Dual generalizations of the concept of cyclicity of codes

In this paper we focus on two generalizations of the notion of cyclicity of codes: polycyclic codes and sequential codes. We establish a duality between these two generalizations and also show connections between them and other well-known generalizations of cyclicity such as the notions of negacyclicity and constacyclicity. In particular, it is shown that a code $C$ is sequential and polycyclic if and only if $C$ and its dual C⊥ are both sequential if and only if $C$ and its dual C⊥ are both polycyclic. Furthermore, any one of these equivalent statements characterizes the family of constacyclic codes.