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 $aS_e + bF_e$, where $a,b\in\mathbb{N}$ and $S_e$ and $F_e$ generate the Picard group $\mathrm{Pic}( \mathcal{H}_e)$. 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.

翻译：本文给出了定义在Hirzebruch曲面$\mathcal{H}_e$上、由除子$aS_e + bF_e$参数化的代数几何码$C_e(a,b)$的对偶的显式形式，其中$a,b\in\mathbb{N}$，且$S_e$和$F_e$生成Picard群$\mathrm{Pic}( \mathcal{H}_e)$。特别地，我们计算了$C_e(a,b)^\perp$最小距离的一个下界。本研究的主要工具之一是我们在论文开头提供的$C_e(a,b)$码的新显式形式。我们还研究了$C_e(a,b)$的某种删截，恢复了之前从环面曲面得到的其他AG码。最后，我们给出了码$C_e(a,b)$之间正交包含的充分条件，并由此构造了CSS量子码。