Homomorphic encoders of profinite abelian groups

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 subgroups. It is known that each order controllable profinite abelian group is topologically isomorphic to a direct product of cyclic groups (see \cite{FHS:2017,kiehlmann}). Here we improve this result and prove that under mild conditions an order controllable group $G$ contains a {set} $\{g_n : n\in\N\}$ that topologically generates $G$, and whose elements $g_n$ have all finite support. As a consequence, we obtain that if $G$ is an order controllable, shift invariant, group code over an abelian group $H$, then $G$ possesses a canonical generator set. Furthermore, our construction also yields that $G$ is algebraically conjugate to a full group shift. Some connections to coding theory are also highlighted.