Integral closures and weight functions over finite fields

Authors:
Douglas A. Leonard;Ruud Pellikaan
Affiliations:
Department of Discrete and Statistical Sciences, Allison Lab, Auburn University, Auburn, AL 36849-5307, USA;Department of Mathematics and Computing Science, Technical University of Eindhoven, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands
Venue:
Finite Fields and Their Applications
Year:
2003

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Abstract

Curves and surfaces of type I are generalized to integral towers of rank r. Weight functions with values in N^r and the corresponding weighted total-degree monomial orderings lift naturally from one domain R"j"-"1 in the tower to the next, R"j, the integral closure of R"j"-"1[x"j]/. The qth power algorithm is reworked in this more general setting to produce this integral closure over finite fields, though the application is primarily that of calculating the normalizations of curves related to one-point AG codes arising from towers of function fields. Every attempt has been made to couch all the theory in terms of multivariate polynomial rings and ideals instead of the terminology from algebraic geometry or function field theory, and to avoid the use of any type of series expansion.