On $m$-dimensional toric codes

Toric codes are a class of $m$-dimensional cyclic codes introduced recently by J. Hansen. They may be defined as evaluation codes obtained from monomials corresponding to integer lattice points in an integral convex polytope $P \subseteq \R^m$. As such, they are in a sense a natural extension of Reed-Solomon codes. Several authors have used intersection theory on toric surfaces to derive bounds on the minimum distance of some toric codes with $m = 2$. In this paper, we will provide a more elementary approach that applies equally well to many toric codes for all $m \ge 2$. Our methods are based on a sort of multivariate generalization of Vandermonde determinants that has also been used in the study of multivariate polynomial interpolation. We use these Vandermonde determinants to determine the minimum distance of toric codes from rectangular polytopes and simplices. We also prove a general result showing that if there is a unimodular integer affine transformation taking one polytope $P_1$ to a second polytope $P_2$, then the corresponding toric codes are monomially equivalent (hence have the same parameters). We use this to begin a classification of two-dimensional toric codes with small dimension.