ISSN 2148-838Xhttp://dx.doi.org/10.13069/jacodesmath.617244 J. Algebra Comb. Discrete Appl. 6(3) • 163–172 Received: 20 October 2018 Accepted: 21 August 2019 Journal of Algebra Combinatorics Discrete Structures and Applications A note on constacyclic and skew constacyclic codes over the ring Zp[u, v]/〈u2 −u, v2 −v, uv −vu〉∗ Research Article Tushar Bag, Habibul Islam, Om Prakash, Ashish K. Upadhyay Abstract: For odd prime p, this paper studies (1 + (p−2)u)-constacyclic codes over the ring R = Zp[u,v]/〈u2 − u,v2 − v,uv − vu〉. We show that the Gray images of (1 + (p − 2)u)-constacyclic codes over R are cyclic and permutation equivalent to a quasi cyclic code over Zp. We derive the generators for (1 + (p − 2)u)-constacyclic and principally generated (1 + (p − 2)u)-constacyclic codes over R. Among others, we extend our results for skew (1 + (p − 2)u)-constacyclic codes over R and exhibit the relation between skew (1 + (p− 2)u)-constacyclic codes with the other linear codes. Finally, as an application of our study, we compute several non trivial linear codes by using the Gray images of (1 + (p− 2)u)-constacyclic codes over this ring R. 2010 MSC: 94B05, 94B15 , 94B60 Keywords: Constacyclic codes, Skew constacyclic codes, Gray map, Quasi-cyclic codes 1. Introduction In algebraic coding theory, one of the main goals is to produce good error-correcting linear codes by means of larger minimum distance and code rate. Towards this, the cyclic code is one of important class of linear codes and researchers have been studying it for last six decades due to their successful applications in the theory of error-correcting codes. The constacyclic code is one of the prominent generalization of cyclic codes by which many good error-correcting codes can be developed over finite fields and rings. In 2006, Qian et al. [12] introduced constacyclic codes over F2 + uF2. Later, Abualrub and Siap [1] also studied structural properties of constacyclic codes over F2 + uF2. In 2011, Karadeniz and Yildiz [9] studied (1 + v)-constacyclic codes over F2 + uF2 + vF2 + uvF2 and constructed some new optimal binary codes as the Gray images of (1 + v)-constacyclic codes over F2 + uF2 + vF2 + uvF2. In 2015, Ashraf and ∗ The research was supported by the University Grant Commission (UGC), Govt. of India. Tushar Bag, Habibul Islam, Om Prakash (Corresponding Author), Ashish K. Upadhyay; Department of Math- ematics, Indian Institute of Technology Patna, Patna–801 103, Bihar, India (email:tushar.pma16@iitp.ac.in, habibul.pma17@iitp.ac.in, om@iitp.ac.in, upadhyay@iitp.ac.in). 163 https://orcid.org/0000-0002-7613-8351 https://orcid.org/0000-0002-2196-1586 https://orcid.org/0000-0002-6512-4229 https://orcid.org/0000-0001-6307-6799 T. Bag et al. / J. Algebra Comb. Discrete Appl. 6(3) (2019) 163–172 Mohammed [2] studied (1 + 2u)-constacyclic codes over Z4 + uZ4. Recently, researchers [11, 13, 14] have studied constacyclic codes over the extension ring Z4 + uZ4 of Z4 and obtained several new linear codes over Z4 from Gray images of these codes. For more works on the topic, we refer [3, 4, 6–8]. Motivated by the above works, we study λ = (1 + (p− 2)u)-constacyclic codes over the finite non- chain ring R = Zp[u,v]/〈u2 −u,v2 −v,uv−vu〉 and find some good codes over Zp, where Zp denotes the finite field with p elements for odd prime p. This ring can also be seen as Zp + uZp + vZp + uvZp, where u2 = u,v2 = v,uv = vu. The paper is organized as follows: In Section 2, we define two Gray maps over the ring R. Section 3 contains some results