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Construction of rational surfaces yielding good codes

Alain Couvreur 1, * 
* Corresponding author
1 TANC - Algorithmic number theory for cryptology
LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau], Inria Saclay - Ile de France
Abstract : 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 "à 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 F_7 and a [91,18,53] code over F_9 are discovered. These codes beat the best known codes up to now.
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Submitted on : Thursday, December 16, 2010 - 2:09:08 PM
Last modification on : Friday, February 4, 2022 - 3:31:02 AM
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Alain Couvreur. Construction of rational surfaces yielding good codes. Finite Fields and Their Applications, 2011, 17 (5), pp.424-441. ⟨10.1016/j.ffa.2011.02.007⟩. ⟨inria-00547454⟩



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