Homomorphic encoders of profinite abelian groups II

TL;DR

This paper studies the structure of order controllable group codes over finite abelian groups, proving they have finite canonical generating sets and are algebraically conjugate to full group shifts.

Contribution

It introduces the concept of order controllability for group codes and proves such codes have finite generators and are algebraically conjugate to full shifts.

Findings

Order controllable group codes have finite canonical generating sets.

Such codes are algebraically conjugate to full group shifts.

The paper provides a structural characterization of these codes.

Abstract

Let $\{G_i :i\in\N\}$ be a family of finite Abelian groups. We say that a subgroup $G\leq \prod\limits_{i\in \N}G_i$ is \emph{order controllable} if for every $i\in \mathbb{N}$ there is $n_i\in \mathbb{N}$ such that for each $c\in G$, there exists $c_1\in G$ satisfying that $c_{1|[1,i]}=c_{|[1,i]}$, $supp (c_1)\subseteq [1,n_i]$, and order$(c_1)$ divides order$(c_{|[1,n_i]})$. In this paper we investigate the structure of order controllable group codes. It is proved that if $G$ is an order controllable, shift invariant, group code over a finite abelian group $H$, then $G$ possesses a finite canonical generating set. Furthermore, our construction also yields that $G$ is algebraically conjugate to a full group shift.