On strongly controllable group codes and mixing group shifts: solvable groups, translation nets, and algorithms

TL;DR

This paper characterizes shift groups in strongly controllable group codes, explores their relation to Latin squares and translation nets, and provides algorithms for constructing such codes via their state groups.

Contribution

It introduces a simple characterization of shift groups, links Latin square structures to group codes, and offers algorithms to construct strongly controllable Latin group codes.

Findings

Shift groups can be characterized simply.

Strongly controllable Latin group codes have solvable shift groups.

An algorithm to find the state group and construct codes is provided.

Abstract

The branch group of a strongly controllable group code is a shift group. We show that a shift group can be characterized in a very simple way. In addition it is shown that if a strongly controllable group code is labeled with Latin squares, a strongly controllable Latin group code, then the shift group is solvable. Moreover the mathematical structure of a Latin square (as a translation net) and the shift group of a strongly controllable Latin group code are closely related. Thus a strongly controllable Latin group code can be viewed as a natural extension of a Latin square to a sequence space. Lastly we construct shift groups. We show that it is sufficient to construct a simpler group, the state group of a shift group. We give an algorithm to find the state group, and from this it is easy to construct a stronlgy controllable Latin group code.