I'm working through Henning Stichtenoth's Algebraic Function Fields and Codes as part research for my undegraduate dissertation, and I'm unsure of the proof of a particular proposition (proposition 2.3.2), which is omitted in this book.
This concerns rational algebraic geometry codes - Working in the rational function field over $\mathbb{F}_q$, we let $D = P_1 + ... + P_n$ be a divisor given as the formal sum of a finite number of degree 1 places of the function field, and $G$ a divisor with support which has no intersection with that of $D$. Then the rational AG code associated with these divisors is $C_\mathscr{L}(D, G) := \{(x(P_1), ..., x(P_n)) : x \in \mathscr{L}(G)\}$.
The proposition in question states that, letting $k$ denote the dimension of such a code, we have $k = 0 \iff \mathrm{deg} \, G < 0$.
One direction of the proof is simple: If $\mathrm{deg} \, G < 0$, then $\mathrm{dim} \, \mathscr{L}(G) = 0$, and so $\mathscr{L}(G) = \{0\}$. Then $C_\mathscr{L}(D, G)$ is just $0$ evaluated at every place $P_1, ..., P_n$, and so it equals $\{(0, ..., 0)\}$, which has dimension 0.
For the other direction, I'm pretty much stumped. I've spoken to my supervisor, and we spent about 30 mins trying to understand this, and didn't get anywhere of consequence. Assuming $k = 0$, we know $k = \mathrm{dim} \, \mathscr{L}(G) - \mathrm{dim} \, \mathscr{L}(G - D)$, and so $\mathscr{L}(G) = \mathscr{L}(G - D)$ (since $G - D \leq G$, and therefore $\mathscr{L}(G - D) \subseteq \mathscr{L}(G)$, so we have nested vector spaces of equal dimension). But I don't think this necessarily tells us anything of interest about the degree of $G$. We know that the rational function field has genus zero, and so rational AG codes are MDS, but all I've been able to get out of this is $\mathrm{deg} \, G \geq -1$.
From the prose surrounding this proposition, the proof should be quite simple, just using results from the previous section, but I can't get anywhere with it unless I can assume $\mathrm{deg} \, G < n$, which is not assumed to be true here. Any help would be greatly appreciated :)
For the other direction, I think intuition of error-correcting codes is what helps, as opposed to algebraic geometry. The key fact is that an $[n,k]$ linear code forms a subspace of dimension $k$ of $\mathbb{F}_q^n$.
If $k=0$, then the code has dimension zero. This means that the set of possible code words is a $0$-dimensional subspace — that is, it only consists of the zero vector. In notation matching your post, $C_{\mathcal{L}}(D,G) = \{ (0,\dots,0) \}$.
Then I think from there, it’s just walking back along your implications to get to get $\deg(G) < 0$.