Toric surface codes and the periodicity of polytopes

Abstract

Toric codes are error-correcting codes that are derived from toric varieties, which hold a unique correspondence to integral convex polytopes. In this paper, we focus on integral convex polytopes $P \subseteq \mathbb{R}^2$ and the toric codes they define. We begin by studying period-1 polytopes -- polytopes satisfying the property $L(tP)$ = $tL(P)$ for all $t \in \mathbb{Z}^+$, where $tP$ is the $t$-dilate of $P$, and we prove an explicit formula for the minimum distance of toric codes associated to a particular class of period-1 polytopes. We also apply the methods of Little and Schwarz, using Vandermonde matrices, to compute the minimum distance of another class of period-1 polytopes.