Linear Skew Constacyclic Codes

Zq (Zq + uZq )-Linear Skew Constacyclic Codes arXiv:1803.09316v1 [cs.IT] 25 Mar 2018

Ahlem Melakhessou ∗, Nuh Aydin †Kenza Guenda

‡

Abstract In this paper, we study skew constacyclic codes over the ring Zq R where R = Zq + uZq , q = ps for a prime p and u2 = 0. We give the definition of these codes as subsets of the ring Zαq Rβ . Some structural properties of the skew polynomial ring R[x, θ] are discussed, where θ is an automorphism of R. We describe the generator polynomials of skew constacyclic codes over R and Zq R. Using Gray images of skew constacyclic codes over Zq R we obtained some new linear codes over Z4 . Further, we have generalized these codes to double skew constacyclic codes over Zq R.

1

Introduction

One of the most important problems in coding theory is to construct codes with as large a minimum Hamming distance as possible. Many algebraic methods have been employed to achieve this goal. Cyclic codes and their various generalizations such as constacyclic codes and quasi-cyclic (QC) codes have played a key role in this quest. One particularly useful generalization of cyclic codes has been the class of quasi-twisted (QT) codes that produced hundreds of new codes with best known parameters [4, 8, 9, 11, 12, 17, 18] recorded in the database [24]. Yet another generalization of cyclic codes, called skew cyclic codes, was introduced in [16] which have been the subject of an increasing research activity over the past decade. On the other hand, codes over rings received much attention in the past few decades. Consequently, algebraic structures of cyclic, skew cyclic, and constacyclic codes over various rings are determined ( [2, 3, 5, 14, 15, 20–22, 25, 29, 32, 33]). Recently, P. Li et al. [27] gave the structure of (1 + u)-constacyclic codes over the ring Z2 Z2 [u] and Aydogdu et al. [6] studied Z2 Z2 [u]-cyclic and constacyclic codes. Further, Jitman et al. [26] considered the structure of skew constacyclic over finite chain rings. ∗

[email protected], Department of Mathematics, University of Batna, Algeria, [email protected], Department of Mathematics and Statistics, Kenyon College,United States. ‡ [email protected], Department of Electrical and Computer Engineering, University of Victoria, PO Box 1700, STN CSC, Victoria, BC, Canada V8W 2Y2 †

1

The aim of this paper is to introduce and study skew constacyclic codes over the ring Zq (Zq + uZq ) where q is a prime power and u2 = 0. We determine the structural properties of these codes. The paper is organized as follows. We first give some basic results about the ring R = Zq + uZq , where q = ps , p is a prime and u2 = 0, and linear codes over Zq R. In Section 3, we construct the non-commutative ring R[x; θ], where the structure of this ring depends on the elements of the commutative ring R and an automorphism θ of R. In Section 4, we give some results on skew constacyclic codes over the ring R and we determine Gray images of skew constacyclic codes over R. In Section 5, we study the algebraic structure of skew constacyclic codes over the ring Zq R. A necessary and sufficient condition for a skew constacyclic code over Zq R to contain its dual is given, and we determine Gray images of skew constacyclic codes over Zq R. These codes are then further generalized to double skew constacyclic codes. Finally, using Gray images of skew constacyclic codes over Zq R we obtained some new linear codes over Z4 .

2

Preliminaries

Let (α, β) denote n = α + 2β where α and β are positive integers. Consider R = Zq + uZq , where q = ps , p is a prime and u2 = 0. The ring R is isomorphic to the quotient ring Zq [u]/ hu2 i. R is not a chain ring, whereas it is a local ring with the maximal ideal hu, pi. Each element r of R can be expressed uniquely as r = a + ub, where a, b ∈ Zq . Let Cβ be a linear code of length β over R. Define C0 = {a ∈ Zβq ; a + ub ∈ Cβ for some b ∈ Zβq } and C1 = {b ∈ Zβq ; a + ub ∈ Cβ for some a ∈ Zβq }. Obviously C0 and C1 are linear codes over Zq . By the definition of C0 and C1 we have that Cβ can be uniquely expressed as Cβ = C0 + uC1.

(1)

Moreover, from (1) and the definition of the dual code we have that Cβ⊥ = C0⊥ + uC1⊥ . Consider the ring Zq and the ring R = Zq + uZq , where u2 = 0. We construct the ring

2

(2)

