Golomb Rulers
Golomb Rulers
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.
References
References
[2] T. Rokicki and G. Dogon. "Possibly Optimal Golomb Rulers Calculated for 160 to 40,000 Marks." (Jun 27, 2023) cube20.org/golomb.
External Links
External Links
Permanent Citation
Permanent Citation
Ed Pegg Jr
"Golomb Rulers"
http://demonstrations.wolfram.com/GolombRulers/
Wolfram Demonstrations Project
Published: July 10, 2023