Homomorphic encoders of profinite abelian groups II

Abstract

Let $\{G_i:i\in \mathbb{N}\}$ be a family of finite Abelian groups. We say that a subgroup $G\le \prod _{i\in \mathbb{N}}{G}_i$ 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.

arXiv commentPortions of this work previously appeared as arXiv:2103.13135, which has been split