Golomb Ruler

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Golomb Ruler

• Tingting Meng

April 10th
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What is Golomb Ruler?

• Rulers for which there is a single way to measure
  each distance.
• Array (a1 , a2, , an) such that aj ai
  distinct, ? i, j ?1,n. n means n-mark ruler

0
1
6
4
2
NO
0
1
6
4
YES
0
1
3
7
YES
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Optimal Golomb Ruler

• Optimal Golomb Ruler (OGR) shortest Golomb
  ruler for a given number of marks
• no other n-mark Golomb rulers having smaller
  largest mark an, i.e., the "length" of the
  optimal n-mark ruler.
• The number of distances in an optimal n-mark
  ruler is n(n-1)/2.

0
1
6
4
YES
0
1
3
7
NO
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Perfect Golomb Ruler

• A perfect Golomb ruler can measure all distances
  up to its length an.
• No perfect Golomb ruler exists for five or more.

0
1
6
4
1
2
3
4
5
6
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Where Golomb Ruler?

• X-ray crystallography
• Data encryption
• Coding theory
• Phase array radio antennae design
• Linear arrays of sensors
• Pulse phase modulation communication

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Why Golomb Ruler?

• W. Babcock discovered the credit of Golomb ruler
  in 1954. By placing each pair of channels inside
  the frequency spectrum at a distinct distance,
  the intermodulation distortion which appeared in
  third and fifth order consecutive radio bands can
  be lessened greatly or even eliminated.
• In radio astronomy, astronomers use arrays of
  radio telescopes in a single line to collect data
  on particular stars or celestial regions.  By
  placing the telescopes at the marks of a Golomb
  ruler, more non-redundant information can be
  extracted .
• In information theory, Golomb rulers are used for
  error detection and correction.  Golomb rulers of
  the same length are used to generate
  self-orthogonal codes, codes that do not share
  common differences.  Such codes allow for easier
  error checking since redundancies between codes
  will show up as errors and may then be resent or
  corrected.

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How to generate OGR?
NO
0 1 2
0 1
YES. Min Length 3
0 1 3
0 2
YES. Min Length 3
0 2 3
NO
0 3
e.g. Optimal Golomb Ruler with mark n3
0
NO
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Current State of Golomb ruler

• Creating Golomb rulers is easy, but finding the
  optimal Golomb rulers (OGR) for a specified order
  is computationally very challenging.
• OGR-24 has been discovered and
• Current problem OGR-25 is currently underway
  (as of 8 December 2005)
• http//www.distributed.net/ogr/

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Question?
If you are confused like this kitty is,
please ask questions ( y )