On algebras admitting a complete set of near weights, evaluation codes, and Goppa codes

Abstract

In 1998 Høholdt, van Lint and Pellikaan introduced the concept of a “weight function” defined on a ${\mathbb{F}_q}$-algebra and used it to construct linear codes, obtaining among them the algebraic geometry (AG) codes supported on one point. Later, in 1999, it was proved by Matsumoto that all codes produced using a weight function are actually AG codes supported on one point. Recently, “near weight functions” (a generalization of weight functions), also defined on a ${\mathbb{F}_q}$-algebra, were introduced to study codes supported on two points. In this paper we show that an algebra admits a set of m near weight functions having a compatibility property, namely, the set is a “complete set”, if and only if it is the ring of regular functions of an affine geometrically irreducible algebraic curve defined over ${\mathbb{F}_q}$ whose points at infinity have a total of m rational branches. Then the codes produced using the near weight functions are exactly the AG codes supported on m points. A bound for the minimum distance of these codes is presented with examples which show that in some situations it compares better than the usual Goppa bound.