On semigroups, Grobner basis and algebras admitting a complete set of near weights

In 1998 Hoholdt, van Lint and Pellikaan introduced the concept of a weight function defined on an -algebra and used it to construct linear codes and find bounds for their minimum distances, studying the case where is a finite field. Later, this concept was generalized to that of near weight functions, and Carvalho and Silva characterized the -algebras which admit a so-called complete set of near weight functions as being the ring of regular functions of an irreducible affine curve defined over which has a total of exactly branches at the points at infinity. In the present paper we show how one can use a certain subsemigroup of which appears naturally in the study of such algebras to obtain a Grobner basis for the defining ideals of the curves. We also prove a converse of the main result, obtaining such an -algebra from certain subsemigroups of and polynomials constructed from them. The results in this work extend to near weight functions similar results obtained by Geil, Pellikaan, Matsumoto and Miura for weight functions.