Zq R = {(e, r)|e ∈ Zq , r ∈ R}. The ring Zq R is not an R-module under the operation of standard multiplication. To make Zq R an R-module, we follow the approach in [3]. First define the map η : R → Zq a + ub 7→ a. Then, for d ∈ R, the multiplication ∗ by d ∗ (e, r) = (η(d)e, dr). This multiplication can be naturally generalized to the ring Zαq Rβ as follows. For any d ∈ R and v = (e0 , e1 , . . . , eα−1 , r0 , r1 , . . . , rβ−1 ) ∈ Zαq Rβ let d ∗ v = (η(d)e0 , η(d)e1, . . . , η(d)eα−1 , dr0 , dr1 , . . . , drβ−1 ), where (e0 , e1 , . . . , eα−1 ) ∈ Zαq and (r0 , r1 , . . . , rβ−1 ) ∈ Rβ . The following results are analogous to the ones obtained in [3, 5] for the ring Z2 (Z2 + uZ2 ). Lemma 2.1 The ring Zαq Rβ is an R-module under the above definition. Lemma 2.1 allows us to give the next definition. Definition 2.2 A non-empty subset C of Zαq Rβ is called a Zq R-linear code if it is an Rsubmodule of Zαq Rβ . We note that the ring Rβ is isomorphic to Z2β q as an additive group. Hence if C is a Zq R-linear code then it is isomorphic to a group of the form 1 Zqk0 +k2 × Z2k q . 1 Definition 2.3 If C ⊆ Zαq Rβ is a Zq R-linear code, group isomorphic to Zqk0 +k2 × Z2k q , then C is called a Zq R-additive code of type (α, β; k0, k1 , k2 ) where k0 , k1 , k2 and k3 are as defined above.

The following results and definitions are analogous to the ones obtained in [3]. Let C be a Zq R-linear code. Let Cα be the canonical projection of C on the first α coordinates and Cβ on the last β coordinates. Since the canonical projection is a linear map, Cα and Cβ are linear codes over Zq and R of length α and β, respectively. A code C is called separable if C is the direct product of Cα and Cβ , i.e., C = Cα × Cβ . 3

We introduce an inner product for two vectors ′ ′ v = (v0 , . . . , vα−1 , v0′ , . . . , vβ−1 ), w = (w0 , . . . , wα−1 , w0′ , . . . , wβ−1 ) ∈ Zαq Rβ

defined by hv, wi = u

α−1 X

vi wi +

β−1 X

v´j w´j .

j=0

i=0

Let C be a Zq R-linear code. The dual of C is defined by C ⊥ = {v ∈ Zαq Rβ |hv, wi = 0, ∀w ∈ C} , and C ⊥ = Cα⊥ × Cβ⊥ ,

(3)

where C is separable.

3

Skew Polynomial Ring R[x; θ]

In this section we construct the non-commutative ring R[x; θ]. The structure of this ring depends on the elements of the commutative ring R and an automorphism θ of R. Note that an automorphism θ of R must fix every element of Zq , hence it satisfies θ(a+ub) = a+θ(u)b. Therefore, it will be determined by its action on u. Let θ(u) = r + us for some r, s ∈ Zq . Then, θ(a + ub) = a + br + ubs = c + du for some c, d ∈ Zq . So let θ be an automorphism of R and let m be its order. The skew polynomial ring R[x; θ] is the set of polynomials over R where the coefficients are written on the left. The addition of these polynomials is defined in the standard manner while multiplication, which we will denote by ∗, is defined using the distributive law and the rule x ∗ a = θ(a)x. The set R[x; θ] with respect to addition and multiplication defined above forms a noncommutative ring called the skew polynomial ring. An element g(x) ∈ R[x; θ] is said to be a right divisor (resp. left divisor) of f (x) if there exists q(x) ∈ R[x; θ] such that f (x) = q(x) ∗ g(x) ( resp. f (x) = g(x) ∗ q(x)). In this case, f (x) is called a left multiple (resp. right multiple) of g(x). Lemma 3.1 [30, Lemma 1] Let f (x), g(x) ∈ R[x; θ] be such that the leading coefficient of g(x) is a unit. Then there exist q(x), r(x) ∈ R[x; θ] such that f (x) = q(x) ∗ g(x) + r(x), where r(x) = 0 or deg r(x) < deg g(x). 4

Definition 3.2 [30, Definition 1] A polynomial f (x) ∈ Z(R[x; θ]) of R[x; θ] is said to be central polynomial if f (x) ∗ r(x) = r(x) ∗ f (x), for all r(x) ∈ R[x; θ]. Theorem 3.3 The center Z(R[x; θ]) of R[x; θ] is the set Zps [xm ] where m is the order of θ ∈ Aut(R). Proof. The proof is similar to Theorem 2.1 in [21].



Corollary 3.4 [30, Corollary 2] Let f (x) = xβ − 1. Then f (x) ∈ Z(R[x; θ]) if and only if m | β. Further, xβ − (λ0 + uλ1) ∈ Z(R[x; θ]) if and only if m | β and (λ0 + uλ1 ) is fixed by θ.

4

Skew (λ0 + uλ1)−Constacyclic Codes over R

To study skew constacyclic codes over R, we first consider some structural properties of R[x; θ]/hxβ − (λ0 + uλ1 )i. The Corollary 3.4, shows that the polynomial (xβ − (λ0 + uλ1 )) is in the center Z(R[x; θ]) of the ring R[x; θ], hence generates a two-sided ideal if and only if m | β and (λ0 + uλ1 ) is fixed by θ. Therefore, in this case, R[x; θ]/hxβ − (λ0 + uλ1 )i is a well-defined residue class ring. If m ∤ β then the quotient space R[x; θ]/hxβ − (λ0 + uλ1 )i is a left R[x; θ]-module with multiplication defined as r(x) ∗ (f (x) + (xβ − (λ0 + uλ1 )) = r(x) ∗ f (x) + (xβ − (λ0 + uλ1 )), for any r(x), f (x) ∈ R[x; θ], not necessarily a ring. Next we define skew (λ0 + uλ1 )-constacyclic codes over the ring R. Definition 4.1 A subset Cβ of Rβ is called a skew (λ0 + uλ1 )-constacyclic code of length β if 1. Cβ is an R-submodule of Rβ . 2. If (c0 , c1 , . . . , cβ−1 ) ∈ Cβ , then ((λ0 + uλ1 )θ(cβ−1 ), θ(c0 ), . . . , θ(cβ−2 )) ∈ Cβ .

