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

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dc.creatorCarvalho, Cicero
dc.creatorSilva, Ercilio
dc.date2008-07-21
dc.date.accessioned2026-07-07T09:51:46Z
dc.date.available2026-07-07T09:51:46Z
dc.descriptionIn 1998 Hoholdt, van Lint and Pellikaan introduced the concept of a ``weight function'' defined on a F_q-algebra and used it to construct linear codes, obtaining among them the algebraic-geometric (AG) codes supported on one point. Later 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 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 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 formula 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.
dc.description17 pages
dc.identifierhttps://arxiv.org/abs/0807.3198
dc.identifierhttp://arxiv.org/abs/0807.3198
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/165364
dc.subjectInformation Theory
dc.subjectH.1.1; E.4
dc.titleOn algebras admitting a complete set of near weights, evaluation codes and Goppa codes
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