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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 themaximum number of $\mathbb{F}_q$-zeros of polynomials in $\mathcal{L}_P$ interms of the Minkowski length of $P$ and $q$, the size of the field.Consequently, this produces lower bounds for the minimum distance of toriccodes defined by evaluating elements of $\mathcal{L}_P$ at the points of thealgebraic torus $(\mathbb{F}_q^*)^3$. Our approach is based on understandingfactorizations of polynomials in $\mathcal{L}_P$ with the largest possiblenumber of non-unit factors. The related combinatorial result that we obtain isa description of Minkowski sums of lattice polytopes contained in $P$ with thelargest possible number of non-trivial summands.

$\mathbb{F}_q$-zeros of sparse trivariate polynomials and toric 3-fold codes