5

In particular, if λ0 + uλ1 = 1, then Cβ is a skew cyclic code over R, and we have the classical cyclic codes when θ = Id and λ0 + uλ1 = 1. In polynomial representation, a codeword (c0 , c1 , . . . , cβ−1 ) of a skew (λ0 + uλ1 )-constacyclic code is the vector of coefficients of the corresponding polynomial c0 + c1 x + . . . + cβ−1 xβ−1 ∈ R[x; θ]/hxβ − (λ0 + uλ1 )i.

Theorem 4.2 Let Cβ be a linear code of length β over R, and let Cβ = C0 + uC1, where C0 and C1 are linear codes of length β over Zq . Then Cβ is a skew (λ0 + uλ1 )-constacyclic code with respect to the automorphism θ if and only if C0 and C1 are λ0 -constacyclic code and (λ0 + λ1 )-constacyclic code over Zq , respectively, where λ0 , λ1 ∈ Z∗ps . Proof. Let (a0 , a1 , . . . , aβ−1 ) ∈ C0 and (b0 , b1 , . . . , bβ−1 ) ∈ C1 . Assume that ri = ai + ubi for i = 0, 1, . . . , β − 1. We have ((λ0 + uλ1 )θ(rβ−1 ), θ(r0 ), . . . , θ(rβ−2 )) = (λ0 θ(rβ−1 ), . . . , θ(rβ−2 )) + u (λ1 θ(rβ−1 ), . . . , θ(rβ−2 )) = (λ0 cβ−1 , c0 , . . . , cβ−2 ) + u((λ0 + λ1 )dβ−1 , d0, . . . , dβ−2 ), where θ(ai + ubi ) = ci + di u for some ci , di ∈ Zq (i = 0, . . . , β − 1), since Cβ is a skew (λ0 + uλ1 )-constacyclic code over R. Hence (λ0 cβ−1 , c0 , . . . , cβ−2 ) ∈ C0 and ((λ0 + λ1 )dβ−1 , d0 , . . . , dβ−2 ) ∈ C1 , then C0 and C1 are λ0 -constacyclic code and (λ0 +λ1 )-constacyclic code over Zq , respectively. On the other hand, suppose that C0 is a λ0 -constacyclic code over Zq and C1 is a (λ0 +λ1 )constacyclic code over Zq and (r0 , r1 , . . . , rβ−1 ) ∈ Cβ , where θ(ai + ubi ) = ci + di u ∈ Cβ for some ci , di ∈ Zq (i = 0, . . . , β − 1), then (c0 , c1 , . . . , cβ−1 ) ∈ C0 and (d0 , d1, . . . , dβ−1 ) ∈ C1 . Note that ((λ0 + uλ1 )θ(rβ−1 ), θ(r0 ), . . . , θ(rβ−2 )) = (λ0 θ(rβ−1 ), . . . , θ(rβ−2 )) + u (λ1 θ(rβ−1 ), . . . , θ(rβ−2 )) = (λ0 cβ−1 , c0 , . . . , cβ−2 ) + u((λ0 + λ1 )dβ−1 , d0, . . . , dβ−2 ) ∈ C0 + uC1 = Cβ , so Cβ is a skew (λ0 + uλ1)-constacyclic code over R.

6



Corollary 4.3 Let Cβ = C0 + uC1 be a linear code of length β over R. Then Cβ is a skew (λ0 +uλ1 )-constacyclic code if and only if its dual code Cβ⊥ is a skew (λ0 +uλ1 )−1 -constacyclic code over R. ′ ′ Proof. Let v ′ = (v0′ , v1′ , . . . , vβ−1 ) ∈ Cβ and w ′ = (w0′ , w1′ , . . . , wβ−1 ) ∈ Cβ⊥ . Since


′ (λ0 + uλ1 )θβ−1 (v1′ ), (λ0 + uλ1 )θβ−1 (v2′ ), . . . , (λ0 + uλ1 )θβ−1 (vβ−1 ), θβ−1 (v0′ ) ∈ Cβ ,

we have

′ 0 = h((λ0 + uλ1)θβ−1 (v1′ ), (λ0 + uλ1 )θβ−1 (v2′ ), . . . , (λ0 + uλ1 )θβ−1 (vβ−1 ), θβ−1 (v0′ )), w ′i ′ ′ = h((λ0 + uλ1)θβ−1 (v1′ ), . . . , (λ0 + uλ1 )θβ−1 (vβ−1 ), θβ−1 (v0′ )), (w0′ , . . . , wβ−1 )i ′ ′ ), θβ−1 ((λ0 + uλ1 )−1 v0′ )), (w0′ , w1′ , . . . , wβ−1 )i = (λ0 + uλ1 )h(θβ−1 (v1′ ), . . . , θβ−1 (vβ−1 β−1 P ′ ′ θβ−1 (vi′ )wi−1 ). = (λ0 + uλ1 )(θβ−1 ((λ0 + uλ1)−1 v0′ )wβ−1 + i=1

