Toric surface codes and Minkowski sums

Toric codes are evaluation codes obtained from an integral convex polytope $P \subset \R^n$andfinitefield$\F_q$. They are, in a sense, a natural extension of Reed-Solomon codes, and have been studied recently by J. Hansen and D. Joyner. In this paper, we obtain upper and lower bounds on the minimum distance of a toric code constructed from a polygon $P\subset {\mathbb{R}}^2$ by examining Minkowski sum decompositions of subpolygons of $P$. Our results give a simple and unifying explanation of bounds of Hansen and empirical results of Joyner; they also apply to previously unknown cases.