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Proceedings of the London Mathematical Society Advance Access originally published online on November 6, 2008
Proceedings of the London Mathematical Society 2009 98(3):714-740; doi:10.1112/plms/pdn042
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© 2008 London Mathematical Society

{delta}-sequences and evaluation codes defined by plane valuations at infinity

Carlos Galindo

Departamento de Matemáticas
Universitat Jaume I
Campus Riu Sec s/n
12071 Castellón
Spain

Francisco Monserrat

IUMPA
Universidad Politécnica de Valencia
Camino de Vera s/n
46022 Valencia
Spain
framonde@mat.upv.es

Received 23 August 2007. Revision received 9 July 2008.

We introduce the concept of {delta}-sequence. A {delta}-sequence {Delta} generates a well-ordered semigroup S in Z2 or R. We explain how to construct (and to compute parameters of) the dual code of any evaluation code associated with a weight function defined by {Delta} from the polynomial ring in two indeterminates to a semigroup S as above. We prove that this is a simple procedure that can be understood by considering a particular class of valuations of function fields of surfaces, called plane valuations at infinity. We also give algorithms to construct an unlimited number of {delta}-sequences of the different existing types, and so this paper helps know and use a new, large set of codes.

2000 Mathematics Subject Classification 94B27, 14B05, 11T71.

The research is supported by Spain Ministry of Education MTM2007-64704, JCyL VA025A07 and Bancaixa P1-1A2005-08.


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