On the structure of cyclic codes over 픽qRS and applications in quantum and LCD codes constructions

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2020.2966542, IEEE Access Date of publication xxxx 00, 0000, date of current version xxxx 00, 0000. Digital Object Identifier 10.1109/ACCESS.2017.DOI On the structure of cyclic codes over Fq RS and applications in quantum and LCD codes constructions HAI Q. DINH1,2 , TUSHAR BAG3 , ASHISH KUMAR UPADHYAY4 , RAMAKRISHNA BANDI5 , WARATTAYA CHINNAKUM6 1 Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam. (e-mail: dinhquanghai@tdtu.edu.vn) 2 Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam. (e-mail: dinhquanghai@tdtu.edu.vn) 3 Department of Mathematics, Indian Institute of Technology Patna, Patna 801103, India. (e-mail: tushar.pma16@iitp.ac.in) 4 Department of Mathematics, Indian Institute of Technology Patna, Patna 801103, India. (e-mail: upadhyay@iitp.ac.in) 5 Department of Mathematics, International Institute of Information Technology Naya Raipur, Atal Nagar 493661, India. (e-mail: ramakrishna@iiitnr.edu.in ) 6 Centre of Excellence in Econometrics, Chiang Mai University, Thailand. (e-mail: warattaya_chin@hotmail.co.th) Corresponding author: Hai Q. Dinh (e-mail: dinhquanghai@tdtu.edu.vn). ABSTRACT Let p be an odd prime, q = pm , R = Fq +uFq with u2 = 1, and S = Fq +uFq +vFq +uvFq with u2 = 1, v 2 = 1, uv = vu. In this paper, Fq RS-cyclic codes over Fq RS are studied. As an application, we present a construction of quantum error-correcting codes (QECCs) from the Fq RS-cyclic codes over Fq RS, which provides new QECCs. We also consider linear complementary dual (LCD) codes from the Fq RS-cyclic codes over Fq RS. Among others, we construct a Gray map over Fq RS and discuss the Gray images of Fq RS-cyclic codes over Fq . INDEX TERMS Cyclic codes, Mixed alphabet codes, QECCs, LCD codes. I. INTRODUCTION The family of cyclic codes is one of the most important families of codes which was introduced by Prange [41], [47] in 1954. Due to their rich algebraic structure and ease in implementation, these codes are widely used. The study of cyclic codes over finite rings has seen rapid growth after the work of Hammons et al. [27]. The properties of cyclic codes and their constructions over various finite rings are well explored in the literature. Since the last two decades, researchers have started studying codes over mixed alphabets. The study of linear codes over mixed alphabets was initiated by Brouwer et al. [17] in 1998. In [17], the authors studied mixed alphabet codes as Z2 -submodule over Zr2 Zs3 . However, thereafter not much work has been done on mixed alphabet codes. Recently, codes over mixed alphabets have caught attention of the researchers. In 2010, Borges et al. [15] studied Z2 Z4 -additive codes and the corresponding Z2 Z4 -linear codes. In this work, they discussed the standard form for generator matrices, parity-check matrices of Z2 Z4 -additive codes and established the relation between them. They also talked about the automorphism groups of these codes. In 2013, Aydogdu et al. [10] studied the structure of Z2 Z2s -additive codes, which generalize Z2 Z4 -additive codes. In that paper, the authors presented some fundamental parameters, standard form of generator and parity-check matrices of Z2 Z4 -additive codes. Furthermore, they also provided two bounds on the minimum distance of Z2 Z4 -additive codes. In a similar line, Aydogdu et al. [8] generalized these results of [10], [15] over Zpr Zps . In 2014, Abualrub et al. [1] studied Z2 Z4 -additive cyclic codes. They showed that dual of a Z2 Z4 -cyclic code is also a Z2 Z4 -cyclic code, and studied infinite family of MDS codes. Then Borges et al. [16] obtained some important properties of Z2 Z4 -additive cyclic codes, and introduced generator polynomials of Z2 Z4 -additive cyclic and dual Z2 Z4 -additive cyclic codes. They also presented the parameters of Z2 Z4 additive cyclic codes in terms of the degrees of the generator polynomials of these codes. Then in [5], Aydogdu et al. studied MacWilliams identity over Z2 Z2 [u]-additive codes, and constructed some optimal binary codes from this study. In [44], Srinivasulu et al. studied Z2 (Z2 + uZ2 )-additive cyclic codes and presented their generators and minimal spanning sets. They also determined the generators of duals Z2 (Z2 + uZ2 )-additive cyclic codes for odd code-lengths. After that, Aydogdu et al. [6] studied Z2 Z2 [u]-cyclic and constacyclic codes. They presented the 1 VOLUME 4, 2016 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2020.2966542, IEEE Access Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS generator polynomials, minimal spanning sets, their sizes and binary images of Z2 Z2 [u]-cyclic and constacyclic codes under a Gray map. Recently, there are several papers on mixed alphabets such as Z2 Z2 [u3 ] [9]; Z2 Z2 [u] [32]. 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 double cyclic codes over Z4 . After that, extending this double cyclic code structure, Mostafanasab [40] introduced the triple cyclic codes over Z2 . Recently, Wu et al. [48] and Aydogdu et al. [7] independently studied Z2 Z2 Z4 -additive cyclic codes and Z2 Z4 Z8 -cyclic codes, respectively. In all of the aforementioned works, researchers mainly focused on exploring the structural properties of mixed alphabet codes such as generator matrices, parity check matrices, generating polynomials, minimal generating sets, generating polynomials for dual codes etc. There are hardly any papers on the applications of mixed alphabets codes. In this work, our main goal is to study mixed alphabet Fq RS-cyclic codes as well as finding the tools to apply them in some of the recent topics of research like construction of quantum errorcorrecting codes (QECC) and linear complementary dual (LCD) codes. To do that, in the first part of our article, we study the properties of Fq RS-cyclic codes, and in the later part (Sections 6 and 7), we discuss the construction of QECCs and LCD codes from Fq RS-cyclic codes. In this article, we first study linear codes and then cyclic code over Fq RS, where R = Fq + uFq , u2 = 1 and S = Fq + uFq + vFq + uvFq , u2 = 1, v 2 = 1, uv = vu and q = pm for odd prime p and positive integer m. As an application of this work, using the structure of cyclic code over Fq RS, we construct QECCs and also LCD codes over Fq . The paper is organized as follows: In Section 2, we describe the basic terminology to understand cyclic codes over Fq RS and their properties. In Section 3, a construction of linear codes over R and S have been discussed. In Section 4, we define a Gray map, through which we discuss some properties of codes over Fq RS. In Section 5, cyclic codes over R, S and Fq RS are studied. As an application