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.

In this paper, we investigate the structure and properties of skew negacyclic codes and skew quasi-negacyclic codes over the ring [Formula: see text] Some structural properties of [Formula: see text] are discussed, where [Formula: see text] is an automorphism of [Formula: see text] A skew quasi-negacyclic code of length [Formula: see text] with index [Formula: see text] over [Formula: see text] is viewed both as in the conventional row circulant form and also as an [Formula: see text]-submodule of [Formula: see text], where [Formula: see text] is the Galois extension ring of degree [Formula: see text] over [Formula: see text] and [Formula: see text] is an automorphism of [Formula: see text] A sufficient condition for one generator skew quasi-negacyclic codes to be free is determined. Some distance bounds for free one generator skew quasi-negacyclic codes are discussed. Furthermore, given the decomposition of a skew quasi-negacyclic code, we provide the decomposition of its dual code. As a result, a characterization of xself-dual skew quasi-negacyclic codes over [Formula: see text] is provided. By using computer search we obtained a number of new linear codes over [Formula: see text] from skew negacyclic and skew quasi-negacyclic codes over [Formula: see text].

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In this paper, we give conditions for the existence of Hermitian selfdual Θ−cyclic and Θ−negacyclic codes over the finite chain ring F q + uF q . By defining a Gray map from R = F q + uF q to F 2 q , we prove that the Gray images of skew cyclic codes of odd length n over R with even characteristic are equivalent to skew quasi-twisted codes of length 2n over F q of index 2. We also extend an algorithm of Boucher and Ulmer [9] to construct self-dual skew cyclic codes based on the least common left multiples of non-commutative polynomials over F q + uF q .