Skew Cyclic codes over $\F_q+u\F_q+v\F_q+uv\F_q$

In this paper, we study skew cyclic codes over the ring $R=\F_q+u\F_q+v\F_q+uv\F_q$, where $u^{2}=u,v^{2}=v,uv=vu$, $q=p^{m}$ and $p$ is an odd prime. We investigate the structural properties of skew cyclic codes over $R$ through a decomposition theorem. Furthermore, we give a formula for the number of skew cyclic codes of length $n$ over $R.$


Introduction
Cyclic codes form an important subclass of linear block codes, studied from the fifties onward. Their clear algebraic structures as ideals of a quotient ring of a polynomial ring makes for an easy encoding. A landmark paper [11] has shown that some important binary nonlinear codes with excellent error-correcting capabilities can be identified as images of linear codes over Z 4 under the Gray map.
Recently, in [3], D. Boucher et al. gave skew cyclic codes defined by using the skew polynomial ring with an automorphism θ over the finite field with q elements. The definition generalizes the concept of cyclic codes over non-commutative polynomial rings. Soon afterwards, D. Boucher et al. studied skew constacyclic codes in [5]. Later, in [4], some important results on the duals of the skew cyclic codes over F q [x; θ] are given. In [12], I. Siap et al. presented the structure of skew cyclic codes of arbitrary length. Further, S. Jitman et al. in [10] defined skew constacyclic codes over the skew polynomial ring with coefficients from finite rings. In [1], T. Abualrub and P. Seneviratne studied skew cyclic codes over ring Moreover, J. Gao [6] and F. Gursoy et al. [8] presented skew cyclic codes over F p + vF p and F q + vF q with different automorphisms, respectively. In [7], J. Gao et al. also studied skew generalized quasi-cyclic codes over finite fields.
In this article, we mainly study skew cyclic codes over ring R = F q + uF q + vF q + uvF q , where u 2 = u, v 2 = v, uv = vu and q = p m .
In our work, the automorphism θ on the ring R is defined to be where b i ∈ F q , and i = 0, 1, 2, 3. In fact, for any a 1 η 1 + a 2 η 2 + a 3 η 3 + a 4 η 4 ∈ R, we have Note that if m is even, the order of the ring automorphism | θ | is m, otherwise, 2m.
The material is organized as follows. In Section 2, we show the basics of codes over ring R that we need for further reference. Section 3 derives the structure of linear codes over R. In Section 4, we introduce skew cyclic codes over ring R and give the structural properties of skew cyclic codes over R through a decomposition theorem. Section 5, we give a example to illustrate the discussed results.

Preliminary
Let F q be a finite field with q elements, where q = p m , p is an odd prime. Throughout, we let R denote the commutative ring F q + uF q + vF q + uvF q , where u 2 = u, v 2 = v, and uv = vu. Let η 1 = 1 − u − v + uv, η 2 = uv, η 3 = u − uv, η 4 = v − uv. It is easy to verify that η 2 i = η i , η i η j = 0, and 4 k=1 η k = 1, where i, j = 1, 2, 3, 4, and i = j. According to [2], we have R = η 1 R ⊕ η 2 R ⊕ η 3 R ⊕ η 4 R. By calculating, we can easily obtain that η i R ∼ = F q , i = 1, 2, 3, 4. Therefore, for any r ∈ R, r can be expressed uniquely as We recall the definition of the Gray map over R in [13] Φ : This map can be naturally extended to the case over R n .
For any element r = a + bu + cv + duv ∈ R, we define the Lee weight of r as w L (r) = w H (a, a + b, a + c, a + b + c + d), where w H denotes the ordinary Hamming weight for q-ary codes. The Lee distance of r ∈ R can be similarly defined.
From the definition of the Gray map Φ, we can easily check that Φ is F q -linear and it is also a distance-reserving isometry from (R n , d L ) to (F 4n q , d H ), where d L and d H denote the Lee and Hamming distance in R n and F 4n q , respectively.

