Abstract
The linear recurrence relations satisfied by finitely many sequences of finite length over a ground field are described by homogeneous ideals in the polynomial ring in two variables by using Macaulay's theory of inverse systems. The class of these ideals is shown to be precisely the class of homogeneous primary ideals the associated prime of which is the irrelevant maximal ideal. In the case of a single sequence, the classical Berlekamp-Massey algorithm for linear feedback shift register synthesis can be applied to obtain a minimal Gröbner basis of the ideal. The case of multiple sequences is reduced to the case of single sequences by ideal intersection, and the set of all linear recurrence relations of minimal order for the given sequences is generated by the low degree polynomials of the Gröbner basis.
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References
Becker, Th., Weispfenning, V.: Gröbner Bases — A Computational Approach to Commutative Algebra. Berlin, Heidelberg, New York: Springer 1993
Cox, D., Little, J., O'Shea, D.: Ideals, Varieties, and Algorithms. Berlin, Heidelberg, New York: Springer 1992
Ding, C.: Proof of Massey's conjectured algorithm. In: Guenther, C. G. (ed.) Advances in Cryptology — EUROCRYPT '88, 345–349. Lecture Notes in Computer Science, Vol. 330. Berlin, Heidelberg, New York: Springer 1988
Feng, G. L., Tzeng, K. K.: A Generalization of the Berlekamp-Massey Algorithm for Multi-sequence Shift-Register Synthesis with Applications to Decoding Cyclic Codes. IEEE Trans. Inform. Theory 37, Sept., 1274–1287 (1991)
Köthe, G., Topologische Lineare Rüme 1. Berlin, Heidelberg, New York: Springer 1960
Lidl, R., Niederreiter, H.: Finite Fields. Reading, MA: Addison-Wesley 1983
Massey, J. L.: Shift-Register Synthesis and BCH Decoding. IEEE Trans. Inform. Theory 15, Jan, 122–127 (1969)
Macaulay, M. S.: The Algebraic Theory of Modular Systems. New York: Stechert-Hafner 1964
Macaulay, M. S.: Some Properties of Enumeration in the Theory of Modular Systems. Proc. London Math. Soc. 26, 531–555 (1927)
Oberst, U.: Multidimensional Constant Linear Systems. Acta Applicandae Mathematicae 20, 1–175 (1990)
Oberst, U.: On the Minimal Number of Trajectories Determining a Multidimensional System. Math. Control, Signals, Systems 6, 264–288 (1993)
Pauer, F.: Gröbnerbasen und ihre Anwendungen. In: Chatterji, S. D., et al. (ed.) Jahrbuch Uberblicke Mathematik, 127–149. Braunschweig: Vieweg 1991
Sakata, S.: N-Dimensional Berlekamp-Massey Algorithm for Multiple Arrays and Construction of Multivariate Polynomials with Preassigned Zeros. In: Mora, T. (ed.) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes '88, 356–376. Lecture Notes in Computer Science, Vol. 357. Berlin, Heidelberg, New York: Springer 1989
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Althaler, J., Dür, A. Finite linear recurring sequences and homogeneous ideals. AAECC 7, 377–390 (1996). https://doi.org/10.1007/BF01293596
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DOI: https://doi.org/10.1007/BF01293596
Keywords
- Linear Recurrence Relation
- Annihilator Ideal
- Inverse System
- Shift Register Synthesis Problem
- Berlekamp-Massey Algorithm