on Gray images of λ = (1+(p−2)u)-constacyclic codes over the ring R. In Section 4, we derive the generators for λ = (1+(p−2)u)-constacyclic and one-generator λ = (1+(p−2)u)-constacyclic codes, respectively over R. In Section 5, we extend our results for skew λ = (1 + (p− 2)u)-constacyclic codes over R and finally, in Section 6 some non trivial examples are included by using our Gray maps. 2. Preliminaries For an odd prime p, let R = Zp + uZp + vZp + uvZp, where u2 = u,v2 = v,uv = vu. Then R is a commutative non-chain semi-local ring with maximal ideals 〈u,v〉 and 〈u, 1 − v〉. Recall that a non empty subset C of Rn is called a linear code of length n over R if it forms an R-submodule of Rn, and elements of C are referred as codewords. A linear code C of length n over R is said to be a λ-constacyclic code if C is closed under the constacyclic shift operator Υ : Rn −→ Rn, defined by Υ(c0,c1, . . . ,cn−1) = (λcn−1,c0, . . . ,cn−2), where λ is a unit in R. Note that a constacyclic code is a cyclic code for λ = 1 and a negacyclic code for λ = −1. By identifying a codeword c = (c0,c1, . . . ,cn−1) ∈ Rn to a polynomial c(x) = c0 + c1x + · · · + cn−1xn−1 in R[x] 〈xn−λ〉, a linear code C is a λ-constacyclic code of length n over R if and only if it is an ideal of the ring R[x]〈xn−λ〉. For the rest of this article, we denote λ = (1 + (p− 2)u). Here, we define two new Gray maps over R. The first Gray map is φ1 : R → Z2p define by φ1(a + ub + vc + uvd) = (a + b + c + d, (p− 1)(a + b + c + d)), (1) where a,b,c,d ∈ Zp. It is easy to see that φ1 is a Zp-linear map and can be extended component-wise as follows. φ1 : R n → Z2np (r0,r1, . . . ,rn−1) 7−→ (a0 + b0 + c0 + d0, . . . ,an−1 + bn−1 + cn−1 + dn−1, (p− 1)(a0 + b0 +c0 + d0), . . . , (p− 1)(an−1 + bn−1 + cn−1 + dn−1)), where ri = ai + ubi + vci + uvdi ∈ R and ai,bi,ci,di ∈ Zp for i = 0, 1, . . . ,n− 1. The next Gray map is φ2 : R → Z3p define by φ2(a + ub + vc + uvd) = (b + c + d, (p− 1)(2a + b + c + d),a), (2) where a,b,c,d ∈ Zp. This φ2 is also a Zp-linear map and can be extended component-wise like the φ1-Gray map. Note that these two Gray maps are Zp-linear but not bijection, similar to the Gray map in [13]. The Lee weight of any element a+ub+vc+uvd ∈ R is defined as wL(r) = wH(φi(a+ub+vc+uvd)), i = 1, 2, where wH denotes the Hamming weight over Zp. Lee weight for r = (r0,r1, . . . ,rn−1) ∈ Rn is defined by wL(r) = ∑n−1 i=0 wL(ri). The Lee distance between any two elements r ′,r′′ ∈ Rn is dL(r′,r′′) = wL(r′−r′′) and the minimum Lee distance of C is defined as dL(C) = min{dL(r′,r′′) | r′ 6= r′′; r′,r′′ ∈ C}. By this discussion, one can check that φ1,φ2 are distance preserving Zp-linear maps from (Rn,dL) to (Z2np ,dH) and (Z3np ,dH), respectively, where dH denotes the minimum Hamming distance of codes over Zp. 164 T. Bag et al. / J. Algebra Comb. Discrete Appl. 6(3) (2019) 163–172 3. Gray images of (1 + (p−2)u)-constacyclic codes In this section, we explore the connection between cyclic, quasi cyclic and (1 + (p−2)u)-constacyclic codes via the Gray maps φ1 and φ2, defined in the previous section. Proposition 3.1. Let Υ be the (1 + (p− 2)u)-constacyclic shift on Rn and ρ be the cyclic shift on Z2np . If φ1 is the Gray map from Rn to Z2np as defined in equation (1), then φ1Υ=ρφ1. Proof. Let r = (r0,r1, . . . ,rn−1) ∈ Rn, where ri = ai + ubi + vci + uvdi and ai,bi,ci,di ∈ Zp for i = 0, 1, . . . ,n− 1. Then φ1(r) =(a0 + b0 + c0 + d0, . . . ,an−1 + bn−1 + cn−1 + dn−1, (p− 1)(a0 + b0 + c0 + d0), . . . , (p− 1)(an−1 + bn−1 + cn−1 + dn−1)). Applying