Then Cβ⊥ is a skew (λ0 + uλ1)−1 -constacyclic code. On the other hand, as β is a multiple of the order of θ, it follows that 0 = θ(0) ′ + = θ((λ0 + uλ1 )(θβ−1 ((λ0 + uλ1 )−1 v0′ )wβ−1 ′ = (λ0 + uλ1 )(v0′ θβ ((λ0 + uλ1 )−1 wβ−1 )+

β−1 P i=1

β−1 P

′ )) θβ−1 (vi′ )wi−1

i=1

′ vi′ θβ−1 wi−1 )

′ ′ ′ )i. )), (v0′ , v1′ , . . . , vβ−1 ), θ(w0′ ), . . . , θ(wβ−2 = (λ0 + uλ1 )h((λ0 + uλ1 )−1 θ(wβ−1

Therefore ′ ′ ((λ0 + uλ1)−1 θ(wβ−1 ), θ(w0′ ), . . . , θ(wβ−2 )) ∈ Cβ⊥ .



We are now ready to consider the generator polynomials of skew constacyclic codes over R. Theorem 4.4 A code Cβ of lenght β over R is a skew (λ0 + uλ1 )-constacyclic code if and

 only if Cβ is a left R[x; θ]-submodule of R[x; θ]/ xβ − (λ0 + uλ1 ) . Proof. Let c(x) = c0 + c1 x + . . . + cβ−1 xβ−1 ∈ Cβ , then

x ∗ c(x) = (λ0 + uλ1 )θ(cβ−1 ) + θ(c0 )x + . . . + θ(cβ−2 )xβ−1 = ((λ0 + uλ1 )θ(cβ−1 ), θ(c0 ), . . . , θ(cβ−2 )) ∈ Cβ . By iteration and linearity one can get r(x)∗c(x) ∈ Cβ , for all r(x) ∈ R[x, θ]/hxβ −(λ0 +uλ1 )i.

 This shows that Cβ is a left R[x; θ]-submodule of R[x; θ]/ xβ − (λ0 + uλ1 ) . Conversely, 7



 suppose that Cβ is a left R[x; θ]-submodule of R[x; θ]/ xβ − (λ0 + uλ1 ) , then we have that x ∗ c(x) ∈ Cβ . Thus, Cβ is skew (λ0 + uλ1 )−constacyclic code.  The following proofs of the next two Theorems are analogous to the proofs of Theorem 4 and Theorem 5 of [30] given for the ring Z4 + uZ4 ; for that we omit them. Theorem 4.5 [30, Theorem 4] If Cβ is a skew (λ0 + uλ1 )-constacyclic code of length β over R containing a minimum degree polynomial gβ (x) whose leading coefficient is a unit, then Cβ is a free code such that Cβ = hgβ (x)i and gβ (x)|xβ − (λ0 + uλ1 ). Further, Cβ has a basis {gβ (x), x ∗ gβ (x), . . . , xβ−deg(gβ (x))−1 ∗ gβ (x)} and |Cβ | = |R|β−deg(gβ (x)) . The converse of the above theorem is given by the following. Theorem 4.6 [30, Theorem 5] Let Cβ be a free, principally generated skew (λ0 + uλ1 )constacyclic code of length β over R. Then there exists a minimal degree polynomial gβ (x) ∈ Cβ having its leading coefficient a unit such that Cβ = hgβ (x)i and gβ (x)|xβ − (λ0 + uλ1 ). The following definition is analogous to the ring Zq . Definition 4.7 [23, Definition 2] Let C be a linear code over Fpd of length n = Nl and let λ ∈ F∗pd . If for any codeword (c0,0 , c0,1 , . . . , c0,l−1, . . . , cN −1,0 , cN −1,1 , . . . , cN −1,l−1 ) ∈ C, we have that (λcN −1,0 , λcN −1,1 , . . . , λcN −1,l−1 , . . . , c0,0 , c0,1 , . . . , c0,l−1 ) ∈ C, then we say that C is a λ-quasi-twisted (QT) code of length n. If l is the least positive integer satisfies that n = Nl, then C is said to be a λ-quasi-twisted code with index l. Define the Gray map Ψ by Ψ : Rβ → Z2β q (r0 , r1 , . . . , rβ−1 ) 7→ (b0 , b1 , . . . , bβ−1 , a0 + b0 , a1 + b1 , . . . , aβ−1 + bβ−1 ) where ri = ai + ubi ∈ R, ai , bi ∈ Zq for i = 0, 1, . . . , β − 1. The next Proposition follows from the definition of QT codes. Proposition 4.8 If Cβ is a skew (λ0 +uλ1)-constacyclic code of length β over R, then Ψ(C) is a QT code of index 2 and length 2β over Zq . 8

