Any strongly controllable group system or group shift or any linear block code is a linear system whose input is a generator group

TL;DR

This paper demonstrates that any strongly controllable group system or linear block code can be viewed as a linear system with a generator group, providing new insights into their structure and harmonic theory.

Contribution

It introduces the concept of generator groups for group systems, establishing their role in representing strongly controllable systems and linear block codes, and explores their harmonic properties.

Findings

Any strongly controllable group system is a linear system with a generator group.

Generator groups contain elementary systems that form a tile-like structure over the input set.

New structural results and harmonic theory for block codes are derived from generator groups.

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 $t\in\mathbf{Z}$. In the setting of group systems, a natural definition of a linear system is a homomorphism from a group of inputs to an output group system $A$. We show that any group can be the input group of a linear system and some group system. In general the kernel of the homomorphism is nontrivial. We show that any $\ell$-controllable complete group system $A$ is a linear system whose input group is a generator group $({\mathcal{U}},\circ)$, deduced from $A$, and then the kernel is always trivial. The input set ${\mathcal{U}}$ is a set of tensors, a double Cartesian product space of sets $R_{0,k}^t$, with indices $k$, for $0\le k\le\ell$, and time $t$, for $t\in\mathbf{Z}$. $R_{0,k}^t$ 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 an 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. Any elementary system is sufficient to define a unique generator group up to isomorphism, and therefore is sufficient to construct a linear system and group system as well. Any linear block code is a strongly controllable group system. Then we can obtain new results on the structure of block codes using the generator group. There is a harmonic theory of group systems which we study using the generator group.