The predictable degree property and row reducedness for systems over a finite ring

Motivated by applications in communications, we consider linear discrete-time systems over the finite ring Zpr. We solve the open problem of deriving a theory of row-reduced representations for these systems. We introduce a less restrictive form of representation than the adapted form introduced by Fagnani and Zampieri. We call this form the “composed form”. We define the concept of “p-predictable degree property” and “p-row-reduced”. We demonstrate that these concepts, coupled with the composed form, provide a natural setting that extends several classical results from the field case to the ring case. In particular, the classical rank test on the leading row coefficient matrix is generalized. We show that any annihilator of B of a pre-specified degree is uniquely parametrized with finitely many coefficients in terms of a kernel representation in p-row-reduced composed form. The underlying theory is the theory of “reduced p-basis” for submodules of that is developed in this paper. We show how to construct a p-row-reduced kernel representation in composed form by constructing a reduced p-basis for the module B⊥.