Enumeration of Golomb rulers

Generally a ruler is marked in equal increments, e.g., a 12 inch ruler has 12 markings,\ud each 1 inch apart. In this paper we discuss a special type of ruler discovered by\ud Soloman W. Golomb. We define a Golomb ruler to be a ruler of length d with n\ud markings where the distance between any two markings is unique. This concept is\ud useful in the field of electrical engineering.\ud We use the fact that a Golomb ruler can be represented as a lattice point in Rm,\ud where the A;-th coordinate represent the k-th measure, to study it from a geometric\ud point of view. It turns out that if we relax the definition of a Golomb ruler to allow\ud real number markings then the set of all Golomb ruler of m measure and length d is\ud a polytope in Rm equipped with a hyperplane arrangement. We prove results about\ud the combinatorial structure of these special type of polytopes and derive from them\ud structural theorems about the enumeration of Golomb rulers