Linear codes over R
In this section, we mainly show some familiar structural properties of R. The proofs of the following theorems can be found in [13], so we omit them here.
If A i (i = 1, 2, 3, 4) are codes over R, we denote their direct sum by Definition 3.1 Let C be a linear code of length n over R, we define that It is clear that C i (i = 1, 2, 3, 4) are linear codes over F n q . Furthermore, Throughout the paper C i (i = 1, 2, 3, 4) will be reserved symbols referring to these special subcodes.
According to Definition 3.1 and [13], we have the following theorem.
According to the definition of the Gray map Φ, we can easily obtain the following theorem. Let C = η 1 C 1 ⊕ η 2 C 2 ⊕ η 3 C 3 ⊕ η 4 C 4 be a linear code of length n over R. Since C is a F q -module, then we have the following lemma. Lemma 3.1 If G i are generator matrices of q-ary linear codes C i (i = 1, 2, 3, 4), respectively, then the generator matrix of C is In light of the definition of Gray map Φ, we can easily obtain the following proposition.
Proposition 3.1 If C is a linear code of length n over R with generator matrice G, then we have 4 Skew Cyclic codes over F q + uF q + vF q + uvF q In this section, we assume C 3 and C 4 are equivalent. Before studying skew cyclic codes over R, we define a skew polynomial ring R[X; θ] and skew cyclic codes over R. Next, we determine the structural properties of skew cyclic codes over R through a decomposition theorem.
Definition 4.1 We define the skew polynomial ring as R[x; θ] = {a 0 + a 1 x + · · · + a n x n |a i ∈ R, i = 0, 1, · · · , n}, where the coefficients are written on the left of the variable x. The multiplication is defined by the basic rule (ax i )(bx j ) = aθ i (b)x i+j , and the addition is defined to be the usual addition rule of polynomials.
It is easily checked that the ring R[x; θ] is not commutative unless θ is the identity automorphism on R.
Definition 4.2 A nonempty subset C of R n is called a skew cyclic code of length n if C satisfies the following conditions: (1) C is a submodule of R n ; (2) if r = (r 0 , r 1 , · · · , r n−1 ) ∈ C, then skew cyclic shift ρ(r) = (θ(r n−1 ), θ(r 0 ), · · · , θ(r n−2 )) ∈ C. Theorem 4.1 Let C = η 1 C 1 ⊕ η 2 C 2 ⊕ η 3 C 3 ⊕ η 4 C 4 be a linear code of length n over R, where C i (i = 1, 2, 3, 4) are codes over F q of length n. Then C is a skew cyclic code with respect to the automorphism θ if and only if C i are skew cyclic codes over F q with respect to the automorphism θ.
According to [4,Corollary 18], we know that the dual code of every skew cyclic code over F q is also skew cyclic. By using this connection and Theorem 4.1, we get the following corollary.
Corollary 4.1 If C is a skew cyclic code over R, then the dual code C ⊥ is also skew cyclic.
The following theorem determines the generator polynomials of a skew cyclic code of length n over R.
be a skew cyclic code of length n over R and suppose that g i (x) are generator polynomials of C i (i=1, 2, 3, 4) respectively. .

Theorem 4.3
Let C i (i = 1, 2, 3, 4) be skew cyclic codes over F q and g i (x) be the monic generator polynomials of these codes respectively, then there is a unique polynomial g(x) ∈ R[x; θ] such that C = g(x) and g(x) is a right divisor of x n −1, where g(x) = 4 i=1 η i g i (x). Proof By Theorem 4.2, we know C = η 1 g 1 (x), η 2 g 2 (x), η 3 g 3 (x), η 4 g 4 (x) . We take g(x) = η 1 g 1 (x) + η 2 g 2 (x) + η 3 g 3 (x) + η 4 g 4 (x), obviously, we have g(x) ⊆ C. On the other hand, one can check that η i g i (x) = η i g(x)(i = 1, 2, 3, 4), which implies C ⊆ g(x) . Hence C = g(x) . Since g i (x) are monic right divisors of x n − 1 ∈ F q [x; θ], then there exist This implies g(x) is a right divisor of x n − 1. Let g(x) = g 0 + g 1 x + · · · + g t x t and h(x) = h 0 + h 1 x + · · · + h n−t x n−t be polynomials in F q [x; θ] such that x n − 1 = h(x)g(x) and C be the skew cyclic code generated by g(x) in F q [x; θ]/(x n − 1), according to Corollary 18 in [4], then the dual code of C is a skew cyclic code generated by h(x) = h n−t + θ(h n−t−1 )x + · · · + θ n−t (h 0 )x n−t . Therefore we have the following corollary. Corollary 4.3 Let C i be skew cyclic codes over F q and g i (x) be their generator polynomial such that x n − 1 = h i (x)g i (x) in F q [x; θ]. If C is a skew cyclic code over R, then In light of previous introduction, we know that the order of θ is even. Therefore, we always assume that n be odd in the rest of the paper.
Theorem 4.4 [6] Let n be odd and C be a skew cyclic code of length n, then C is equivalent to a cyclic code of length n over R.
By Theorem 4.4, we can determine the number of distinct skew cyclic codes of odd length n over R.
θ i ] is irreducible, then the number of distinct skew cyclic codes of length n over R is equal to the number of ideals in R[x]/(x n − 1), i.e. r i=1 (s i + 1) 4 .

Application Examples
In this section, we will exhibit a example of skew cyclic codes and their Gray images over GF (9). Before giving a example, we first give the definition of Plotkin Sum.
Let C ⊕ P D denote the Plotkin sum of two linear codes C and D, also called (u|u + v) construction, where u ∈ C, v ∈ D. For more information on the Plotkin sum, one can see a good survey [9].
In the following, we assume G i are generator matrices of 9-ary linear codes C i for i = 1, 2, 3, 4, respectively. Let C = η 1 C 1 ⊕ η 2 C 2 ⊕ η 3 C 3 ⊕ η 4 C 4 be a linear code of length n over R, then its Gray image Φ(C) is none other than (C 1 ⊕ P C 2 ) ⊕ P (C 3 ⊕ P C 4 ).