ρ on both sides, we get ρφ1(r) =((p− 1)(an−1 + bn−1 + cn−1 + dn−1),a0 + b0 + c0 + d0, . . . ,an−1 + bn−1 + cn−1 + dn−1, (p− 1)(a0 + b0 + c0 + d0), . . . , (p− 1)(an−2 + bn−2 + cn−2 + dn−2)). On the other hand, φ1Υ(r) =((p− 1)(an−1 + bn−1 + cn−1 + dn−1),a0 + b0 + c0 + d0, . . . ,an−2 + bn−2 + cn−2 + dn−2,an−1 + bn−1 + cn−1 + dn−1, (p− 1)(a0 + b0 + c0 + d0), . . . , (p− 1)(an−2 + bn−2 + cn−2 + dn−2)). Therefore, φ1Υ=ρφ1. The derived relation in Proposition 3.1, will help us to find the Gray images of (1 + (p − 2)u)- constacyclic codes of R. In that way, we have the following result. Theorem 3.2. The φ1-Gray image of (1 + (p−2)u)-constacyclic code of length n over R is a cyclic code of length 2n over Zp. Proof. Let C be a (1 + (p−2)u)-constacyclic code of length n over R. Then Υ(C) = C, applying φ1 on both sides, we get φ1Υ(C) = φ1(C). By Proposition 3.1, ρφ1(C) = φ1Υ(C) = φ1(C). Therefore, φ1(C) is a cyclic code of length 2n over Zp. Definition 3.3. Let a ∈ Zmnp , where a = (a1|a2| . . . |am−1|am) and each ai ∈ Znp for i = 1, 2, . . . ,m. Let ηm be a map from Zmnp to Z mn p defined by ηm(a) = (ρ(a1)|ρ(a2)| . . . |ρ(am)), where ρ is the cyclic shift from Znp to Z n p and ′ |′ is the usual vector concatenation. A linear code C of length mn over Zp is called a QC or quasi cyclic code of index m if ηm(C) = C. Similar to Proposition 3.1, here we derive a relation based on the Gray map φ2, which will help us to find the φ2-Gray images of (1 + (p− 2)u)-constacyclic codes of R. Proposition 3.4. Let Υ be the (1 + (p−2)u)-constacyclic shift on Rn, φ2, the Gray map from Rn to Z3np defined in equation (2), and η3, the map defined in the preliminary section. Then φ2Υ = δη3φ2, where the permutation δ on Z3np is defined as δ(x1,x2, . . . ,x3n) = (xβ(1),xβ(2), . . . ,xβ(3n)) with the permutation β = (1,n + 1) on the set {1, 2, . . . , 3n}. Proof. Let r = (r0,r1, . . . ,rn−1) ∈ Rn, where ri = ai + ubi + vci + uvdi and ai,bi,ci,di ∈ Zp for i = 0, 1, . . . ,n− 1. Then φ2(r) =(b0 + c0 + d0, . . . ,bn−1 + cn−1 + dn−1, (p− 1)(2a0 + b0 + c0 + d0), . . . , (p− 1)(2an−1 + bn−1 + cn−1 + dn−1),a0, . . . ,an−1). 165 T. Bag et al. / J. Algebra Comb. Discrete Appl. 6(3) (2019) 163–172 Now, applying η3 on both sides, we get η3φ2(r) =(bn−1 + cn−1 + dn−1,b0 + c0 + d0, . . . ,bn−2 + cn−2 + dn−2, (p− 1)(2an−1 + bn−1 + cn−1 + dn−1), (p− 1)(2a0 + b0 + c0 + d0), . . . , (p− 1)(2an−2 + bn−2 + cn−2 + dn−2),an−1,a0, . . . ,an−2). On the other hand, φ2Υ(r) =((p− 1)(2an−1 + bn−1 + cn−1 + dn−1),b0 + c0 + d0, . . . ,bn−2 + cn−2 + dn−2, bn−1 + cn−1 + dn−1, (p− 1)(2a0 + b0 + c0 + d0), . . . , (p− 1)(2an−2 + bn−2 + cn−2 + dn−2),an−1,a0, . . . ,an−2). Now, applying δ on η3φ2(r), we get δη3φ2(r) = φ2Υ(r). Therefore, φ2Υ = δη3φ2. In Theorem 3.2, we presented the φ1-Gray images of (1 + (p−2)u)-constacyclic codes over R. Similar to that, using Proposition 3.4, we present the φ2-Gray image of (1 + (p− 2)u)-constacyclic code over R as below. Theorem 3.5. The φ2-Gray image of (1 + (p−2)u)-constacyclic code of length n over R is permutation equivalent to a QC code of index 3 over Zp. Proof. Let C be a (1 + (p− 2)u)-constacyclic code of length n over R. Then Υ(C) = C. By applying φ2, we have φ2(Υ(C)) = φ2(C). Now, by Proposition 3.4, φ2(Υ(C)) = δη3(φ2(C)) = φ2(C), Therefore, φ2(C) is permutation equivalent to a QC code of length 3n and index 3 over Zp. Now, we present the permutation version Φπ of φ1 defined as Φπ(r0,r1, . . . ,rn−1) =(a0 + b0 + c0 + d0, (p− 1)(a0 + b0 + c0 + d0),a1 + b1 + c1 + d1, (p− 1)(a1 + b1 + c1 + d1), . . . ,an−1 + bn−1 + cn−1 + dn−1, (p− 1)(an−1 + bn−1 + cn−1 + dn−1)), where ri = ai + ubi + vci + uvdi ∈ R and ai,bi,ci,di ∈ Zp for i = 0, 1, . . . ,n− 1. Using similar arguments used in the proof of [11, Theorem 4.3], one can easily show the following result. Proposition 3.6. For any r ∈ Rn, we have Φπρ(r) = ρ2Φπ(r). The following corollary is a direct consequence of Proposition 3.6. Corollary 3.7. Let C be a cyclic code of length n over R. Then Φπ(C) is equivalent to a quasi cyclic code of length 2n and index 2 over Zp. 4. Generators of (1 + (p−2)u)-constacyclic codes In this section, we derive the generators of λ = (1 + (p− 2)u)-constacyclic codes of length n over R, when gcd(n,p) = 1. We start with the approach shown in [2, 11]. Let n be an odd integer. Then Ψ : Rn = R[x] 〈xn − 1〉 −→ Rn,λ = R[x] 〈xn −λ〉 166 T. Bag et al. / J. Algebra Comb. Discrete Appl. 6(3) (2019) 163–172 defined by Ψ(c(x)) = c(λx), is a ring isomorphism. By this isomorphism it is evident that I is an ideal of the ring Rn if and only if Ψ(I) is an ideal of ring Rn,λ. Note that λn = 1 when n is an even integer and λn = λ when n is an odd integer. Then the map µ : Rn → Rn defined by µ(c0,c1, . . . ,cn−1) = (c0,λc1,λ 2c2, . . . ,λ n−1cn−1). (3) corresponds to the map Ψ in the polynomial form. Then it is easy to see that C is a cyclic code of odd length n over R if and only if µ(C) is a λ-constacyclic code of length n over R. Theorem 4.1. [10, Theorem 3.4] Let C be a cyclic code of length n over R. If gcd(n,p) = 1, then C = 〈g(x) + ua1(x) + uvr1(x),va2(x) + uva3(x)〉, where a1(x) | g(x) | (xn − 1) and a3(x) | a2(x) | g(x) | (xn − 1). Theorem 4.1 gives the generators of cyclic codes of length n over R, see [10]. Now, extending this result for λ-constacyclic codes of length n over R, we present the generators for λ-constacyclic codes of length n over R, where gcd(n,p) = 1. Theorem 4.2. Let C be a λ-constacyclic code of length n over R. If gcd(n,p) = 1, then C is an ideal of Rn,λ which is generated by C = 〈g(x̄) + ua1(x̄) + uvr1(x̄),va2(x̄) + uva3(x̄)〉, where x̄ = λx and g(x),ai(x),r1(x) are polynomials in Zp[x]/〈xn − 1〉, satisfying the conditions a1(x) | g(x) | (xn − 1) and a3(x) | a2(x) | g(x) | (xn − 1). Using the φ1-Gray map, we give the one-generated λ-constacyclic code of length n over R as follows: Theorem 4.3. Let C = 〈a(x) + ub(x) + vc(x) + uvd(x)〉 be a λ-constacyclic code of length n over R, where a(x),b(x),c(x),d(x) ∈ Zp[x]. Then φ1(C) is a cyclic code of length 2n over Zp generated by the polynomial (a(x) + b(x) + c(x) + d(x)) + xn[(p− 1)(a(x) + b(x) + c(x) + d(x))]. Proof. The polynomial version of the Gray map φ1 is φ1 : R[x]/〈xn −λ〉−→ Zp[x]/〈xn − 1〉×Zp[x]/〈xn − 1〉 defined by φ1(a(x) + ub(x) + vc(x) + uvd(x)) =(a(x) + b(x) + c(x) + d(x), (p− 1)a(x) + b(x) + c(x) + d(x)). Rest of this proof is straightforward. Note that for ri(x) ∈ Zp[x], we have φ1((r1 + ur2 + vr3 + uvr4)(a1 + ub2 + vc3 + uvd4)) = r1[(a + b + c + d), (p− 1)(a + b + c + d)] + r2[(a + b + c + d), (p− 1)(a + b + c + d)] + r3[(a + b + c + d), (p− 1)(a + b + c + d)] + r4[(a + b + c + d), (p− 1)(a + b + c + d)], where [(a(x) + b(x) + c(x) + d(x)), (p−1)(a(x) + b(x) + c(x) + d(x))] represents the element (a(x) + b(x) + c(x) + d(x)) + xn[(p− 1)(a(x) + b(x) + c(x) + d(x))] in Zp[x]/〈x2n − 1〉. Example 4.4. Let n = 4, p = 5 and the one generated (1 + 3u)-constacyclic code be C = 〈(1 + u + v + uv) + (1 + uv)x + (u + uv)x2 + (v + uv)x3〉. By Theorem 4.3, φ1(C) is a cyclic code of length 8 over Z5 generated by the polynomial 3x7 + 3x6 + 3x5 + x4 + 2x3 + 2x2 + 2x + 4 with minimum Lee distance 8. 