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Differential approach for the study of duals of algebraic-geometric codes on surfaces
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 on a curve is a differential code. 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.Comment: 21 page
Construction of Rational Surfaces Yielding Good Codes
In the present article, we consider Algebraic Geometry codes on some rational
surfaces. The estimate of the minimum distance is translated into a point
counting problem on plane curves. This problem is solved by applying the upper
bound "\`a la Weil" of Aubry and Perret together with the bound of Homma and
Kim for plane curves. The parameters of several codes from rational surfaces
are computed. Among them, the codes defined by the evaluation of forms of
degree 3 on an elliptic quadric are studied. As far as we know, such codes have
never been treated before. Two other rational surfaces are studied and very
good codes are found on them. In particular, a [57,12,34] code over
and a [91,18,53] code over are discovered, these
codes beat the best known codes up to now.Comment: 20 pages, 7 figure
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