of this study, in Section 6 and Section 7, we construct QECCs and LCD codes, respectively, with worked out examples. All the computations are done using Magma Computing Software. II. PRELIMINARIES Let Fq denote the finite field of characteristic p with q elements, where q = pm for odd prime p and positive integer m. The set Fnq of all ordered n-tuples over Fq forms a vector space with the usual component-wise addition and scalar multiplication of vectors. A non-empty subset C of Fnq is called a code of length n over Fq , and it is called a linear code if C is a subspace of Fnq . An element of C is called a codeword. Throughout this article we use the word code to refer a linear code. By wH (C), we denote the Hamming weight of a code C, which is defined as the smallest Hamming weight of all of 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 as dH (C) = min{dH (x, y)| x, y ∈ C, x 6= y}. The Euclidean inner product of x and y in Fnq is defined as x · y = x0 y0 + x1 y1 + · · · + xn−1 yn−1 . The dual code C ⊥ of C is defined as C ⊥ = {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 F [x] c(x) = c0 + c1 x + · · · + cn−1 xn−1 ∈ hxnq −1i . By this identification, it can be easily shown that a linear code C of length n over Fq is a cyclic code if and only if it is an ideal F [x] of the ring hxnq −1i . We can extend these concepts to linear code, dual code, cyclic code over finite commutative rings depending upon the structure of the rings. Let R be any finite commutative ring. Then we refer a nonempty subset C as a linear code of length n over R if it forms an R-submodule of Rn . Continuing our discussion we extend previous discussion to codes over product of finite commutative rings. From now onward, we denote R = Fq + uFq , with u2 = 1 and S = Fq + uFq + vFq + uvFq , with u2 = 1, v 2 = 1, uv = vu, where q = pm for odd prime p and positive integer 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) ∈ Fq RS we define the following S-scalar multiplication on Fq RS as • : S × Fq RS −→ Fq RS such that s • (x, y, z) = (η(s)x, τ (s)y, sz). This is a well-defined multiplication. It can be extended β γ component-wise over Fα q × R × S as follows: β γ α β γ • : S × (Fα q × R × S ) −→ Fq × R × S 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 , β γ z1 , . . . , zγ−1 ) ∈ Fα q × R × S . By this multiplication, α β γ Fq × R × S forms an S-module. 2 VOLUME 4, 2016 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2020.2966542, IEEE Access Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS β γ Definition 2.1. A non-empty subset C of Fα q × R × S is said to be a Fq RS-linear code of length (α, β, γ) if C is an β γ S-submodule of Fα q ×R ×S . Let t = (x0 , x1 , . . . , xα−1 , y0 , y1 , . . . , yβ−1 , z0 , z1 , . . . , ′ , z0′ , z1′ , zγ−1 ) and t′ = (x′0 , x′1 , . . . , x′α−1 , y0′ , y1′ , . . . , yβ−1 ′ β γ . . . , zγ−1 ) ∈ Fα q × R × S . Now we define inner product as, γ−1 β−1 α−1 X X X zi zi′ . yi yi′ + xi x′i + v t · t′ = uv j=0 i=0 k=0 Let C be a Fq RS-linear code of length (α, β, γ). Then the dual code of C is defined as β γ ′ C ⊥ = {t′ ∈ Fα q × R × S | t · t = 0, ∀t ∈ C}. β γ Definition 2.2. An S-submodule C of Fα q × R × S is said to be a Fq RS-cyclic code of length (α, β, γ) if for any c = (x0 , x1 , . . . , xα−1 , y0 , y1 , . . . , yβ−1 , z0 , z1 , . . . , zγ−1 ) ∈ C its cyclic shift ℘(c) := (xα−1 , x0 , x1 , . . . , xα−2 , yβ−1 , y0 , y1 , . . . , yβ−2 , zγ−1 , z0 , z1 , . . . , zγ−2 ) ∈ C. Proposition 2.3. Let C be a Fq RS-cyclic code of length (α, β, γ). Then C ⊥ is also a Fq RS-cyclic code of length (α, β, γ). Proof. Let C be a Fq RS-cyclic code of length (α, β, γ) and ′ ′ t′ = (x′0 , x′1 , . . . , x′α−1 , y0′ , y1′ , . . . , yβ−1 , z0′ , z1′ , . . . , zγ−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′ ) = ′ ′ ′ , y0′ , y1′ , . . . , yβ−2 , zγ−1 , z0′ , z1′ , (x′α−1 , x′0 , x′1 , . . . , x′α−2 , yβ−1 ′ ⊥ . . . , zγ−2 ) ∈ C . Now by the above defined inner product we get, t · ℘(t′ ) = uv{x0 x′α−1 + x1 x′0 + · · · + xα−1 x′α−2 } ′ ′ + y1 y0′ + · · · + yβ−1 yβ−2 } +v{y0 yβ−1 ′ ′ ′ +{z0 zγ−1 + z1 z0 + · · · + zγ−1 zγ−2 }. As C is a Fq RS-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 product of ℘l−1 (t) and t′ , we get ℘l−1 (t) · t′ = 0, where ℘ l−1 (t) · t ′ = uv{x1 x′0 + x2 x′1 + · · · + x0 x′α−1 } ′ +v{y1 y0′ + y2 y1′ + · · · + y0 yβ−1 } ′ +{z1 z0′ + z2 z1′ + · · · + z0 zγ−1 }. Comparing the coefficients from both sides we get, x1 x′0 + x2 x′1 + · · · + xα−1 x′α−2 + x0 x′α−1 = 0, ′ ′ y1 y0′ + y2 y1′ + · · · + yβ−1 yβ−2 + y0 yβ−1 = 0, ′ ′ z1 z0′ + z2 z1′ + · · · + zγ−1 zγ−2 + z0 zγ−1 = 0. Therefore, t · ℘(t′ ) = 0. Hence ℘(t′ ) ∈ C ⊥ , implying C ⊥ is a Fq RS-cyclic code of length (α, β, γ).  Fq [x] hxα −1i R[x] hxβ −1i S[x] hxγ −1i Let Sα,β,γ = × × and f = (a0 , a1 , . . . , aα−1 , b0 , b1 , . . . , bβ−1 , c0 , c1 , . . . , cγ−1 ) ∈ β γ Fα q × R × S . Then f can be identified by an element in Sα,β,γ as f (x) = = (a0 + a1 x + · · · + aα−1 xα−1 , b0 + b1 x + · · · + bβ−1 xβ−1 , c0 + c1 x + · · · + cγ−1 xγ−1 ) (a(x), b(x), c(x)), β which gives the one-to-one identification between Fα q ×R × γ S and Sα,β,γ . Let s(x) = s0 + s1 x + · · · + sδ xδ ∈ S[x] and (a(x), b(x), c(x)) ∈ Sα,β,γ . Then the above defined Sscalar multiplication induces the multiplication ⋆ in Sα,β,γ as 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, with respect to the multiplication ⋆, Sα,β,γ forms an S[x]-module. Proposition 2.4. A code C is a Fq RS-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 Fq RS-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 be f (x) = (a(x), r(x), s(x)). Note that x ⋆ f (x) = (aα−1 + a0 x + · · · + aα−2 xα−1 , rβ−1 + r0 x + · · · + rβ−2 xβ−1 , sγ−1 + s0 x + · · · + sγ−2 xγ−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) ∈ C for 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 S. Recall that R = Fq + uFq , where u2 = 1 and S = Fq + uFq + vFq + uvFq , where u2 = 1, v 2 = 1, uv = vu. For a, b ∈ Fq , any r ∈ R is of the form r = a + ub = η1 â + η2 b̂, where â, b̂ ∈ Fq such that â = (a − b), b̂ = (a + b) and η1 = 1−u , 2 η2 = 1+u . 