$\mathbb{Z}_{q}(\mathbb{Z}_{q}+u\mathbb{Z}_{q})-$ linear skew constacyclic codes

Abstract

In this paper, we study skew constacyclic codes over the ring ZqR where R = Zq + uZq, q = p s for a prime p and u 2 = 0. We give the definition of these codes as subsets of the ring Z α q R β. 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 ZqR, also we determine their minimal spanning sets and their sizes. Further, by using the Gray images of skew constacyclic codes over ZqR we obtained some new linear codes over Z4. Finally, we have generalized these codes to double skew constacyclic codes over ZqR.

Skew cyclic codes over $$\mathbb {Z}_{4}+u\mathbb {Z}_{4}+v\mathbb {Z}_{4}$$

“…More recently, in [14] skew-cyclic codes over the ring Z 4 + uZ 4 , where u 2 = 1 have been studied. Also, the authors in [12] studied skew-constacyclic codes over the ring Z q (Z q + uZ q ), where q = p s for a prime p and u 2 = 0. In [7], the structures of cyclic codes over the ring S = Z 8 + uZ 8 + vZ 8 , where u 2 = u, v 2 = v, uv = vu = 0 were determined.…”