Karthick Gowdhaman

Cruz Mohan
email: bcruzmohan@gmail.com

Chinnapillai Durairajan

Selda Çalkavur
email: dselda.calkavur@kocaeli.edu.tr

Patrick Solé
email: patrick.sole@telecom-paris.fr

Chinnapillai Durairajan • D Selda

Skew Cyclic and Skew Constacyclic Codes over a mixed alphabet

Keywords: Linear codes, Skew-cyclic codes, Gray map, Skew Constacylic codes Mathematics Subject Classification (2000) 94B05, 94B15, 94B35, 94B60

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Introduction

Constacyclic codes constitute a remarkable generalization of cyclic codes and form an important class of linear codes in the coding theory. They were introduced by Berlekamp to construct the analogue of BCH codes in the Lee metric [START_REF] Berlekamp | Negacyclic codes for the Lee metric[END_REF].

In 2007, Boucher et al. [START_REF] Boucher | Skew cyclic codes[END_REF] was the first one to study cyclic codes using a non commutative ring over the skew polynomials ring F[x; θ] where θ denotes the automorphism of the finite field F and they produced many numerical examples which improved the tables of best known codes. The advantage of skew polynomials ring is that the polynomial x n -1 has more factors in skew polynomials ring than in commutative rings. Later, in [START_REF] Boucher | Skew constacyclic codes over Galois rings[END_REF] Boucher et al. generalised this idea to skew constacyclic codes. In 2011, Siap et al. [START_REF] Siap | Skew cyclic codes of arbitrary length[END_REF] studied skew cyclic codes of arbitrary length, and established a strong connection with classical cyclic and quasi-cyclic codes. In 2012, Abualrub et al. [START_REF] Abualrub | On θ-cyclic codes over F 2 + vF 2[END_REF] studied θ-cyclic codes over the non-chain ring F 2 + uF 2 , v 2 = v with respect to Euclidean and Hermitian inner products, and Jitman et al. [START_REF] Jitman | Skew constacyclic codes over finite chain ring[END_REF] studied skew constacyclic codes over finite chain rings, and gave the generators of Euclidean and Hermitian dual codes. Later, these codes over non-chain rings are extensively studied. For instance, the rings F 3 + vF 3 in [START_REF] Ashraf | Skew cyclic codes F 3 + vF 3[END_REF], F q + vF q , v 2 = v in [START_REF] Gursoy | Construction of skew cyclic codes over Fq + vFq[END_REF], F q + uF q + vF q , u 2 = u, v 2 = v, uv = vu = 0 in [START_REF] Ashraf | Skew cyclic codes Fq + uFq + vFq[END_REF] are considered as alphabet for skew cyclic codes. Also, Yao et al. [START_REF] Yao | On Skew cyclic codes over Fq + uFq + vFq + uvFq[END_REF] and Dertli & Cengellenmis [START_REF] Dertli | Skew cyclic codes over Fq + uFq + vFq + uvFq[END_REF] studied these codes over F q + uF q + vF q + uvF q , u 2 = u, v 2 = v, uv = vu. In 2017, Gao et al. [START_REF] Gao | Skew constacyclic codes over the ring Fq + vFq[END_REF] obtained the structure of skew constacyclic codes over the non-chain ring F q + vF q , v 2 = v and they obtained skew (-1 + 2v)-constacyclic codes. Islam and Prakash have determined the structural properties of skew constacyclic codes over F q + uF q + vF q + uvF q , u 2 = u, v 2 = v, uv = vu in [START_REF] Islam | Skew cyclic and skew (α 1 + uα 2 + vα 3 + uvα 4 )-constacyclic codes over Fq + uFq + vFq + uvFq[END_REF], and F q + uF q + vF q , u 2 = u, v 2 = v, uv = vu = 0 in [START_REF] Islam | A note on skew constacyclic codes over Fq + uFq + vFq[END_REF]. In 2019, Bhardwaj and Raka [START_REF] Bhardwaj | Skew constacyclic codes over a non-chain ring Fq[END_REF] studied the skew constacyclic codes over the ring F q [u, v] f (u), g(v), uv -vu by using two non trivial automorphisms.

In another tack, Borges et al. [START_REF] Borges | Z 2 Z 4 -linear codes: generator matrices and duality[END_REF] introduced Z 2 Z 4 -linear codes that is to say codes over the mixed alphabet Z 2 Z 4 , where some coordinates are binary and the rest is quaternary. They have obtained their dual codes as well as their generator matrices. In continuation, Fernandez-Cordoba et al. [START_REF] Fernandez-Cordoba | Z 2 Z 4 -linear codes: Rank and kernel[END_REF] determined the rank and kernel of Z 2 Z 4 -linear codes. These codes found an engineering application in the area of steganography [START_REF] Rifa-Pous | Z 2 Z 4 -additive perfect codes in steganography[END_REF]. Additive codes over the mixed alphabet Z 2 Z 2 s were considered in [START_REF] Aydogdu | The structure of Z 2 Z 2 s -additive codes: Bounds on the minimum distance[END_REF]. Then, the mixed alphabet Z p Z p s , and, more generally Z p r Z p s was studied in [START_REF] Aydogdu | On Z p r Z p s -additive codes[END_REF][START_REF] Shi | On ZpZ p k -additive codes and their duality[END_REF][START_REF] Yao | ZpZ p s -additive cyclic codes are asymptotically good[END_REF][START_REF] Yao | Asymptotically good Z p r Z p s -additive cyclic codes[END_REF]. On the other hand, to the progress of cyclic codes on mixed alphabets, in 2014, Abualrub et al. [START_REF] Abualrub | Z 2 Z 4 -Additive cyclic codes[END_REF] defined

