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

TL;DR

This paper investigates the control properties of abelian group codes generated by finite state machines, showing that controllability is achieved only when the state group is a direct product of prime order cyclic groups.

Contribution

It characterizes when abelian group codes from finite state machines are controllable, specifically linking controllability to the structure of the state group as a product of prime order groups.

Findings

Controllability occurs only if the state group is isomorphic to a product of cyclic groups of prime order.

The study extends binary convolutional codes to arbitrary finite abelian groups using group extensions.

Provides conditions under which group codes are controllable based on the structure of the state group.

Abstract

Finite State Machine (FSM) model is widely used in the construction of binary convolutional codes. If Z_2={0,1} is the binary mod-2 addition group and (Z_2)^n is the n-times direct product of Z_2, then a binary convolutional encoder, with rate (k/n)< 1 and memory m, is a FSM with (Z_2)^k as inputs group, (Z_2)^n as outputs group and (Z_2)^m as states group. The next state mapping nu:[(Z_2)^k x (Z_2)^m] --> (Z_2)^m is a surjective group homomorphism. The encoding mapping omega:[(Z_2)^k x (Z_2)^m] --> (Z_2)^n is a homomorphism adequately restricted by the trellis graph produced by nu. The binary convolutional code is the family of bi-infinite 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 x S) as the extension of U by S, the next state homomorphism nu:(U x S) --> S needs to be surjective and the encoding homomorphism omega:(U x S) --> Y has restrictions given by the trellis graph produced by nu. The code produced by a WSHE is known as group code. In this work we will study the case when the extension (U x S) is abelian with U being Z_p, 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 (Z_p)^j, for some positive integer j.