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

Abstract

In 1998 Høholdt, van Lint and Pellikaan introduced the concept of a weight function defined on an ${\mathbb {F}}$-algebra and used it to construct linear codes and find bounds for their minimum distances, studying the case where ${\mathbb {F}}$ is a finite field. Later, this concept was generalized to that of near weight functions, and Carvalho and Silva characterized the ${\mathbb {F}}$-algebras which admit a so-called complete set of $m$ near weight functions as being the ring of regular functions of an irreducible affine curve defined over ${\mathbb {F}}$ which has a total of exactly $m$ branches at the points at infinity. In the present paper we show how one can use a certain subsemigroup of ${\mathbb {N}}_0^m$ which appears naturally in the study of such algebras to obtain a Gröbner basis for the defining ideals of the curves. We also prove a converse of the main result, obtaining such an ${\mathbb {F}}$-algebra from certain subsemigroups of ${\mathbb {N}}_0^m$ 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.