On the algebraic structure of (M,σ,δ)-skew codes

In this paper, we are interested in linear codes invariant under the (σ,δ)-action of a given matrix M, that we call (M,σ,δ)-skew codes. Various subclasses of these codes, depending on the (σ,δ)-cyclical property of M, are derived. In particular, we distinguish between three fundamental types: (M,σ,δ)-cyclic, (M,σ,δ)-quasi-cyclic and (M,σ,δ)-multi-cyclic. These different subclasses allow us to obtain some of the well known types of linear codes as special cases depending on the choice of M. We observe that the (σ,δ)-similarity in Mn(Fq) gives rise to various relations between these codes. As common in the study of cyclic codes and their generalizations, we use the skew polynomial ring Fq[x,σ,δ] to study (M,σ,δ)-skew codes. As an application of the theory developed here, we construct some optimal linear codes and MDS codes over different alphabets with additional desirable properties such as self orthogonality and linear complementary duality (LCD).