167 T. Bag et al. / J. Algebra Comb. Discrete Appl. 6(3) (2019) 163–172 Now, we study about a permutation over Zp, referred as Nechaev’s permutation, which is defined as π(c0,c1, . . . ,c2n−1) = (cτ(0),cτ(1), . . . ,cτ(2n−1)), where n is odd and τ = (1,n + 1)(3,n + 3) · · ·(2i + 1,n + 2i + 1) · · ·(n− 2, 2n− 2) is a permutation on the set {0, 1, . . . , 2n− 1}. Proposition 4.5. Let µ be the map defined in the equation (3). If π is the Nechaev permutation and n is odd, then φ1µ = πφ1. Proof. Let r = (r0,r1, . . . ,rn−1) ∈ Rn, where ri = ai + ubi + vci + uvdi ∈ R and ai,bi,ci,di ∈ Zp for i = 0, 1, . . . ,n− 1. Since n is odd, we have µ(r) = (r0, (1 + (p− 2)u)r1,r2, (1 + (p− 2)u)r3,r4, . . . , (1 + (p− 2)u)rn−2,rn−1). Also, φ1(r) = (a0 + b0 + c0 + d0, . . . ,an−1 + bn−1 + cn−1 + dn−1, (p − 1)(a0 + b0 + c0 + d0), . . . , (p− 1)(an−1 + bn−1 + cn−1 + dn−1)). Therefore, φ1µ(r) =(a0 + b0 + c0 + d0, (p− 1)(a1 + b1 + c1 + d1),a2 + b2 + c2 + d2, . . . ,an−1 + bn−1 + cn−1 + dn−1, (p− 1)(a0 + b0 + c0 + d0),a1 + b1 + c1 + d1, . . . , (p− 1)(an−1 + bn−1 + cn−1 + dn−1)). On the other hand, πφ1(r) =(a0 + b0 + c0 + d0, (p− 1)(a1 + b1 + c1 + d1),a2 + b2 + c2 + d2, . . . ,an−1 + bn−1 + cn−1 + dn−1, (p− 1)(a0 + b0 + c0 + d0),a1 + b1 + c1 + d1, . . . , (p− 1)(an−1 + bn−1 + cn−1 + dn−1)). Hence, φ1µ = πφ1. Corollary 4.6. Let π be the Nechaev permutation and n be an odd. If Λ is the Gray image of a cyclic code of length n over R, then π(Λ) is a cyclic code. Proof. Let C be a cyclic code of length n over R and Λ = φ1(C). Then by Proposition 4.5, φ1µ(C) = πφ1(C) = π(Λ). Also, from our discussion in the beginning of this section, µ(C) is a (1 + (p − 2)u)- constacyclic code. Hence, by Theorem 3.2, φ1(µ(C)) = π(Λ) is a cyclic code. Analogous to Nechaev permutation, we present another permutation over Zp, defined as χ(c1, . . . ,c3n) = (c%(1), . . . ,c%(3n)), where % = (2,n + 2)(4,n + 4) · · ·(n− 1, 2n− 1) is a permutation on the set {1, 2, . . . , 3n}. Proposition 4.7. Let µ and χ be the maps defined above. Then φ2µ = χφ2. Moreover, if n is odd and α is the φ2-Gray image of a cyclic code of length n over R, then α is permutation equivalent to a QC code of length 3n and index 3 over Zp. Proof. It can be easily shown using similar procedure adopted in the proof of [3, Proposition 4]. 5. Skew (1 + (p−2)u)-constacyclic codes over R In Section 3, we have derived some relations to study the (1 + (p−2)u)-constacyclic codes over R in terms of cyclic and quasi cyclic codes over Zp. Extending these discussion, here we obtain some relations to study the skew (1 + (p− 2)u)-constacyclic codes over R as an extension in noncommutative set up. 168 T. Bag et al. / J. Algebra Comb. Discrete Appl. 6(3) (2019) 163–172 Definition 5.1. Let Aut(R) be the set of all automorphisms defined over the ring R and θ ∈ Aut(R). The set R[x; θ] = {r0 +r1x+· · ·+rn−1xn−1 | ri ∈ Rn} forms a ring under the usual addition of polynomial and the multiplication defined as (rxi)(sxj) = rθi(s)xi+j. This ring is called skew polynomial ring over R. Also, it is a non-commutative ring unless θ is the identity. We define a non-trivial automorphism θ : R −→ R θ(0) = 0, θ(1) = 1, θ(u) = v, θ(v) = u; i.e., θ(a + ub + vc + uvd) = a + vb + uc + uvd for a,b,c,d ∈ Zp. Note that the order of the automorphism is 2. For the rest of this section, we consider this automorphism θ over R. Definition 5.2. A subset C of Rn is called a skew λ-constacyclic code of length n over R if C satisfies the following conditions: i) C is a