2 It is easy to check that ηi2 = ηi , ηi ηj = 0 and η1 +η2 = 1, for i, j = 1, 2; i 6= j. Therefore, R = η1 R ⊕ η2 R and any r ∈ R can be expressed as r = η1 r1 + η2 r2 , where r1 , r2 ∈ Fq . We define a Gray map ψR : R → F2q given as ψR (r) = (r1 , r2 ). This map can be extended Rβ → F2β component wise as (r0 , r1 , . . . , rβ−1 ) 7→ q 3 VOLUME 4, 2016 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2020.2966542, IEEE Access Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS (r0,1 , r1,1 , . . . , rβ−1,1 , r0,2 , r1,2 , . . . , rβ−1,2 ), where r = (r0 , r1 , . . . , rβ−1 ) ∈ Rβ and ri = η1 ri,1 + η2 ri,2 for i = 0, 1, . . . , β − 1. Then for ri = η1 ri,1 + η2 ri,2 ∈ R, we define the Lee weight of ri as wL (ri ) = wH (ψR (ri )), where wH denotes the Hamming weight over Fq . The Lee distance between ri and ri′ ∈ R is defined as dL (ri , ri′ ) = wL (ri − ri′ ) = wH (ψR (ri − ri′ )). Then it is easy to show that, the Gray map ψR is a distance preserving Fq -linear map from Rβ (Lee distance) to F2β q (Hamming distance). Let Bβ be a linear code of length β over R. Then we define Bβ,1 = {r1 ∈ Fβq | η1 r1 + η2 r2 ∈ Bβ ; r2 ∈ Fβq }, Bβ,2 = {r2 ∈ Fβq | η1 r1 + η2 r2 ∈ Bβ ; r1 ∈ Fβq }. Therefore, Bβ,i are linear codes of length β over Fq for i = 1, 2. So a linear code Bβ of length β over R can be uniquely expressed as Bβ = η1 Bβ,1 ⊕ η2 Bβ,2 , then | Bβ |=| Bβ,1 || Bβ,2 |. By definition of the Gray map we have ψR (Bβ ) = Bβ,1 ⊗ Bβ,2 , then using the fact ψR is distance preserving, 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) = ζ1 x̂ + ζ2 ŷ + ζ3 ŵ + ζ4 ẑ, where x̂, ŷ, ŵ, ŵ ∈ Fq such that x̂ = (x + y + w + z), ŷ = (x + y − w − z), ŵ = (x − y + w − z), ẑ = (x − y − w + z) and 1 1 ζ2 = (1 + u − v − uv), ζ1 = (1 + u + v + uv), 4 4 1 1 ζ3 = (1 − u + v − uv), ζ4 = (1 − u − v + uv). 4 4 P4 2 It is easy to see that ζi = ζi , ζi ζj = 0 and i=1 ζi = 1, where i, j = 1, 2, 3, 4; i 6= j. Therefore, S = ζ1 S ⊕ ζ2 S ⊕ ζ3 S ⊕ ζ4 S. Thus, any s ∈ S can be uniquely expressed as s = ζ1 s1 + ζ2 s2 + ζ3 s3 + ζ4 s4 , 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 S γ → F4γ q component wise as Let Cγ be a linear code of length γ over S, then we define Cγ,1 = Cγ,2 = Cγ,3 = Cγ,4 = {s1 ∈ Fγq | ζ1 s1 + ζ2 s2 + ζ3 s3 + ζ4 s4 and s2 , s3 , s4 ∈ Fγq }, {s2 ∈ Fγq | ζ1 s1 + ζ2 s2 + ζ3 s3 + ζ4 s4 and s1 , s3 , s4 ∈ Fγq }, {s3 ∈ Fγq | ζ1 s1 + ζ2 s2 + ζ3 s3 + ζ4 s4 and s1 , s2 , s4 ∈ Fγq }, {s4 ∈ Fγq | ζ1 s1 + ζ2 s2 + ζ3 s3 + ζ4 s4 and s1 , s2 , s3 ∈ Fγq }. ∈ Cγ ∈ Cγ ∈ Cγ ∈ Cγ Arguing as above, any linear code Cγ of length γ over S can be written asQ Cγ = ζ1 Cγ,1 ⊕ ζ2 Cγ,2 ⊕ ζ3 Cγ,3 ⊕ ζ4 Cγ,4 4 then | Cγ |= j=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 Fq RS. Any arbitrary element of Fq RS can be written as (a, r, s) = (a, η1 r1 +η2 r2 , ζ1 s1 +ζ2 s2 +ζ3 s3 +ζ4 s4 ), where a ∈ Fq , r ∈ R and s ∈ S. Define a Gray map Ψ : Fq RS → F7q as follows: Ψ(a, r, s) = (a, r1 , r2 , s1 , s2 , s3 , s4 ). This is also a Fq -linear map and can be extended componentwise in the following way: β γ α+2β+4γ Ψ : Fα q R S −→ Fq 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 ), β where (a0 , a1 , . . . , aα−1 ) ∈ Fα q , (r0 , r1 , . . . , rβ−1 ) ∈ R γ and (s0 , s1 , . . . , sγ−1 ) ∈ S such that each ri = η1 ri,1 + η2 ri,2 ∈ R and sj = ζ1 sj,1 + ζ2 sj,2 + ζ3 sj,3 + ζ4 sj,4 ∈ S, for i = 0, 1, . . . , β − 1 and j = 0, 1, . . . , γ − 1. (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 ), β γ Similar to [48], for any element (a′ , r′ , s′ ) ∈ Fα q ×R ×S ′ ′ ′ ′ ′ ′ we define the Lee weight of (a , r , s ) as wL (a , r , s ) = wH (a′ ) + wL (r′ ) + wL (s′ ), where wH denote the Hamming weight and wL denote the Lee weight. The Lee distance β γ ′ ′′ between x′ , x′′ ∈ Fα q × R × S is defined as dL (x , x ) = ′ ′′ ′ ′′ wL (x − x ) = wH (Ψ(x − x )). where s = (s0 , s1 , . . . , sγ−1 ) ∈ S γ and sj = ζ1 sj,1 + ζ2 sj,2 + ζ3 sj,3 + ζ4 sj,4 for j = 0, 1, . . . , γ − 1. As above, for sj = ζ1 sj,1 + ζ2 sj,2 + ζ3 sj,3 + ζ4 sj,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 map ψS is a distance preserving Fq -linear map from S γ (Lee distance) to F4γ q (Hamming distance). Proposition 4.1. Let Ψ be the Gray map defined above. Then, 1) Ψ is a Fq -linear distance preserving map from β γ α+2β+4γ Fα (Hamming disq R S (Lee distance) to Fq tance). 2) C is a Fq RS-linear code of length (α, β, γ) over Fq RS, then Ψ(C) is a [α + 2β + 4γ, k, dH ] linear code over Fq , where dL = dH . 4 VOLUME 4, 2016 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2020.2966542, IEEE Access Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS Proof. (1.) Let x′ = (a′ , r′ , s′ ), x′′ = (a′′ , r′′ , s′′ ) ∈ β γ Fα q R S , where a′ = (a′0 , a′1 , . . . , a′α−1 ), a′′ = (a′′0 , a′′1 , . . . , a′′α−1 ) ∈ Fα q; r′ = η1 r′1 + η2 r′2 , r′′ = η1 r′′1 + η2 r′′2 ∈ Rβ , and s′ = ζ1 s′1 + ζ2 s′2 + ζ3 s′3 + ζ4 s′4 , s′′ = ζ1 s′′1 + ζ2 s′′2 + ζ3 s′′3 + ζ4 s′′4 ∈ S γ , such that for i = 1, 2 and j = 1, 2, 3, 4 = ′′ ′′ ′′ (ri,0 , ri,1 , . . . , ri,β−1 ) = Ψ℘(a0 , a1 , . . . , aα−1 , r0 , r1 , . . . , rβ−1 , s0 , s1 , . . . , sγ−1 ) = Ψ(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, χg Ψ(c) ′ ′ ′ r′i = (ri,0 , ri,1 , . . . , ri,β−1 ), r′′i and sj = ζ1 sj,1 + ζ2 sj,2 + ζ3 sj,3 + ζ4 sj,4 ∈ S for i = 0, 1, . . . , β − 1; j = 0, 1, . . . , γ − 1. Then Ψ℘(c) β ∈R , s′j = (s′j,0 , s′j,1 , . . . , s′j,γ−1 ), s′′j = (s′′j,0 , s′′j,1 , . . . , s′′j,γ−1 ) ∈ S γ . Then Ψ(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 ) = Ψ(x′ ) + Ψ(x′′ ), and Ψ(cx′ ) = (ca′ , cr′1 , cr′2 , cs′1 , cs′2 , cs′3 , cs′4 ) = cΨ(x′ ), where c ∈ Fq . Therefore, Ψ is a Fq -linear map. For the other part, using the fact that Ψ is a Fq -linear map we get, dL (x′ , x′′ ) = wL (x′ − x′′ ) = wH (Ψ(x′ − x′′ )) = dH (Ψ(x′ ), Ψ(x′′ )). Hence, the result follows. (2.) It is not difficult to show upon noticing that Ψ is a Fq -linear distance preserving and bijective map.  such that d = Definition 4.2. Suppose d ∈ Fmn q (d1 , d2 , · · · , dn−1 , dn ), where di ∈ Fm q ; i = 1, 2, . . . , n. defined to Fm Let ρ be the cyclic shift from Fm q q by ρ(x0 , x1 , . . . , xm−1 ) = (xm−1 , x0 , . . . , xm−2 ). Define χ : Fmn −→ Fmn by, χ(d1 , d2 , . . . , dn ) = q q (ρ(d1 ), ρ(d2 ), · · · , ρ(dn )). Then a code C is called a quasicyclic code of index n if χ(C) = C. 1 2 n Suppose d′ ∈ Fm × Fm × · · · × Fm such that q q q ′ ′ ′ ′ ′ ′ i d = (d1 , d2 , · · · , dn−1 , dn ), where di ∈ Fm q ;i = mi mi 1, 2, . . . , n. Let ρ be the cyclic shift from Fq to Fq defined by ρ(x0 , x1 , . . . , xmi −1 ) = (xmi −1 , x0 , . . . , xmi −2 ). Define m2 mn m2 mn 1 1 −→ Fm χ g : Fm q × Fq × · · · × F q q × Fq × · · · × F q ′ ′ ′ ′ ′ ′ by, χg (d1 , d2 , · · · , dn ) = (ρ(d1 ), ρ(d2 ), · · · , ρ(dn )), then a code C is called a generalized quasi-cyclic code of index n if χg (C) = C. = χg Ψ(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, Ψ℘=χg Ψ.  