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# Topic: Golomb ruler

###### In the News (Fri 26 Apr 19)

 Golomb ruler - Wikipedia, the free encyclopedia In mathematics, a Golomb ruler, named after Solomon W. Golomb, is a set of marks at integer positions along an imaginary ruler such that no two pairs of marks are the same distance apart. 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. One practical use of Golomb rulers is in the design of phased array radio antennae such as radio telescopes. en.wikipedia.org /wiki/Golomb_ruler   (609 words)

 Co to linijka Golomba?   (Site not responding. Last check: 2007-10-07) Golomb rulers can also play a significant role in combinatorics, coding theory and communications, and Dr. Golomb was one of the first to analyze them for use in these areas. The length of this ruler is 11, and it happens to be one of the two shortest such rulers with five marks. Golomb rulers are usually characterized by their differences, rather than absolute distances as in the above diagram. republika.pl /zimniak/golomb.html   (483 words)

 DC Projects - Optimal Golomb Ruler (OGR)   (Site not responding. Last check: 2007-10-07) Golomb rulers refer to a spacing technique that is used in a variety of areas such as astronomy (placement of antennas), xray sensing devices (placement of sensors), and myriad other fields such as data encryption. A sample Golomb Ruler with 4 marks is 0-1-3-7. These are the shortest golomb rulers for a given number of marks. www.dcparadise.org /01bdc193120e63d6d/01bdc193120e66173/index.asp   (276 words)

 Graceful Graphs Golomb rulers with more than 4 divisions (not perfect Golomb rulers) are those which measure unique lengths. Clearly, a ruler with divisions of lengths 1, 2, 4, 8, 16, etc. is a Golomb ruler. For example, if we have a ruler with division lengths of 1, 2, 3, and 27, then the 27 portion can be placed anywhere in the ruler and not change the Golombicity of the ruler. kevingong.com /Math/GracefulGraphs.html   (2204 words)

 Golomb Rulers - The Search For 20 and 21! Golomb rulers are named after Dr. Solomon W. Golomb, a professor of Mathematics with a special interest in combinatorial analysis, number theory, coding theory and communications. Golomb rulers can play a significant role in combinatorics, coding theory and communications, and Dr. Golomb was one of the first to analyze them for use in these areas. As an interesting aside, note that --to date-- their have been no new golomb rulers found and documented with 17-100 marks that are better than those found by Atkinson in 1987 (with two exceptions: Lloyd Miller found two better rulers using cyclic techniques he learned from Kris Coolsaet's CDS Home Pages). members.aol.com /golomb20/intro.htm   (1581 words)

 Golomb ruler -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-10-07) One practical use of Golomb rulers is in the design of (Click link for more info and facts about phased array) phased array radio antennae such as (Astronomical telescope that picks up electromagnetic radiations in the radio-frequency range from extra-terrestrial sources) radio telescopes. Antennae in an [0,1,4,6] Golomb ruler configuration can often be seen at (Click link for more info and facts about cell site) cell sites. A search for optimal Golomb rulers of order 25 is currently underway (as of April 2005). www.absoluteastronomy.com /encyclopedia/g/go/golomb_ruler.htm   (611 words)

 Math Games: Rulers, Arrays, and Gracefulness Golomb rulers were named as a result of his investigations. Rulers with 20 to 22 marks were found with an intense computer search by Mark Garry and David Vanderschel. The rulers to beat are maintained by James Shearer, and consist mainly of constructions via affine and projective planes. www.maa.org /editorial/mathgames/mathgames_11_15_04.html   (1796 words)

 DCCentral [Optimal Golomb Ruler] Golomb rulers were discovered by Solomon W. Golomb, a mathematics professor with an interest in coding, combinational analysis, and mathematical puzzles. An optimal Golomb ruler with four marks is 0-1-4-6. Optimal Golomb rulers — also referred to as OGR — are used in numerous fields of study, such as laser technology, crystallography, and radio astronomy. library.thinkquest.org /C007645/english/2-golomb-0.htm   (301 words)

