Derived Subgroups and Controllability of Group Codes

"Convolutional codes over groups are defined by Loeliger and Mittelholzer in [6] as being controllable and observable group codes. This means the convolutional codes over groups deserve be studied from both algebraic and system theoretic point of view. We first give a fast review of algebraic facts which will be used to describe the encoder of a group code. Then, we use these facts together with system basics of group codes to show that the controllability of these codes depends on the existence of a class of normal series of subgroups from its state group."