Homomorphic Encoders of Profinite Abelian Groups II

Abstract

Let {Gi:i∈N} be a family of finite Abelian groups. We say that a subgroup G≤∏i∈NGi is order controllable if for every i∈N, there is ni∈N such that for each c∈G, there exists c1∈G satisfying c1|[1,i]=c|[1,i], supp(c1)⊆[1,ni], and order (c1) divides order (c|[1,ni]). 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.