 Modular and Regular Golomb Rulers Golomb rulers of the same length are used to generate self-orthogonal codes, codes that do not share common differences. For a Golomb ruler with n marks to be perfect it must be of length n choose 2, because if it were shorter then some distance would be measured twice since there are less distances than measurements. Modular Golomb Rulers, sometimes called Circular Golomb Rulers, are a particular case of Golomb Rulers that have a near optimal construction. cgm.cs.mcgill.ca /~athens/cs507/Projects/2003/JustinColannino   (1902 words)

 [No title] A Golomb Ruler is defined as a ruler which includes a predetermined number of marks (say, "N") placed at integer multiples of some fixed unit and measuring the maximum number of distinct differences possible (equal to N*(N-1)/2). As such, one condition of a Golomb Ruler is that no single distance can be measured twice (i.e., using two different sets of marks) because to do so would mean that the total number of distances that could be measured would be less than N*(N-1)/2. For larger rulers however (e.g., 15, or 17 and higher) a revised version is more efficient; it uses the average of the marks 'once-removed' from the middle two marks. members.aol.com /golomb20/gvant/gvantdoc.txt   (2599 words)

 American Scientist Online - Collective Wisdom   (Site not responding. Last check: 2007-10-07) A ruler on which no two pairs of marks measure the same distance is called a Golomb ruler, after Solomon W. Golomb of the University of Southern California, who described the concept 25 years ago. The 0-1-4-6 example is a perfect Golomb ruler, in that all integer intervals from 1 to the length of the ruler are represented. On rulers with more than four marks, perfection is not possible; the best you can do is an optimal Golomb ruler, which for a given number of marks is the shortest ruler on which no intervals are duplicated. www.amsci.org /template/AssetDetail/assetid/20836?&print=yes   (3690 words)

 MATHEWS: Golomb Rulers   (Site not responding. Last check: 2007-10-07) A Golomb ruler is defined as a ruler which has marks at integer locations such that the distance between any two marks of the ruler is unique. Normally the first mark of the ruler is set on position 0 and the position of the right-most mark is called the length of the ruler. Golomb rectangles are introduced by J. Robinson as two-dimensional arrays N X M of ones and zeros such that the difference between the positions of every pair of ones in the array, considered as vectors, are distinct. www.wschnei.de /number-theory/golomb-rulers.html   (651 words)

 Golomb Example 1  As an example, a Golomb ruler with M = 4 can be constructed by placing mark 1 at distance 0 on the ruler, mark 2 at distance 1, mark 3 at distance 4, and mark 4 at distance 6. This corresponds to a ruler of length 6 which is the optimal Golomb ruler for M = 4. This iterative process is continued, with progressively longer rulers, until the optimal length ruler is found. www.cs.ubc.ca /~sillito/papers/thesis/final/node7.html   (778 words)

 Graceful Graphs: Connections to Golomb Rulers and Other Studies   (Site not responding. Last check: 2007-10-07) The Golomb ruler can be described as a problem in radio astronomy: Using as few telescopes as possible (telescopes are expensive), construct a line of telescopes such that for every distance between 1 and N (for some N), there is a pair of telescopes that far apart from each other. Such a labelling is what Golomb called a graceful labelling of the graph, which is then said to be graceful. However, Golomb was not the first to study graceful labellings; Alexander Rosa studied them under the notion of alpha- and beta-valuations. www.qbrundage.com /michaelb/pubs/graceful/golomb.html   (294 words)

 Golomb rulers   (Site not responding. Last check: 2007-10-07) A Golomb ruler is a set of positive integers, containing zero, so that no two differences between any pair of these numbers are the same. To check whether a given set is a Golomb ruler we may use a technique very similar to checking whether a set is a CDS: we construct a difference table and check whether all elements of that table are different. In fact, it is easily proved that any Golomb ruler of length L is a CDS when we choose a modulus n larger than two times L - and sometimes a smaller modulus works as well. www.inference.phy.cam.ac.uk /cds/part16.htm   (556 words)

