ . As C is a FqRS-cyclic code,we get

(aα−1, a0, a1, . . . , aα−2, bβ−1, b0, b1, . . . , bβ−2,

cγ−1, c0, c1, . . . , cγ−2) ∈ C,

which implies (aα−1, a0, . . . , aα−2) ∈ Aα , (bβ−1, b0, . . . ,bβ−2) ∈ Bβ and (cγ−1, c0, c1, . . . , cγ−2) ∈ Cγ . Therefore,Aα,Bβ and Cγ are cyclic codes of length α, β and γ overFq,R and S, respectively.

Conversely, suppose Aα,Bβ and Cγ are cyclic codesof length α, β and γ over Fq,R and S. Let (a′0, a

′

1, . . . ,

a′α−1) ∈ Aα , (b′

0, b′

1, . . . , b′

β−1) ∈ Bβ and (c′0, c′

1, . . . , c′

γ−1)∈ Cγ , then (a′α−1, a

′

0, . . . , a′

α−2) ∈ Aα , (b′

β−1, b′

0, . . . , b′

β−2)∈ Bβ and (c′γ−1, c

′

0, c′

1, . . . , c′

γ−2) ∈ Cγ . Therefore,

(a′α−1, a′

0, a′

1, . . . , a′

α−2, b′

β−1, b′

0, b′

1, . . . , b′

β−2,

c′γ−1, c′

0, c′

1, . . . , c′

γ−2) ∈ Aα ⊗ Bβ ⊗ Cγ = C.

Hence, C is a FqRS-cyclic code of length (α, β, γ ). �By Theorems 10, 15 and 24, we have the following

corollary.Corollary 25: Suppose C = Aα⊗Bβ⊗Cγ is a FqRS-linear

code of length (α, β, γ ), where Aα,Bβ and Cγ are linearcodes of length α, β and γ over Fq,R and S, respectively.Then C is a FqRS-cyclic code of length (α, β, γ ) if and only ifAα, Bβ,i and Cγ,j are cyclic codes of length α, β and γ overFq,R and S, where i = 1, 2; j = 1, 2, 3, 4.In Theorem 20, we have studied the generator polynomial

for a FqRS-cyclic code of length (α, β, γ ). Now here westudy the generator polynomial for a separable FqRS-cycliccode of length (α, β, γ ) as follows.Theorem 26: Let C = Aα ⊗ Bβ ⊗ Cγ be a FqRS-cyclic

code of length (α, β, γ ), where Aα = 〈a(x)〉, Bβ = 〈r(x)〉and Cγ = 〈s(x)〉. Then C = 〈a(x)〉 ⊗ 〈r(x)〉 ⊗ 〈s(x)〉.Example 27: Let α = 4, β = 5 and γ = 8. Denote

Sα,β,γ =F81[x]〈x4−1〉

×R[x]〈x5−1〉

×S[x]〈x8−1〉

, where R = F81 +

uF81(u2 = 1) and S = F81 + uF81 + vF81 + uvF81(u2 =1, v2 = 1, uv = vu). Take F81 =

F3[x]〈x4+2x3+2〉

and w is a zero

of the polynomial x4 + 2x3 + 2 in F81, then

x4 − 1 = (x + 1)(x + 2)(x + w20)(x + w60) ∈ F81[x].

Let a(x) = (x + 2)(x + w20). Then Aα = 〈a(x)〉 is a cycliccode of length 4 over F81.

x5−1 = (x+2)(x+w8)(x+w24)(x+w56)(x+w72)∈F81[x].

