Toric-code estimate from the degree of the evaluation set

Let $X=X(\Delta)$ be a complete toric variety of dimension $r$, let $G$ be a $T$-invariant Cartier divisor, let $P_G$ be its associated polytope, and let $M$ be the lattice associated to $X$. Let $\psi_G$ be the support function associated to $G$. For the toric evaluation code $C_L=C_L(G,P,X)$, write $n$ for its length, $k$ for its dimension, and $d$ for its minimum distance. Suppose $X$ is nonsingular and projective, $\psi_G$ is strictly convex, and

$$\operatorname{deg}(P)>\operatorname{deg}(G^r).$$

Toric-code degree conjecture. If $q$ is sufficiently large, then every $f\in L(G)=H^0(X,\operatorname{\mathcal O}(G))$ has at most $n=\operatorname{deg}(P)$ zeros in $X(\mathbb{F}_q)$. Consequently,

$$k\geq \dim H^0(X,\operatorname{\mathcal O}(G))=|P_G\cap M|,$$

and

$$d\geq n-r!\cdot |P_G\cap M|.$$

Moreover, if $n>r!|P_G\cap M|$, then

$$\dim H^0(X,\operatorname{\mathcal O}(G))=|P_G\cap M|.$$

The claim gives simultaneous estimates for the dimension and minimum distance of toric codes under strict convexity and a degree inequality. The supplied text explicitly records that the analogous statement is false when $X$ is singular, so this formulation is refuted outside its stated nonsingularity hypothesis; no refutation of the stated nonsingular version is supplied.