for a finite field $mathbb{f}_q$, the bivariate skew polynomial ring $mathbb{f}_q[x,y;rho,theta]$ has been used to study codes cite{xh}. in this paper, we give some characterizations of the ring $r[x,y;rho,theta]$, where $r$ is a commutative ring. we investigate 2-d skew $(lambda_1,lambda_2)$-constacyclic codes in the ring $r[x,y;rho,theta]/langle x^l-lambda_1,y^s-lambda_2rangle_{mathit{l}}.$ a...

For a finite field $mathbb{F}_q$, the bivariate skew polynomial ring $mathbb{F}_q[x,y;rho,theta]$ has been used to study codes cite{XH}. In this paper, we give some characterizations of the ring $R[x,y;rho,theta]$, where $R$ is a commutative ring. We investigate 2-D skew $(lambda_1,lambda_2)$-constacyclic codes in the ring $R[x,y;rho,theta]/langle x^l-lambda_1,y^s-lambda_2rangle_{mathit{l}}.$ A...

In this paper, we study skew cyclic and skew constacyclic codes over the ring R = Fq + uFq + vFq + uvFq where q = p^m; p is an odd prime, u^2 = u; v^2 = v; uv = vu. We show that Gray image of a skew cyclic code of length n is a skew quasi-cyclic code of length 4n over Fq of index 4. We also study the structural properties of skew cyclic and skew constacyclic codes over R. Further, generating po...

In this paper, we study the structure of cyclic, quasi-cyclic, constacyclic codes and their skew codes over the finite ring R = Z3+vZ3+v 2 Z3, v 3 = v. The Gray images of cyclic, quasi-cyclic, skew cyclic, skew quasicyclic and skew constacyclic codes over R are obtained. A necessary and sufficient condition for cyclic (negacyclic) codes over R that contains its dual has been given. The paramete...

In this paper, skew cyclic and skew constacyclic codes over finite non-chain ring R = F_q+uF_q+vF_q, where q= p^m, p is an odd prime and u^{2}=u, v^{2}=v, uv=vu=0 are studied. We show that Gray image of a skew cyclic code of length n over R is a skew quasi-cyclic code of length 3n over F_q of index 3. Structural properties of skew cyclic and skew constacyclic over R are obtained. Further, gener...

In this paper, we study skew constacyclic codes over the ring ZqR where R = Zq + uZq, q = p s for a prime p and u2 = 0. We give the definition of these codes as subsets of the ring ZqR . 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 ZqR. Using Gray images ...

In this paper, we study the structure of cyclic, quasi cyclic, constacyclic codes and their skew codes over the finite ring R. The Gray images of cyclic, quasi cyclic, skew cyclic, skew quasi cyclic and skew constacyclic codes over R are obtained. A necessary and sufficient condition for cyclic (negacyclic) codes over R that contains its dual has been given. The parameters of quantum error corr...

An significant milestone study in coding theory recognized to be the paper written by Hammons at al. [1]. Fields are useful area for constructing codes but after the study [1] finite ring have received a great deal of attention. Most of the studies are concentrated on the case with codes over finite chain rings. However, optimal codes over nonchain rings exist (e.g see [2].) In [3], et al. stud...

Skew polynomial rings over finite fields ([7] and [10]) and over Galois rings ([8]) have been used to study codes. In this paper, we extend this concept to finite chain rings. Properties of skew constacyclic codes generated by monic right divisors of x − λ, where λ is a unit element, are exhibited. When λ = 1, the generators of Euclidean and Hermitian dual codes of such codes are determined tog...

We introduce circulant matrices that capture the structure of a skew-polynomial ring F[x; θ] modulo the left ideal generated by a polynomial of the type x − a. This allows us to develop an approach to skew-constacyclic codes based on such circulants. Properties of these circulants are derived, and in particular it is shown that the transpose of a certain circulant is a circulant again. This rec...