Let r1(x) = (x+w8)(x+w24) and r2(x) = (x+w56)(x+w72).Then Bβ,1 = 〈r1(x)〉 and Bβ,2 = 〈r2(x)〉 are cyclic codes of

length 5 over F81. Therefore, Bβ = 〈r(x)〉 is a cyclic code oflength 5 over R, where

r(x) = η1r1(x) + η2r2(x)

=12(1− u)(x2 + w46x + w32)

+12(1+ u)(x2 + w14x + w48)

=12{2x2 + x(w46

+ w14)+ (w32+ w48)

+ux(w14− w46)+ u(w48

− w32)}

=12{2x2 + x + w70

+ uw35x + uw55}

= x2 + (2+ uw75)x + (w30+ uw15).

x8 − 1 = (x + 1)(x + 2)(x + w10)(x + w20)(x + w30)

×(x + w50)(x + w60)(x + w70) ∈ F81[x].

Let

s1(x) = (x + 1)(x + w10), s2(x) = (x + 2)(x + w20),

s3(x) = x + w30)(x + w50), s4(x) = (x + w60)(x + w70).

Then Cγ,j(x) = 〈sj(x)〉 are cyclic codes of length 8 over F81,for j = 1, 2, 3, 4. Therefore, Cγ = 〈s(x)〉 is a cyclic code oflength 8 over S, where

s(x) = ζ1s1(x) + ζ2s2(x)+ ζ3s3(x) + ζ4s4(x)

=14(1+ u+ v+ uv)(x2 + w20x + w10)

+14(1+ u− v− uv)(x2 + w10x + w60)

+14(1− u+ v− uv)(x2 + w20x + 1)

+14(1− u− v+ uv)(x2 + x + w50)

= x2 + (uw70+ vw20

+ uvw30)x

+(uw20+ 2v+ uvw10).

Thus, C = 〈(a(x)|0|0), (0|r(x)|0), (0|0|s(x))〉 = 〈a(x)〉 ⊗〈r(x)〉 ⊗ 〈s(x)〉 is a separable FqRS-cyclic code of length(4, 5, 8), where a(x), r(x) and s(x) are as above. �

VI. APPLICATIONProperties of mixed alphabet cyclic codes has been studiedin [47]. In the next two sections we mainly focus on theapplications of separable FqRS-cyclic codes.

A. QUANTUM ERROR-CORRECTING CODES (QECCS)FROM FqRS-CYCLIC CODESBy [3], let H be a Hilbert space of dimension q over thecomplex numbers C. Define H⊗n to be n-fold tensor productof the Hilbert space H , that is, H⊗n = H ⊗ H ⊗ · · · ⊗H (n-times). Then H⊗n is a Hilbert space of dimension qn.A QECC of length n and dimension k over Fq is defined tobe a Hilbert subspace of H⊗n having dimension qk . A QECCwith length n, dimension k and minimum distance d over Fqis denoted by [[n, k, d]]q.

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One of the important advancements in the construc-tion of codes is the construction of QECCs from classicalerror-correcting codes. Errors prompted by inevitable inter-action with environments - such as decoherence, quantumnoise and other inaccuracies are among the prominent imped-iments leading to erroneous in quantum information. QECCssecures the quantum information from being exploited bysuch inaccuracy. Chronologically speaking, the first QECCwas studied independently by Steane [44] and Shor [45].However the construction of QECCs from classical codes,their existence proofs, and correction methods were givenby Calderbank et al. [18]. Later many QECCs have beenconstructed using ideas of [18] over finite fields andfinite rings (See [2], [3], [11], [13], [24]–[26], [29], [30], [32],[33], [41]).Theorem 28 [18] (CSS Construction): Let C1 and C2 be

[n, k1, d1] and [n, k2, d2] linear codes over GF(q) respec-tively with C⊥2 ⊆ C1. Furthermore, let d = min{d1, d2}. Thenthere exists a QECC, C with parameters [[n, k1+k2−n, d]]q.In particular, if C⊥1 ⊆ C1, then there exists a QECC withparameters [[n, 2k1 − n, d1]]q

