Golomb ruler
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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[1] and Babcock.[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 proven 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 finding the optimal Golomb rulers for a specified order is computationally very challenging. Distributed.net has completed distributed massively parallel searches for optimal order-24,[4] order-25[5] and order-26[6][7] Golomb rulers, confirming the suspected candidates[8][9]. Distributed.net also has plans to find optimal Golomb rulers of order-27 and order-28. However, they are not expected to take as long as the previous projects due to the discovery of an improved algorithm.[10] Currently the search for the order-27 ruler is underway, the expected time to finish it is about seven years[11].
One practical use of Golomb rulers is in the design of phased array radio antennas such as radio telescopes. Antennas in a [0,1,4,6] Golomb ruler configuration can often be seen at cell sites.[dubious ]
Currently, the complexity of finding optimal Golomb rulers of arbitrary order n is unknown, but it is believed to be an NP-hard problem.[3]
Contents |
[edit] Known optimal Golomb rulers
The following table contains all known optimal Golomb rulers, excluding those with marks in the reverse order.
| order | length | marks |
|---|---|---|
| 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 |
| 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 |
| 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 |
| 8 | 34 | 0 1 4 9 15 22 32 34 |
| 9 | 44 | 0 1 5 12 25 27 35 41 44 |
| 10 | 55 | 0 1 6 10 23 26 34 41 53 55 |
| 11 | 72 | 0 1 4 13 28 33 47 54 64 70 72 0 1 9 19 24 31 52 56 58 69 72 |
| 12 | 85 | 0 2 6 24 29 40 43 55 68 75 76 85 |
| 13 | 106 | 0 2 5 25 37 43 59 70 85 89 98 99 106 |
| 14 | 127 | 0 4 6 20 35 52 59 77 78 86 89 99 122 127 |
| 15 | 151 | 0 4 20 30 57 59 62 76 100 111 123 136 144 145 151 |
| 16 | 177 | 0 1 4 11 26 32 56 68 76 115 117 134 150 163 168 177 |
| 17 | 199 | 0 5 7 17 52 56 67 80 81 100 122 138 159 165 168 191 199 |
| 18 | 216 | 0 2 10 22 53 56 82 83 89 98 130 148 153 167 188 192 205 216 |
| 19 | 246 | 0 1 6 25 32 72 100 108 120 130 153 169 187 190 204 231 233 242 246 |
| 20 | 283 | 0 1 8 11 68 77 94 116 121 156 158 179 194 208 212 228 240 253 259 283 |
| 21 | 333 | 0 2 24 56 77 82 83 95 129 144 179 186 195 255 265 285 293 296 310 329 333 |
| 22 | 356 | 0 1 9 14 43 70 106 122 124 128 159 179 204 223 253 263 270 291 330 341 353 356 |
| 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 |
| 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 |
| 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 |
| 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 |
[edit] See also
[edit] Notes
- ^ S. Sidon, "Ein Satz über trigonometrische Polynome und seine Anwendungen in der Theorie der Fourier-Reihen", Mathematische Annalen 106 (1932), pp. 536–539
- ^ Wallace C. Babcock. "Intermodulation Interference in Radio Systems/Frequency of Occurrence and Control by Channel Selection", Bell System Technical Journal 31 (1953), pp. 63–73.
- ^ a b "Modular and Regular Golomb Rulers". http://cgm.cs.mcgill.ca/~athens/cs507/Projects/2003/JustinColannino.
- ^ "stats.distributed.net - OGR-24 Overall Project Stats". http://stats.distributed.net/projects.php?project_id=24. Retrieved on 2008-03-27.
- ^ "stats.distributed.net - OGR-25 Overall Project Stats". http://stats.distributed.net/projects.php?project_id=25. Retrieved on 2008-09-22.
- ^ http://n0cgi.distributed.net/cgi/planarc.cgi?user=bovine
- ^ http://www.distributed.net/projects.php
- ^ "distributed.net - .plan archives". http://n0cgi.distributed.net/cgi/planarc.cgi?user=nugget&plan=2004-11-01.10:24. Retrieved on 2008-03-27.
- ^ "distributed.net - .plan archives 2". http://n0cgi.distributed.net/cgi/planarc.cgi?user=bovine&plan=2008-10-25.23:14. Retrieved on 2008-10-26.
- ^ http://n0cgi.distributed.net/cgi/planarc.cgi?user=bovine&plan=2008-10-26.09:52
- ^ bovine's plan, 24-Feb-2009 17:26
[edit] References
- Gardner, Martin (March 1972). "Mathematical games". Scientific American: 108–112.