R-submodule of Rn; ii) If c = (c0,c1, . . . ,cn−1) ∈ C, then σθ,λ(c) = (θ(λcn−1),θ(c0), . . . ,θ(cn−2)) ∈ C. In polynomial representation of a skew λ-constacyclic code of length n over R, identifying a codeword c = (c0,c1, . . . ,cn−1) ∈ Rn by a polynomial c(x) = c0 + c1x + · · ·+ cn−1xn−1 in R[x; θ]/〈xn−λ〉, we have the following result. Theorem 5.3. Let C be a linear code of length n over R. Then C is a skew λ-constacyclic code of length n over R if and only if C is a left R[x; θ]-submodule of R[x; θ]/〈xn −λ〉. In Proposition 3.1 and Proposition 3.4, we derived relations using λ-constacyclic shift over R to study the φ1 and φ2-Gray images of λ-constacyclic codes over R. Similarly, here also we derive some relations using skew λ-constacyclic shift over R which will help us to study φ1 and φ2-Gray images of skew λ-constacyclic codes over R. These results can be seen as extension of Proposition 3.1 and Proposition 3.4. Proposition 5.4. If σθ,λ is the skew constacyclic shift on Rn and φ1,φ2 are the Gray maps defined in equation (1) and equation (2), respectively, then 1. ρφ1 = φ1σθ,λ. 2. δη3φ2 = φ2σθ,λ, where ρ,δ and η3 are as in Proposition 3.1 and Proposition 3.4. Proof. 1. From Proposition 3.1, we have ρφ1(r) =((p− 1)(an−1 + bn−1 + cn−1 + dn−1),a0 + b0 + c0 + d0, . . . ,an−1 + bn−1 + cn−1 + dn−1, (p− 1)(a0 + b0 + c0 + d0), . . . , (p− 1)(an−2 + bn−2 + cn−2 + dn−2)). Also, σθ,λ(r) = (θ(λrn−1),θ(r0),θ(r1), . . . ,θ(rn−2)). Applying φ1 on both sides, we get φ1σθ,λ(r) =((p− 1)(an−1 + bn−1 + cn−1 + dn−1),a0 + b0 + c0 + d0, . . . ,an−1 + bn−1 + cn−1 + dn−1, (p− 1)(a0 + b0 + c0 + d0), . . . , (p− 1)(an−2 + bn−2 + cn−2 + dn−2)). Hence, ρφ1 = φ1σθ,λ. 169 T. Bag et al. / J. Algebra Comb. Discrete Appl. 6(3) (2019) 163–172 2. From Proposition 3.4, we have δηφ2(r) =((p− 1)(2an−1 + bn−1 + cn−1 + dn−1),b0 + c0 + d0, . . . ,bn−2+ cn−2 + dn−2,bn−1 + cn−1 + dn−1, (p− 1)(2a0 + b0 + c0 + d0), . . . , (p− 1)(2an−2 + bn−2 + cn−2 + dn−2),an−1,a0, . . . ,an−2). Also, σθ,λ(r) = (θ(λrn−1),θ(r0),θ(r1), . . . ,θ(rn−2)). Applying φ2 on both sides, we get φ2σθ,λ(r) =((p− 1)(2an−1 + bn−1 + cn−1 + dn−1),b0 + c0 + d0, . . . ,bn−2+ cn−2 + dn−2,bn−1 + cn−1 + dn−1, (p− 1)(2a0 + b0 + c0 + d0), . . . , (p− 1)(2an−2 + bn−2 + cn−2 + dn−2), an−1,a0, . . . ,an−2). Hence, δηφ2 = φ2σθ,λ. In Theorem 3.2 and Theorem 3.5, we discussed the φ1 and φ2-Gray images of λ-constacyclic codes over R. Extending these results for skew λ-constacyclic codes over R, we present the φ1 and φ2-Gray image of a skew λ-constacyclic code over R as below: Theorem 5.5. Let φ1,φ2 be the Gray maps defined in equation (1) and equation (2), respectively. Then 1. φ1-Gray image of a skew λ-constacyclic code of length n over R is a cyclic code of length 2n over Zp. 2. φ2-Gray image of a skew λ-constacyclic code of length n over R is permutation equivalent to a quasi-cyclic code of length 3n and index 3 over Zp. Proof. 1. Let C be a skew (1 + (p − 2)u)-constacyclic code of length n over R. Then σθ,λ(C) = C. Now, applying φ1 on both sides, we get φ1σθ,λ(C) = φ1(C). By Proposition 5.4, ρφ1(C) = φ1σθ,λ(C) = φ1(C). Therefore, φ1(C) is a cyclic code of length 2n over Zp. 2. Let C be a skew (1 + (p − 2)u)-constacyclic code of length n over R. Then σθ,λ(C) = C. Now, applying φ2 on both sides, we get φ2σθ,λ(C) = φ2(C). Also, by Proposition 5.4, φ2(σθ,λ(C)) = δη(φ2(C)) = φ2(C). Thus, φ2(C) is permutation equivalent to a QC code of length 3n and index 3 over Zp. 6. Examples For better understanding of our study, we present some non trivial examples which are computed by