Herewe recall the dual containing property for cyclic codesfrom [18].Lemma 29 [18]: Let C be a cyclic code of length n with

generator polynomial m(x) over Fq. Then C contains its dualcode if and only if xn−1 ≡ 0 (mod m(x)m∗(x)), where m∗(x)is the reciprocal polynomial of m(x).Now we present the9-image of FqRS-linear codes, which

will take a crucial role in our construction of QECCs fromFqRS-cyclic codes.Proposition 30: Let C be a FqRS-linear code of length

(α, β, γ ). Then 9(C) = Aα ⊗ Bβ,1 ⊗ Bβ,2 ⊗ Cγ,1 ⊗Cγ,2 ⊗ Cγ,3 ⊗ Cγ,4 is a linear code of length (α +2β + 4γ ) over Fq, where Aα,Bβ,i and Cγ,j are codes oflength α, β and γ over Fq, for i = 1, 2; j = 1, 2, 3, 4.Moreover, |9(C)| = |Aα||Bβ,1||Bβ,2||Cγ,1||Cγ,2||Cγ,3||Cγ,4|and dH (9(C)) = min{dH (Aα), dH (Bβ,i), dH (Cγ,j)}, wherei = 1, 2; j = 1, 2, 3, 4.

Proof: Follows from the definition of ⊗ and 9(C). �Now we give our main result to construct QECCs from

FqRS-cyclic codes.Theorem 31: Let C = Aα⊗Bβ⊗Cγ be a FqRS-cyclic code

of length (α, β, γ ). If A⊥α ⊆ Aα,B⊥β,i ⊆ Bβ,i and C⊥γ,j ⊆ Cγ,j,for i = 1, 2; j = 1, 2, 3, 4, then there exists a QECC withparameters [[(α + 2β + 4γ ), 2k − (α + 2β + 4γ ), dH ]],where dH denotes the Hamming distance and k denotes thedimension of the code 9(C), respectively.

Proof: As 9(C) = Aα ⊗ Bβ,1 ⊗ Bβ,2 ⊗ Cγ,1 ⊗ Cγ,2 ⊗Cγ,3 ⊗ Cγ,4, it is easy to check that 9(C)⊥ = A⊥α ⊗ B⊥β,1 ⊗B⊥β,2⊗C

⊥

γ,1⊗C⊥

γ,2⊗C⊥

γ,3⊗C⊥

γ,4. Now let A⊥α ⊆ Aα,B⊥β,i ⊆Bβ,i and C⊥γ,j ⊆ Cγ,j, for i = 1, 2; j = 1, 2, 3, 4 holds. Then9(C)⊥ ⊆ 9(C). Thus, by Theorem 28, there exists a QECCwith parameters [[(α+2β+4γ ), 2k−(α+2β+4γ ), dH ]]q,where dH denotes the Hamming distance of the code 9(C)and k denotes the dimension of the code 9(C). �

Example 32: Let α = 10, β = 15, γ = 20 and Sα,β,γ =F5[x]〈x10−1〉

×R[x]〈x15−1〉

×S[x]〈x20−1〉

, where R = F5+uF5(u2 = 1) and

S = F5 + uF5 + vF5 + uvF5(u2 = 1, v2 = 1, uv = vu).

x10 − 1 = (x + 1)5(x + 4)5 ∈ F5[x].

Let a(x) = (x+1)2(x+4). Then Aα = 〈a(x)〉 is a cyclic codeof length 10 over F5 with parameters [10, 7, 3].

x15 − 1 = (x + 4)5(x2 + x + 1)5 ∈ F5[x].

Let r1(x) = (x + 4)2(x2 + x + 1) and r2(x) = (x2 + x +1)2. Then Bβ,1(x) = 〈r1(x)〉 and Bβ,2(x) = 〈r2(x)〉 are cycliccodes having same parameters [15, 12, 3]. Therefore, Bβ =〈η1r1(x), η2r2(x)〉 is a cyclic code of length 15 over R.

x20 − 1 = (x + 1)5(x + 2)5(x + 3)5(x + 4)5 ∈ F5[x].

