GRÖBNER BASES , PADÉ APPROXIMATION , AND DECODING OF LINEAR CODES

This paper shows how Gröbner basis techniques can be used in coding theory, especially in the construction and decoding of linear codes. A simple algorithm is given for computing the reduced Gröbner basis of the vanishing ideal of a given set of finitely many points, and it is used for finding Padé approximation of any polynomial (given implicitly), which is a major step in decoding. A new method is given for construction of a large class of linear codes that can also be decoded efficiently. These codes include as special cases many of the well known codes such as Reed-Solomon codes, Hermitian codes and, more generally, all one-point algebraic geometry codes.

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