3rd International Workshop on Linear Systems (LinSys2007)

The predictable degree property and a parametrization for annihilators of a behavior over a finite ring
Margreta Kuijper, Raquel Pinto, Jan Willem Polderman

We consider linear discrete-time systems over the finite ring Z_{p^r}. Such systems are relevant in the communications area, notably coded modulation systems. More specifically, syndrome formers and encoders for convolutional codes over Z_{p^r} correspond to kernel and image representations of Z_{p^r}-behaviors and motivate our work.

In this talk we focus on the predictable degree property for polynomial kernel representations. This property has enjoyed considerable attention in the literature, both system theoretic (Wedderburn, Wolovich, Kailath) and coding theoretic (Forney et al.). In the field case this property is equivalent to row-reducedness and leads to minimality concepts and useful parametrizations of annihilators of a behavior.

How to come up with a sensible definition in the Z_{p^r}-case that gives rise to equally useful results? It turns out that this is not straightforward. In the talk we show that the adapted kernel representation that was introduced by Fagnani and Zampieri in the 90's is not suitable for this purpose. Instead we define another, less restrictive representation that fits the bill. We give a construction procedure and derive a parametrization result for annihilators of a Z_{p^r}-behavior that parallels the field case.