Let s1(x) = (x+ 1)2(x+ 2), s2(x) = (x+ 2)2(x+ 4), s3(x) =(x + 1)2(x + 3) and s4(x) = (x + 1)(x + 3)2. Then Cγ,i(x) =〈si(x)〉 and Cγ,j(x) = 〈sj(x)〉 are cyclic codes of length 20with same parameters [20, 17, 3], where i = 1, 2 and j =3, 4. Therefore, Cγ = 〈ζ1s1(x), ζ2s2(x), ζ3s3(x), ζ4s4(x)〉 is acyclic code of length 20 over S.Thus, 9(C) is a linear code with parameters [120, 99, 3]

overF5. Note that a(x)a∗(x) divides x10−1; ri(x)r∗i (x) dividesx15 − 1, for i = 1, 2; and sj(x)s∗j (x) divides x

20− 1, for j =

1, 2, 3, 4. Hence, by Lemma 29, we getA⊥α ⊆ Aα,B⊥β,i ⊆ Bβ,iand C⊥γ,j ⊆ Cγ,j, i = 1, 2; j = 1, 2, 3, 4. Then by Theo-rem 31, we get 9(C)⊥ ⊆ 9(C). Therefore, by Theorem 31,there exists a QECC with parameters [[120, 96, 3]]5. �Example 33: Let α = 7, β = 14, γ = 21 and Sα,β,γ =

F7[x]〈x7−1〉

×R[x]〈x14−1〉

×S[x]〈x21−1〉

, where R = F7 + uF7 (u2 = 1)

and S = F7 + uF7 + vF7 + uvF7 (u2 = 1, v2 = 1, uv = vu).

x7 − 1 = (x + 6)7 ∈ F7[x].

Let a(x) = (x + 6)2. Then Aα = 〈a(x)〉 is a cyclic code oflength 7 over F7 with parameters [7, 5, 3].

x14 − 1 = (x + 1)7(x + 6)7 ∈ F7[x].

Let r1(x) = (x + 1)2(x + 6) and r2(x) = (x + 1)(x + 6)2.Then Bβ,i(x) = 〈ri(x)〉 are cyclic codes of length 14 havingsame parameters [14, 11, 3], where i = 1, 2. Therefore, Bβ =〈η1r1(x), η2r2(x)〉 is a cyclic code of length 14 over R.

x21 − 1 = (x + 3)7(x + 5)7(x + 6)7 ∈ F7[x].

Let s1(x) = (x + 3)(x + 5)2, s2(x) = (x + 5)(x + 6)2,s3(x) = s4(x) = (x + 3)(x + 6)2. Then Cγ,j(x) =〈sj(x)〉 are cyclic codes of length 21 having same param-eters [21, 18, 3], for j = 1, 2, 3, 4. Therefore, Cγ =

〈ζ1s1(x), ζ2s2(x), ζ3s3(x), ζ4s4(x)〉 is a cyclic code oflength 21 over S.Thus, 9(C) is a linear code with parameters [119, 99, 3]

over F7. Because a(x)a∗(x) divides x7− 1, ri(x)r∗i (x) dividesx14 − 1, for i = 1, 2; and sj(x)s∗j (x) divides x

21− 1, for j =

1, 2, 3, 4; it follows from Lemma 29 that A⊥α ⊆ Aα,B⊥β,i ⊆Bβ,i and C⊥γ,j ⊆ Cγ,j, i = 1, 2; j = 1, 2, 3, 4. Then by

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TABLE 1. New QECCs from cyclic codes over Fp.

TABLE 2. New QECCs from cyclic codes over R.