Z 2 Z 4 -additive cyclic codes as Z 4 [x]-submodule of Z 2 [x]/ x r -1 × Z 4 [x]/
x s -1 and derived the unique set of generators, and minimal spanning set for these codes where s is an odd integer. Also, Borges et al. [START_REF] Borges | Z 2 Z 4 -additive cyclic codes, generator polynomials and dual codes[END_REF] found generator polynomials and duals for Z 2 Z 4 -additive cyclic codes. After introducing the new mixed alphabets Z 2 Z 2 [u]additive codes, where u 2 = 0 in [START_REF] Aydogdu | On Z 2 Z 2 [u]-additive codes[END_REF], Aydogdu et al. [START_REF] Aydogdu | The Z 2 Z 2 [u]-cyclic and constacyclic codes[END_REF] were also investigated constacyclic codes over mixed alphabets by defining them as

Z 2 [u][x]-submodules of Z 2 [x]/ x α -1 × Z 2 [u][x]/ x β -(1 + u) .
They obtained some optimal binary linear codes as the Gray images of Z 2 Z 2 [u]-cyclic codes. Meanwhile, [START_REF] Srinivasulu | The Z 2 (Z 2 + uZ 2 )-additive cyclic codes and their duals[END_REF] studied the algebraic properties of Z 2 Z 2 [u]-additive cyclic and constacyclic codes with the unit 1 + u, respectively. Therefore, in continuation of these studies, the expected generalization should be Z 2 r Z 2 s [u]additive cyclic and constacyclic codes, where u 2 = 0.

Motivated by the two kinds of above studies, here we consider a mixed alphabet R = F q R 1 R 2 where R 1 = F q + uF q with u 2 = u and R 2 = F q + uF q + vF q with u 2 = u, v 2 = v, uv = vu = 0 and study θ t -cyclic and (θ t , α)-cyclic codes over R. We completely determine the algebraic structure of these codes. We study their q-ary images under Gray maps, and give some numerical examples in short lengths.

The material is organized as follows. The next section collects some background material. Section 3 studies Gray maps. Section 4 is dedicated to skew cyclic codes, and Section 5 to skew constacyclic codes. Section 6 concludes the article.

Preliminaries

Let F q denote the finite field of characteristic p with q elements, where q = p m for odd prime p and positive integer m. The set F n q of all ordered n-tuples over F q forms a vector space with the usual component-wise addition and scalar multiplication of vectors. A non-empty subset C of F n q is called a code of length n over F q and it is called a linear code if C is a subspace of F n q . From now onward, we denote R 1 = F q + uF q , with u 2 = u and R 2 = F q + uF q + vF q , with u 2 = u, v 2 = v, uv = vu = 0, where q = p m for odd prime p and positive integer m. Note that R 1 and R 2 are finite non-chain rings. Let a + ub + vc be an element of R 2 . Then we define two maps η and δ as follows:

η : R 2 → F q , δ : R 2 → R 1 , η(a + ub + vc) = a, δ(a + ub + vc) = a + ub,
It is clear that η and δ are ring homomorphisms. We consider the ring R,

R = F q R 1 R 2 = {(x, y, z) | x ∈ F q , y ∈ R 1 and z ∈ R 2 }
We define a R 2 -multiplication in this ring as follows, :

R 2 × R → R r (x, y, z) = (η(r)x, δ(r)y, rz)
This is a well defined multiplication and it can be extended componentwise to

R γ = F n1 q × R n2 1 × R n3 2 by : R 2 × R γ → R γ r (x 1 , • • • , x n1 , y 1 , • • • , y n2 , z 1 , • • • , z n3 ) = (η(r)x 1 , • • • , η(r)x n1 , δ(r)y 1 , • • • , δ(r)y n2 , rz 1 , • • • , rz n3 ) where (x 1 , • • • , x n1 , y 1 , • • • , y n2 , z 1 , • • • , z n3 ) ∈ R γ . By this multiplication, R γ is a R 2 module. A non- empty subset C of R γ is said to be a R-linear code of length (n 1 , n 2 , n 3 ) if C is an R 2 -submodule of R γ .
Now we define the inner product as follows,

c, c = n1 1 x i x i + n2 1 y j y j + n3 1 z k z k where c = (x 1 , • • • , x n1 , y 1 , • • • , y n2 , z 1 , • • • , z n3 ), c = (x 1 , • • • , x n1 , y 1 , • • • , y n2 , z 1 , • • • , z n3 ) are in R γ .
Let C be a R-linear code of length (n 1 , n 2 , n 3 ). Then the dual code of C is defined as,

C ⊥ = {c ∈ R γ | c, c = 0 ∀ c ∈ C}

Decomposition and Properties of Gray Maps

Recall that, R 1 = F q + uF q , with u 2 = u. Consider the idempotent orthogonal elements e 1 = u and e 2 = 1 -u, then

R 1 = e 1 R 1 ⊕ e 2 R 1 ∼ = e 1 F q ⊕ e 2 F q
∵ e 1 e 2 = 0, e 2 1 = e 1 , e 1 + e 2 = 1. Hence, R 1 = {ae 1 + be 2 | a, b in F q }. We now define the Gray map,

ϕ 1 : R 1 → F 2 q ϕ 1 (ae 1 + be 2 ) = (a, b)
It can be extended to the length n by

ϕ 1 : R n 1 → F 2n q ϕ 1 ((a 1 , • • • , a n )e 1 + (b 1 , • • • , b n )e 2 ) = (a 1 , • • • , a n , b 1 , • • • , b n )
Note that it is a linear map. We define the Gray weight of a codeword in R 1 as

wt G (ae 1 + be 2 ) = wt H (a, b)
where wt H denotes the Hamming weight. If x, y lie in R n 1 , then their mutual distance is given by

d G (x, y) = n 1 wt G (x i -y i ) = 2n 1 wt H (ϕ 1 (x) -ϕ 1 (y)) = d H (ϕ 1 (x), ϕ 1 (y)).
Hence ϕ 1 is a weight preserving map. A non-empty subset C of R n i is said to be a linear code of length

n if C is R i submodule of R n i . For i ∈ {1, 2}, A i ⊆ R 1 A 1 ⊕ A 2 = {a 1 + a 2 | a i ∈ A i } and A 1 ⊗ A 2 = {(a 1 , a 2 ) | a i ∈ A i }.
Let C e be a linear code of length n over R 1 . Then we define,