Proof. Let Cβ be a skew constacyclic code in R[x; θ]/hxβ − (λ0 + uλ1 )i and let (b0 , b1 , . . . , bβ−1 , a0 + b0 , a1 + b1 , . . . , aβ−1 + bβ−1 ) ∈ Ψ(C). Then there is a codeword (b0 , b1 , . . . , bβ−1 , a0 + b0 , a1 + b1 , . . . , aβ−1 + bβ−1 ) ∈ Cβ such that Ψ(r0 , r1 , . . . , rβ−1 ) = (b0 , b1 , . . . , bβ−1 , a0 + b0 , a1 + b1 , . . . , aβ−1 + bβ−1 ). Since Cβ is skew (λ0 + uλ1 )-constacyclic, we have ((λ0 + uλ1)θ(rβ−1 ), θ(r0 ), . . . , θ(rβ−2 )) ∈ Cβ and Ψ((λ0 + uλ1 )θ(rβ−1 ), θ(r0 ), . . . , θ(rβ−2 )) = (λ1 aβ−1 + λ0 kbβ−1 , kb0 , . . . , kbβ−2 , (λ0 + λ1 )aβ−1 + λ0 kbβ−1 , a0 + kb0 , . . . , aβ−2 + kbβ−2 ). Hence, Ψ(C) is (λ0 + uλ1 )-QT with index 2 of length 2β. 

5

Zq R -Linear Skew (λ0 + uλ1)−Constacyclic Codes

In this section, we study skew (λ0 + uλ1 )-constacyclic codes over the ring Zq R.

Definition 5.1 Let θ be an automorphism of R. A linear code C over Zαq Rβ is called skew constacyclic if C satisfies the following two conditions. 1. C is an R-submodule of Zαq Rβ . 2. If (e0 , e1 , . . . , eα−1 , r0 , r1 , . . . , rβ−1 ) ∈ C then (eα−1 , e0 , . . . , eα−2 , (λ0 + uλ1 )θ(rβ−1 ), θ(r0 ), . . . , θ(rβ−2 )) ∈ C. In polynomial representation, each codeword c = (e0 , e1 , . . . , eα−1 , r0 , r1 , . . . , rβ−1 ) of a skew constacyclic code can be represented by a pair of polynomials

c(x) =

e0 + e1 x + . . . + eα−1 xα−1 , r0 + r1 x + . . . + rβ−1 xβ−1

!

= (e(x), r(x)) ∈ Zq [x]/hxα −1i×R[x; θ]/hxβ −(λ0 +uλ1)i.

9

Let h(x) = h0 + h1 x + . . . + ht xt ∈ R [x] and let (f (x), g(x)) ∈ Zq [x]/hxα − 1i × R[x; θ]/hxβ − (λ0 + uλ1 )i. The multiplication is defined by the basic rule h(x) ∗ (f (x), g(x)) = (η(h(x))f (x), h(x)g(x)) where η(h(x)) = η(h0 ) + η(h1 )x + . . . + η(ht )xt . Theorem 5.2 Let C be a linear code over Zq R of length (α, β), and let C = Cα × Cβ , where Cα is linear code over Zq of length α and Cβ is linear code over R of length β. Then C is a skew (λ0 + uλ1 )-constacyclic code with respect to the automorphism θ if and only if Cα is a cyclic code over Zq and Cβ is a skew (λ0 + uλ1)-constacyclic code over R with respect to the automorphism θ. Proof. Let (e0 , e1 , . . . , eα−1 ) ∈ Cα and let (r0 , r1 , . . . , rβ−1 ) ∈ Cβ . If C is a skew constacyclic code, then (eα−1 , e0 , . . . , eα−2 , (λ0 + uλ1 )θ(rβ−1 ), θ(r0 ), . . . , θ(rβ−2 )) ∈ C, which implies that (eα−1 , e0 , . . . , eα−2 ) ∈ Cα and ((λ0 + uλ1 )θ(rβ−1 ), θ(r0 ), . . . , θ(rβ−2 )) ∈ Cβ . Hence, Cα is a cyclic code over Zq and Cβ is a skew (λ0 + uλ1 )-constacyclic code over R with respect to the automorphism θ. On the other hand, suppose that Cα is a cyclic code over Zq and Cβ is a skew constacyclic code over R. Note that (eα−1 , e0 , . . . , eα−2 ) ∈ Cα and ((λ0 + uλ1 )θ(rβ−1 ), θ(r0 ), . . . , θ(rβ−2 )) ∈ Cβ , so C is a skew constacyclic code over Zq R.



