Golomb ruler

Golomb ruler of order 4 and length 6. This ruler is both optimal and perfect.

In mathematics, a Golomb ruler is a set of marks at integer positions along an imaginary 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.

The Golomb ruler was named for Solomon W. Golomb and discovered independently by Sidon (1932)^[1] and Babcock (1953).^[2] 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.^[3]

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 proven that no perfect Golomb ruler exists for five or more marks.^[4] A Golomb ruler is optimal if no shorter Golomb ruler of the same order exists. Creating Golomb rulers is easy, but finding 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-27 Golomb rulers, each time confirming the suspected candidate ruler.^[5][6][7][8] In February 2014, distributed.net began the search to find optimal Golomb rulers (OGRs) of order-28.

Currently, the complexity of finding OGRs of arbitrary order n (where n is given in unary) is unknown. In the past there was some speculation that it is an NP-hard problem.^[4] 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

$A = \{ a_1,a_2,...,a_m \} \quad a_1 < a_2 < ... < a_m$

is a Golomb ruler if and only if

$\forall i,j,k,l \in \left\{1,2,...,m\right\}, a_i-a_j=a_k-a_l \iff i=k \and j=l.$^[10]

The order of such a Golomb ruler is $m$ and its length is $a_m - a_1$. The canonical form has $a_1 = 0$ and, if $m > 2$, $a_2 - a_1 < a_m - a_{m-1}$. Such a form can be achieved through translation and reflection.

Golomb rulers as functions

An injective function

$f:\left\{1,2,...,m\right\}\to\left\{0,1,...,n\right\}$

with $f(1)=0$ and $f(m)=n$ is a Golomb ruler if and only if

$\forall i,j,k,l \in \left\{1,2,...,m\right\}, f(i)-f(j)=f(k)-f(l) \iff i=k \and j=l.$^[11]:236

The order of such a Golomb ruler is $m$ and its length is $n$. The canonical form has

$f(2)<f(m)-f(m-1)$ if $m>2$.

Optimality

A Golomb ruler of order m with length n may be optimal in either of two respects:^[11]:237

• it may be optimally dense, exhibiting maximal m for the specific value of n
• it may be optimally short, exhibiting minimal n for the specific value of m

The general term optimal Golomb ruler is used to refer to the second type of optimality.

Practical applications

Example of a conference room with proportions of a [0, 2, 7, 8, 11] Golomb ruler, making it configurable to 10 different sizes.^[12]

Information theory and error correction

Golomb rulers are used within Information Theory related to error correcting codes.^[13]

Radio frequency selection

Golomb rulers are used in the selection of radio frequencies to reduce the effects of intermodulation interference with both terrestrial^[14] and extraterrestrial^[15] applications.

Radio antenna placement

Golomb rulers are used in the design of phased arrays of radio antennas. Antennas in a [0,1,4,6] Golomb ruler configuration can often be seen at AM tower or cell sites. 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]

Current Transformers

Multi-ratio current transformers use Golomb rulers to place transformer tap points.

Methods of construction

A number of construction methods produce asymptotically optimal Golomb rulers.

Erdős–Turan construction

The following construction, due to Paul Erdős and Pál Turán, produces a Golomb ruler for every odd prime p.^[12]

$2pk+(k^2\,\bmod\,p),k\in[0,p-1]$

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.