C e1 = {y 1 ∈ F n q | e 1 y 1 + e 2 y 2 ∈ C e , for some y 2 ∈ F n q } C e2 = {y 2 ∈ F n q | e 1 y 1 + e 2 y 2 ∈ C e , for some y 1 ∈ F n q } Therefore any linear code C e over R 1 can be represented as C e = e 1 C e1 ⊕ e 2 C e2 and ϕ 1 (C e ) = C e1 ⊗ C e2 . Hence C e1 and C e2 are F q -linear codes. Also note that ϕ 1 (C ⊥ e ) = ϕ 1 (C e ) ⊥ . Recall that, R 2 = F q + uF q + vF q , with u 2 = u, v 2 = v, uv = vu = 0. Let o 1 = (1 -u -v), o 2 = u, o 3 = v be idempotent orthogonal elements in R 2 , then R 2 = o 1 R 2 ⊕ o 2 R 2 ⊕ o 3 R 2 ∼ = o 1 F q ⊕ o 2 F q ⊕ o 2 F q ∵ o i o j = 0 (i = j), o 2 i = o i , o 1 + o 2 + o 3 = 1.
Hence any element in R 2 can be written as ao 1 + bo 2 + o 3 c 3 . We now define a weight preserving linear Gray map ϕ 2 ,

ϕ 2 : R 2 → F 3 q ϕ 2 (ao 1 + bo 2 + co 3 ) = (a, b, c)
It can be extended to length n by the formula

ϕ 2 ((a 1 , • • • , a n )o 1 + (b 1 , • • • , b n )o 2 + (c 1 , • • • , c n )o 2 ) = (a 1 , • • • , a n , b 1 , • • • , b n , c 1 , • • • , c n ).
We define the Gray weight of a codeword in R 2 as

wt G (ao 1 + bo 2 + co 3 ) = wt H (a, b, c)
where wt H denotes the Hamming weight. If x, y are in R n 2 , then their Gray distance is given by

d G (x, y) = n 1 wt G (x i -y i ) = 3n 1 wt H (ϕ 2 (x) -ϕ 2 (y)) = d H (ϕ 2 (x), ϕ 2 (y)). For i ∈ {1, 2, 3}, A i ⊆ R 2 A 1 ⊕ A 2 ⊕ A 3 = {a 1 + a 2 + a 3 | a i ∈ A i } and A 1 ⊗ A 2 ⊗ A 3 = {(a 1 , a 2 , a 3 ) | a i ∈ A i }.
Let C o be a linear code of length n over R 2 . We define,

C o1 = {z 1 ∈ F n q | o 1 z 1 + o 2 z 2 + o 3 z 3 ∈ C o , for some z 2 , z 3 ∈ F n q } C o2 = {z 2 ∈ F n q | o 1 z 1 + o 2 z 2 + o 3 z 3 ∈ C o , for some z 1 , z 3 ∈ F n q } C o3 = {z 3 ∈ F n q | o 1 z 1 + o 2 z 2 + o 3 z 3 ∈ C o , for some z 1 , z 2 ∈ F n q }
Then any linear code C o over R 2 can be represented as

C o = o 1 C o1 ⊕ o 2 C o2 ⊕ o 3 C o3 and ϕ 2 (C o ) = C o1 ⊗ C o2 ⊗ C o3 , where C o1 , C o2 and C o3 are F q -linear codes. Also note that ϕ 2 (C ⊥ o ) = ϕ 2 (C o ) ⊥ .
Henceforth, we define the Gray map ϕ on R using the maps defined previously,

ϕ : R → F 6 q ϕ(x, y, z) = (x, ϕ 1 (y), ϕ 2 (z)) now we can extend this map to R γ , ϕ(x 1 , • • • , x n1 , y 1 , • • • , y n2 , z 1 , • • • , z n3 ) = (x 1 , • • • , x n1 , ϕ 1 (y 1 ), • • • , ϕ 1 (y n2 ), ϕ 2 (z 1 ), • • • , ϕ 2 (z n3 ))
then the Gray weight of an element in R γ can be denoted by wt

G (α) = wt H (ϕ(α)) Any linear code C of R γ can be represented by C = C 1 ⊗ C e ⊗ C o ,
where C 1 , C e and C o are linear code over F q , R 1 and R 2 . Let G Fq be the generator matrix for linear code over F q . The generator matrix G R1 for a linear code over R 1 is denoted by

G R1 = e 1 G e1 e 2 G e2
where G ei is the generator matrix for the linear code C ei , for i = {1, 2}. The generator matrix G R2 for the linear code over R 2 is

G R2 =   o 1 G o1 o 2 G o2 o 3 G o3  
where G oi is the generator matrix for the linear code C oi , for i = {1, 2, 3}. Using the generator matrices above, we can say that the generator matrix G for the linear code over R is,

G =   G Fq 0 0 0 G R1 0 0 0 G R2   Note that the minimum distance of C is min{d H (C 1 ), d H (ϕ 1 (C e )), d H (ϕ 2 (C o ))}.
The following Theorem provides the weight preserving nature of the Gray map, Theorem 1 The Gray map ϕ defined above is linear and weight preserving.

Proof Let x = (x 1 , x 2 , x 3 ), x = (x 1 , x 2 , x 3 ) be in R γ where x 1 , x 1 ∈ F n1 q , x 2 , x 2 ∈ R n2 1 , x 3 , x 3 ∈ R n3 2 ϕ(x + x ) = ϕ(x 1 + x 1 , x 2 + x 2 , x 3 + x 3 ) = (x 1 + x 1 , ϕ 1 (x 2 + x 2 ), ϕ 2 (x 3 + x 3 )) = (x 1 , ϕ 1 (x 2 ), ϕ 2 (x 3 )) + (x 1 , ϕ 1 (x 2 ), ϕ 2 (x 3 ))(∵ ϕ 1 and ϕ 2 are linear) = ϕ(x) + ϕ(x )
Using the linear map ϕ,

d G (x, x ) = wt G (x -x ) = wt H (ϕ(x) -ϕ(x )) = d H (ϕ(x), ϕ(x ))
Hence ϕ is a weight preserving linear map.