Using Proposition 4.3, we have the following result. Theorem 4.4. Let C be a linear code of length (α, β, γ) over Fq RS. Then the Ψ-Gray image of Fq RS-cyclic code of length (α, β, γ) is a generalized quasi-cyclic code of index 7 over Fq . In particular, when α = β = γ, the Ψ-Gray image of a Fq RS-cyclic code of length (α, β, γ) is a quasi-cyclic code of index 7 over Fq . V. CYCLIC CODES OVER Fq , R, S AND Fq RS In this section, we study cyclic codes of length α, β and γ over the rings Fq , R and S, respectively. Then using the structure of the cyclic codes over Fq , R and S, we discuss the cyclic codes of length (α, β, γ) over Fq RS. A. CYCLIC CODES OVER Fq Theorem 5.1. [28, Theorem 12.9] Let A be a cyclic code of length α over Fq . Then there exists a unique monic polynomial a(x) ∈ Fq [x]/hxα − 1i such that Aα = ha(x)i and a(x) | (xα − 1). Moreover, the dimension of Aα is r = α − deg(a) with {a(x), xa(x), · · · , xr−1 a(x)} as a basis. B. CYCLIC CODES OVER R AND S Proposition 4.3. Let χg and Ψ be the maps defined above, and ℘ be the cyclic shift over Fq RS. Then Ψ℘=χg Ψ. Theorem 5.2. Let Bβ = η1 Bβ,1 ⊕ η2 Bβ,2 be a linear code of length β over R. Then Bβ is a cyclic code of length β over R if and only if Bβ,i are cyclic codes of length β over Fq , for i = 1, 2. Proof. Let c = (a0 , a1 , . . . , aα−1 , r0 , r1 , . . . , rβ−1 , s0 , s1 , . . . , β γ sγ−1 ) ∈ Fα q R S , where each ri = η1 ri,1 + η2 ri,2 ∈ R Proof. Let (x0 , x1 , . . . , xβ−1 ) ∈ Bβ,1 and (y0 , y1 , . . . , yβ−1 ) ∈ Bβ,2 . Then z = (z0 , z1 , . . . , zβ−1 ) ∈ Bβ , where zi = Based on Definition 4.2, we have the following result. 5 VOLUME 4, 2016 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2020.2966542, IEEE Access Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS η1 xi + η2 yi , and xi , yi ∈ Fq for i = 0, 1, . . . , β − 1. Suppose Bβ 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 , η1 x0 + η2 y0 , . . . , η1 xβ−2 + η2 yβ−2 ) = η1 (xβ−1 , x0 , . . . , xβ−2 ) + η2 (yβ−1 , y0 , . . . , yβ−2 ). Therefore, by the direct sum decomposition of the linear code Bβ we get (xβ−1 , x0 , . . . , xβ−2 ) ∈ Bβ,1 and (yβ−1 , y0 , . . . , yβ−2 ) ∈ Bβ,2 . Hence, Bβ,i are cyclic codes of length β over Fq , for i = 1, 2. ′ ) ∈ Bβ such that Conversely, let z′ = (z0′ , z1′ , . . . , zβ−1 each zi′ = η1 x′i + η2 yi′ , where x′i , yi′ ∈ Fq for i = 0, 1, . . . , β − 1. Then x′ = (x′0 , x′1 , . . . , x′β−1 ) ∈ Bβ,1 ′ and y′ = (y0′ , y1′ , . . . , yβ−1 ) ∈ Bβ,2 . Suppose that Bβ,i are cyclic codes of length β over Fq , for i = 1, 2. Then ρ(x′ ) ∈ B1 and ρ(y′ ) ∈ B2 . Note that η1 ρ(x′ ) + η2 ρ(y′ ) ′ ′ = η1 (x′β−1 , x′0 , . . . , x′β−2 ) + η2 (yβ−1 , y0′ , . . . , yβ−2 ) ′ ′ ) = (η1 x′β−1 + η2 yβ−1 , η1 x′0 + η2 y0′ , . . . , η1 x′β−2 + η2 yβ−2 ′ ′ = (zβ−1 , z0′ , . . . , zβ−2 ) = ρ(z′ ) ∈ Bβ . Therefore, Bβ is a cyclic code of length β over R.  Using the direct sum decomposition of dual linear code Bβ⊥ over R, and arguing as above we can show the direct sum decomposition of the dual cyclic code over R as follows. Corollary 5.3. Let Bβ be a cyclic code of length β over R. ⊥ ⊥ is also a cyclic code of Then its dual Bβ⊥ = η1 Bβ,1 ⊕ η2 Bβ,2 ⊥ length β over R if and only if Bβ,i are cyclic codes of length β over Fq , for i = 1, 2. In Theorem 5.2, we have presented the direct sum decomposition of a cyclic code Bβ over R. Here we give the generator polynomial for such codes. Theorem 5.4. Let Bβ = η1 Bβ,1 ⊕ η2 Bβ,2 be a cyclic code of length β over R and ri (x) be the generator monic polynomial of the cyclic code Bβ,i , for i = 1, 2. Then P2 1) Bβ = hη1 r1 (x), η2 r2 (x)i and |Bβ | = q 2β− i=1 deg(ri ) , 2) Bβ = hr(x)i, where r(x) = η1 r1 (x) + η2 r2 (x) such that r(x) | (xβ − 1). Proof. (1.) Suppose Bβ is a cyclic code of length β over R. Then by Theorem 5.2, we get that Bβ,i are cyclic codes of F [x] length β over Fq , implying Bβ,i = hri (x)i ⊆ hxβq −1i for i = 1, 2. As Bβ = η1 Bβ,1 ⊕ η2 Bβ,2 , we can write Bβ = {r(x) | r(x) = η1 r1 (x) + η2 r2 (x); ri (x) ∈ Bβ,i , i = 1, 2}, which implies Bβ ⊆ hη1 r1 (x), η2 r2 (x)i ⊆ hxR[x] β −1i . On the other hand, let η1 r1 (x)h1 (x) + η2 r2 (x)h2 (x) ∈ hη1 r1 (x), η2 r2 (x)i, where h1 (x), h2 (x) ∈ hxR[x] β −1i . Suppose that the coefficients of hi (x), i = 1, 2 are hij = gj + ugj′ ∈ R. Using the decomposition of elements in R, we can write hij = η1 ĝj + η2 gˆj ′ , where ĝj , gˆj ′ ∈ Fq . Thus, we get η1 h1 (x) = η1 g1 (x) and η2 h2 (x) = η2 g2 (x), where g1 (x) = ĝ0 +ĝ1 x+· · ·+ĝβ−1 xβ−1 and g2 (x) = gˆ0 ′ + gˆ1 ′ x+ F [x] · · ·+gβ−1 ˆ ′ xβ−1 ∈ hxβq −1i . Therefore, hη1 r1 (x), η2 r2 (x)i ⊆ Bβ . Hence, Bβ = hη1 r1 (x), η2 r2 (x)i. For the other part, note that |Bβ | = |Bβ,1 ||Bβ,2 |. Therefore, |Bβ | = q β−deg(r1 ) × q β−deg(r2 ) = 2β−(deg(r1 )+deg(r2 )) q . (2.) Let ri (x) be the generator monic polynomial of the cyclic code Bβ,i , for i = 1, 2. By the first part, Bβ = hη1 r1 (x), η2 r2 (x)i. Suppose r(x) = η1 r1 (x) + η2 r2 (x) such that B̂ = hr(x)i. Then r(x) ∈ Bβ , implying B̂ ⊆ Bβ . Also as ηi (η1 r1 (x) + η2 r2 (x)) = ηi ri (x) for i = 1, 2, implies Bβ ⊆ B̂. Therefore Bβ = B̂ = hr(x)i, where r(x) = η1 r1 (x) + η2 r2 (x). As ri (x) is the monic generator polynomial of the cyclic code Bβ,i , ri (x) divides xβ − 1 such that there is fi (x) ∈ Fq [x] with xβ −1 = ri (x)fi (x), for i = 1, 2. Thus, xβ −1 = hxβ −1i (η1 + η2 )(xβ − 1) = (η1 r1 (x)f1 (x) + η2 r2 (x)f2 (x)) = (η1 f1 (x) + η2 f2 (x))(η1 r1 (x) + η2 r2 (x)), for i = 1, 2. Therefore, xβ − 1 = (η1 f1 (x) + η2 f2 (x))r(x). Hence, r(x) | (xβ − 1).  Using Corollary 5.3 and Theorem 5.4, we present the following result. Corollary 5.5. Let Bβ = η1 Bβ,1 ⊕ η2 Bβ,2 be a cyclic code of length β over R. Suppose ri (x) are the monic generator polynomials of the cyclic codes Bi and fi∗ (x) are the reciprocal polynomials of fi (x) such that xβ − 1 = fi (x)ri (x), for i = 1, 2. Then P2 1) Bβ⊥ = hη1 f1∗ (x), η2 f2∗ (x)i and |Bβ⊥ | = q i=1 deg(ri )) , 2) Bβ⊥ = hf ′ (x)i, where f ′ (x) = η1 f1∗ (x) + η2 f2∗ (x). We demonstrate our results, presenting the following examples. Example 5.6. Let β = 5 and R = F49 +uF49 , where u2 = 1. 7 [x] and w is a zero of the polynomial Take F49 = hx2F+6x+3i 2 x + 6x + 3 in F49 , then x5 − 1 = (x + 6)(x2 + w27 x + 1)(x2 + w45 x + 1) ∈ F49 [x]. Let r1 (x) = x2 + w27 x + 1 and r2 (x) = x2 + w45 x + 1. Then Bβ,1 = hr1 (x)i and Bβ,2 = hr2 (x)i are cyclic codes of length 5 over F49 . Therefore, Bβ = η1 Bβ,1 ⊕ η2 Bβ,2 is a cyclic code of length 5 over R. Thus, Bβ = hη1 r1 (x), η2 r2 (x)i = hr(x)i, where r(x) = η1 r1 (x) + η2 r2 (x). Then η1 r1 (x) + η2 r2 (x) 1 1 = (1 − u)(x2 + w27 x + 1) + (1 + u)(x2 + w45 x + 1) 2 2 1 = {2x2 + (w27 + w45 )x + u(w45 − w27 )x + 2} 2 1 = {2x2 + x + uw44 x + 2} 2 = x2 + 4x + 4uw44 x + 1 = x2 + (4 + uw28 )x + 1. 