Any strongly controllable group system or group shift or any linear block code is isomorphic to a generator group

TL;DR

This paper proves that any strongly controllable group system, including linear block codes and group shifts, is isomorphic to a generator group, revealing a fundamental structure that unifies various algebraic systems.

Contribution

It establishes an isomorphism between strongly controllable group systems and generator groups, introducing a tensor-based representation and elementary systems with a tile-like structure.

Findings

Any strongly controllable group system is isomorphic to a generator group.

The generator group is represented as a tensor space with a tile-like elementary system.

The results unify algebraic systems like linear codes and group shifts under a common framework.

Abstract

Consider any sequence of finite groups $A^t$, where $t$ takes values in an integer index set $\mathbf{Z}$. A group system $A$ is a set of sequences with components in $A^t$ that forms a group under componentwise addition in $A^t$, for each $t\in\mathbf{Z}$. As shown previously, any strongly controllable complete group system $A$ can be decomposed into generators. We study permutations of the generators when sequences in the group system are multiplied. We show that any strongly controllable complete group system $A$ is isomorphic to a generator group $({\mathcal{U}},\circ)$. The set ${\mathcal{U}}$ is a set of tensors, a double Cartesian product space of sets $G_k^t$, with indices $k$, for $0\le k\le\ell$, and time $t$, for $t\in\mathbf{Z}$. $G_k^t$ is a set of unique generator labels for the generators in $A$ with nontrivial span for the time interval $[t,t+k]$. We show the generator group contains a unique elementary system, an infinite collection of elementary groups, one for each $k$ and $t$, defined on small subsets of ${\mathcal{U}}$, in the shape of triangles, which form a tile like structure over ${\mathcal{U}}$. There is a homomorphism from each elementary group to any elementary group defined on smaller tiles of the former group. The group system $A$ may be constructed from either the generator group or elementary system. These results have application to linear block codes, any algebraic system that contains a linear block code, group shifts, and harmonic theory in mathematics, and systems theory, coding theory, control theory, and related fields in engineering.