Toric Surfaces and Codes, Techniques and Examples

Abstract. We treat toric surfaces and their application to construction of error-correcting codes and determination of the parameters of the codes, surveying and expanding the results of [4]. For any integral convex polytope in R there is an explicit construction of a unique error-correcting code of length (q − 1) over the finite field Fq. The dimension of the code is equal to the number of integral points in the polytope. The code can be considered as obtained by evaluation of rational functions on a (not uniguely determined) toric surface associated to the given polytope. Intersection theory on the toric surface will in two different ways be applied to bound the minimal distance of the code. In some cases we even obtain the precise minimal distance of the code. The techniques are illustrated by several examples