Corollary 5.3 Let C = Cα × Cβ be a skew (λ0 + uλ1 )−constacyclic code over Zq R. Then the dual code C ⊥ = Cα⊥ × Cβ⊥ of C is a skew (λ0 + uλ1 )−1 -constacyclic code over Zq R. Proof. From Equation (3), we have C ⊥ = Cα⊥ × Cβ⊥ . According to Corollary (4.3), we have Cβ⊥ skew (λ0 + uλ1 )−1 -constacyclic code over R. Hence the dual codes C ⊥ is skew (λ0 + uλ1 )−1 -constacyclic over Zq R.  If λ0 + uλ1 = 1, then C is called a skew cyclic code, and if θ = Id, then the code is just constacyclic. 10

Theorem 5.4 Let C be a linear code over Zq R of length (α, β), and let C = Cα × Cβ , where Cα is linear code over Zq of length α and Cβ is linear code over R of length β. Then C is a skew cyclic code with respect to the automorphism θ if and only if Cα is a cyclic code over Zq and Cβ is a skew cyclic code over R with respect to the automorphism θ. Proof. Let (e0 , e1 , . . . , eα−1 ) ∈ Cα and let (r0 , r1 , . . . , rβ−1 ) ∈ Cβ . If C is a skew cyclic code, then (eα−1 , e0 , . . . , eα−2 , θ(rβ−1 ), θ(r0 ), . . . , θ(rβ−2 )) ∈ C, which implies that (eα−1 , e0 , . . . , eα−2 ) ∈ Cα , and (θ(rβ−1 ), θ(r0 ), . . . , θ(rβ−2 )) ∈ Cβ . Hence, Cα is a cyclic code over Zq and Cβ is a skew cyclic code over R with respect to the automorphism θ. On the other hand, suppose that Cα is a cyclic code over Zq and Cβ is a skew cyclic code over R. Note that (eα−1 , e0 , . . . , eα−2 ) ∈ Cα and (θ(rβ−1 ), θ(r0 ), . . . , θ(rβ−2 )) ∈ Cβ , so C is a skew cyclic code over Zq R.



Lemma 5.5 A code C = Cα ×Cβ of length (α, β) over Zq R is a skew (λ0 +uλ1 )-constacyclic code if and only if C is left R[x; θ]-submodule of Zq [x]/hxα − 1i × R[x; θ]/hxβ − (λ0 + uλ1 )i. Proof. Let C be a skew (λ0 + uλ1 )-constacyclic code. Let c ∈ C, and let the associated polynomial of c be c(x) = (e(x), r(x)). As x ∗ c(x) is a skew constacyclic shift of c, x ∗ c(x) ∈ C. Then, by linearity of C, r(x) ∗ c(x) ∈ C for any r(x) ∈ R[x, θ]. Thus C is left R[x; θ]-submodule of Zq [x]/hxα − 1i × R[x; θ]/hxβ − (λ0 + uλ1 )i. Conversely, suppose that C is a left R[x; θ]-submodule of Zq [x]/hxα − 1i × R[x; θ]/hxβ − (λ0 + uλ1 )i, then we have that x ∗ c(x) ∈ C. Thus, C is a skew (λ0 + uλ1 )−constacyclic code. 

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Lemma 5.6 [30, Lemma 1] Let g(x), h(x) ∈ R[x; θ]. If g(x) ∗ h(x) is a monic central element of R[x; θ], then g(x) ∗ h(x) = h(x) ∗ g(x). Theorem 5.7 Let C = Cα × Cβ be a linear code over Zq R of length (α, β), and let Cα be a principally generated free cyclic code of length α over Zq having a monic generator polynomial g1 (x) such that xα − 1 = h1 (x) ∗ g1 (x), and let Cβ be principally generated free skew (λ0 + uλ1 )-constacyclic code of length β over R with respect to the automorphism θ having a monic generator polynomial g2 (x) such that xβ − (λ0 + uλ1 ) = h2 (x) ∗ g2 (x). Then the code C generated by g(x) = (g1 (x), g2 (x)) is a skew constacyclic code. Moreover, {g(x), x ∗ g(x), . . . , xl−1 ∗ g(x)} is a spanning set of C, where l = deg(h(x)) and h(x) = lcm{h1 (x), h2 (x)}. Proof. Let xα − 1 = h1 (x) ∗ g1 (x) and xβ − (λ0 + uλ1 ) = h2 (x) ∗ g2 (x), for some monic polynomials h1 ∈ Zq [x] and h2 (x) ∈ R[x; θ]. Let h(x) = lcm{h1 (x), h2 (x)}. Then h(x) ∗ g(x) = h(x) ∗ (g1 (x), g2 (x)) = 0 as h(x) ∗ gi (x) = h′ (x) ∗ hi (x) ∗ gi (x) = 0 for i = 0, 1. Now let v(x) ∈ C, then v(x) = r(x) ∗ g(x) for some r(x) ∈ R[x; θ]. We have r(x) = q(x) ∗ h(x) + t(x), where t(x) = 0 or deg(t(x)) < deg(h(x)). Then v(x) = r(x) ∗ g(x) = t(x) ∗ g(x) = 0. From t(x) = 0 or deg(t(x)) < deg(h(x)), the result follows.

5.1



The Gray Images of Skew Constacyclic Codes over Zq R

We define a Gray map Φ from Zαq Rβ to Zα+2β by q Φ : Zαq Rβ → Zα+2β q (e0 , . . . , eα−1 , r0 , . . . , rβ−1 ) 7→ (e0 , e1 , . . . , eα−1 , b0 , b1 , . . . , bβ−1 , a0 + b0 , a1 + b1 , . . . , aβ−1 + bβ−1 ) where ri = ai + ubi for i = 0, 1, . . . , β − 1.

