Classifying toric 3-fold codes of dimensions 4 and 5

Abstract

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.