6 VOLUME 4, 2016 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2020.2966542, IEEE Access Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS By direct computation, we get Similar to Example 5.6, here we present an example to illustrate the above results of cyclic codes over S. x5 − 1 = (x3 + (3 + uw4 )x2 + (4 + uw28 )x + 6) Thus, r(x) | (x5 − 1). Also | Bβ |= 4910−4 = 496 . For the dual code of Bβ , note that Example 5.11. Let γ = 4 and S = F9 + uF9 + vF9 + uvF9 , where u2 = 1, v 2 = 1, uv = vu. Clearly x2 + 2x + 2 is irreducible in F3 , fix w to be a zero of the polynomial x2 + 3 [x] , then 2x + 2 in F9 . We view F9 = hx2F+2x+2i f1 (x) = (x + 6)(x2 + w45 x + 1), x4 − 1 = (x + 1)(x + 2)(x + w2 )(x + w6 ) ∈ F9 [x]. × (x2 + (4 + uw28 )x + 1). Let f2 (x) = (x + 6)(x2 + w27 x + 1). s2 (x) = (x + 1)(x + w2 ), s1 (x) = (x + 1)(x + 2), Then, we get f1∗ (x) = 6x3 + w27 x2 + w3 x + 1, s3 (x) = (x + 2)(x + w2 ), f2∗ (x) = 6x3 + w45 x2 + w21 x + 1. Then Cγ,j (x) = hsj (x)i are cyclic codes of length 4 over F9 , for j = 1, 2, 3, 4. Therefore, Cγ = ζ1 Cγ,1 ⊕ ζ2 Cγ,2 ⊕ ζ3 Cγ,3 ⊕ ζ4 Cγ,4 is a cyclic code of length 4 over S, where ζj ; j = 1, 2, 3, 4 are as above. ⊥ Thus, Bβ,i = hfi∗ (x)i are dual cyclic codes of length 5 ⊥ ⊥ over F49 , for i = 1, 2. Therefore, Bβ⊥ = η1 Bβ,1 ⊕ η2 Bβ,2 ⊥ is a dual cyclic code of length 5 over R and Bβ = hη1 f1∗ (x), η2 f2∗ (x)i = hf ′ (x)i, where f ′ (x) = η1 f1∗ (x) + η2 f2∗ (x) = 6x3 + (4 + uw28 )x2 + (3 + uw4 )x + 1. Also | Bβ⊥ |= 494 .  Using similar arguments as in the case of cyclic codes over R, we obtain the following results on cyclic codes over S. Theorem 5.7. Let Cγ = ζ1 Cγ,1 ⊕ ζ2 Cγ,2 ⊕ ζ3 Cγ,3 ⊕ ζ4 Cγ,4 be a linear code of length γ over S. Then Cγ is a cyclic code of length γ over S if and only if Cγ,j are cyclic codes of length γ over Fq , where j = 1, 2, 3, 4. Corollary 5.8. Let Cγ = ζ1 Cγ,1 ⊕ ζ2 Cγ,2 ⊕ ζ3 Cγ,3 ⊕ ζ4 Cγ,4 be a cyclic code of length γ over S. Then the dual Cγ⊥ = ⊥ ⊥ ⊥ ⊥ ζ1 Cγ,1 ⊕ ζ2 Cγ,2 ⊕ ζ3 Cγ,3 ⊕ ζ4 Cγ,4 is a cyclic code of length ⊥ γ over S if and only if Cγ,j are cyclic codes of length γ over Fq , for j = 1, 2, 3, 4. Theorem 5.9. Let Cγ = ζ1 Cγ,1 ⊕ ζ2 Cγ,2 ⊕ ζ3 Cγ,3 ⊕ ζ4 Cγ,4 be a cyclic code of length γ over S and sj (x) be generator monic polynomial of Cγ,j , for j = 1, 2, 3, 4. Then 1) Cγ = hζ1 s1 (x), ζ2 s2 (x), ζ3 s3 (x), ζ4 s4 (x)i and | Cγ |= q 4γ−(deg(s1 )+deg(s2 )+deg(s3 )+deg(s4 )) . 2) Cγ = hs(x)i, where s(x) = ζ1 s1 (x) + ζ2 s2 (x) + ζ3 s3 (x) + ζ4 s4 (x) such that s(x) | (xγ − 1). Corollary 5.10. Let Cγ = ζ1 Cγ,1 ⊕ζ2 Cγ,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 gj∗ (x) are the reciprocal polynomials of gj (x) such that xγ −1 = gj (x)sj (x), for j = 1, 2, 3, 4. Then 1) Cγ⊥ = hζ1 g1∗ (x), ζ2 g2∗ (x), ζ3 g3∗ (x), ζ4 g4∗ (x)i and | Cγ⊥ |= q (deg(s1 )+deg(s2 )+deg(s3 )+deg(s4 )) . 2) Cγ⊥ = hg ′ (x)i, where g ′ (x) = ζ1 g1∗ (x) + ζ2 g2∗ (x) + ζ3 g3∗ (x) + ζ4 g4∗ (x). s4 (x) = (x + w2 )(x + w6 ). Thus, Cγ = hs(x)i, where s(x) = ζ1 s1 (x) + ζ2 s2 (x) + ζ3 s3 (x) + ζ4 s4 (x) 1 = (1 + u + v + uv)(x + 1)(x + 2) 4 1 + (1 + u − v − uv)(x + 1)(x + w2 ) 4 1 + (1 − u + v − uv)(x + 2)(x + w2 ) 4 1 + (1 − u − v + uv)(x + w2 )(x + w6 ) 4 = 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 dual code of Cγ , note that g1 (x) = x2 + 1, g2 (x) = x2 + w3 x + w2 , g3 (x) = x2 + w5 x + w6 , g4 (x) = x2 + 2. Then the reciprocal polynomials are as follows, g1∗ (x) = x2 + 1, g2∗ (x) = w2 x2 + w3 x + 1, g3∗ (x) = w6 x2 + w5 x + 1, g4∗ (x) = 2x2 + 1. ⊥ (x) = hgj (x)i are dual cyclic codes of length 4 Then Cγ,j ⊥ ⊥ over F9 , for j = 1, 2, 3, 4 and Cγ⊥ = ζ1 Cγ,1 ⊕ ζ2 Cγ,2 ⊕ ⊥ ⊥ ζ3 Cγ,3 ⊕ ζ4 Cγ,4 is a dual cyclic code of length 4 over S. Therefore, Cγ⊥ = hg ′ (x)i, where g ′ (x) = ζ1 g1∗ (x) + ζ2 g2∗ (x) + ζ3 g3∗ (x) + ζ4 g4∗ (x) = (2 + uw6 + vw2 + 2uv)x2 + (w2 + u + 2v + w6 uv)x + (2 + 2u + 2v + uv). Also | Cγ⊥ |= 98 .  7 VOLUME 4, 2016 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2020.2966542, IEEE Access Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS C. Fq RS-CYCLIC CODES In Theorem 5.1, Theorem 5.4 and Theorem 5.9, we have discussed the generator polynomials for cyclic codes over Fq , R and S respectively. Now we present the generator polynomials for Fq RS-cyclic codes as follows. Theorem 5.12. Let C be a Fq RS-cyclic code. Then C = h(a(x)|0|0), (0|r(x)|0), (l1 (x)|l2 (x)|s(x))i, where a(x)|(xα − 1), r(x)|(xβ − 1), s(x)|(xγ − 1) and l1 (x) ∈ Fq [x]/hxα − 1i, l2 (x) ∈ R[x]/hxβ − 1i. Proof. From Theorems 5.1, 5.4 and 5.9, we have Aα = ha(x)i, Bβ = hr(x)i and Cγ = hs(x)i such that a(x)|(xα − 1), r(x)|(xβ −1), s(x)|(xγ −1). Then the proof follows from Theorem 3.1 of [48].  A Fq RS-linear code C of length (α, β, γ) is called a separable code if C = A′α ⊗ Bβ′ ⊗ Cγ′ , while considering A′α , Bβ′ , and Cγ′ as punctured codes of C by deleting the coordinates outside the α, β and γ components, respectively. Lemma 5.13. Let C = h(a(x)|0|0), (0|r(x)|0), (l1 (x)|l2 (x)| s(x))i be a Fq RS-cyclic code. Then 1) 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 = hgcd(a(x), l1 (x))i, Bβ′ = hgcd(r(x), l2 (x))i, and Cγ′ = hs(x)i. Proof. Proof is parallel to that of Lemmas 3.2, 3.3 and 3.4 of [48].  Lemma 5.14. Let C = h(a(x)|0|0), (0|r(x)|0), (l1 (x)|l2 (x)| s(x))i be a Fq RS-cyclic code. Then 1) 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 [48].  The following Lemma is a direct consequence of Lemma 5.14. Lemma 5.15. Let C = h(a(x)|0|0), (0|r(x)|0), (l1 (x)|l2 (x)| s(x))i be a Fq RS-cyclic code. Then the following are equivalent: 1) C is separable; 2) a(x)|l1 (x), r(x)|l2 (x); 3) C = h(a(x)|0|0), (0|r(x)|0), (0|0|s(x))i. if Aα , Bβ and Cγ are cyclic codes of length α, β and γ over Fq , R and S, respectively. Proof. Let C be a Fq RS-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 ) ∈ Bβ and (c0 , c1 , . . . , cγ−1 ) ∈ Cγ . As C is a Fq RS-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 over Fq , R and S, respectively. Conversely, suppose Aα , Bβ and Cγ are cyclic codes of 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 Fq RS-cyclic code of length (α, β, γ).  By Theorems 5.2, 5.7 and 5.16, we have the following corollary. Corollary 5.17. Suppose C = Aα ⊗ Bβ ⊗ Cγ is a Fq RSlinear code of length (α, β, γ), where Aα , Bβ and Cγ are linear codes of length α, β and γ over Fq , R and S, respectively. Then C is a Fq RS-cyclic code of length (α, β, γ) if and only if Aα , Bβ,i and Cγ,j are cyclic codes of length α, β and γ over Fq , R and S, where i = 1, 2; j = 1, 2, 3, 4. In Theorem 5.12, we have studied the generator polynomial for a Fq RS-cyclic code of length (α, β, γ). Now here we study the generator polynomial for a separable Fq RScyclic code of length (α, β, γ) as follows. Theorem 5.18. Let C = Aα ⊗ Bβ ⊗ Cγ be a Fq RS-cyclic code of length (α, β, γ), where Aα = ha(x)i , Bβ = hr(x)i and Cγ = hs(x)i. Then C = ha(x)i ⊗ hr(x)i ⊗ hs(x)i. A′α = hgcd(a(x), 0)i = ha(x)i = Aα , Example 5.19. Let α = 4, β = 5 and γ = 8. Denote R[x] S[x] F81 [x] Sα,β,γ = hx 4 −1i × hx5 −1i × hx8 −1i , where R = F81 + 2 uF81 (u = 1) and S = F81 + uF81 + vF81 + uvF81 (u2 = F3 [x] 1, v 2 = 1, uv = vu). Take F81 = hx4 +2x 3 +2i and w is a zero 4 3 of the polynomial x + 2x + 2 in F81 , then Bβ′ = hgcd(r(x), 0)i = hr(x)i = Bβ , x4 − 1 = (x + 1)(x + 2)(x + w20 )(x + w60 ) ∈ F81 [x]. Cγ′ = hs(x)i = Cγ . Let a(x) = (x + 2)(x + w20 ). Then Aα = ha(x)i is a cyclic code of length 4 over F81 . Theorem 5.16. Suppose C = Aα ⊗Bβ ⊗Cγ is a Fq RS-linear code of length (α, β, γ), where Aα , Bβ and Cγ are linear codes of length α, β and γ over Fq , R and