On Groups with a Shift Structure: the Schreier Matrix and an Algorithm

A time-invariant group trellis is essentially a group shift or time-invariant group code. A group with a shift structure is the branch group of a strongly controllable time-invariant group trellis. We show that the Loeliger and Mittelholzer definition of a group with a shift structure is equivalent to a second definition by using a controllable Schreier matrix, a certain normal series written as a matrix having a special property of the diagonal terms. This suggests a third definition of a group with a shift structure. We use the third definition to derive properties that the group must have, and then give an algorithm to find the shift structure of the group.