Theorem 3.1, we get9(C)⊥ ⊆ 9(C). Hence, by Theorem 31,there exists a QECC with parameters [[119, 79, 3]]7. �Example 34: Let α = 12, β = 15, γ = 30 and Sα,β,γ =

F25[x]〈x12−1〉

×R[x]〈x15−1〉

×S[x]〈x30−1〉

, where R = F25+ uF25 (u2 = 1),

and S = F25 + uF25 + vF25 + uvF25 (u2 = 1, v2 = 1,uv = vu). Take F25 =

F5[x]〈x2+4x+2〉

and w is a zero of the

polynomial x2 + 4x + 2 in F25, then

x12 − 1 = (x + 1)(x + 2)(x + 3)(x + 4)(x + w2)(x + w4)

×(x + w8)(x + w10)(x + w14)(x + w16)

×(x + w20)(x + w22) ∈ F25[x].

Let a(x) = (x+w4)(x+w8)(x+w10). Then Aα = 〈a(x)〉 is acyclic code of length 12 over F25 with parameters [12, 9, 3].

x15 − 1 = (x + 4)5(x + w4)5(x + w20)5 ∈ F25[x].

Let r1(x) = (x + w4)2(x + 4) and r2(x) = (x + w20)2

(x + 4). Then Bβ,1(x) = 〈r1(x)〉 and Bβ,2(x) = 〈r2(x)〉 arecyclic codes of length 15 having same parameters [15, 12, 3].Therefore, Bβ = 〈η1r1(x), η2r2(x)〉 is a cyclic code oflength 15 over R.

x30 − 1 = (x + 1)5(x + 4)5(x + w4)5(x + w8)5

×(x + w16)5(x + w20)5 ∈ F25[x].

Let s1(x) = (x + w8)(x + 1)2(x + 4), s2(x) = (x + w4)(x + 2)2(x + 1), s3(x) = (x + w16)2(x + w20)(x + 4)and s4(x) = (x + w20)2(x + w4)(x + 1). ThenCγ,j(x) = 〈sj(x)〉 are cyclic codes of length 30 havingsame parameters [30, 26, 3] for j = 1, 2, 3, 4. Therefore,Cγ = 〈ζ1s1(x), ζ2s2(x), ζ3s3(x), ζ4s4(x)〉 is a cyclic code oflength 30 over S.Thus, 9(C) is a linear code with parameters [162, 137, 3]

overF25. Clearly, a(x)a∗(x) divides x12−1; ri(x)r∗i (x) divides

x15 − 1, for i = 1, 2; and sj(x)s∗j (x) divides x30 − 1,for j = 1, 2, 3, 4. Hence, by Lemma 29, we get A⊥α ⊆Aα,B⊥β,i ⊆ Bβ,i and C⊥γ,j ⊆ Cγ,j, i = 1, 2; j = 1, 2, 3, 4.Then by Theorem 31, we get 9(C)⊥ ⊆ 9(C). Therefore, byTheorem 31, there exists a QECC with parameters[[162, 112, 3]]25. �In TABLEs 1, 2, 3, and 4, we present some new QECCs

with better parameters from our study of cyclic codes overFp,R, S and FpR, respectively. We denote A,B,C,D,E,FandG, to represent the numbers 10, 11, 12, 13, 14, 15 and 16,respectively.

For simpicity in calculation, in TABLEs 2 and 4, we takethe generator r1(x) = r2(x) for the corresponding cycliccodes Bβ,i for i = 1, 2. Similarly, in TABLE 3, we takethe generator s1(x) = s2(x) = s3(x) = s4(x) for the cor-responding cyclic codes Cγ,j for j = 1, 2, 3, 4. We write thecoefficients of the generator polynomials in decreasing order,for example, we write 120C4F to represent the polynomialx5 + 2x4 + 12x2 + 4x + 15.In TABLE 4, the fifth column presents parameters of

Gray images over FpR, which is the restriction on 9

over FpR.

