  For a given lattice polytope $P$ in $\mathbb{R}^3$, consider the space
$\mathcal{L}_P$ of trivariate polynomials over a finite field $\mathbb{F}_q$,
whose Newton polytopes are contained in $P$. We give an upper bound for the
maximum number of $\mathbb{F}_q$-zeros of polynomials in $\mathcal{L}_P$ in
terms of the Minkowski length of $P$ and $q$, the size of the field.
Consequently, this produces lower bounds for the minimum distance of toric
codes defined by evaluating elements of $\mathcal{L}_P$ at the points of the
algebraic torus $(\mathbb{F}_q^*)^3$. Our approach is based on understanding
factorizations of polynomials in $\mathcal{L}_P$ with the largest possible
number of non-unit factors. The related combinatorial result that we obtain is
a description of Minkowski sums of lattice polytopes contained in $P$ with the
largest possible number of non-trivial summands.
