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

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.