Toric Surface Codes and Minkowski Sums

Toric codes are evaluation codes obtained from an integral convex polytope P ⊂ R and finite field Fq. They are, in a sense, a natural extension of Reed-Solomon codes, and have been studied recently in [6], [8], [9], and [12]. In this paper, we obtain upper and lower bounds on the minimum distance of a toric code constructed from a polygon P ⊂ R by examining Minkowski sum decompositions of subpolygons of P . Our results give a simple and unifying explanation of bounds in [9] and empirical results in [12]; they also apply to previously unknown cases.