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Finding a minimal polynomial vector set of a vector of nD arrays

  • Shojiro Sakata
Submitted Contributions
Part of the Lecture Notes in Computer Science book series (LNCS, volume 539)

Abstract

We propose an algorithm for finding efficiently a minimal set of correlated linear recurrences capable of generating a given vector of finite n-dimensional (nD) arrays. The output of the algorithm is a Gröbner basis of a module over the multivariate polynomial ring, provided that the size of the given arrays is sufficiently large in comparison with the degrees of the characteristic polynomials of the correlated linear recurrences found by the method. This algorithm is also an extension of the Berlekamp-Massey algorithm for finding a minimal polynomial set of an nD array. Although the algorithm has a close connection with the nD Berlekamp-Massey algorithm for multiple nD arrays, the former will find a minimal set of compound linear recurrences which relate all the nD arrays of the given vector while the latter finds a minimal set of linear recurrences which are in common to all the given nD arrays.

Keywords

Total Order Polynomial Vector Linear Recurrence Apply Algebra Algebraic Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    E. R. Berlekamp, Nonbinary BCH decoding. Algebraic Coding Theory, McGraw-Hill Publ. Comp., Chapters 7 and 10, 1968.Google Scholar
  2. [2]
    J. L. Massey, Shift-register synthesis and BCH decoding. IEEE Trans. Information Theory, vol.IT-15, pp.122–127, 1969.Google Scholar
  3. [3]
    B. Buchberger, An algorithmic method in polynomial ideal theory. (N. K. Bose, ed.) Recent Trends in Multidimensional Systems Theory, D. Reidel Publ. Comp., pp.184–232, 1985.Google Scholar
  4. [4]
    F. Mora, H. M. Möller, New constructive methods in classical ideal theory. J. of Algebra, vol.100, no.1, pp.138–178, 1986.Google Scholar
  5. [5]
    A. Furukawa, T. Sasaki, H. Kobayashi, Gröbner basis of a module over K[x 1,..., x n] and polynomial solutions of a system of linear equations. (B. W. Char, ed.) Proc. SYMSAC'86, Waterloo, Canada, pp.222–224, 1986.Google Scholar
  6. [6]
    B. Wall, Computation of syzygies solution of linear systems over a multivariate polynomial ring. RISC-LINZ Series no.88-86.0, 1988.Google Scholar
  7. [7]
    S. Sakata, Finding a minimal set of linear recurring relations capable of generating a given finite two-dimensional array. J. of Symbolic Computation, vol.5, pp.321–337, 1988.Google Scholar
  8. [8]
    S. Sakata, Synthesis of two-dimensional linear feedback shift registers. (L. Huguet, A. Poli, eds.) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Lecture Notes in Computer Science, Springer Verlag: Proc. of AAECC-5, Menorca, Spain, pp.394–407, 1989.Google Scholar
  9. [9]
    S. Sakata, N-dimensional Berlekamp-Massey algorithm for multiple arrays and construction of multivariate polynomials with preassinged zeros. (T. Mora, ed.) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Lecture Notes in Computer Science, Springer Verlag: Proc. of AAECC-6, Rome, Italy, pp.356–376, 1989.Google Scholar
  10. [10]
    S. Sakata, Extension of the Berlekamp-Massey algorithm to N dimensions. Information and Computation, vol.84, no.2, pp.207–239, 1990.Google Scholar
  11. [11]
    S. Sakata, Two-dimensional shift register synthesis and Gröbner bases for polynomial ideals over an integer residue ring. (H. F. Mattson, Jr., T. Mora, eds.) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes: Proc. of AAECC-7, Toulouse, France, 1989 (to appear).Google Scholar
  12. [12]
    S. Sakata, A Gröbner basis and a minimal polynomial set of a finite nD array. (S. Sakata, ed.) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes: Proc. of AAECC-8, Tokyo, Japan, 1990 (to appear).Google Scholar
  13. [13]
    P. Rocha and J. C. Willems, Canonical computational forms for AR 2-D systems. Multidimensional Systems and Signal Processing, vol.1, no.3, pp.251–278, 1990.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Shojiro Sakata
    • 1
  1. 1.Department of Knowledge-Based Information EngineeringToyohashi University of TechnologyTempaku, ToyohashiJapan

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