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Let λ1 , λ2 , . . . , λl be non-zero constants in Zq . A linear code of length n = ml will be called a generalized QT code, which is in turn a special case of multi-twisted (MT) codes [13], if for any codeword (c1 , c2 , . . . , c(m−1)l , c(m−1)l+1 , . . . , cml ) ∈ C we have that (λ1 cml , λ2 cml−1 , . . . , λl c(m−1)l+1 , c1 , . . . , c(m−1)l+1 ) ∈ C.

Theorem 5.8 Let C be a skew constacyclic code in Zq [x]/hxα −1i×R[x; θ]/hxβ −(λ0 +uλ1 )i 1. If α = β, then Φ(C) is a (λ0 + uλ1 )-QT code with index 2 and length 3α. 2. If α 6= β, then Φ(C) is a generalized (λ0 + uλ1 )-QT code with index 2 and length (α, 2β). Proof. and let

Let C be a skew constacyclic code in Zq [x]/hxα − 1i × R[x; θ]/hxβ − (λ0 + uλ1 )i

(e0 , e1 , . . . , eα−1 , b0 , b1 , . . . , bβ−1 , a0 + b0 , a1 + b1 , . . . , aβ−1 + bβ−1 ) ∈ Φ(C). Then there is a codeword (e0 , e1 , . . . , eα−1 , b0 , b1 , . . . , bβ−1 , a0 +b0 , a1 +b1 , . . . , aβ−1 +bβ−1 ) ∈ C such that Φ(e0 , e1 , . . . , eα−1 , r0 , r1 , . . . , rβ−1 ) = (e0 , e1 , . . . , eα−1 , b0 , b1 , . . . , bβ−1 , a0 +b0 , a1 +b1 , . . . , aβ−1 +bβ−1 ). Since C is skew constacyclic, we have (eα−1 , e0 , . . . , eα−2 , (λ0 + uλ1 )θ(rβ−1 ), θ(r0 ), . . . , θ(rβ−2 )) ∈ C and Φ(eα−1 , e0 , . . . , eα−2 , (λ0 + uλ1 )θ(rβ−1 ), θ(r0 ), . . . , θ(rβ−2 )) = (eα−1 , . . . , eα−2 , λ1 aβ−1 + λ0 kbβ−1 , kb0 , . . . , kbβ−2 , (λ0 + λ1 )aβ−1 + λ0 kbβ−1 , a0 + kb0 , . . . , aβ−2 + kbβ−2 ). If α = β, then n = 3α, thus Φ(C) is (λ0 + uλ1)-QT with index 2. Otherwise Φ(C) is a generalized (λ0 + uλ1 )-QT code with index 2 and length (α, 2β). 

5.2

Double Skew Constacyclic Codes over Zq R

In this subsection, we study double skew constacyclic codes over Zq R. Let n1 = α + 2β and n2 = α′ + 2β ′ be integers such that n = n1 + n2 . We consider a partition of the set of the n coordinates into two subsets of n1 and n2 coordinates, respectively, so that C is a subset of ′ ′ Zαq Rβ × Zαq Rβ . 13

Definition 5.9 A linear code C of length n over Zq R is called a double skew constacyclic code if C satisfies the following conditions. 1. C is a linear code. 2. If (e0 , e1 , . . . , eα−1 , r0 , r1 , . . . , rβ−1 , e′0 , e′1 , . . . , e′α′ −1 , r0′ , r1′ , . . . , rβ′ ′ −1 ) ∈ C then (eα−1 , . . . , eα−2 , λθ(rβ−1 ), . . . , θ(rβ−2 ), e′α′ −1 , . . . , e′α′ −2 , λθ(rβ′ ′ −1 ), . . . , θ(rβ′ ′ −2 )) ∈ C, where λ = λ0 + uλ1 . ′

Denote by Rα,β,α′ ,β ′ the ring Zq [x]/hxα − 1i × R[x; θ]/hxβ − (λ0 + uλ1 )i × Zq [x]/hxα − 1i × ′ R[x; θ]/hxβ − (λ0 + uλ1 )i. In polynomial representation, each codeword c = (e0 , e1 , . . . , eα−1 , r0 , r1 , . . . , rβ−1 , e′0 , e1 , . . . , e′α′ −1 , r0′ , r1′ , . . . , rβ′ ′ −1 ) of a skew constacyclic code can be represented by four polynomials 

Let

  c(x) =  

e0 + e1 x + . . . + eα−1 xα−1 , r0 + r1 x + . . . + rβ−1 xβ−1 , ′ e′0 + e′1 x + . . . + e′α′ −1 xα −1 , ′ r0′ + r1′ x + . . . + rβ′ ′ −1 xβ −1



   = (e(x), r(x), e′ (x), r ′ (x)) ∈ Rα,β,α′ ,β ′ . 

h(x) = h0 + h1 x + · · · + ht xt ∈ R [x] and let (f (x), g(x), f ′(x), g ′ (x)) ∈ Rα,β,α′ ,β ′ . We define a multiplication by h(x) ∗ (f (x), g(x)) = (η(h(x))f (x), h(x)g(x), η(h(x))f ′ (x), h(x)g ′ (x)), where η(h(x)) = η(h0 ) + η(h1 )x + . . . + η(ht )xt . This gives us the following theorem.

