Behavioral models over rings--minimal representations and applications to coding and sequences

In this paper we consider polynomial kernel representations for behaviors. For behaviors over fields it is well-known that minimal representations, i.e. representations with minimal row degrees, are exactly those representations for which the polynomial matrix is row-reduced. In this paper we consider behaviors over a particular type of ring, namely Z_{p^r}, where p is a prime number and r is a positive integer. As a starting point in this investigation we focus on minimal partial realizations. These are equivalent to shortest linear recurrence relations. We present an algorithm that computes a parametrization of all shortest linear recurrence relations for a finite sequence in Z_{p^r}. For this we extend well-known techniques developed by Reeds and Sloane in the 80's with methods from the theory of behavioral modeling.