The Semidirect Product Z2 by a Finite Group S is Bad for Non Abelian Codes

In this work we study the time invariant trellis group codes with non abelian trellis section group B which is the semidirect product of the additive group Z2 = {0, 1} by a finite group S. We will show that when S is abelian then the code has free distance limitations, and on the other hand, when S is non abelian the code is non controllable. Therefore, there are not convolutional codes with non abelian trellis section isomorphic to the semidirect product Z2 by S.