The following theorem gives the parameters of the Gray image of a linear code.

Theorem 2 Let C ⊆ R γ be (n 1 + n 2 + n 3 , d G ) linear code then ϕ(C) is (n 1 + 2(n 2 ) + 3(n 3 ), d H ) linear code over F q , where d G = d H .
Proof The proof can extended from the proof of Theorem 1.

The following Theorem characterizes ϕ(C),

Theorem 3 Let C ⊆ R γ be linear, then ϕ(C) = C 1 ⊗ i=2 i=1 C ei ⊗ j=3 j=1 C oj , | C |=| C 1 | i=2 i=1 | C ei | j=3 j=1 | C oj |. Proof Let ϕ(x, y, z) = (x, ϕ 1 (y), ϕ 3 (z)) = (a 1 , . . . , a 6 ) ∈ ϕ(C) ⊆ F 6 q . Note that ϕ is bijective and C = C 1 ⊗ C e ⊗ C o is linear. So a 1 = x ∈ C 1 . Also note that ϕ 1 (y) = (a 2 , a 3 ), ϕ 2 (z) = (a 4 , a 5 , a 6 ). Since ϕ i s are bijective, a 2 e 1 + a 3 e 2 ∈ e 1 C e1 ⊕ e 2 C e2 = C e . Hence (a 2 , a 3 ) ∈ C e1 ⊗ C e2 and similarly (a 4 , a 5 , a 6 ) ∈ C o1 ⊗ C o2 ⊗ C o2 .
The converse holds in a similar way. The second part of the statement follows from the fact that ϕ is bijective.

The following Theorem furnishes the decompostion of the dual of the linear code C.

Theorem 4 Let C = C 1 ⊗ C e ⊗ C o be a linear code over R then C ⊥ = C ⊥ 1 ⊗ C ⊥ e ⊗ C ⊥ o where C ⊥ 1 , C ⊥ e and C ⊥ o
are duals for the respective linear codes. . Now, we have ϕ(x, y, z) = (x, ϕ 1 (y), ϕ 2 (z)), ϕ(x , y , z ) = (x , ϕ 1 (y ), ϕ 2 (z )), then the inner product is given by ϕ(x, y, z), ϕ(x , y , z ) = (x, ϕ 1 (y), ϕ 2 (z)), (x , ϕ 1 (y ), ϕ 2 (z )) = (x, 0, 0), (x , 0, 0) + (0, ϕ 1 (y), 0), (0, ϕ 1 (y ), 0)

Proof Let C ⊥ = {c ∈ R γ | c, c = 0 for all c ∈ C} = {(x , y , z ) ∈ R γ |x ∈ F n1 q , y ∈ R n2 1 , z ∈ R n3 2 }. Let c = (x, y, z) ∈ C = C 1 ⊗ C e ⊗ C o . Then c, c = xx + yy + zz = 0. So x ∈ C ⊥ 1 , y ∈ C ⊥ e , z ∈ C ⊥ o and so C ⊥ ⊆ C ⊥ 1 ⊗ C ⊥ e ⊗ C ⊥ o . Since |C ⊥ | = |C ⊥ 1 ||C ⊥ e ||C ⊥ o |,
+ (0, 0, ϕ 2 (z)), (0, 0, ϕ 2 (z )) = 0 (∵ ϕ 1 (C ⊥ ) = ϕ 1 (C) ⊥ , ϕ 2 (C ⊥ ) = ϕ 2 (C) ⊥ ) So, ϕ(C ⊥ ) ⊆ ϕ(C) ⊥ .
Since the cardinality is same, the statement holds.

The following result provides the self duality nature of the linear code and its Gray image,

Corollary 1 Let C ⊆ R γ be linear, then C is self-dual iff ϕ(C) is self-dual. Further ϕ(C) is a self- orthogonal over F q iff C is self-orthogonal.
Proof Let C be a self-dual linear code of length n over R. That is C = C ⊥ . Then ϕ(C) = ϕ(C ⊥ ), and hence by Theorem 5, we have ϕ(C) = (ϕ(C)) ⊥ . Thus ϕ(C) is a self-dual linear code of length n 1 + 2n 2 + 3n 3 over F q . Conversely, let ϕ(C) be a self-dual linear code of length n 1 + 2n 2 + 3n 3 over F q . Then ϕ(C) = (ϕ(C)) ⊥ , and hence by Theorem 5, we have ϕ(C) = ϕ(C ⊥ ). Since ϕ is bijection, C = C ⊥ . Therefore, C is a self-dual linear code over R γ . Similarly the self orthogonal case holds.

Skew Cyclic R-codes

Let θ t be a non-trivial Frobenius automorphism defined by

θ t : F q → F q , θ t (a) = a p t ,
where t divides m. It can be extended to R 1 and R 2 by

θ t (a + ub) = θ t (a) + uθ t (b), θ t (a + ub + vc) = θ t (a) + uθ t (b) + vθ t (b).
Since t|m, the order of automorphism θ t is m t . We define a polynomial ring

R i [x, θ t ] (1 ≤ i ≤ 2) as follows, R i [x, θ t ] = {a 1 + • • • + a n x n |a j ∈ R i , 1 ≤ j ≤ n} Clearly R i [x, θ t ]
is a ring with respect to usual addition and the multiplication defined by

ax m bx n = aθ m t (a)x m+n
Note that it is a non commutative ring unless θ t is an identity map. A non empty set C is said to be linear code of length

n i over R i if it is a R i submodule of R ni i .
Using the above polynomial rings above, we extend the polynomial ring to R by

R[x, θ t ] = {(a(x), b(x), c(x)) : a(x) ∈ F q [x], b(x) ∈ R 1 [x], c(x) ∈ R 2 [x]}. It can be seen that R[x, θ t ] is a R 2 [x; θ t ]
submodule with respect to usual addition and multiplication defined by : R 2

