Dual of Algebraic Geometry codes from Hirzebruch surfaces

In this paper, we give an explicit form for the dual of the algebraic geometry code $C_e(a,b)$ defined on an Hirzebruch surface ${\mathcal{H}}_e$ and parametrized by the divisor $a{S}_e+b{F}_e$, where $a,b\in \mathbb{N}$ and $S_e$ and $F_e$ generate the Picard group $\mathrm{P}\mathrm{i}\mathrm{c}({\mathcal{H}}_e)$. Notably, we compute a lower bound for the minimum distance of $C_e(a,b{)}^{\perp }$. One of the main ingredient for our study is a new explicit form of the code $C_e(a,b)$ which we provide at the beginning of the paper. We also investigate some puncturing of $C_e(a,b)$, recovering other previously studied AG codes from toric surfaces. Finally, we provide a sufficient condition for orthogonal inclusions between the codes $C_e(a,b)$, and construct CSS quantum codes from them.