Conference "Arithmetic, Geometry and Coding Theory"
March 14 - 18, 2011
CIRM, Marseille, France
Organisers: Yves Aubry (Université de la Méditerranée, IML), Christophe Ritzenthaler (Université de la Méditerranée, IML), Alexey Zykin (Mathematical Department of HSE, Laboratoire Poncelet, IITP)
The topic of this talk is Algebraic Geometry codes from surfaces. It is motivated by two observations. First, in the literature, many explicit examples of good codes from surfaces arise from rational surfaces. However, this property of rationality is never used for studying them. Second, the usual approach consists in constructing codes from "well known" surfaces, i.e. surfaces whose geometry is well understood and then check whether these codes are good.
In this talk, the strategy is the converse. We first look for the properties which should be expected from a surface in order to produce good codes. Then, we construct rational surfaces satisfying these particular properties by blowing up the projective plane at some closed points. The property of rationality is deeply used in the study of the corresponding codes since the estimate of the minimum distance is reduced to a point counting problem on families of plane curves. By this manner, codes beating the best codes of Andries Brouwer tables have been discovered.