On the theory of graded structures
Journal of Symbolic Computation
Extension of the Berlekamp-Massey algorithm to N dimensions
Information and Computation
The concept of Gröbner algebras
Journal of Symbolic Computation
Decoding of codes defined by a single point on a curve
IEEE Transactions on Information Theory - Part 1
The minimum distance of codes in an array coming from telescopic semigroups
IEEE Transactions on Information Theory - Part 1
On Koetter’s Algorithm and the Computation of Error Values
Designs, Codes and Cryptography
Evaluation codes and plane valuations
Designs, Codes and Cryptography
Algebraic-geometry codes, one-point codes, and evaluation codes
Designs, Codes and Cryptography
Improvements to evaluation codes and new characterizations of Arf semigroups
AAECC'03 Proceedings of the 15th international conference on Applied algebra, algebraic algorithms and error-correcting codes
Generalized Sudan's list decoding for order domain codes
AAECC'07 Proceedings of the 17th international conference on Applied algebra, algebraic algorithms and error-correcting codes
Extended norm-trace codes with optimized correction capability
AAECC'07 Proceedings of the 17th international conference on Applied algebra, algebraic algorithms and error-correcting codes
Improved evaluation codes defined by plane valuations
Finite Fields and Their Applications
On the Structure of Order Domains
Finite Fields and Their Applications
Evaluation codes defined by finite families of plane valuations at infinity
Designs, Codes and Cryptography
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The key concept for generalizing the Berlekamp-Massey algorithm is the existence of an order function, a map from a ring to the nonnegative integers which determines a filtration of the ring with one-dimensional quotients. In this article I show that an order function determines a unique valuation on the function field, which has a residue field equal to the base field. The geometry of these valuations for several monomial orderings on a polynomial ring is discussed, and an ordering is constructed which does not correspond to any monomial ordering. The geometric description allows us to define order functions on general surfaces and higher dimensional varieties.