Duality between Multidimensional Convolutional Codes and Systems

Multidimensional convolutional codes arise as a generalization of “classical” onedimensional codes. We will introduce m-dimensional convolutional codes of length n as submodules of Dn where D is the polynomial ring in m variables over a finite field. Besides their coding theoretic significance, they can also be regarded as the annihilating modules of systems of partial difference equations, the latter being studied in much detail in discrete-time multidimensional systems theory. We will apply the duality theorem of Oberst [5] to this particular case and employ the duality to investigate certain first-order representations of onedimensional convolutional codes. Dedicated to Diederich Hinrichsen on the occasion of his 60th birthday ∗Supported in part by NSF grant DMS-96-10389. Paul Weiner would like to thank the Center for Applied Mathematics at Notre Dame for a fellowship which financially supported the presented research.