[x] × R[x, θ t ] → R[x, θ t ] (ax s ) (b 1 x i , b 2 x j , b 3 x k ) = (η(a)x s b 1 x i , δ(a)x s b 2 x j , ax s b 3 x k ) = (η(a)θ s t (b 1 )x s+i , δ(a)θ s t (b 2 )x s+j , aθ s t (b 3 )x s+k )
However under associative and distributive laws, the multiplication can be extended to R γ [x;

θ t ] = Fq[x;θt] x n 1 -1 × R1[x;θt] x n 2 -1 × R2[x;θt] x n 3 -1 as follows, : R 2 [x; θ t ] × R γ [x; θ t ] → R γ [x; θ t ] r(x) (f 1 (x)+ x n1 -1 , f 2 (x)+ x n2 -1 , f 3 (x)+ x n3 -1 ) = (η(r(x))f 1 (x)+ x n1 -1 , δ(r(x))f 2 (x)+ x n2 -1 , r(x)f 3 (x) + x n3 -1 ) Definition 1 [14] A non-trivial R-submodule C of R n is called a θ t -cyclic code if for any c = (c 0 , c 1 , . . . , c n-1 ) ∈ C, σ 1 (c) = (θ t (c n-1 ), θ t (c 0 ), . . . , θ t (c n-2 )) ∈ C. The operator σ 1 is called a θ t -cyclic shift operator on R n . Definition 2 A non-trivial R 2 -submodule C of R γ is called a θ t -cyclic code if for any c = (c 0,1 , c 1,1 , . . . , c n1-1,1 , c 0,e , c 1,e , . . . , c n2-1,e , c 0,o , c 1,o , . . . , c n3-1,o ) ∈ C, σ(c) = (θ(c n1-1,1 ), θ(c 0,1 ) , . . . , θ(c n1-2,1 ), θ(c n2-1,e ), θ(c 0,e ), . . . , θ(c n2-2,e ), θ(c n3-1,o ), θ(c 0,o ), . . . , θ(c n3-2,o )) ∈ C. The operator σ is called as θ t -cyclic shift operator on R n .
The following result yields the relationship between the θ t -cyclic codes over R and F q , Then σ(z) = (θ(z n1-1,1 ), θ(z 0,1 ), . . . , θ(z n1-2,1 ), θ(z n2-1,e ), θ(z 0,e ), . . . , θ(z n2-2,e ), θ(z n3-1,o ) , θ(z 0,o ), . . . , θ(z n3-2,o )) = (σ(z 1 ), σ(z e ), σ(z o )) ∈ C. From this we can conclude that, We recall the following Theorem from [START_REF] Ashraf | Skew cyclic codes Fq + uFq + vFq[END_REF].

Theorem 6 Let C = C 1 ⊗ C e ⊗ C o ⊆ R γ be linear. Then C is a θ t -
Theorem 7 [4] Let C o = o 1 C o1 ⊕ o 2 C o2 ⊕ o 3 C o3 be a linear code over R 2 of length n 3 then C o is θ t -cyclic code iff C oi (1 ≤ i ≤ 3) is a θ t -cyclic code of length n 3 over F q .
The analogue of this result in our setting is as follows.

Theorem 8 [START_REF] Gursoy | Construction of skew cyclic codes over Fq + vFq[END_REF] Let

C e = e 1 C e1 ⊕ e 2 C e2 be a linear code over R 1 of length n 2 then C e is θ t -cyclic code iff C ei (1 ≤ i ≤ 2) is a θ t -cyclic code of length n 2 over F q . Theorem 9 C = C 1 ⊗ C e ⊗ C o be a linear code of length γ = n 1 + n 2 + n 3 , then C is θ t -cyclic iff C 1 , C ei , C oj (1 ≤ i ≤ 2, 1 ≤ j ≤ 3) are θ t -cyclic code of length n 1 , n 2 , n 3 over F q respecively.
Proof We obtain the proof on combining proofs of Theorem 6,7 and 8.

These notions are well-behaved with respect to duality as the next result shows.

Theorem 10 If C is a θ t -cyclic code of length n, then its dual C ⊥ is also a θ t -cyclic code. [START_REF] Boucher | Coding with skew polynomial rings[END_REF] and once again by using Theorem 9, C ⊥ becomes a θ t -cylic code.

Proof From Theorem 9, C 1 , C ei , C oj (1 ≤ i ≤ 2, 1 ≤ j ≤ 3) are θ t -cyclic codes over F q . Then C ⊥ 1 , C ⊥ ei , C ⊥ oj (1 ≤ i ≤ 2, 1 ≤ j ≤ 3) are θ t -cyclic codes over F q from
Recall the following result from [START_REF] Siap | Skew cyclic codes of arbitrary length[END_REF].

Lemma 1 [START_REF] Siap | Skew cyclic codes of arbitrary length[END_REF] Let C be a θ t -cyclic code of length n over F q . Then there exists a polynomial f (

x) ∈ F q [x; θ t ] such that C = f (x) and x n -1 = g(x)f (x) in F q [x; θ t ].
By assuming o(θ t )|n, the counterpart follows;

