Toric Surface Codes and Minkowski Length of Polygons

Abstract

In this paper we prove new lower bounds for the minimum distance of a toric surface code $\mathcal{C}_P$ defined by a convex lattice polygon $P\subset\mathbb{R}^2$. The bounds involve a geometric invariant $L(P)$, called the full Minkowski length of P. We also show how to compute $L(P)$ in polynomial time in the number of lattice points in P.