Mathematics – Algebraic Geometry
Scientific paper
2010-08-13
Finite Fields and their Applications, Volume 17(5), 2011, Pages 424-441
Mathematics
Algebraic Geometry
20 pages, 7 figures
Scientific paper
10.1016/j.ffa.2011.02.007
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 $\mathbf{F}_7$ and a [91,18,53] code over $\mathbf{F}_9$ are discovered, these codes beat the best known codes up to now.
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