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THE DUAL MINIMUM DISTANCE OF ARBITRARY DIMENSIONAL ALGEBRAIC–GEOMETRIC CODES
, 905
"... Abstract. In this article, the minimum distance of the dual C ⊥ of a functional code C on an arbitrary dimensional variety X over a finite field Fq is studied. The approach consists in finding minimal configurations of points on X which are not in “general position”. If X is a curve, the result impr ..."
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Abstract. In this article, the minimum distance of the dual C ⊥ of a functional code C on an arbitrary dimensional variety X over a finite field Fq is studied. The approach consists in finding minimal configurations of points on X which are not in “general position”. If X is a curve, the result improves in some situations the wellknown Goppa designed minimum distance. AMS Classification: 14J20, 94B27, 14C20.
THE DUAL MINIMUM DISTANCE OF ARBITRARY–DIMENSIONAL ALGEBRAIC–GEOMETRIC CODES
, 2011
"... Abstract. In this article, the minimum distance of the dual C ⊥ of a functional code C on an arbitrary–dimensional variety X over a finite field Fq is studied. The approach is based on problems à la Cayley–Bacharach and consists in describing the minimal configurations of points on X which fail to i ..."
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Abstract. In this article, the minimum distance of the dual C ⊥ of a functional code C on an arbitrary–dimensional variety X over a finite field Fq is studied. The approach is based on problems à la Cayley–Bacharach and consists in describing the minimal configurations of points on X which fail to impose independent conditions on forms of some fixed degree m. If X is a curve, the result improves in some situations the wellknown Goppa designed distance.
DIFFERENTIAL APPROACH FOR THE STUDY OF DUALS OF ALGEBRAICGEOMETRIC CODES ON SURFACES
, 2010
"... Abstract. The purpose of the present article is the study of duals of functional codes on algebraic surfaces. We give a direct geometrical description of them, using differentials. Even if this geometrical description is less trivial, it can be regarded as a natural extension to surfaces of the resu ..."
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Abstract. The purpose of the present article is the study of duals of functional codes on algebraic surfaces. We give a direct geometrical description of them, using differentials. Even if this geometrical description is less trivial, it can be regarded as a natural extension to surfaces of the result asserting that the dual of a functional code CL(D,G) on a curve is the differential code CΩ(D,G). We study the parameters of such codes and state a lower bound for their minimum distance. Using this bound, one can study some examples of codes on surfaces, and in particular surfaces with Picard number 1 like elliptic quadrics or some particular cubic surfaces. The parameters of some of the studied codes reach those of the best known codes up to now.