B. LINEAR COMPLEMENTARY DUAL CODESA linear code C is called LCD or linear complementarydual code if C ∩ C⊥ = {0}. LCD codes were first intro-duced by Massey [38]. This family of codes have showneffectiveness against side-channel attacks(SCA) and faultinjection attacks(FIA) to improve the security related infor-mation on sensitive devices [19]. Authors have exploredproperties of LCD codes with various conditions andstructures in [34], [35], [48].

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TABLE 3. New QECCs from cyclic codes over S.

TABLE 4. New QECCs from cyclic codes over FpR.

In this section, we briefly discuss the LCD codes over Fq,R and FqRS and give some examples for better understandingof our study.Lemma 35 [38]: Let C be a cyclic code over Fq generated

by f (x). Then C is LCD if and only if f (x) is self-reciprocal.Proposition 36: Let Bβ = 〈η1r1(x), η2r2(x)〉 be a cyclic

code over R. Then Bβ is a LCD code over R if and only ifri(x) are self-reciprocal polynomials over Fq, for i = 1, 2.Proof: Let Bβ be a LCD code over R, i.e. Bβ ∩ B⊥β = {0}.

Note that

Bβ ∩ B⊥β = η1(Bβ,1 ∩ B⊥

β,1)⊕ η2(Bβ,2 ∩ B⊥

β,2).

Then by the definition of ⊕, we get Bβ ∩ B⊥β = {0},whenever both Bβ,1 ∩ B⊥β,1 = {0} and Bβ,2 ∩ B

⊥

β,2 = {0},in other words, whenever both Bβ,1 and Bβ,2 are LCD codesover Fq.Hence, by Lemma 35, ri(x) are self-reciprocal polynomials

over Fq, for i = 1, 2.Conversely, let ri(x) be a self-reciprocal polynomial

over Fq, for i = 1, 2. As ri(x) is the monic generator poly-nomial of Bβ,i, for i = 1, 2, by Lemma 35 we get, Bβ,1 andBβ,2 are LCD codes over Fq. Thus, Bβ,1 ∩ B⊥β,1 = {0} andBβ,2 ∩ B⊥β,2 = {0}. Also as Bβ = η1Bβ,1 ⊕ η2Bβ,2,

Bβ ∩ B⊥β = (η1Bβ,1 ⊕ η2Bβ,2) ∩ (η1B⊥β,1 ⊕ η2B⊥

β,2)

= η1(Bβ,1 ∩ B⊥β,1)⊕ η2(Bβ,2 ∩ B⊥

β,2) = {0}.

Hence, Bβ is a LCD code over R. �Using similar arguments we can prove the following result

over S.Proposition 37: Let Cγ = 〈ζ1s1(x), ζ2s2(x), ζ3s3(x),

ζ4s4(x)〉 be a cyclic code over S. Then Cγ is a LCD code overS if and only if sj(x) are self-reciprocal polynomials over Fq,for j = 1, 2, 3, 4.Proposition 38: Let C be a FqRS-cyclic code of length

(α, β, γ ). Then 9(C) = Aα ⊗ Bβ,1 ⊗ Bβ,2 ⊗ Cγ,1 ⊗ Cγ,2 ⊗Cγ,3 ⊗ Cγ,4 is a LCD code of length (α + 2β + 4γ ) over Fqif and only if Aα,Bβ,i and Cγ,j are LCD codes of length α, βand γ over Fq, for i = 1, 2; j = 1, 2, 3, 4.

Proof: Note that 9(C) ∩9(C)⊥

= (Aα ⊗ Bβ,1 ⊗ Bβ,2 ⊗ Cγ,1 ⊗ · · · ⊗ Cγ,4)

∩(A⊥α ⊗ B⊥

β,1 ⊗ B⊥

β,2 ⊗ C⊥

γ,1 ⊗ · · · ⊗ C⊥

γ,4)

= (Aα ∩ A⊥α )⊗ (Bβ,1 ∩ B⊥β,1)⊗ (Bβ,2 ∩ B⊥β,2)

⊗(Cγ,1 ∩ C⊥γ,1)⊗ · · · ⊗ (Cγ,4 ∩ C⊥γ,4).

