Golomb Ruler

A Golomb ruler with marks is a set of distinct nonnegative integers , called "marks," such that the positive differences , computed over all possible pairs of different integers , ..., with are distinct.

Up to a translation, finite Sidon sets are exactly Golomb rulers.

No perfect Golomb ruler, i.e., a Golomb ruler that uniquely measures all distances up to its length, exists for five or more marks (Golomb 1972; Gardner 1983, p. 154).

Let be the largest integer in an -mark Golomb ruler. Then an -mark Golomb ruler is optimal if

1. There is no other -mark Golomb ruler with a smaller largest mark , and

2. The ruler is written in canonical form as the "smaller" of the equivalent rulers and , where "smaller" means the first differing entry is less than the corresponding entry in the other ruler.

In such a case, is called the "length" of the optimal -mark ruler.

Thus, (0, 1, 3) is the unique optimal 3-mark Golomb ruler modulo reversal (i.e., (0, 2, 3) is considered the same ruler).

For example, the set (0, 1, 3, 7) is a 4-mark Golomb ruler since its differences are (, , , , , ), all of which are distinct. However, the unique optimal Golomb 4-mark ruler is (0, 1, 4, 6), which measures the distances (1, 2, 3, 4, 5, 6). As a further example, it turns out that the length of an optimal 6-mark Golomb ruler is 17. In fact, there are a total of four distinct 6-mark Golomb rulers, all of length 17, one of which is given by (0, 1, 4, 10, 12, 17).

Rulers may be indicated either by giving the distances at which the marks occur, e.g., (0, 1, 3, 7) in the above example, or by the differences between successive marks, which in this case would be .

The lengths of the optimal -mark Golomb rulers for , 3, 4, ... are 1, 3, 6, 11, 17, 25, 34, ... (OEIS A003022, Vanderschel and Garry). Although the lengths of the optimal -mark Golomb rulers are not known for , the known 21- through 28-mark rulers were proved optimal by the Golomb ruler search project between 1998 and 2022.

Proving optimality of Golomb rulers is notoriously difficult. For example, in 1967, J. P. Robinson and A. J. Bernstein found the 24-mark ruler (0, 9, 33, 37, 38, 97, 122, 129, 140, 142, 152, 191, 205, 208, 252, 278, 286, 326, 332, 353, 368, 384, 403, 425), which was verified to be optimal on Nov. 1, 2004 by a 4-year computation on distributed.net that performed an exhaustive search through rulers (distributed.net 2004). In 1984, M. D. Atkinson and A. Hassenklover found the 25-mark ruler (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). A follow-up eight-year distributed.net project for the 25-mark ruler, assisted by people, announced on Oct. 25, 2008 that the 25-mark ruler was optimal.

The OGR-28 search took approximately 8.5 years and involved more than 65,000 volunteers from over 80 countries. It verified 524,091,443 search stubs at least twice independently and established that the previously known 28-mark ruler of length 585 is optimal (distributed.net 2022).

Reviews of construction methods for Golomb rulers are given by Dimitromanolakis (2002), Drakakis (2009), and Rokicki and Dogon. As noted by Rokicki and Dogon, the construction developed by Singer (1938), together with subsequent constructions of Bose (1942) and Bose and Chowla (1962-63), produces rulers that are guaranteed to be near-optimal and often appear to be optimal. With the exception of orders 7, 8, 9, 13, 15, and 16, these finite field constructions have produced all known optimal rulers (cf. Pegg 2016).

The numbers of inequivalent optimal -mark Golomb rulers for , 3, ... are 1, 1, 1, 2, 4, 5, 1, 1, 1, ... (OEIS A036501), and the number of distances in an optimal -mark ruler is given by the triangular number , so for , 2, ..., the first few are 0, 1, 3, 6, 10, 15, ... (OEIS A000217).

The following table gives a complete list of optimal Golomb rulers for small . A more complete table is maintained by J. B. Shearer.

		optimal rulers
2	1	(0, 1)
3	1	(0, 1, 3)
4	1	(0, 1, 4, 6)
5	2	(0, 1, 4, 9, 11), (0, 2, 7, 8, 11)
6	4	(0, 1, 4, 10, 12, 17), (0, 1, 4, 10, 15, 17), (0, 1, 8, 12, 14, 17),
		(0, 1, 8, 11, 13, 17)
7	5	(0, 1, 4, 10, 18, 23, 25), (0, 2, 3, 10, 16, 21, 25), (0, 1, 11, 16, 19, 23, 25),
		(0, 1, 7, 11, 20, 23, 25), (0, 2, 7, 13, 21, 22, 25)
8	1	(0, 1, 4, 9, 15, 22, 32, 34)