S respectively. Then C is a Fq RS-cyclic code of length (α, β, γ) if and only x5 −1 = (x+2)(x+w8 )(x+w24 )(x+w56 )(x+w72 ) ∈ F81 [x]. Thus, for a separable code, we get Let r1 (x) = (x + w8 )(x + w24 ) and r2 (x) = (x + w56 )(x + w72 ). Then Bβ,1 = hr1 (x)i and Bβ,2 = hr2 (x)i are cyclic 8 VOLUME 4, 2016 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2020.2966542, IEEE Access Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS codes of length 5 over F81 . Therefore, Bβ = hr(x)i is a cyclic code of length 5 over R, where r(x) = η1 r1 (x) + η2 r2 (x) 1 = (1 − u)(x2 + w46 x + w32 ) 2 1 + (1 + u)(x2 + w14 x + w48 ) 2 1 2 46 = {2x + x(w + w14 ) + (w32 + w48 ) 2 + ux(w14 − w46 ) + u(w48 − w32 )} 1 = {2x2 + x + w70 + uw35 x + uw55 } 2 = 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) = hsj (x)i are cyclic codes of length 8 over F81 , for j = 1, 2, 3, 4. Therefore, Cγ = hs(x)i is a cyclic code of length 8 over S, where s(x) = ζ1 s1 (x) + ζ2 s2 (x) + ζ3 s3 (x) + ζ4 s4 (x) 1 = (1 + u + v + uv)(x2 + w20 x + w10 ) 4 1 + (1 + u − v − uv)(x2 + w10 x + w60 ) 4 1 + (1 − u + v − uv)(x2 + w20 x + 1) 4 1 + (1 − u − v + uv)(x2 + x + w50 ) 4 = x2 + (uw70 + vw20 + uvw30 )x + (uw20 + 2v + uvw10 ). Thus, C = h(a(x)|0|0), (0|r(x)|0), (0|0|s(x))i = ha(x)i ⊗ hr(x)i ⊗ hs(x)i is a separable Fq RS-cyclic code of length (4, 5, 8), where a(x), r(x) and s(x) are as above.  VI. APPLICATION Properties of mixed alphabet cyclic codes has been studied in [48]. In the next two sections we mainly focus on the applications of separable Fq RS-cyclic codes. A. QUANTUM ERROR-CORRECTING CODES (QECCs) FROM Fq RS-CYCLIC CODES By [3], let H be a Hilbert space of dimension q over the complex numbers C. Define H ⊗n to be n-fold tensor product of the Hilbert space H, that is, H ⊗n = H ⊗ H ⊗ · · · ⊗ H(ntimes). Then H ⊗n is a Hilbert space of dimension q n . A QECC of length n and dimension k over Fq is defined to be a Hilbert subspace of H ⊗n having dimension q k . A QECC with length n, dimension k and minimum distance d over Fq is denoted by [[n, k, d]]q . One of the important advancements in the construction of codes is the construction of QECCs from classical errorcorrecting codes. Errors prompted by inevitable interaction with environments - such as decoherence, quantum noise and other inaccuracies are among the prominent impediments leading to erroneous in quantum information. QECCs secures the quantum information from being exploited by such inaccuracy. Chronologically speaking, the first QECC was studied independently by Shor [45] and Steane [46]. However the construction of QECCs from classical codes, their existence proofs, and correction methods were given by Calderbank et al. in [18]. Later many QECCs have been constructed using ideas of [18] over finite fields and finite rings (See [2], [3], [11], [13], [24]–[26], [29], [30], [33], [34], [42]). Theorem 6.1. [18] (CSS Construction) Let C1 and C2 be [n, k1 , d1 ] and [n, k2 , d2 ] linear codes over GF(q) respectively with C2⊥ ⊆ C1 . Furthermore, let d = min{d1 , d2 }. Then there exists a QECC, C with parameters [[n, k1 + k2 − n, d]]q . In particular, if C1⊥ ⊆ C1 , then there exists a QECC with parameters [[n, 2k1 − n, d1 ]]q Here we recall the dual containing property for cyclic codes from [18]. Lemma 6.2. [18] Let C be a cyclic code of length n with generator polynomial m(x) over Fq . Then C contains its dual code 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 the Ψ-image of Fq RS-linear codes, which will take a crucial role in our construction of QECCs from Fq RS-cyclic codes. Proposition 6.3. Let C be a Fq RS-linear code of length (α, β, γ). Then Ψ(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 of length α, β and γ over Fq , for i = 1, 2; j = 1, 2, 3, 4. Moreover, |Ψ(C)| = |Aα ||Bβ,1 ||Bβ,2 ||Cγ,1 ||Cγ,2 ||Cγ,3 ||Cγ,4 | and dH (Ψ(C)) = min{dH (Aα ), dH (Bβ,i ), dH (Cγ,j )}, where i = 1, 2; j = 1, 2, 3, 4. Proof. Follows from the definition of ⊗ and Ψ(C).  Now we give our main result to construct QECCs from Fq RS-cyclic codes. Theorem 6.4. Let C = Aα ⊗Bβ ⊗Cγ be a Fq RS-cyclic code ⊥ ⊥ of length (α, β, γ). If A⊥ α ⊆ Aα , Bβ,i ⊆ Bβ,i and Cγ,j ⊆ Cγ,j , for i = 1, 2; j = 1, 2, 3, 4, then there exists a QECC with parameters [[(α + 2β + 4γ), 2k − (α + 2β + 4γ), dH ]], where dH denotes the Hamming distance and k denotes the dimension of the code Ψ(C), respectively. Proof. As Ψ(C) = Aα ⊗ Bβ,1 ⊗ Bβ,2 ⊗ Cγ,1 ⊗ Cγ,2 ⊗ Cγ,3 ⊗ 9 VOLUME 4, 2016 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2020.2966542, IEEE Access Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS ⊥ ⊥ Cγ,4 , it is easy to check that Ψ(C)⊥ = A⊥ α ⊗ Bβ,1 ⊗ Bβ,2 ⊗ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ Cγ,1 ⊗ Cγ,2 ⊗ Cγ,3 ⊗ Cγ,4 . Now let Aα ⊆ Aα , Bβ,i ⊆ Bβ,i ⊥ ⊆ Cγ,j , for i = 1, 2; j = 1, 2, 3, 4 holds. Then and Cγ,j ⊥ Ψ(C) ⊆ Ψ(C). Thus, by Theorem 6.1, there exists a QECC with parameters [[(α + 2β + 4γ), 2k − (α + 2β + 4γ), dH ]]q , where dH denotes the Hamming distance of the code Ψ(C) and k denotes the dimension of the code Ψ(C).  Example 6.5. Let α = 10, β = 15, γ = 20 and Sα,β,γ = F5 [x] R[x] S[x] 2 hx10 −1i × hx15 −1i × hx20 −1i , where R = F5 + uF5 (u = 1) 2 2 and S = F5 + uF5 + vF5 + uvF5 (u = 1, v = 1, uv = vu). x10 − 1 = (x + 1)5 (x + 4)5 ∈ F5 [x]. Let a(x) = (x + 1)2 (x + 4). Then Aα = ha(x)i is a cyclic code of 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) = hr1 (x)i and Bβ,2 (x) = hr2 (x)i are cyclic codes having same parameters [15, 12, 3]. Therefore, Bβ = hη1 r1 (x), η2 r2 (x)i 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]. 2 2 Let s1 (x) = (x + 1) (x + 2), s2 (x) = (x + 2) (x + 4), s3 (x) = (x + 1)2 (x + 3) and s4 (x) = (x + 1)(x + 3)2 . Then Cγ,i (x) = hsi (x)i and Cγ,j (x) = hsj (x)i are cyclic codes of length 20 with same parameters [20, 17, 3], where i = 1, 2 and j = 3, 4. Therefore, Cγ = hζ1 s1 (x), ζ2 s2 (x), ζ3 s3 (x), ζ4 s4 (x)i is a cyclic code of length 20 over S. Thus, Ψ(C) is a linear code with parameters [120, 99, 3] over F5 . Note that a(x)a∗ (x) divides x10 − 1; ri (x)ri∗ (x) divides x15 − 1, for i = 1, 2; and sj (x)s∗j (x) divides x20 − 1, for j = 1, 2, 3, 4. Hence, by Lemma 6.2, we get A⊥ α ⊆ ⊥ ⊥ Aα , Bβ,i ⊆ Bβ,i and Cγ,j ⊆ Cγ,j , i = 1, 2; j = 1, 2, 3, 4. Then by Theorem 6.4, we get Ψ(C)⊥ ⊆ Ψ(C). Therefore, by Theorem 6.4, there exists a QECC with parameters [[120, 96, 3]]5 .  