  A toric code is an error-correcting code determined by a toric variety or its
associated integral convex polytope. We investigate $4$- and $5$-dimensional
toric $3$-fold codes, which are codes arising from polytopes in $\mathbf{R}^3$
with four and five lattice points, respectively. By computing the minimum
distances of each code, we fully classify the $4$-dimensional codes. We further
present progress toward the same goal for dimension $5$ codes. In particular,
we classify the $5$-dimensional toric $3$-fold codes arising from polytopes of
width 1.
