Abstract
In this paper, the relationship between a Gröbner basis and a minimal polynomial set of a finite nD array is discussed. A minimal polynomial set of a finite nD array is determined by the nD Berlekamp-Massey algorithm. It is shown that a minimal polynomial set is not always a Gröbner basis even if the uniqueness condition is satisfied, and a stronger sufficient condition for a minimal polynomial set to be a Gröbner basis is presented. Furthermore, a simple test whether a given set of polynomials is a Gröbner basis is proposed based on the theory of nD linear recurring arrays. The observations will be important in applying the nD Berlekamp-Massey algorithm to decode some kinds of nD cyclic codes and algebraic geometry codes.
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References
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© 1991 Springer-Verlag Berlin Heidelberg
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Sakata, S. (1991). A Gröbner basis and a minimal polynomial set of a finite nD array. In: Sakata, S. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1990. Lecture Notes in Computer Science, vol 508. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-54195-0_58
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DOI: https://doi.org/10.1007/3-540-54195-0_58
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