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. Very good codes are found, in particular a [57, 12, 34] code over F7 and a [91, 18, 53] code over F9 are discovered, these codes beat the best known codes up to now. From these best codes and using classical operations on codes it is possible to generate plenty of other codes beating the parameters of best known codes up to now. MSC: 94B27, 14J26, 11G25, 14C20.

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Alain Couvreur
• Keywords: CORR 2010, Education, Plane Curves, Present Article, Rational Surfaces