The approximate eigenvector associated with a modulation code

Let S be a constrained system, described in terms of a labelled graph M of finite type. Furthermore, let C be an irreducible constrained system consisting of the collection of possible code sequences of some sliding-block decodable modulation code for S. It is known that this code could then be obtained by state-splitting, using a suitable approximate eigenvector. In this paper we show that the collection of all approximate eigenvectors that could be used in such a construction of C contains a unique minimal element. Moreover, we show how to construct its linear span from knowledge of M and C only, thus providing a lower bound on the components of such vectors. For illustration we discuss an example showing that sometimes arbitrarily large approximate eigenvectors are required to obtain the best code (in terms of decoding-window size) although a small vector is also available