Linear Recurring Arrays, Linear Systems and Multidimensional Cyclic Codes over Quasi-Frobenius Rings

Abstract

This paper generalizes the duality between polynomial modules and their inverse systems (Macaulay), behaviors (Willems) or zero sets of arrays or multi-sequences from the known case of base fields to that of commutative quasi-Frobenius (QF) base rings or even to QF-modules over arbitrary commutative Artinian rings. The latter generalization was inspired by the work of Nechaev et al. who studied linear recurring arrays over QF-rings and modules. Such a duality can be and has been suggestively interpreted as a Nullstellensatz for polynomial ideals or modules. We also give an algorithmic characterization of principal systems. We use these results to define and characterize n-dimensional cyclic codes and their dual codes over QF rings for n>1. If the base ring is an Artinian principal ideal ring and hence QF, we give a sufficient condition on the codeword lengths so that each such code is generated by just one codeword. Our result is the n-dimensional extension of the results by Calderbank and Sloane, Kanwar and Lopez-Permouth, Z. X. Wan, and Norton and Salagean for n=1.