using the φ1 and φ2-Gray images of the λ-constacyclic codes over R. All computations of this section are carried out by Magma software [5]. Example 6.1. Let p = 3,λ = 1 + u and n = 8. Following Theorem 4.2, assume that C = 〈(1 + u)x5 + (2 + 2u)x3 + (1 + u)x2 + 2 + u + uv, (v + uv)x3 + 2vx2 + vx + v + 2uv〉 be the (1 + u)-constacyclic code of length 8 over R = Z3[u,v]/〈u2 −u,v2 −v,uv −vu〉. Therefore, the φ2-Gray image of C is a [24, 8, 4] linear code over Z3. Example 6.2. Let p = 17,λ = 1 + 15u and n = 9. Further, assume C = 〈(1 + 15u)x5 + 3x4 + (1 + 15u)x3 + 16x2 + (1 + 15u)x + 16 + 16u + uv, (v + 5uv)x3 + 16uvx + 16v + 16uv〉 is a (1 + 15u)-constacyclic code of length 9 over R = Z17[u,v]/〈u2 − u,v2 − v,uv − vu〉. Then φ1(C) and φ2(C) have parameters [18, 10, 5] and [27, 10, 6] over Z17, respectively. 170 T. Bag et al. / J. Algebra Comb. Discrete Appl. 6(3) (2019) 163–172 Table 1. Linear codes as Gray images of (1 + (p− 2)u)-constacyclic codes over Zp λ n h(x̄) k(x̄) φ1(C) φ2(C) 1 + 3u 6 [1, 1 + 3u,u, 4 + 2u, 1 + 4u + uv] [v + 3uv, 2v, 2v + 4uv,v + uv] [12, 5, 2]5 [18, 5, 6]5 1 + 3u 8 [1 + 3u, 3, 2, 4 + uv] [4v, 3 + uv] [16, 8, 2]5 [24, 12, 5]5 1 + 3u 9 [1 + 3u, 4, 0, 1, 4 + 2u, 0, 1 + 2u, 3 + uv] [v, 0, 0,v + 3uv, 0, 0,v + uv] [18, 5, 2]5 [27, 5, 6]5 1 + 5u 4 [1 + 5u, 1, 1 + 4u, 1 + u + uv] [v, 0,v + uv] [8, 4, 2]7 [12, 4, 4]7 1 + 5u 10 [1 + 5u, 0, 0, 0, 6u, 1 + u + uv] [v, 6v + 2uv,v, 6v + 2uv,v + uv] [20, 10, 2]7 [30, 11, 5]7 1 + 9u 5 [1 + 9u, 8, 10u, 8 + 10u + uv] [v,v + 8uv, 3v + 2uv] [10, 5, 2]11 [15, 5, 6]11 1 + 9u 8 [1 + 9u, 2, 8 + 6u, 3 + u, 7 + 8u, 1 + u + uv] [v + 9uv, 10v,v + 8uv, 10v + 10uv] [16, 8, 2]11 [24, 8, 6]11 1 + 11u 4 [1, 9 + 7u, 8 + u] [v + 11uv, 8v + uv] [8, 3, 4]13 [12, 5, 4]13 1 + 11u 7 [1 + 11u, 6, 1 + 11u, 0, 11 + 3u, 11 + 12u] [v, 8v + 10uv, 4v + uv, 8v + 5uv,v + uv] [14, 5, 4]13 [21, 5, 8]13 1 + 15u 6 [1, 1 + 15u, 0, 16 + u, 16 + u] [v + 15uv, 16v + uv] [12, 5, 2]17 [18, 7, 2]17 1 + 15u 4 [1 + 15u, 1, 1 + 14u, 1 + u] [v, 5v + 6uv, 4v + 4uv] [8, 3, 4]17 [12, 3, 6]17 We recall from Theorem 4.2 that generator of a λ-constacyclic code of length n with gcd(n,p) = 1 is given by C = 〈h(x̄),k(x̄)〉, where h(x̄) = g(x̄) + ua1(x̄) + uvr1(x̄),k(x̄) = va2(x̄) + uva3(x̄) with a1(x) | g(x) | (xn − 1) and a3(x) | a2(x) | g(x) | (xn − 1), x̄ = λx. In Table 1, we obtain some linear codes from the λ-constacyclic codes over Zp. First column includes the value of λ, second column is the length of the constacyclic code while third and fourth column gives the coefficients of generator polynomials h(x̄),k(x̄). Lastly, fifth and sixth column shows the parameters of φ1 and φ2-Gray images, respectively. We write coefficients of generator polynomials in decreasing order, for example, we write [u, 0, 1 + uv,v, 1 + u + uv] to represent the polynomial ux4 + (1 + uv)x2 + vx + 1 + u + uv. 7. Conclusion and future works In this paper, we studied λ = (1 + (p − 2)u)-constacyclic codes over R = Zp + uZp + vZp + uvZp for odd prime p. We have constructed two new Gray maps over R and have shown some results based on their definitions. We have derived generators for λ = (1 + (p − 2)u)-constacyclic and one-generated λ = (1 + (p − 2)u)-constacyclic codes over R. Using some permutation maps over Zp, we have shown some results to understand cyclic and quasi cyclic codes by λ = (1 + (p − 2)u)-constacyclic codes over this ring in simpler way. At last, we have discussed skew (1 + (p − 2)u)-constacyclic codes over this ring and extended our results from Section 3. As a future work, finding the