# How can I compute minimal distance of the AG-code on the Hirzebruch surface $\mathbb{F}_3$?

Let $\mathbb{F}_3$ be the Hirzebruch surface (with index $3$) over a finite field $\mathbb{F}_q$ and $\pi\!: \mathbb{F}_3 \to \mathbb{P}^1$ be the unique $\mathbb{P}^1$-fibration on $\mathbb{F}_3$. Also, let $S$, $\Sigma$ be sections of $\pi$ with self-intersections $S^2 = 3$, $\Sigma^2 = -3$. At the same time, $F$, $F^\prime$ are two $\mathbb{F}_q$-conjugate fibers of $\pi$. It is well known that $$T = \mathbb{F}_3 \setminus \big( S \cup \Sigma \cup F \cup F^\prime \big)$$ is a non-split algebraic torus, which is equal to the Weil restriction (of scalars) $\mathbb{Res}_{\mathbb{F}_{q^2}/\mathbb{F}_{q}}(\mathbb{G}_m)$ of the split one-dimensional algebraic torus $\mathbb{G}_m$.

Consider a $T$-invariant $\mathbb{F}_q$-divisor $D$ on $\mathbb{F}_3$ (for example, $D = S$) and the algebraic geometry code $C_q\big( \mathbb{F}_3, D, T(\mathbb{F}_q) \big)$, i.e., the image of the map $$\mathrm{Ev}\!: H^0(\mathbb{F}_3, D) \to \mathbb{F}_q^n,\qquad f \mapsto \big( f(P_1), \cdots\!, f(P_n) \big),$$ where $T(\mathbb{F}_q) = \{P_1, \cdots\!, P_n\}$, $n = q^2-1$.

How can I compute minimal distance $d$ of the AG-code $C_q\big( \mathbb{F}_3, D, T(\mathbb{F}_q) \big)$? I remind that $d = \mathrm{min} \ w\big( \mathrm{Ev}(f) \big)$, where the minimum runs over $f \in H^0(\mathbb{F}_3, D) \setminus \{0\}$ and $$w\big( \mathrm{Ev}(f) \big) = |\{i \mid f(P_i) \neq 0\}|.$$

• It is a great idea to use the same notation for a Hirzebruch surface and for a finite field! – Sasha Mar 18 '18 at 9:43
• Usually the minimal distance of this type of code comes from the functions $f$ for which the curve $f=0$ has the largest number of irreducible components. – Felipe Voloch Mar 18 '18 at 20:42