An 
-mark Golomb ruler 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.
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 exists no other 
-mark
Golomb rulers having 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 the 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 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 in 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 ![[1,2,4]](/images/equations/GolombRuler/Inline24.svg)
.
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 2023.
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 as 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.
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 technique developer by Singer (1938) with some follow-ups by Bose (1942) and Bose and Chowla (1962-63) produces rulers that are guaranteed to be near-optimal, appear to be optimal, and with the exception of six small sizes, have produced all known optimal rulers (cf. Pegg 2016).
The number 
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) |
