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

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, then the shift group is solvable. Moreover the mathematical structure of a Latin square (as a translation net) and of a shift group are shown to be 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 give some algorithms to find shift groups and Latin group codes.