Control in Trellis Codes produced by Finite State Machines with Information Group Zp

A trellis code is the image of a signal mapper from a time invariant group code produced by a Finite State Machine, FSM. Group codes can be described as dynamical systems and good group codes must be necessarily well behaved dynamical systems. For instance good group codes must be controllable and observable, among other properties of well-behaved systems. In this paper we work with trellis codes produced by Finite State Machines over non-abelian groups. The necessity of non-abelian groups on FSM is because there no exist any regular signal mapper between the outputs of a classical binary convolutional encoder and a M −PSK signal set. Also, it has been shown that the capacity of an AWGN channel using abelian group codes is upper bounded by the capacity of the same channel using PSK modulation eventually with different energies per symbol. We will show that when the trellis section group is non-abelian and the input group of the FSM is a cyclic group Zp = {0, 1, . . . , p − 1}, p prime, then the trellis code produced by the FSM is noncontrollable.