Example 6.6. Let α = 7, β = 14, γ = 21 and Sα,β,γ = R[x] S[x] F7 [x] 2 hx7 −1i × hx14 −1i × hx21 −1i , where R = F7 + uF7 (u = 1) 2 2 and S = F7 +uF7 +vF7 +uvF7 (u = 1, v = 1, uv = vu). x7 − 1 = (x + 6)7 ∈ F7 [x]. Let a(x) = (x + 6)2 . Then Aα = ha(x)i is a cyclic code of length 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) = hri (x)i are cyclic codes of length 14 having same parameters [14, 11, 3], where i = 1, 2. Therefore, Bβ = hη1 r1 (x), η2 r2 (x)i 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) = hsj (x)i are cyclic codes of length 21 having same parameters [21, 18, 3], for j = 1, 2, 3, 4. Therefore, Cγ = hζ1 s1 (x), ζ2 s2 (x), ζ3 s3 (x), ζ4 s4 (x)i is a cyclic code of length 21 over S. Thus, Ψ(C) is a linear code with parameters [119, 99, 3] over F7 . Because a(x)a∗ (x) divides x7 − 1, ri (x)ri∗ (x) divides x14 − 1, for i = 1, 2; and sj (x)s∗j (x) divides x21 − 1, for j = 1, 2, 3, 4; it follows from Lemma 6.2 that ⊥ ⊥ A⊥ α ⊆ Aα , Bβ,i ⊆ Bβ,i and Cγ,j ⊆ Cγ,j , i = 1, 2; j = 1, 2, 3, 4. Then by Theorem 6.4, we get Ψ(C)⊥ ⊆ Ψ(C). Hence, by Theorem 6.4, there exists a QECC with parameters [[119, 79, 3]]7 .  Example 6.7. Let α = 12, β = 15, γ = 30 and Sα,β,γ = R[x] S[x] F25 [x] 2 hx12 −1i × hx15 −1i × hx30 −1i , where R = F25 + uF25 (u = 1), and S = F25 + uF25 + vF25 + uvF25 (u2 = 1, v 2 = 5 [x] and w is a zero of the 1, uv = vu). Take F25 = hx2F+4x+2i 2 polynomial x + 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 = ha(x)i is a cyclic 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) = hr1 (x)i and Bβ,2 (x) = hr2 (x)i are cyclic codes of length 15 having same parameters [15, 12, 3]. Therefore, Bβ = hη1 r1 (x), η2 r2 (x)i is a cyclic code of length 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). Then Cγ,j (x) = hsj (x)i are cyclic codes of length 30 having same parameters [30, 26, 3] for j = 1, 2, 3, 4. Therefore, Cγ = hζ1 s1 (x), ζ2 s2 (x), ζ3 s3 (x), ζ4 s4 (x)i is a cyclic code of length 30 over S. Thus, Ψ(C) is a linear code with parameters [162, 137, 3] over F25 . Clearly, a(x)a∗ (x) divides x12 − 1; ri (x)ri∗ (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 6.2, we get A⊥ α ⊆ ⊥ ⊥ Aα , Bβ,i ⊆ Bβ,i and Cγ,j ⊆ Cγ,j , i = 1, 2; j = 1, 2, 3, 4. Then by Theorem 6.4, we get Ψ(C)⊥ ⊆ Ψ(C). Therefore, by Theorem 6.4, 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 over 10 VOLUME 4, 2016 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2020.2966542, IEEE Access Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS Fp , R, S and Fp R, respectively. We denote A, B, C, D, E, F and G, to represent the numbers 10, 11, 12, 13, 14, 15 and 16, respectively. For simpicity in calculation, in TABLEs 2 and 4, we take the generator r1 (x) = r2 (x) for the corresponding cyclic codes Bβ,i for i = 1, 2. Similarly, in TABLE 3, we take the generator s1 (x) = s2 (x) = s3 (x) = s4 (x) for the corresponding cyclic codes Cγ,j for j = 1, 2, 3, 4. We write the coefficients of the generator polynomials in decreasing order, for example, we write 120C4F to represent the polynomial x5 + 2x4 + 12x2 + 4x + 15. In TABLE 4, the fifth column presents parameters of Gray images over Fp R, which is the restriction on Ψ over Fp R. A linear code C is called LCD or linear complementary dual code if C ∩ C ⊥ = {0}. LCD codes were first introduced by Massey [39]. This family of codes have shown effectiveness against side-channel attacks(SCA) and fault injection attacks(FIA) to improve the security related information on sensitive devices [19]. Authors have explored properties of LCD codes with various conditions and structures in [35], [36], [49]. In this section, we briefly discuss the LCD codes over Fq , R and Fq RS and give some examples for better understanding of our study. Lemma 6.8. [39] 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 6.9. Let Bβ = hη1 r1 (x), η2 r2 (x)i be a cyclic code over R. Then Bβ is a LCD code over R if and only if ri (x) are self-reciprocal polynomials over Fq , for i = 1, 2. Proof. Let Bβ be a LCD code over R, i.e. Bβ ∩ Bβ⊥ = {0}. Note that Bβ ∩ = η1 (Bβ,1 ∩ ⊥ Bβ,1 ) ⊕ η2 (Bβ,2 ∩ ⊥ Bβ,2 ). Bβ ∩ Bβ⊥ = ⊥ Bβ,2 ∩ Bβ,2 = Then by the definition of ⊕, we get {0}, ⊥ whenever both Bβ,1 ∩ Bβ,1 = {0} and {0}, in other words, whenever both Bβ,1 and Bβ,2 are LCD codes over Fq . Hence, by Lemma 6.8, 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 polynomial of Bβ,i , for i = 1, 2, by Lemma 6.8 we get, Bβ,1 and Bβ,2 ⊥ are LCD codes over Fq . Thus, Bβ,1 ∩ Bβ,1 = {0} and Bβ,2 ∩ ⊥ Bβ,2 = {0}. Also as Bβ = η1 Bβ,1 ⊕ η2 Bβ,2 , ⊥ ⊥ Bβ ∩ Bβ⊥ = (η1 Bβ,1 ⊕ η2 Bβ,2 ) ∩ (η1 Bβ,1 ⊕ η2 Bβ,2 ) ⊥ ⊥ = η1 (Bβ,1 ∩ Bβ,1 ) ⊕ η2 (Bβ,2 ∩ Bβ,2 ) = {0}. Hence, Bβ is a LCD code over R. Proposition 6.11. Let C be a Fq RS-cyclic code of length (α, β, γ). Then Ψ(C) = Aα ⊗ Bβ,1 ⊗ Bβ,2 ⊗ Cγ,1 ⊗ Cγ,2 ⊗ Cγ,3 ⊗ Cγ,4 is a LCD code of length (α + 2β + 4γ) over Fq if and only if Aα , Bβ,i and Cγ,j are LCD codes of length α, β and γ over Fq , for i = 1, 2; j = 1, 2, 3, 4. Proof. Note that Ψ(C) ∩ Ψ(C)⊥ = (Aα ⊗ Bβ,1 ⊗ Bβ,2 ⊗ Cγ,1 ⊗ · · · ⊗ Cγ,4 ) ⊥ ⊥ ⊥ ⊥ ∩ (A⊥ α ⊗ Bβ,1 ⊗ Bβ,2 ⊗ Cγ,1 ⊗ · · · ⊗ Cγ,4 ) ⊥ ⊥ = (Aα ∩ A⊥ α ) ⊗ (Bβ,1 ∩ Bβ,1 ) ⊗ (Bβ,2 ∩ Bβ,2 ) B. LINEAR COMPLEMENTARY DUAL CODES. Bβ⊥ Proposition 6.10. Let Cγ = hζ1 s1 (x), ζ2 s2 (x), ζ3 s3 (x), ζ4 s4 (x)i be a cyclic code over S. Then Cγ is a LCD code over S if and only if sj (x) are self-reciprocal polynomials over Fq , for j = 1, 2, 3, 4.  Using similar arguments we can prove the following result over S. ⊥ ⊥ ⊗ (Cγ,1 ∩ Cγ,1 ) ⊗ · · · ⊗ (Cγ,4 ∩ Cγ,4 ). Therefore, Ψ(C) ∩ Ψ(C)⊥ = {0} if and only if Aα ∩ A⊥ α = ⊥ ⊥ {0}, Bβ,i ∩ Bβ,i = {0}, and Cγ,j ∩ Cγ,j = {0}, for i = 1, 2; j = 1, 2, 3, 4. Hence, Ψ(C) is a LCD code if and only if Aα , Bβ,i and Cγ,j are LCD codes over Fq , for i = 1, 2; j = 1, 2, 3, 4.  Example 6.12. 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) are self-reciprocal. Therefore, Bγ,1 = hr1 (x)i and Bβ,2 = hr2 (x)i are LCD codes of length 6 over F7 . Hence, Bβ = hη1 r1 (x), η2 r2 (x)i is a LCD code over R.  Example 6.13. Let R = F3 + uF3 , where u2 = 1 and S = F3 + uF3 + vF3 + uvF3 where u2 = 1, v 2 = 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 = ha(x)i 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 = hbi (x)i are LCD codes over F3 , for i = 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 and s4 (x) = (x2 + 1)3 . As sj (x) are self reciprocal, Cγ,j = hsj (x)i are LCD codes over F3 , for j = 1, 2, 3, 4. Hence, Ψ(C) is a LCD code of length 66 over F3 .  