Theorem 11 Let C = C 1 ⊗ C e ⊗ C o be a θ t -cyclic code of length n over R and order of θ t divides n. Then C = B 1 , B e , B o where B 1 = (f 1 (x), 0, 0) , B e = (0, f e (x), 0) , and B o = (b 1 (x), b e (x), f o (x)) , such that C 1 = f 1 (x) , C e = f e (x) , C o = f o (x) , b 1 (x) ∈ C 1 and b 2 (x) ∈ C e . Proof Let C = C 1 ⊗ C e ⊗ C o be a θ t -cyclic code of length γ = n 1 + n 2 + n 3 over R. Then by Thereom 6, C 1 , C e , C o are θ t -cyclic codes of length n i over F q , R 1 and R 2 . Define a homomorphism from C to R as follows, ψ : C → R ψ(c 1 (x), c e (x), c o (x)) = (0, 0, c o (x)) Define ker(ψ) = {(c 1 (x), c e (x), 0) : c 1 (x) ∈ C 1 , c e (x) ∈ C e } I = {(c 1 (x), c e (x)) ∈ F q [x; θ t ] × R 1 [x; θ t ] : (c 1 (x), c e (x), 0) ∈ ker(ψ)}. Clearly I = I 1 × I e forms a submodule of F q [x; θ t ] × R 1 [x; θ t ].
Therefore there exist a polynomial f 1 (x) and f e (x) in F q [x; θ t ] and R 1 [x; θ t ] respectively, generating I 1 and I e with f 1 (x)|x n1 -1 and f e (x)|x ne -1. Thus I = (f 1 (x), 0), (0, f e (x)) , then for any (c 1 (x), c e (x), 0) ∈ ker(ψ), (c 1 (x), c e (x)) = v(x) (f 1 (x), 0), (0, f e (x)) for some v(x) ∈ R 1 [x; θ t ]. Finally it leads to ker(ψ) = (f 1 (x), 0, 0), (0, f e (x), 0) . The fact that C is a submodule implies that ψ(C) is a submodule. By using the first isomorphism theorem,

C/ker(ψ) ∼ = ψ(C). Let (b 1 (x), b e (x), f o (x)) ∈ C then ψ(b 1 (x), b e (x), f o (x)) = (0, 0, f o (x)). From this any θ t -cyclic code of length n can be represented by C = (f 1 (x), 0, 0)(0, f e (x), 0), (b 1 (x), b e (x), f o (x)) where f 1 (x)|(x n1 - 1), f e (x)|(x n2 -1) and f o (x)|(x n3 -1). Further, we have C is θ t -cyclic then C k where k ∈ {1, e 1 , e 2 , o 1 , o 2 , o 3 } is skew θ t -cyclic code over F q with respective lengths. From Theorem 3, |C| =| C 1 | i=2 i=1 | C ei | j=3 j=1 | C oj | since each C k is θ t -cyclic it is generated by a polynomial f k (x) thus | C |= q γ-6 i=1 k where γ = n 1 + 2(n 2 ) + 3(n 3 ).
The following Theorem provides the generator polynomials for θ t -cylic codes over F q .

Theorem 12 Let C = C 1 ⊗ C e ⊗ C o be a skew cyclic code over R of length γ = n 1 + n 2 + n 3 . Then there exists a polynomial (i) f 1 (x) ∈ F q [x; θ t ] such that C 1 = f 1 (x) and x n1 -1 = g 1 (x)f 1 (x). (ii) f e (x) ∈ R 1 [x; θ t ] such that C e = f e (x) and x n2 -1 = g e (x)f e (x) where f e (x) = 2 i=1 e i f ei (x). (iii) f o (x) ∈ R 2 [x; θ t ] such that C o = f o (x) and x n3 -1 = g o (x)f o (x) where f o (x) = 3 i=1 o i f oi (x).
Proof Let C be a θ t -cyclic code of length γ = n 1 + n 2 + n 3 . From Theorem 6, we have that C 1 , C e and C o are θ t -cyclic codes. Using Lemma 1, (i) follows.

The the proof of (ii) is as follows, let C e = e 1 C e1 ⊕ e 2 C e2 be a θ t -cyclic code of length n 2 over R 1 . Thereom 7 says that, C e1 and C e2 are θ t -cyclic codes of length n 2 over F q . Lemma 1 says that we have

C i = f ei (x) and x n2 -1 = g ei (x)f ei (x) in F q [x; θ t ] for i ∈ {1, 2}. Then e i f ei (x) ∈ C for i ∈ {1, 2}. Also for any f e (x) ∈ C, we have f e (x) = 2 i=1 e i h ei (x)f ei (x) where h ei (x) ∈ F q [x; θ t ] for i ∈ {1, 2}. Thus f e (x) ∈ e 1 f e1 (x), e 2 f e2 (x) . Therefore, C = e 1 f e1 (x), e 2 f e2 (x) . As x n2 -1 = g ei (x)f ei (x) in F q [x; θ t ] for i ∈ {1, 2}. Let f e (x) = e 1 f e1 (x) + e 2 f e2 (x) ∈ R 1 [x; θ t ]. Then f e (x) ∈ C. On the other hand e i f ei (x) = e i f e (x) ∈ f e (x) for i = 1, 2. Consequently, C = f e (x) . Further, [ 2 i=1 e i g ei (x)]f e (x) = 2 i=1 e i g ei (x)f ei (x) = 2 i=1 e i (x n2 -1) = x n2 -1. Then x n2 -1 = g e (x)f e (x) in R 1 [x; θ t ],
where g e (x) = 2 i=1 e i g ei (x). Thus (ii) is follows. (iii) is similar to the proof of (ii).

Skew Constacyclic Code over R

In this section, we study skew θ t -constacyclic codes over R. We choose a unit element α ∈ R * 2 such that α satisfying the condition

α 2 = 1, (α = 1, -1, • • • ). Definition 3 Let α i ∈ F p t /{0}, a linear code C ⊆ R γ [x, θ] is called skew α = α 1 +uα 2 +vα 3 -constacyclic code if it is invariant under the cyclic shift operator λ α that is whenever c = (x 0 , x 1 , • • • , x n1-1 , y 0 , y 1 , • • • , y n2-1 , z 0 , z 1 , • • • , z n3-1 ) ∈ C λ α (c) = (α 1 θ t (x n1-1 ), θ t (x 0 ), • • • , θ t (x n1 ), (α 1 + uα 2 )θ t (y n1-1 ), θ t (y 1 ), • • • , θ t (y n2-2 ), (α 1 + uα 2 + vα 3 )θ t (z n3-1 ), θ t (z 0 ), • • • , θ t (z n3-2 )) ∈ C
The following two results translate symmetry conditions into algebraic constraints. The proof of the first result is immediate and omitted.

