Projective toric codes, affine Reed-Muller-type codes and squarefree evaluation 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 $\mathbb{R}^s$. Given a finite field $\mathbb{F}_q$, we determine the minimum distance of the projective toric code of $\mathcal{P}$. Then, we show a formula, in terms of the degree and a fixed monomial order, to compute the $r$-th generalized Hamming weights of an affine Reed--Muller-type code, and give a lower bound for this number which is easier to compute. We determine the minimum distance and the 2nd generalized Hamming weight of a squarefree evaluation code over an affine torus.