Example 6.14. Let R = F9 + uF9 , where u2 = 1 and S = F9 + uF9 + vF9 + uvF9 where u2 = 1, v 2 = 1, uv = vu. 3 [x] and w Let α = 4, β = 7 and γ = 16. Take F9 = hx2F+2x+2i is a zero of the polynomial x2 + 2x + 2 in F9 , then x4 − 1 = (x + 1)(x + 2)(x + w2 )(x + w6 ) ∈ F9 [x], 11 VOLUME 4, 2016 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2020.2966542, IEEE Access Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS TABLE 1. New QECCs from cyclic codes over Fp . α 9 10 12 24 24 32 36 42 48 Coefficient of a(x) 12 11 116 115C392 113B5AA4 1AF 6B15F 2 14 14 11183 Parameters of Aα [9, 8, 2]3 [10, 9, 2]5 [12, 10, 3]13 [24, 18, 5]13 [24, 17, 6]13 [32, 24, 6]17 [36, 35, 2]17 [42, 41, 2]7 [48, 44, 4]17 New QECCs [[9, 7, 2]]3 [[10, 8, 2]]5 [[12, 8, 3]]13 [[24, 12, 5]]13 [24, 10, 6]13 [[32, 16, 6]]17 [[36, 34, 2]]17 [[42, 40, 2]]7 [[48, 40, 4]]17 Existing QECCs [[9, 3, 2]]3 (ref. [43]) [[10, 6, 2]]5 (ref. [22]) [[12, 4, 3]]13 (ref. [23]) [[24, 12, 4]]13 (ref. [23]) [[24, 4, 6]]13 (ref. [23]) [[32, 12, 6]]17 (ref. [23]) [[36, 30, 2]]17 (ref. [23]) [[42, 38, 2]]7 (ref. [22]) [[48, 30, 4]]17 (ref. [23]) TABLE 2. New QECCs from cyclic codes over R. β Coefficient of ri (x) Parameters of Bβ,i Parameters of ψR(Bβ ) New QECCs Existing QECCs 9 12 18 20 20 48 72 90 12 14 11 12 1034 13 13 14 [9, 8, 2]3 [12, 11, 2]17 [18, 17, 2]3 [20, 19, 2]5 [20, 17, 3]5 [48, 47, 2]5 [72, 71, 2]5 [90, 89, 2]5 [18, 16, 2]3 [24, 22, 2]17 [36, 34, 2]3 [40, 38, 2]5 [40, 34, 3]5 [96, 94, 2]5 [144, 142, 2]5 [180, 178, 2]5 [[18, 14, 2]]3 [[24, 20, 2]]17 [[36, 32, 2]]3 [[40, 36, 2]]5 [[40, 28, 3]]5 [[96, 92, 2]]5 [[144, 140, 2]]5 [[180, 176, 2]]5 [[18, 12, 2]]3 (ref. [43]) [[24, 18, 2]]17 (ref. [23]) [[36, 30, 2]]3 (ref. [23]) [[40, 32, 2]]5 (ref. [3]) [[40, 24, 3]]5 (ref. [38]) [[96, 90, 2]]5 (ref. [37]) [[144, 138, 2]]5 (ref. [37]) [[180, 172, 2]]5 (ref. [38]) TABLE 3. New QECCs from cyclic codes over S. γ 20 20 24 78 Coefficient of sj (x) 14 1103 1405 1A7A Parameters of Cγ,j [20, 19, 2]5 [20, 17, 3]5 [24, 21, 3]7 [78, 75, 3]13 Parameters of ψS(Cγ ) [80, 76, 2]5 [80, 68, 3]5 [96, 84, 3]7 [312, 300, 3]17 New QECCs [[80, 72, 2]]5 [[80, 56, 3]]5 [[96, 72, 3]]7 [[312, 288, 3]]13 Existing QECCs [[66, 52, 2]]5 (ref. [13]) [[80, 54, 3]]5 (ref. [13]) [[96, 60, 3]]7 (ref. [20]) [[312, 282, 3]]13 (ref. [20]) TABLE 4. New QECCs from cyclic codes over Fp R. α β Coefficient of a(x) Coefficient of ri (x) 5 5 56 110 10 10 56 44 14 131 12143201 11 11 1144 11201044 13 Parameters of Gray images over Fp R [25, 23, 2]5 [25, 17, 3]5 [168, 147, 3]5 [198, 195, 2]5 x7 − 1 = (x + 2)(x3 + wx2 + w7 x + 2) 3 3 2 5 × (x + w x + w x + 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 + w7 x + 2)(x3 + w3 x2 + w5 x + 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 that Aα , Bβ,i and Cγ,j are LCD codes over F9 , for i = 1, 2; j = 1, 2, 3, 4. Hence, Ψ(C) is a LCD code of length 82 over F9 .  New QECCs Existing QECCs [[25, 19, 2]]5 [[25, 9, 3]]5 [[168, 126, 3]]5 [[198, 192, 2]]5 [[25, 15, 2]]5 (ref. [43]) [[25, 5, 3]]5 (ref. [43]) [[168, 96, 2]]5 (ref. [4]) [[198, 132, 2]]5 (ref. [4]) VII. CONCLUSION In this paper, we first discussed the cyclic codes over R and S, then using these structures we studied the concatenated structure of Fq RS-cyclic codes. We defined a Gray map over Fq RS and discussed some properties of Fq RS-cyclic codes. As an application of our study, we constructed quantum codes from Fq RS-cyclic codes. We also studied LCD codes as another application of the Fq RS-cyclic codes. This study can be generalized over product of finite rings. In our recent work [12], we gave necessary and sufficient conditions of dual-containing property for a skew cyclic, skew negacyclic, self-dual skew constacyclic codes over a finite ring. So taking direct product of cyclic, constacyclic, skew cyclic and skew constacyclic codes over finite rings, one can also construct QECCs. 12 VOLUME 4, 2016 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2020.2966542, IEEE Access Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS ACKNOWLEDGEMENT T. 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. Apart of this paper was written during a stay of H.Q. Dinh in the Vietnam Institute For Advanced Study in Mathematics (VIASM), he would like to thank the members of VIASM for their hospitality. H.Q. Dinh and W. Chinnakum are grateful to the Centre of Excellence in Econometrics, Chiang Mai University, Thailand, for partial financial support. This research is partially supported by the Research Administration Centre, Chaing Mai University, Thailand. REFERENCES [1] T. Abualrub, I. Siap and N. Aydin, Z2 Z4 -Additive cyclic and codes, IEEE Trans. Inf. Theory 60(3), 1508–1514, 2014. [2] M. Ashraf and G. Mohammad, Construction of quantum codes from cyclic codes over Fp + v Fp , Int. J. Inf. Coding Theory 3(2), 137–144, 2015. [3] M. Ashraf and G. Mohammad, Quantum codes from cyclic codes over Fq + uFq + v Fq + uv Fq , Quantum Inf. 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DINH is a professor in Applied Mathematics at the Department of Mathematical Sciences, Kent State University, USA. After completed his B.Sc. (1998), M.Sc. (2000), and Ph.D. (2003) in Mathematics at Ohio University, USA, he worked one year as a visiting professor at North Dakota State University, USA. Since 2004, Prof. Dinh has been working at Kent State University as a tenure professor in mathematics. Prof. Dinh research interests include Algebra and Coding Theory. Since 2004, he has published more than 80 papers at high level SCI(E) research journals such as Journal of Algebra, Journal of Pure and Applied Algebra, IEEE Transactions in Information Theory, IEEE Communication Letters, Finite Fields and Their Applications, Applicable Algebra in Engineering Communication and Computing, Discrete Applied Mathematics. Prof. Dinh has been a well known invited/keynote speaker at numerous international conferences and mathematics colloquium. Other than universities in the US, he also gave many honorary tutorial lectures at international universities in China, Indonesia, Kuwait, Mexico, Singapore, Thailand, Vietnam. WARATTAYA CHINNAKUM is an assistant professor in the Faculty of Economics of Chiang Mai University. She works on the macroeconomic theory, economic development, econometrics and economic for public policy. Also, she is a member of the Centre of Excellence in Econometrics.Her research has explored a wide range of topics including economic development, financial econometrics and tourism economics. TUSHAR BAG has completed his BSc degree from Ramakrishna Mission Vidymandira under University of Calcutta and MSc degree from IIT Kanpur, India. Tushar Bag is pursuing his PhD from Department of Mathematics, IIT Patna, India. His main research focus is on algebraic coding theory and codes over rings. Till now he has published 12 papers. ASHISH KUMAR UPADHYAY received a BSc and a MSc degree in Mathematics from the University of Allahabad, India. He received his PhD degree from Indian Institute of Science in 2005. He is currently working as an Associate Professor in Dept of Mathematics, IIT Patna. His research interests include Algebraic coding Theory and Algebraic Topology. RAMAKRISHNA BANDI is an assistant professor in the Department of Mathematics, Dr SPM International Institute of Information Technology Naya Raipur. He received PhD from Indian Institute of Technology Roorkee, Roorkee. His area of interests are Algebraic Coding Theory ans Information Theory. 14 VOLUME 4, 2016 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/.