Therefore,9(C)∩9(C)⊥ = {0} if and only if Aα∩A⊥α = {0},Bβ,i ∩ B⊥β,i = {0}, and Cγ,j ∩ C

⊥γ,j = {0}, for i = 1, 2;

j = 1, 2, 3, 4. Hence, 9(C) is a LCD code if and only ifAα,Bβ,i and Cγ,j are LCD codes over Fq, for i = 1, 2;j = 1, 2, 3, 4. �Example 39: Let R = F7+ uF7, where u2 = 1 and β = 6.

x6 − 1 = (x+1)(x+2)(x+3)(x + 4)(x + 5)(x + 6) ∈ F7[x].

Let r1(x) = (x + 3)(x + 5) = x2 + x + 1 and r2(x) = (x + 2)(x + 4) = x2 + x + 1. Then both r1(x) and r2(x) areself-reciprocal. Therefore, Bγ,1 = 〈r1(x)〉 and Bβ,2 =〈r2(x)〉 are LCD codes of length 6 over F7. Hence,Bβ = 〈η1r1(x), η2r2(x)〉 is a LCD code over R. �Example 40: Let R = F3 + uF3, where u2 = 1 and

S = F3+ uF3+ vF3+ uvF3 where u2 = 1, v2 = 1, uv = vu.Take α = 4, β = 7 and γ = 12. Then

x4 − 1 = (x + 1)(x + 2)(x2 + 1) ∈ F3[x].

Let a(x) = (x + 1)(x2 + 1). As a(x) is self-reciprocal,Aα = 〈a(x)〉 is a LCD code over F3.

x7 − 1 = (x + 2)(x6 + x5 + x4 + x3 + x2 + x + 1) ∈ F3[x].

Let r1(x) = r2(x) = (x6+x5+x4+x3+x2+x+1). As ri(x)are self-reciprocal, Bβ,i = 〈bi(x)〉 are LCD codes over F3, fori = 1, 2.

x12 − 1 = (x + 1)3(x + 2)3(x2 + 1)3 ∈ F3[x].

Let s1(x) = (x+ 1), s2(x) = (x2+ 1), s3(x) = (x2+ 1)2 ands4(x) = (x2 + 1)3. As sj(x) are self reciprocal, Cγ,j = 〈sj(x)〉are LCD codes over F3, for j = 1, 2, 3, 4. Hence, 9(C) is aLCD code of length 66 over F3. �Example 41: Let R = F9 + uF9, where u2 = 1 and

S = F9+ uF9+ vF9+ uvF9 where u2 = 1, v2 = 1, uv = vu.

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Let α = 4, β = 7 and γ = 16. Take F9 =F3[x]

〈x2+2x+2〉and w is

a zero of the polynomial x2 + 2x + 2 in F9, then

x4 − 1 = (x + 1)(x + 2)(x + w2)(x + w6) ∈ F9[x],

x7 − 1 = (x + 2)(x3 + wx2 + w7x + 2)

×(x3 + w3x2 + w5x + 2) ∈ F9[x],

x16 − 1 = (x + 1)(x + 2)(x + w)(x + w2)(x + w3)(x + w5)

×(x + w6)(x + w7)(x2 + w)(x2 + w3)

×(x + w5)(x + w7) ∈ F9[x].

