Classifying toric surface codes of dimension $7$

Toric codes are a class of error-correcting codes coming from a lattice polytope defining a toric variety. Previous authors have completed classifications of these toric surface codes with dimension up to $k = 6$, and we classify toric surface codes with dimension $k = 7$ while building on their methods. We first determine that there are $22$ polygons, up to lattice equivalence, which yield codes of dimension $7$. We further show that these $22$ classes generate monomially inequivalent codes for sufficiently large finite fields.