### Golomb Rulers

#### by James Buckland

Golomb rulers are mathematical structures (rulers, really) which almost perfectly demonstrate my earlier article on the oft-delayed values of pure mathematics. A perfect Gombol ruler will be a set of markings which can measure every consecutive integer distance below its total length. For example: the gombol ruler of order four (four markings) and length six (in total, six units long) contains, within it, the measurements:

which are consecutive, optimal, and non-repeating. This is a fascinating mathematical structure — particularly the proof that a perfect (non-repeating) Golomb ruler with more than four marks on it cannot exist.

However, it is the *applications, *as duly noted in Bill Rankin’s 1993 thesis on the topic (section 1.2), that are of particular interest — indeed, it has applications in radio telecommunications, x-ray crystallography, radio arrays, and anything else which requires the deliberate and efficient *mechanical* use of integer-valued nodes on wavelengths — that is, the need for an apparent continuum to be split into integer values. In addition, Golomb rulers can be used by information theory to produce efficient error-correcting codes (transmissions with additional information about their coherent communication) as a sort of hash function; a self-orthogonal structure which reacts predictably to any errors in communication.

In conclusion — Golomb rulers are wicked cool.