Symbolic Hamburger-Noether expressions of plane curves and applications to AG codes
Mathematics of Computation
Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics)
On the Structure of Order Domains
Finite Fields and Their Applications
Finite Fields and Their Applications
Improved evaluation codes defined by plane valuations
Finite Fields and Their Applications
Evaluation codes defined by finite families of plane valuations at infinity
Designs, Codes and Cryptography
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We apply tools coming from singularity theory, as Hamburger---Noether expansions, and from valuation theory, as generating sequences, to explicitly describe order functions given by valuations of 2-dimensional function fields. We show that these order functions are simple when their ordered domains are isomorphic to the value semigroup algebra of the corresponding valuation. Otherwise, we provide parametric equations to compute them. In the first case, we construct, for each order function, families of error correcting codes which can be decodified by the Berlekamp---Massey---Sakata algorithm and we give bounds for their minimum distance depending on minimal sets of generators for the above value semigroup.