Set of marks along a ruler such that no two pairs of marks are the same distance apart
In mathematics, a Golomb ruler is a set of marks at integer positions along a ruler such that no two pairs of marks are the same distance apart. The number of marks on the ruler is its order, and the largest distance between two of its marks is its length. Translation and reflection of a Golomb ruler are considered trivial, so the smallest mark is customarily put at 0 and the next mark at the smaller of its two possible values. Golomb rulers can be viewed as a one-dimensional special case of Costas arrays.
The Golomb ruler was named for Solomon W. Golomb and discovered independently by Sidon (1932)[1] and Babcock (1953). Sophie Piccard also published early research on these sets, in 1939, stating as a theorem the claim that two Golomb rulers with the same distance set must be congruent. This turned out to be false for six-point rulers, but true otherwise.[2]
There is no requirement that a Golomb ruler be able to measure all distances up to its length, but if it does, it is called a perfect Golomb ruler. It has been proved that no perfect Golomb ruler exists for five or more marks.[3] A Golomb ruler is optimal if no shorter Golomb ruler of the same order exists. Creating Golomb rulers is easy, but proving the optimal Golomb ruler (or rulers) for a specified order is computationally very challenging.
Distributed.net has completed distributed massively parallel searches for optimal order-24 through order-28 Golomb rulers, each time confirming the suspected candidate ruler.[4][5][6][7][8]
Currently, the complexity of finding optimal Golomb rulers (OGRs) of arbitrary order n (where n is given in unary) is unknown.[clarification needed] In the past there was some speculation that it is an NP-hard problem.[3] Problems related to the construction of Golomb rulers are provably shown to be NP-hard, where it is also noted that no known NP-complete problem has similar flavor to finding Golomb rulers.[9]
Definitions
Golomb rulers as sets
A set of integers where is a Golomb ruler if and only if
[10]
The order of such a Golomb ruler is and its length is . The canonical form has and, if , . Such a form can be achieved through translation and reflection.
Golomb rulers are used in the design of phased arrays of radio antennas. In radio astronomy one-dimensional synthesis arrays can have the antennas in a Golomb ruler configuration in order to obtain minimum redundancy of the Fourier component sampling.[16][17]
The following construction, due to Paul Erdős and Pál Turán, produces a Golomb ruler for every odd prime p.[12]
Known optimal Golomb rulers
The following table contains all known optimal Golomb rulers, excluding those with marks in the reverse order. The first four are perfect.
^ * The optimal ruler would have been known before this date; this date represents that date when it was discovered to be optimal (because all other rulers were proved to not be smaller). For example, the ruler that turned out to be optimal for order 26 was recorded on 10 October 2007, but it was not known to be optimal until all other possibilities were exhausted on 24 February 2009.
^Sidon, S. (1932). "Ein Satz über trigonometrische Polynome und seine Anwendungen in der Theorie der Fourier-Reihen". Mathematische Annalen. 106: 536–539. doi:10.1007/BF01455900. S2CID 120087718.
^ a b"distributed.net - OGR-24 completion announcement". 2004-11-01.
^ a b"distributed.net - OGR-25 completion announcement". 2008-10-25.
^ a b"distributed.net - OGR-26 completion announcement". 2009-02-24.
^ a b"distributed.net - OGR-27 completion announcement". 2014-02-25.
^ a b"Completion of OGR-28 project". Retrieved 23 November 2022.
^Meyer C, Papakonstantinou PA (February 2009). "On the complexity of constructing Golomb rulers". Discrete Applied Mathematics. 157 (4): 738–748. doi:10.1016/j.dam.2008.07.006.
^Dimitromanolakis, Apostolos. "Analysis of the Golomb Ruler and the Sidon Set Problems, and Determination of Large, Near-Optimal Golomb Rulers" (PDF). Retrieved 2009-12-20.
^ a bDrakakis, Konstantinos (2009). "A Review Of The Available Construction Methods For Golomb Rulers". Advances in Mathematics of Communications. 3 (3): 235–250. doi:10.3934/amc.2009.3.235.
^ a bErdős, Paul; Turán, Pál (1941). "On a problem of Sidon in additive number theory and some related problems". Journal of the London Mathematical Society. 16 (4): 212–215. doi:10.1112/jlms/s1-16.4.212.
^Robinson J, Bernstein A (January 1967). "A class of binary recurrent codes with limited error propagation". IEEE Transactions on Information Theory. 13 (1): 106–113. doi:10.1109/TIT.1967.1053951.
^Babcock, Wallace C. (1953). "Intermodulation Interference in Radio Systems" (excerpt). Bell System Technical Journal. 32: 63–73. doi:10.1002/j.1538-7305.1953.tb01422.x. Archived (PDF) from the original on 2011-07-07. Retrieved 2011-03-14.
^Fang, R. J. F.; Sandrin, W. A. (1977). "Carrier frequency assignment for nonlinear repeaters". Comsat Technical Review (abstract). 7: 227. Bibcode:1977COMTR...7..227F.
^Thompson, A. Richard; Moran, James M.; Swenson, George W. (2004). Interferometry and Synthesis in Radio Astronomy (Second ed.). Wiley-VCH. p. 142. ISBN 978-0471254928.
^Arsac, J. (1955). "Transmissions des frequences spatiales dans les systemes recepteurs d'ondes courtes" [Transmissions of spatial frequencies in shortwave receiver systems]. Optica Acta (in French). 2 (112): 112–118. Bibcode:1955AcOpt...2..112A. doi:10.1080/713821025.
^ a b c dRulers, Arrays, and Gracefulness Ed Pegg Jr. November 15, 2004. Math Games.
^ a b c d e f g h i j k l m n o p q rShearer, James B (19 February 1998). "Table of lengths of shortest known Golomb rulers". IBM. Archived from the original on 25 June 2017.
^"In Search Of The Optimal 20 & 21 Mark Golomb Rulers (archived)". Mark Garry, David Vanderschel, et al. 26 November 1998. Archived from the original on 1998-12-06.