A Gröbner basis and a minimal polynomial set of a finite nD array
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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.
KeywordsUniqueness Condition Characteristic Polynomial Total Order Cyclic Code Infinite Array
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