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Abstract: In the present article, we consider Algebraic Geometry codes on some rationalsurfaces. The estimate of the minimum distance is translated into a pointcounting problem on plane curves. This problem is solved by applying the upperbound -\`a la Weil- of Aubry and Perret together with the bound of Homma andKim for plane curves. The parameters of several codes from rational surfacesare computed. Among them, the codes defined by the evaluation of forms ofdegree 3 on an elliptic quadric are studied. As far as we know, such codes havenever been treated before. Two other rational surfaces are studied and verygood codes are found on them. In particular, a 57,12,34 code over$\mathbf{F} 7$ and a 91,18,53 code over $\mathbf{F} 9$ are discovered, thesecodes beat the best known codes up to now.



Autor: Alain Couvreur

Fuente: https://arxiv.org/



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