A note on projective toric codes

Let $d\geq 1$ be an integer, and let $\mathcal{P}$ be the convex hull in $\mathbb{R}^s$ of all integral points $\mathbf{e}_{i_1}+\cdots+\mathbf{e}_{i_d}$ such that $1\leq i_1<\cdots< i_d\leq s$, where $\mathbf{e}_i$ is the $i$-th unit vector in $\mathbb{R}^s$. We determine the minimum distance of the projective toric code $\mathcal{C}_\mathcal{P}(d)$ associated to $\mathcal{P}$ using the projective footprint function of a graded ideal.