On the existence of order functions

Abstract

The notions of well-behaving sequences and order funtions is fundamental in the elementary treatment of geometric Goppa codes. The existence of order functions is proved with the theory of Gröbner bases.

On the Structure of Order Domains

“…o obner algebras [2], and the graded structures of 369 1071-5797/02 $35.00 [32] as shown by [27]. The construction of new order domains with the help of the factor ring theorem was found independently in the rank one case in [22][23][24][25]28]. The presentation of order domains was found in the rank one case in [22][23][24][25] and preluded in [34], and shown for arbitrary rank in [27].…”

Generalized Hamming weights of q-ary Reed-Muller codes

“…Let v P be the discrete valuation at P on R. Let ρ(f ) = −v P (f ), so it is the pole order of f at P . Then ρ is a weight function on R. See [16,31,37,38]. In [40,29] bounds are given for the GHW's of algebraic geometry codes involving the gonality sequence.…”

Bounding the number of Fq-rational places in algebraic function fields using Weierstrass semigroups

“…Miura [6] and Pellikaan [7] independently and simultaneously proposed a standard form of defining equations for affine algebraic curves which renders that the subsequent finding of the required f i 's and the computing of f i (P j ) is straightforward. The number of equations in that standard form becomes the minimum if and only if the Weierstrass semigroup Λ of P is telescopic [10].…”