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

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.

[1]  F. Piras.  Some remarks on the inverse systems of polynomial modules , 1998 .

[2]  Shiva Shankar.  The Nullstellensatz for Systems of PDE , 1999 .

[3]  Christiaan Heij,et al.  Exact modelling and identifiability of linear systems , 1992, Autom..

[4]  Ulrich Oberst,et al.  Variations on the fundamental principle for linear systems of partial differential and difference equations with constant coefficients , 1995, Applicable Algebra in Engineering, Communication and Computing.

[5]  A. A. Nechaev,et al.  Linear recurrence sequences over commutative rings , 1992 .

[6]  A. V. Mikhalev,et al.  Linear recurring sequences over modules , 1996 .

[7]  N. J. A. Sloane,et al.  Modular andp-adic cyclic codes , 1995, Des. Codes Cryptogr..

[8]  Peizhong Lu.  A Criterion for Annihilating Ideals of Linear Recurring Sequences over Galois Rings , 2000, Applicable Algebra in Engineering, Communication and Computing.

[9]  Tsit Yuen Lam,et al.  Lectures on modules and rings , 1998 .

[10]  Sandro Zampieri,et al.  A behavioral approach to identifiability of 2D scalar systems , 1997, Autom..

[11]  Alexey S. Kuzmin,et al.  Linear Codes and Polylinear Recurrences over Finite Rings and Modules , 1999, AAECC.

[12]  A Characterization of Zero-Dimensional Annihilators of Tableaus , 1997 .

[13]  Sergio R. López-Permouth,et al.  Cyclic Codes over the Integers Modulopm , 1997 .

[14]  Michael Francis Atiyah,et al.  Introduction to commutative algebra , 1969 .

[15]  Eben Matlis,et al.  Injective modules over Noetherian rings. , 1958 .

[16]  Ramesh Pyndiah,et al.  Near-optimum decoding of product codes: block turbo codes , 1998, IEEE Trans. Commun..

[17]  Ulrich Oberst,et al.  On the minimal number of trajectories determining a multidimensional system , 1993, Math. Control. Signals Syst..

[18]  F. S. Macaulay,et al.  The Algebraic Theory of Modular Systems , 1972 .

[19]  U. Oberst.  Multidimensional constant linear systems , 1990, EUROCAST.

[20]  Graham H. Norton,et al.  On the Structure of Linear and Cyclic Codes over a Finite Chain Ring , 2000, Applicable Algebra in Engineering, Communication and Computing.

[21]  Ulrich Oberst,et al.  Finite Dimensional Systems of Partial Differential or Difference Equations , 1996 .