Construction of good codes from weak Del Pezzo surfaces

TL;DR

This paper develops algebraic geometric codes from weak Del Pezzo surfaces, addressing challenges in parameter computation due to surface singularities and the distinction between Cartier and Weil divisors.

Contribution

It introduces a method to construct codes from weak Del Pezzo surfaces and handles the complexities arising from their singularities.

Findings

Codes constructed from weak Del Pezzo surfaces with explicit parameters

Analysis of the impact of surface singularities on code properties

Method to distinguish Cartier and Weil divisors in code construction

Abstract

We construct algebraic geometric codes from weak del Pezzo surfaces. The codes are associated to the anti-canonical class of the anti-canonical model and to the set of rational points of these models. Since we consider weak Del Pezzo surfaces, the anti canonical model is not smooth any more. This complicates the computation of the parameters of the codes; in particular we need to distinguish the Cartier divisors from the Weil ones.