On the control of generic abelian group codes

Abstract

Group Codes are a generalization of the well known Binary Convolutional Codes. For this reason Group Codes are also called Generalized Convolutional Codes. A classical binary convolutional encoder with rate k/n <; 1 and m memory registers can be described as a Finite State Machine (FSM) in terms of the binary groups Z^k₂, Zⁿ₂ and Z^m₂, and adequate next-state and encoder homomorphisms defined over the direct product Z^k₂⊕Z^m₂. Then the binary convolutional code is the family of bi-infinite sequences produced by the binary convolutional encoder. Since the direct product of groups U ⊕ S can be generalized as an extension U ⊗ S, then the encoder of a group code is a FSM M = (U, S, Y, ν, ω) where U is the inputs group, S is the states group, Y is the outputs group. The next-state homomorphism ν and the encoder homomorphism ω are defined over U ⊗ S. The elements of the group code produced by the FSM are bi-infinite sequences y = {yk}kϵZ with yk ϵ Y. Then, each y can be interpreted as a trajectory of a Dynamical System, hence a group code is a Dynamical System. Therefore a group code will be controllable when it is controllable as a Dynamical System. In this work we present some necessary conditions for the control of group codes produced by FSMs defined on generic abelian extensions U ⊗ S with Z_p = {0, 1, ..., p - 1}, the cyclic group of order p.