 Regular Sampling in : a Golomb Ruler   (Site not responding. Last check: 2007-10-07) This is, in fact, the maximum size for a perfect Golomb ruler, and all rulers greater than this length can not uniquely measure all the integer measures between the start and end of the ruler. Optimal rulers with 19 marks, able to measure 171 distances out on a total ruler of length 246 units, have been found [101]. An example of the Golomb's ruler employed in this analysis is presented in Figure 4.1. almuhit.phys.uvic.ca /~gfl/Astro/thesis/node59.html   (324 words)

 Bryan's Wondrous Website   (Site not responding. Last check: 2007-10-07) The paper researched the previous methods used in determining Golomb Rulers, and then built on a previous Genetic Algorithm based technique by using local search techniques to accelerate convergence. The concept of a "Golomb Ruler" arose from the work of Professor Soloman W. Golomb of the University of Southern California. Golomb Rulers are a class of undirected graph measuring more discrete lengths than the number of marks they carry. www.bfeeney.uklinux.net /golomb.html   (153 words)

 distributed.net OGR Project James B. Shearer has compiled a list of all the best known Golomb rulers up to 150 marks. By detecting that the ruler so far cannot possibly be part of a ruler shorter than the current best known, we can eliminate a lot of unnecessary searching. Although we may find a shorter Golomb ruler partway through, there may be still shorter ones not found yet. hewgill.com /ogr   (827 words)

 Ruler   (Site not responding. Last check: 2007-10-07) Madame X — the cruel, uncrowned ruler of the China seas — promises "gold, love,and adventure" to all women who'll leave their humdrum lives behind. In mathematics, the term "Golomb Ruler" refers to a set of non-negative...An Optimal Golomb Ruler (OGR) is the shortest Golomb Ruler possible for a given... An interactive ruler and compass construction editor for use in teaching thefundamental concepts of geometry to high school students. www.classicgame.org /ruler.html   (943 words)

 Self-Service Science Forum Message   (Site not responding. Last check: 2007-10-07) A Golomb ruler is a device or measuring method for generating measuring devices!?!?. An equivalent golomb ruler would be marked 0 1 4 6, or in 'golomb diff' notation, 0 1 3 2. In this case, the 0 1 4 6 ruler is the optimal 4 mark ruler, since it has the same number of marks as the 0 1 3 7 ruler, and can measure the same number of unique distances as the ruler that is 7 units long, yet it is shorter(less expensive). www2.abc.net.au /science/k2/stn/august2000/posts/topic121794.shtm   (1173 words)

 Untitled Document   (Site not responding. Last check: 2007-10-07) A Golomb Ruler is a set of integers such that no two pairs of numbers from the set have the same difference. Solomon W. Golomb, a professor of Mathematics with interests in number theory, coding theory, and communications. A Golomb ruler is a way to place marks along a line so that each pair of marks measures a unique distance. coe.fgcu.edu /bengel/MegaMath/Hexatrinet.html   (462 words)

 [No title] The algorithm for the Golomb ruler problem is well suited for studying scalability issues. In Figure 1 it is used to verify that the shortest known ruler with 12 respectively 13 marks are indeed optimal. The results for small golomb ruler problems demonstrate an efficient parallel execution time which is at least an order of magnitude smaller than previous results [9, 12]. www.ubka.uni-karlsruhe.de /vvv/ira/1995/40/40.text   (1729 words)

 The Golomb problem This is done by beginning with some initial length L for the ruler and attempting to solve the problem. In fact a significant portion of the work to solve an instance of the Golomb ruler is in proving optimality (i.e. Three small trees (11 marks with length 67, length 68 and length 69) were used in experiments to calibrate the sampling algorithm; that is, to select values for the three parameters. www.cs.ubc.ca /~sillito/papers/thesis/final/node41.html   (272 words)

