On the Design of Optimum Order 2 Golomb Ruler

A Golomb ruler with M marks can be defined as a set {ai} of integers so that all differences δij = aj − ai, i 6= j, are distinct. An order 2 Golomb ruler is a Golomb ruler such that all differences δijk` = |δk` − δij |, {i, j} 6 = {k, `}, are distinct as much as possible. Contruction of optimum order 2 Golomb ruler, i.e., of rulers of minimum length, is a highly combinatorial problem which has applications, e.g., in the design of convolutional self doubly orthogonal codes (CSOC). We propose here a first exact algorithm, different from a pure exhaustive search, for building optimum order 2 Golomb rulers and provide optimal rulers for up to 9 marks and new rulers which improve on the previous ones for up to 20 marks.