Abstract
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.
Similar content being viewed by others
References
Campillo A (1980) Algebroid curves in positive characteristic. Lecture notes in mathematics Vol 613. Springer-Verlag, Berlin
Campillo A, Farrán JI (2002) Symbolic Hamburger–Noether expressions of plane curves and applications to AG codes. Math Comput 71(240):1759–1780
Cossart V, Moreno-Socías G (2003) Irreducibility criterion: a geometric point of view. In: Valuation theory and its applications, vol II (Saskatoon, SK 1999) Fields Inst Commun 33, Amer Math Soc Providence RI, pp 27–42
Cox D, Little J, O’Shea D (1996) Ideals, varieties and algorithms. Springer, Berlin
Delgado F, Galindo C, Nuñez A (2000) Saturation for valuations on two-dimensional regular local rings. Math Z 234:519–550
Geil O, Pellikaan R (2002) On the structure of order domains. Finite Fields Appl 8:369–396
Greuel GM, Pfister G, Schönemann H (2006) SINGULAR 3-0-2, a computer algebra system for polynomial computations. Fachbereich Mathematik der Universität, Kaiserlautern. Available via http://www.singular.uni-kl.de/
Hoholdt T, van Lint JH, Pellikaan R (1998) Algebraic geometry codes. In: Pless VS, Huffman WC, Brualdi RA (eds) Handbook of coding theory, vol. 1, Elsevier, Amsterdam, pp 871–961
O’Sullivan ME, (2001) New codes for the Belekamp–Massey–Sakata algorithm. Finite Fields Appl 7:293–317
Spivakovsky M (1990) Valuations in function fields of surfaces. Amer J Math 112:107–156
Zariski O (1939) The reduction of singularities of an algebraic surface. Ann Math 40:639–689
Zariski O, Samuel P (1960) Commutative algebra, vol II. Springer-Verlag, Berlin
Author information
Authors and Affiliations
Corresponding author
Additional information
communicated by R. Calderbank
Rights and permissions
About this article
Cite this article
Galindo, C., Sanchis, M. Evaluation codes and plane valuations. Des Codes Crypt 41, 199–219 (2006). https://doi.org/10.1007/s10623-006-9011-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-006-9011-z