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{{No footnotes|date=July 2011}}
In [[coding theory]], the '''Lee distance''' is a [[distance]] between two [[String (computer science)|string]]s <math>x_1 x_2 \dots x_n</math> and <math>y_1 y_2 \dots y_n</math> of equal length ''n'' over the ''q''-ary [[alphabet]] {0,&nbsp;1,&nbsp;…,&nbsp;''q''&nbsp;&minus;&nbsp;1} of size ''q''&nbsp;≥&nbsp;2.
It is a [[Metric (mathematics)|metric]], defined as
 
: <math>\sum_{i=1}^n \min(|x_i-y_i|,q-|x_i-y_i|).</math>
 
If ''q''&nbsp;=&nbsp;2 the Lee distance coincides with the [[Hamming distance]].
 
The [[metric space]] induced by the Lee distance is a discrete analog of the [[Elliptic geometry|elliptic space]].
 
==Example==
If ''q''&nbsp;=&nbsp;6, then the Lee distance between 3140 and 2543 is 1&nbsp;+&nbsp;2&nbsp;+&nbsp;0&nbsp;+&nbsp;3&nbsp;=&nbsp;6.
 
==History and application==
The Lee distance is named after [[C. Y. Lee (mathematician)|C. Y. Lee]]. It is applied for phase [[modulation]] while the Hamming distance is used in case of orthogonal modulation.
 
==References==
* {{Citation|first=C. Y.|last=Lee|title=Some properties of nonbinary [[error-correcting codes]]|journal=[[IEEE Transactions on Information Theory|IRE Transactions on Information Theory]]|volume=4|year=1958|pages=77–82|issue=2|doi=10.1109/TIT.1958.1057446}}.
* {{Citation|first=E. R.|last=Berlekamp|authorlink=Elwyn Berlekamp|title=Algebraic Coding Theory|publisher=McGraw-Hill|year=1968}}.
* {{Citation|last1=Deza|first1=E.|first2=M.|last2=Deza|author2-link=Michel Deza|title=Dictionary of Distances|year=2006|publisher=Elsevier|isbn=0-444-52087-2}}.
 
[[Category:Coding theory]]
[[Category:String similarity measures]]

Revision as of 00:21, 26 October 2013

Template:No footnotes In coding theory, the Lee distance is a distance between two strings x1x2xn and y1y2yn of equal length n over the q-ary alphabet {0, 1, …, q − 1} of size q ≥ 2. It is a metric, defined as

i=1nmin(|xiyi|,q|xiyi|).

If q = 2 the Lee distance coincides with the Hamming distance.

The metric space induced by the Lee distance is a discrete analog of the elliptic space.

Example

If q = 6, then the Lee distance between 3140 and 2543 is 1 + 2 + 0 + 3 = 6.

History and application

The Lee distance is named after C. Y. Lee. It is applied for phase modulation while the Hamming distance is used in case of orthogonal modulation.

References

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