Order	Length	Marks	Proven^[*]	Proof discovered by
1	0	0
2	1	0 1
3	3	0 1 3
4	6	0 1 4 6
5	11	0 1 4 9 11 0 2 7 8 11	1967?[18]	John P. Robinson and Arthur J. Bernstein
6	17	0 1 4 10 12 17 0 1 4 10 15 17 0 1 8 11 13 17 0 1 8 12 14 17	1967?[18]	John P. Robinson and Arthur J. Bernstein
7	25	0 1 4 10 18 23 25 0 1 7 11 20 23 25 0 1 11 16 19 23 25 0 2 3 10 16 21 25 0 2 7 13 21 22 25	1967?[18]	John P. Robinson and Arthur J. Bernstein
8	34	0 1 4 9 15 22 32 34	1972[18]	William Mixon
9	44	0 1 5 12 25 27 35 41 44	1972[18]	William Mixon
10	55	0 1 6 10 23 26 34 41 53 55	1972[18]	William Mixon
11	72	0 1 4 13 28 33 47 54 64 70 72 0 1 9 19 24 31 52 56 58 69 72	1972[18]	William Mixon
12	85	0 2 6 24 29 40 43 55 68 75 76 85	1979[18]	John P. Robinson
13	106	0 2 5 25 37 43 59 70 85 89 98 99 106	1981[18]	John P. Robinson
14	127	0 4 6 20 35 52 59 77 78 86 89 99 122 127	1985[18]	James B. Shearer
15	151	0 4 20 30 57 59 62 76 100 111 123 136 144 145 151	1985[18]	James B. Shearer
16	177	0 1 4 11 26 32 56 68 76 115 117 134 150 163 168 177	1986[18]	James B. Shearer
17	199	0 5 7 17 52 56 67 80 81 100 122 138 159 165 168 191 199	1993[18]	W. Olin Sibert
18	216	0 2 10 22 53 56 82 83 89 98 130 148 153 167 188 192 205 216	1993[18]	W. Olin Sibert
19	246	0 1 6 25 32 72 100 108 120 130 153 169 187 190 204 231 233 242 246	1994[18]	Apostolos Dollas, William T. Rankin and David McCracken
20	283	0 1 8 11 68 77 94 116 121 156 158 179 194 208 212 228 240 253 259 283	1997?[18]	Mark Garry, David Vanderschel et al. (web project)
21	333	0 2 24 56 77 82 83 95 129 144 179 186 195 255 265 285 293 296 310 329 333	8 May 1998[19]	Mark Garry, David Vanderschel et al. (web project)
22	356	0 1 9 14 43 70 106 122 124 128 159 179 204 223 253 263 270 291 330 341 353 356	1999[18]	Mark Garry, David Vanderschel et al. (web project)
23	372	0 3 7 17 61 66 91 99 114 159 171 199 200 226 235 246 277 316 329 348 350 366 372	1999[18]	Mark Garry, David Vanderschel et al. (web project)
24	425	0 9 33 37 38 97 122 129 140 142 152 191 205 208 252 278 286 326 332 353 368 384 403 425	13 October 2004[5]	distributed.net
25	480	0 12 29 39 72 91 146 157 160 161 166 191 207 214 258 290 316 354 372 394 396 431 459 467 480	25 October 2008[6]	distributed.net
26	492	0 1 33 83 104 110 124 163 185 200 203 249 251 258 314 318 343 356 386 430 440 456 464 475 487 492	24 February 2009[7]	distributed.net
27	553	0 3 15 41 66 95 97 106 142 152 220 221 225 242 295 330 338 354 382 388 402 415 486 504 523 546 553	19 February 2014[8]	distributed.net

^{^ *} 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 proven 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.

See also

• Costas array
• Sparse ruler
• Perfect ruler
• Sidon sequence
• distributed.net
• BOINC
• Distributed Computing

References

1. 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.
2. Babcock, Wallace C. (1953). "Intermodulation Interference in Radio Systems/Frequency of Occurrence and Control by Channel Selection". Bell System Technical Journal 31: 63–73.
3. Bekir, Ahmad; Golomb, Solomon W. (2007). "There are no further counterexamples to S. Piccard's theorem". IEEE Transactions on Information Theory 53 (8): 2864–2867. doi:10.1109/TIT.2007.899468. MR 2400501..
4. "Modular and Regular Golomb Rulers".
5. "distributed.net - OGR-24 completion announcement". Retrieved 2014-02-25.
6. "distributed.net - OGR-25 completion announcement". Retrieved 2014-02-25.
7. "distributed.net - OGR-26 completion announcement". Retrieved 2014-02-25.
8. "distributed.net - OGR-27 completion announcement". Retrieved 2014-02-25.
9. 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.
10. 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.
11. Drakakis, 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.
12. Erdős, 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.
13. 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.
14. "Intermodulation Interference in Radio Systems" (excerpt). Retrieved 2011-03-14.
15. "Carrier frequency assignment for nonlinear repeaters" (abstract). Bibcode:1977COMTR...7..227F.
16. 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.
17. 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).
18. "table of lengths of shortest known rulers". IBM. Retrieved 2013-11-28.
19. "In Search Of The Optimal 20 & 21 Mark Golomb Rulers (archived)". Mark Garry, David Vanderschel, et al. Archived from the original on 1998-12-06.

• Gardner, Martin (March 1972). "Mathematical games". Scientific American: 108–112.

External links

• James B. Shearer's Golomb ruler pages
• distributed.net: Project OGR
• In Search Of The Optimal 20, 21 & 22 Mark Golomb Rulers
• Golomb rulers up to length of over 200 (via Internet Archive)