Let a(x) = (x + w2)(x + w6), ri(x) = (x3 + wx2 + w7x +2)(x3+w3x2+w5x+2) and sj(x) = (x2+w7)(x2+w5)(x2+w)(x2+w3). Note that a(x), ri(x) and sj(x) are self-reciprocal,for i = 1, 2; j = 1, 2, 3, 4.Thus, arguing as above, we get thatAα,Bβ,i and Cγ,j are LCD codes over F9, for i = 1, 2; j =1, 2, 3, 4. Hence,9(C) is a LCD code of length 82 over F9.�

VII. CONCLUSIONIn this paper, we first discussed the cyclic codes over R andS, then using these structures we studied the concatenatedstructure of FqRS-cyclic codes. We defined a Gray map overFqRS and discussed some properties of FqRS-cyclic codes.As an application of our study, we constructed quantumcodes from FqRS-cyclic codes. We also studied LCD codesas another application of the FqRS-cyclic codes. This studycan be generalized over product of finite rings. In our recentwork [12], we gave necessary and sufficient conditions ofdual-containing property for a skew cyclic, skew negacyclic,self-dual skew constacyclic codes over a finite ring. So takingdirect product of cyclic, constacyclic, skew cyclic and skewconstacyclic codes over finite rings, one can also constructQECCs.

ACKNOWLEDGMENTT. Bag is thankful to the University Grant Commission(UGC), Govt. of India, for financial support under Sr.No. 2061441025 with Ref No. 22/06/2014(i)EU-V. Apartof this paper was written during a stay of H.Q. Dinh inthe Vietnam Institute For Advanced Study in Mathematics(VIASM), he would like to thank the members of VIASM fortheir hospitality. H.Q. Dinh andW. Chinnakum are grateful tothe Centre of Excellence in Econometrics, Chiang Mai Uni-versity, Thailand, for partial financial support. This researchis partially supported by the Research Administration Centre,Chaing Mai University, Thailand.

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HAI Q. DINH received the B.Sc., M.Sc.,and Ph.D. degrees in mathematics from OhioUniversity, USA, in 1998, 2000, and 2003, respec-tively. He worked for one year as a Visiting Pro-fessor with North Dakota State University, USA.Since 2004, he has been with Kent State Univer-sity, USA, as a Tenured Professor of mathemat-ics, where he is currently a Professor of appliedmathematics with theDepartment ofMathematicalSciences. His research interest includes algebra

and coding theory. Since 2004, he has been publishing more than 80 articlesat high level SCI (E) research journals such as Journal of Algebra, Journalof Pure and Applied Algebra, the IEEE TRANSACTIONS IN INFORMATION THEORY,the IEEE COMMUNICATION LETTERS, Finite Fields and Their Applications,Applicable Algebra in Engineering Communication and Computing, andDiscrete Applied Mathematics. He has been a well known invited/keynotespeaker at numerous international conferences and mathematics colloquium.Other than universities in the US, he also gave many honorary tutoriallectures at international universities in China, Indonesia, Kuwait, Mexico,Singapore, Thailand, and Vietnam.

TUSHAR BAG received the B.Sc. degree fromRamakrishnaMission Vidymandira under the Uni-versity of Calcutta and the M.Sc. degree fromIIT Kanpur, India. He is currently pursuing thePh.D. with the Department of Mathematics, IITPatna, India. He has published 12 articles. Hismain research focus is on algebraic coding theoryand codes over rings.

ASHISH KUMAR UPADHYAY received the B.Sc.and M.Sc. degrees in mathematics from the Uni-versity of Allahabad, India, and the Ph.D. degreefrom the Indian Institute of Science, in 2005. He iscurrently an Associate Professor with the Depart-ment of Mathematics, IIT Patna. His researchinterests include algebraic coding theory and alge-braic topology.

RAMAKRISHNA BANDI received the Ph.D.degree from the IIT Roorkee, Roorkee. He is cur-rently an Assistant Professor with the DepartmentofMathematics, International Institute of Informa-tion Technology, Naya Raipur. His area of interestsis algebraic coding theory and information theory.

WARATTAYA CHINNAKUM is currently anAssistant Professor with the Faculty of Economics,Chiang Mai University. She works on the macroe-conomic theory, economic development, econo-metrics, and economic for public policy. She is amember of the Centre of Excellence in Economet-rics. Her research has explored a wide range oftopics including economic development, financialeconometrics, and tourism economics.

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