 Golomb ruler: Definition and Links by Encyclopedian.com - All about Golomb ruler Golomb ruler: Definition and Links by Encyclopedian.com - All about Golomb ruler The shortest Golomb rulers with a given number of points are referred to as optimal Golomb rulers. distributed.net[?] is currently (2002) running a distributed massively parallel[?] search for optimal Golomb rulers. www.encyclopedian.com /go/Golomb-ruler.html   (138 words)

 distributed.net Faq-O-Matic: What is the total number of stubs/nodes in OGR-24 or OGR-25?   (Site not responding. Last check: 2007-10-07) The horizontal axes within those surface plots represent the first two numbers in the stubs that are distributed, that is the "x" and "y" in "24/x-y-?-?-?"; while the height of the plot represent the number of nodes within the completed stub. Keep in mind that the number of nodes within a given stub has absolutely no relation to the "shortness" or "optimality" of the ruler that the project is looking for. The node count merely represents the number of golomb ruler possibilities that were checked within that stub's prefix, of which any single one has a chance of being a golomb ruler that is more optimal than what is currently known. n0cgi.distributed.net /faq/cache/150.html   (222 words)

 Solomon Golomb   (Site not responding. Last check: 2007-10-07) For Solomon Golomb, mathematician and inventor of pentaminoes. If two squares side by side is a "domino", then n squares joined side by side to make a shape is a "polyomino", an idea invented by mathematician Solomon Golomb of USC. There are twelve pentaminoes and six letters in "Golomb", which leads to the nice challenge of spelling "Golomb" using just two letters to make each letter shape. www.scottkim.com /inversions/gallery/golomb.html   (307 words)

 Number Theory Topics For this reason the best and most desired Golomb rulers are defined as follows: between any of his n marks every difference, which can be calculated respective measured, may occur at most one and the length has to be minimal. A perfect Golomb ruler is one, in which every such difference from 1 to the length of the ruler occurs exactly one time. If somebody is interested (but consider the time consuming algorithmic behaviour!), I have developed a searching program for such Golomb rulers, which is available in C and Modula-2: number theory programs. www.lb.shuttle.de /apastron/numbers.htm   (1188 words)

 Golomb Ruler Library References   (Site not responding. Last check: 2007-10-07) Since Golomb rulers require e=1 (called "planar"), we let e=1 and set k=n+1 (n=number of 1st differences on a ruler == number of marks minus 1) giving v=n^2+n+1. The authors list the lengths of golomb rulers found with from 2 through 100 marks (length(100)=8831). A nice feature of this article is a table that details the golomb ruler lengths obtained from PDS's of different "n" via truncation. members.aol.com /golomb20/library.html   (512 words)

 Golomb Rulers - The Search For 20 and 21!   (Site not responding. Last check: 2007-10-07) If you are interested, we have written an introduction to Golomb rulers. His Golomb Ruler's page lists the sources of the rulers as well as their lengths and includes fortran source code. The latest are an AIX Binary compiled by John Caldwell and MultiThreaded clients for BeOS and Solaris compiled by Chuck Sanders. members.aol.com /golomb20   (2473 words)

 New Page 0   (Site not responding. Last check: 2007-10-07) Contruction of optimum order 2 Golomb ruler, i.e., of rulers of minimum length, is a highly combinatorial problem which has applications, e.g., We propose here a first exact algorithm, different from a pure exhaustive search, for building optimum order 2 Golomb rulers and provide optimal rulers for up to 9 marks and new rulers which improve on the previous ones for up to 20 marks. The Golomb Ruler problem consists in finding n integers such that all possible differences are distinct and such that the largest difference is minimum. www.iro.umontreal.ca /labs/orc/Research/Golomb_Rulers/golomb.htm   (272 words)

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