On the Structure of Order Domains

Abstract

The notion of an order domain is generalized. The behaviour of an order domain by taking a subalgebra, the extension of scalars, and the tensor product is studied. The relation of an order domain with valuation theory, Gr .o obner algebras, and graded structures is given. The theory of Gr .o obner bases for order domains is developed and used to show that the factor ring theorem and its converse, the presentation theorem, hold. The dimension of an order domain is related to the rank of its value semigroup. # 2002 Elsevier Science (USA)

On the existence of order functions

“…In Matsumoto-Miura [14] this was done by the same Gröbner bases techniques as in this paper. A generalization of these results were obtained by Geil and the author in [8,9].…”

On the Feng-Rao Bound for Generalized Hamming Weights

“…These seems to be the only order functions that are relevant for coding theoretical purposes. From [6] we have the following definition and theorem.…”

“…Another interpretation was given by Høholdt, van Lint and Pellikaan in [8]. Here they introduced the concept of an order function acting on what is known today as an order domain ( [6]). They reformulated some of the most important results by Feng and Rao into this new setting.…”

The growth of valuations on rational function fields in two variables

“…The construction of such codes appears in [11], in which the generators of the corresponding value monoid are computed for a specific example. An alternative description of this construction is given in [3], and we provide more commentary in an example at the end of this paper. Following this example, we pose an open question.…”