Toward good families of codes from towers of surfaces

In memory of Gilles Lachaud. This article has been submitted with an appendix by Alexander Schmidt previously posted on ArXiv with reference arXiv:1702.03701International audienceWe introduce in this article a new method to estimate the minimum distance of codes from algebraic surfaces. This lower bound is generic, i.e. can be applied to any surface, and turns out to be "liftable" under finite morphisms, paving the way toward the construction of good codes from towers of surfaces. In the same direction, we establish a criterion for a surface with a fixed finite set of closed points $\mathcal P$ to have an infinite tower of $\ell$-\'etale covers in which $\mathcal P$ splits totally. We conclude by stating several open problems. In particular, we relate the existence of asymptotically good codes from general type surfaces with a very ample canonical class to the behaviour of their number of rational points with respect to their $K^2$ and coherent Euler characteristic