Dual of AG Codes from Hirzebruch Surfaces

Duals of AG (Algebraic Geometry) codes on curves are well-understood and can easily be expressed as AG codes on the same curves. However, next to nothing is known for duals of AG codes on surfaces, only that they can be expressed as sums of AG codes on the same surfaces, but the proof is not constructive. As a consequence, only a few duals of AG codes on surfaces are known, and their minimum distances remain difficult to estimate.

The aim of this talk is to give an explicit form for the duals of AG codes from Hirzebruch surfaces. After a visual description of these surfaces and their rational points, I will describe their AG codes before presenting how we manage to compute their duals.