generators of these skew λ = (1 + (p− 2)u)-constacyclic code over R would be interesting. We hope our results would be useful to find some good codes over Zp via these Gray maps over R. Acknowledgment: The authors would like to thank the anonymous referee(s) for their valuable suggestions to improve the presentation of the manuscript. References [1] T. Abualrub, I. Siap, Constacyclic codes over F2 + uF2, J. Franklin Inst. 346(5) (2009) 520–529. [2] M. Ashraf, G. Mohammed, (1 + 2u)–constacyclic codes over Z4 + uZ4 (preprint) (2015). [3] N. Aydin, Y. Cengellenmis, A. Dertli, On some constacyclic codes over Z4[u]/〈u2 − 1〉, their Z4 images, and new codes, Des. Codes Cryptogr. 86(6) (2018) 1249–1255. [4] T. Bag, H. Islam, O. Prakash, A. K. Upadhyay, A study of constacyclic codes over the ring Z4[u]/〈u2− 171 https://doi.org/10.1016/j.jfranklin.2009.02.001 https://arxiv.org/pdf/1504.03445v1.pdf https://doi.org/10.1007/s10623-017-0392-y https://doi.org/10.1007/s10623-017-0392-y https://doi.org/10.1142/S1793830918500568 https://doi.org/10.1142/S1793830918500568 T. Bag et al. / J. Algebra Comb. Discrete Appl. 6(3) (2019) 163–172 3〉, Discrete Math. Algorithms Appl. 10(4) (2018) 1850056. [5] W. Bosma, J. Cannon, Handbook of Magma Functions, Univ. of Sydney 1995. [6] H. Islam, O. Prakash, A study of cyclic and constacyclic codes over Z4 + uZ4 + vZ4, Int. J. Inf. Coding Theory 5(2) (2018) 155–168. [7] H. Islam, T. Bag, O. Prakash, A class of constacyclic codes over Z4[u]/〈uk〉, J. Appl. Math. Comput. 60(1–2) (2019) 237–251. [8] H. Islam, O. Prakash, A note on skew constacyclic codes over Fq + uFq + vFq, Discrete Math. Algorithms Appl. 11(03) (2019) 1950030. [9] S. Karadeniz, B. Yildiz, (1 + v)-constacyclic codes over F2 + uF2 + vF2 + uvF2, J. Franklin Inst. 348(9) (2011) 2625–2632. [10] P. K. Kewat, B. Ghosh, S. Pattanayak, Cyclic codes over Zp[u,v]/ 〈 u2,v2,uv −vu 〉 , Finite Fields Appl. 34 (2015) 161–175. [11] M. Ozen, F. Z. Uzekmek, N. Aydin, N. T. Ozzaim, Cyclic and some constacyclic codes over the ring Z4[u]/〈u2 − 1〉, Finite Fields Appl. 38 (2016) 27–39. [12] J. F. Qian, L. N. Zhang, S. X. Zhu, (1 + u) Constacyclic and cyclic codes over F2 + uF2, Appl. Math. Lett. 19(8) (2006) 820-823. [13] M. Shi, L. Qian, L. Sok, N. Aydin, P. Sole, On constacyclic codes over Z4[u]/〈u2 −1〉 and their Gray images, Finite Fields Appl. 45 (2017) 86–95. [14] H. Yu, Y. Wang, M. Shi, (1 +u)–Constacyclic codes over Z4 +uZ4, Springer Plus 5 (2016) 1325(1–8). 172 https://doi.org/10.1142/S1793830918500568 https://doi.org/10.1142/S1793830918500568 https://www.math.uzh.ch/sepp/magma-2.19.8-cr/Handbook.pdf https://doi.org/10.1504/IJICOT.2018.095017 https://doi.org/10.1504/IJICOT.2018.095017 https://doi.org/10.1007/s12190-018-1211-y https://doi.org/10.1007/s12190-018-1211-y https://doi.org/10.1142/S1793830919500307 https://doi.org/10.1142/S1793830919500307 https://doi.org/10.1016/j.jfranklin.2011.08.005 https://doi.org/10.1016/j.jfranklin.2011.08.005 https://doi.org/10.1016/j.ffa.2015.01.005 https://doi.org/10.1016/j.ffa.2015.01.005 https://doi.org/10.1016/j.ffa.2015.12.003 https://doi.org/10.1016/j.ffa.2015.12.003 https://doi.org/10.1016/j.aml.2005.10.011 https://doi.org/10.1016/j.aml.2005.10.011 https://doi.org/10.1016/j.ffa.2016.11.016 https://doi.org/10.1016/j.ffa.2016.11.016 https://doi.org/10.1186/s40064-016-2717-0 Introduction Preliminaries Gray images of (1+(p-2)u)-constacyclic codes Generators of (1+(p-2)u)-constacyclic codes Skew (1+(p-2)u)-constacyclic codes over R Examples Conclusion and future works References