Theorem 13 Let R n,λ = R[x; θ t ]/ x n -λ . A linear code C of length n over R is (θ t , λ)-cyclic code if and only if C is a left R[x; θ t ]-submodule of R n,λ . Theorem 14 A code C is skew α-cyclic code over R γ = Fq[x,θt] x n 1 -α × R1[x,θt] x n 2 -α × R2[x,θt] x n 3 -α iff C is a left R 2 [x, θ t ] module over R γ . Proof Let C be a skew α-cyclic code. Then by definition x (f (x)|g(x)|h(x)) ∈ C x (f (x)|g(x)|h(x)) = (θ t (f 0 )x + θ t (f 1 )x 2 + • • • + α 1 θ t (f n1-1 ), θ t (g 0 )x + θ t (g 1 )x 2 + • • • + (α 1 + uα 2 )θ t (g n2-1 ), θ t (h 0 )x + θ t (h 1 )x 2 + • • • + (α 1 + uα 2 + vα 3 )θ t (h n3-1 )) ∈ C
And by using linearity of C, r(x)

(g 1 (x)|g 2 (x)|g 3 (x)) ∈ C for some r(x) ∈ R 2 [x, θ t ]. Hence C is an left R 2 [x, θ t ] submodule over R γ . Conversely, assume that C is is an left R 2 [x, θ t ] submodule over R γ , then we have x (f (x)|g(x)|h(x)) ∈ C implies C is skew α-cyclic code. Theorem 15 Let C o ⊆ R n 2 be a skew α = α 1 + uα 2 + vα 3 -cyclic code of length n iff C o1 , C o2 and C o3 are skew α 1 , α 1 + α 2 , α 1 + α 3 -cyclic codes over F q of length n.
(x) ∈ C o , λ α (o 1 (x 0 , x 1 , • • • , x n-1 ) + o 2 (y 0 , y 1 , • • • + y n-1 ) +o 3 (z 0 , z 1 , • • • , z n-1 )) = ((α 1 + uα 2 + vα 3 ) o 1 (θ t (x n-1 ), θ t (x 0 ), • • • , θ t (x n-2 )) + (α 1 + uα 2 + vα 3 ) o 2 (θ t (y n-1 ), θ t (y 0 ), • • • + θ t (y n-2 )) + (α 1 + uα 2 + vα 3 ) o 3 (θ t (z n-1 ), θ t (z 0 ), • • • , θ t (z n-1 ))) =⇒ λ α1 (x) + λ α1+α2 (y) + λ α1+α3 (z) ∈ C o =⇒ λ α1 (x) ∈ C o1 , λ α1+α2 (y) ∈ C o2 , λ α1+α3 (z) ∈ C o3 Hence, C o1 , C o2 and C o3 are skew α 1 , α 1 + α 2 , α 1 + α 3 -cyclic codes over F q of length n.
Conversely, assume that C o1 , C o2 and C o3 are skew α

1 , α 1 + α 2 , α 1 + α 3 -cyclic codes over F q of length n. Let m 0 , m 1 , • • • , m n-1 be an element in C o , where m i = o 1 x i +o 2 y i +o 3 z i such that x = (x 0 , x 2 , • • • , x n-1 ) ∈ C o1 , y = (y 0 , y 2 , • • • , y n-1 ) ∈ C o2 and z = (z 0 , z 2 , • • • , z n-1 ) ∈ C o3 . Then we have λ α1 (x) ∈ C o1 , λ α1+α2 (y) ∈ C o2 and λ α1+α3 (z) ∈ C o3 . So we get, o 1 λ α1 (x) + o 2 λ α1+α2 (y) + o 3 λ α1+α3 (z) = o 1 λ α1 (x 0 , x 1 , • • • , x n-1 ) + o 2 λ α1+α2 (y 0 , y 1 , • • • + y n-1 ) +o 3 λ α1+α3 (z 0 , z 1 , • • • , z n-1 ) ∈ C = λ α (m 0 , m 1 , • • • , m n-1 ) ∈ C Hence, C is skew α-cyclic code over R n
Theorem 16 Let C e be a a skew α = α 1 + uα 2 -cyclic code over R 1 iff C e1 and C e2 are skew α 1 + α 2 and α 1 -cyclic codes over F .

Proof The proof is similar to Theorem 15 taking mod v to the above condition.

Theorem 17 Let C be a skew α = α 1 + uα 2 + vα 3 -cyclic code over R of length γ = n 1 + n 2 + n 3 iff C 1 , C e and C o are α 1 , α 1 + uα 2 and α 1 + uα 2 + vα 3 -cyclic codes over F q , R 1 and R 2 respectively. Proof Let C 1 , C e and C o be α 1 , α 1 + uα 2 and α 1 + uα 2 + vα 3 -cyclic. Consider x = (x 0 , x 1 , • • • , x n1-1 ), y = (y 0 , y 1 , • • • , y n2-1 ) and z = (z 0 , z 1 , • • • , z n3-1 ). Consider α 1 + uα 2 = β. Then we have, (x, y, z) ∈ C =⇒ (λ α1 (x), λ β (y), λ α (y)) ∈ C
Hence C is skew α-cyclic. The converse part holds similarly.

Theorem 18 Let C be a skew α-cyclic code of length γ = n 1 +n 2 +n 3 iff C 1 is skew α 1 -cyclic code of length n 1 , C e1 & C e2 are α 1 + α 2 , α 1 -cyclic codes of length n 2 and C o1 , C o2 & C o3 are skew α 1 , α 1 + α 2 , α 1 + α 3 - cyclic codes over F q of length n 3 .
Proof Using Theorem 15,16,17 the result follows.

