Time and harmonic study of strongly controllable group systems, group shifts, and group codes

In a group trellis, the sequence of branches that split from the identity path and merge to the identity path form two normal chains. The Schreier refinement theorem can be applied to these two normal chains to obtain a Schreier series, a normal chain of the branch group B of the group trellis. It is shown that the components of the shortest length generator sequences of Forney and Trott form a complete system of coset representatives for the Schreier series decomposition of B. These components can be used to construct a natural time domain encoder with the form of a time convolution. The Schreier series and encoder have a natural shift structure related to B, and we find its graph automorphism group. Using the natural shift structure, we find a vector basis group for B which is isomorphic to B. When the basis is formed using generator sequences, the graph automorphism group gives a bound on the number of generator bases. Using the graph automorphism group, the encoder here is compared to an encoder of Forney and Trott. When B is abelian, there is a symmetry and duality between the two encoders which has a transform domain interpretation between a time and spectral domain. When B is nonabelian, the comparison shows that the components of the generator sequences must obey commutative restrictions, and so it is inherently difficult for a branch group to be nonabelian.