Theorem 5.10 A linear code C is a double skew constacyclic code if and only if it is a left R[x; θ]-submodule of Rα,β,α′ ,β ′ .

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Proof. Let c(x) = c0 + c1 x + . . . + cn−1 xn−1 be a codeword of C. Since C is a double skew (λ0 + uλ1 )-constacyclic code, it follows that x ∗ c(x) ∈ C. By linearity of C, r(x) ∗ c(x) ∈ C for any r(x) ∈ R[x; θ]. Therefore C is a left R[x; θ]-submodule of Rα,β,α′ ,β ′ . Conversely, suppose that C is a left R[x; θ]-submodule of Rα,β,α′ ,β ′ , then we have that x ∗ c(x) ∈ C. Thus, C is a double skew (λ0 + uλ1 )-constacyclic code. 

6

New Linear Codes over Z4

Codes over Z4 , sometimes called quaternary codes as well, have a special place in coding theory. Due to their importance, a database of quaternary codes was introduced in [7] and it is available online [19]. Hence we consider the case q = 4 to possible obtain quaternary codes with good parameters. We conducted a computer search, using Magma software [28], to find skew cyclic codes over Z4 (Z4 + uZ4 ) whose Gray images are quaternary linear codes with better parameters than the currently best known codes. We have found ten such codes which are listed in the table below. The automorphism of R = Z4 + uZ4 that we used is θ(a + bu) = a + 3bu = a − bu. In addition to the Gray map given in Section 5.1, there are many other possible linear maps from Z4 + uZ4 to Zℓ4 for various values of ℓ. For example, the following map was used in [10] a + bu → (b, 2a + 3b, a + 3b) which triples the length of the code. We used both of these Gray maps in our computations, and obtained new codes from each map. We first chose a cyclic code Cα over Z4 generated by gα (x). The coefficients of this polynomial is given in ascending order of the terms in the table. Therefore, the entry 31212201, for example, represents the polynomial 3 + x + 2x2 + x3 + 2x4 + 2x5 + x7 . Then we searched for divisors of xβ − 1 in the skew polynomial ring R[x; θ] where R = Z4 + uZ4 and θ(a + bu) = a − bu. For each such divisor gβ (x) we constructed the skew cyclic code over Z4 R generated by (gα (x), gβ (x)) and its Z4 -images under each Gray map described above. As a result of the search, we obtained ten new linear codes over Z4 . They are now added to the database ( [19]) of quaternary codes. In the table below, which Gray map is used to obtain each new code is not explicitly stated, but it can be inferred from the values of α, β and n, the length of the Z4 image. If n = α + 2β, then it is the map given in section 5.1 and if n = α + 3β it is the map described in this section. For example, the second code in the table has length 57 = 15 + 3 · 14. This means that the Gray map that triples the length of a code over R is used to obtain this code.

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When xβ − 1 = g(x) ∗ h(x) we can use either the generator polynomial g(x) or the parity check polynomial h(x) to define the skew cyclic code over R. For the codes given in the table below we used the parity check polynomial because it has smaller degree. In general a linear code C over Z4 has parameters [n, 4k1 2k2 ], and when k2 = 0, C is a free code. In this case C has a basis with k vectors just like a linear code over a field. All of the codes in the table below are free codes, hence we will simply denote their parameters by [n, k, d] where d is the Lee weight over Z4 . Our computational results suggest that considering skew cyclic and skew constacyclic codes over Zq (Zq + uZq ) is promising to obtain codes with good parameters over Zq . Table 1: New quaternary codes

α 15 15 15 15 7 7 7 7 7 7

7

β 14 14 14 14 14 14 14 14 14 14

gα hβ 4 3 2 31212201 x + (u + 1)x + x + (3u + 2)x + 3u + 3 31212201 x4 + (u + 1)x3 + x2 + (3u + 2)x + 3u + 3 3021310231 x3 + 2ux2 + (3u + 3)x + 2u + 3 3021310231 x3 + 3x2 + (3u + 2)x + 1 3121 x4 + (3u + 3)x3 + 3x2 + (u + 2)x + 3u + 3 3121 x4 + (u + 3)x3 + (u + 1)x2 + (u + 2)x + 3u + 3 12311 x3 + (2u + 1)x2 + 3ux + 3u + 3 12311 x3 + (2u + 1)x2 + ux + u + 1 12311 x3 + ux2 + (3u + 3)x + 1 12311 x3 + (u + 2)x2 + x + 1

Z4 Parameters [43, 8, 26] [57, 8, 38] [43, 6, 30] [57, 6, 42] [35, 8, 20] [49, 8, 32] [35, 6, 22] [35, 6, 24] [49, 6, 35] [49, 6, 36]

Conclusion

In this paper skew constacyclic codes are considered over the ring Zq R, where R = Zq + uZq , q is a prime power and u2 = 0 and their algebraic and structural properties are studied. Considering their Gray images, we obtained some new linear codes over Z4 from skew cyclic codes over Zq R. Moreover, these codes are then generalized to double skew constacyclic codes.

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