Theorem 19 Let C be a skew α = α 1 + uα 2 + vα 3 -cyclic code over R of length γ = n 1 + n 2 + n 3 iff C ⊥ 1 , C ⊥ e and C ⊥ o are (α 1 ) -1 , (α 1 + uα 2 ) -1 and (α 1 + uα 2 + vα 3 ) -1 -cyclic. Proof Let C be a skew α-cyclic code, [ Lemma 3.1,[27]] says that C ⊥ is skew (α 1 +uα 2 +vα 3 ) -1 -cyclic code. From Theorem 17, we have C ⊥ 1 , C ⊥ e and C ⊥ o are skew (α 1 ) -1 , (α 1 + uα 2 ) -1 and (α 1 + uα 2 + vα 3 ) -1 -cyclic. Corollary 2 Let C = C 1 ⊗ C e ⊗ C o . be a skew α = α 1 + uα 2 + vα 3 -cyclic code over R of length γ = n 1 + n 2 + n 3 .
Then there exists a polynomial. (i) f 1 (x) ∈ F q [x; θ t ] such that C 1 = f 1 (x) and x n1 -α 1 = g 1 (x)f 1 (x) . (ii) f e (x) ∈ R 1 [x; θ t ] such that C e = f e (x) and x n2 -(α 1 + uα 2 ) = g e (x)f e (x). Proof The proof is similar to proof of Theorem 11.

Example 1 Let q = 9 and F 9 = F 3 [z] with z 2 +1 = 0 then consider the ring R γ = F9[x,θt] x 4 -1 × R1[x,θt] x 5 -1 × R2[x,θt]

x 5 -1 and θ 3 be the Frobenius automorphism that is θ 3 (a) = a 3 where a ∈ F 9 .

x 4 -1 = (x + 1)(x + 2)(x + z)(x + 2z) ∈ F 9 [x, θ 3 ]

x 5 -1 = (x + 2)(x 4 + x 3 + x 2 + x + 1) ∈ F 9 [x, θ 3 ] f 1 (x) = (x + 1) , f e (x) = e 1 (x + 2) + e 2 (x + 2) , f o (x) = o 1 (x + 2) + o 2 (x + 2) + o 3 (x + 2) from Theorem 12 we have f i divides x ni -1 for (i = 1, e, o) thus we have a code with parameter [29, 18, 2] over F 9 . Example 2 Let q = 25 and F 9 = F 3 [z] with z 2 + z + 1 = 0 then consider the ring R γ = F25[x,θt] x 4 -1 × R1[x,θt] x 6 -1 ×

R2[x,θt]

x 4 -1 and θ 5 be the frobemious automorphism that is θ 5 (a) = a 5 where a ∈ F 25 .

x 4 -1 = (x + 2)(x + 3)(x + z)(x + z + 1) ∈ F 25 [x, θ 5 ] x 6 -1 = (x 2 -1)(x 2 + x + 1)(x 2 -x + 1) ∈ F 25 [x, θ 5 ] f 1 (x) = (x + 2) , f e (x) = e 1 (x 2 -1) + e 2 (x 2 -1) , f o (x) = o 1 (x + z + 1) + o 2 (x + z + 1) + o 3 (x + z + 1) from Theorem 12 we have f i divides x ni -1 for (i = 1, e, o) thus we have a code with parameter [START_REF] Qian | 1 + u) constacyclic and cyclic codes over F 2 + uF 2[END_REF][START_REF] Gao | Skew constacyclic codes over the ring Fq + vFq[END_REF][START_REF] Abualrub | Z 2 Z 4 -Additive cyclic codes[END_REF] over F 25 .

Conclusion and open problems

In this note, we have studied the algebraic and metric structure of skew cyclic and skew constacyclic codes over a special mixed alphabet. Thus, our codes have a structure of module over the largest of the three alphabets R 2 . Codes over the product ring F q × R 1 × R 2 would be modules over that larger ring. The two algebraic structures are different and should not be confused.

The present work leads itself to two paths of generalization: consider different mixed alphabets or replace the concepts of cyclicity by that of quasi-cyclicity. The former path seems easier than the latter, in view of the many examples of rings that have been used as alphabets of cyclic codes in recent years. On the other hand, the structure of quasi-cyclic codes is always more subtle than that of cyclic codes.

  the statement holds. The next result shows that the Gray maps is compatible with duality.Theorem 5 If C ⊆ R γ is linear, then ϕ(C ⊥ ) = ϕ(C) ⊥ . Proof Let (x,y, z) ∈ C and (x , y , z ) ∈ C ⊥ where x ∈ C 1 , y ∈ C e , z ∈ C o and x ∈ C ⊥ 1 , y ∈ C ⊥ e , z ∈ C ⊥ o , then (x|y|z, x |y |z ) = 0. Using Theorem 4, C ⊥ 1 , C ⊥ e and C ⊥ o are duals for C 1 , C e and C o

σ(z 1 )

 1 ∈ C 1 , σ(z e ) ∈ C e and σ(z o ) ∈ C o Hence C 1 , C e and C o are θ t -cyclic code of length n i . The converse holds in a similar way.

Proof

  Let C o be a skew α-cyclic code, Let a = xo 1 + yo 2 + zo 3 ∈ C o where x = (x 0 , x 1 , • • • , x n-1 ) ∈ C o1 , y = (y 0 , y 1 , • • • , y n-1 ) ∈ C o2 and x = (z 0 , z 1 , • • • , z n-1 ) ∈ C o3 . Then we have by definiton, λ α

  (iii) f o (x) ∈ R 2 [x; θ t ] such that C o = f o (x) and x n3 -(α 1 + uα 2 + vα 3 ) = g o (x)f o (x).Proof The proof is similar to the proof of Theorem 12. Theorem 20 Let C = C 1 ⊗ C e ⊗ C o be a θ t -constacyclic code of length γ over R. Then C = B 1 , B 2 , B o where B 1 = (f 1 (x), 0, 0) , B 1 = (0, f e (x), 0) , and B 1 = (b 1 (x), b e (x), f o (x)) .

  cyclic code if and only if C 1 , C e and C o are θ t -cyclic codes of length n 1 , n 2 and n 3 over F q , R 1 and R 2 respectively.Proof Let C = C 1 ⊗ C e ⊗ C o be a θ t -cyclic code over R. Let z = (z 1 , z e , z o ) ∈ C that is z = (z 0,1 , z 1,1 , . . . , z n1-1,1 , z 0,e , z1,e , . . . , z n2-1,e , z 0,o , z 1,o , . . . , z n3-1,o ) ∈ C.