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Golomb Rulers

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A Golomb ruler is a rod of minimal integer length with marks so that all distances between marks are distinct. Some distances may be missed. For a perfect Golomb ruler, all the distances are distinct and none are missed; the longest one is {0,1,4,6}.

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In 2022, distributed.net proved that a length of 585 was minimal for 28 marks [1]. In all, 37 Golomb rulers have been proven to be minimal.

Surprisingly, with eight small exceptions, all proven minimal Golomb rulers had been constructed earlier, using a 1938 method by James Springer [2]. It used projective or affine methods shown in the Demonstration Golomb Rulers and Fibonacci Sequences. Due to increasingly large gaps between primes, the Singer method completely fails at 492116 marks. Since a proof of optimality is unknown for 29 marks, 492116 marks will probably not be resolved any time soon.

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Contributed by: Ed Pegg Jr (August 25)
Open content licensed under CC BY-NC-SA

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References:

[1] Wikipedia. "Golomb Ruler." (Jun 27, 2023) en.wikipedia.org/wiki/Golomb_ruler.

[2] T. Rokicki and G. Dogon. "Possibly Optimal Golomb Rulers Calculated for 160 to 40,000 Marks." (Jun 27, 2023) cube20.org/golomb.

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