On an approximate eigenvector associated with a modulation code

Let S be a constrained system of finite type, described in terms of a labeled graph M of finite type. Furthermore, let C be an irreducible constrained system of finite type, consisting of the collection of possible code sequences of some finite-state-encodable, 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 correspondence, 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 arbitrary large approximate eigenvectors are required to obtain the best code (in terms of decoding-window size) although a small vector is also available.