On the control of abelian group codes with information group of prime order

Finite State Machine (FSM) model is widely used in the construction of binary convolutional codes. If Z2 = {0, 1} is the binary mod-2 addition group and Z n 2 is the n-times direct product of Z2, then a binary convolutional encoder, with rate k n < 1 and memory m, is a FSM with Z2 as inputs group, Z n 2 as outputs group and Z m 2 as states group. The next state mapping ν : Z2⊕Z m 2 → Z m 2 is a surjective group homomorphism. The encoding mapping ω : Z2 ⊕ Z m 2 → Z n 2 is a homomorphism adequately restricted by the trellis graph produced by ν. The binary convolutional code is the family of biinfinite sequences produced by the binary convolutional encoder. Thus, a convolutional code can be considered as a dynamical system and it is known that well behaved dynamical systems must be necessarily controllable. The generalization of binary convolutional encoders over arbitrary finite groups is made by using the extension of groups, instead of direct product. In this way, given finite groups U,S and Y , a wide-sense homomorphic encoder (WSHE) is a FSM with U as inputs group, S as states group, and Y as outputs group. By denoting U ⊠ S as the extension of U by S, the next state homomorphism ν : U ⊠ S → S needs to be surjective and the encoding homomorphism ω : U ⊠ S → Y has restrictions given by the trellis graph produced by ν. The code produced by a WSHE is known as group code. In this work we will study the case when the extension U ⊠ S is abelian with U being Zp, p a positive prime number. We will show that this class of WSHEs will produce controllable codes only if the states group S is isomorphic with Zp, for some positive integer j. keywords Finite State Machine, Group Code, Dynamical System, Control.