Lee distance

Lee distance
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Lee_distance".

content

1 Definition and basic properties
2 Example
3 History and application
4 References

Definition and basic properties

In coding theory, the Lee distance is a distance between two strings $x 1 x 2 ... x n$ and $y 1 y 2 ... y n$ of equal length n over the q-ary alphabet {0,1,…,q-1} of size $q\geq 2$ . It is metric, defined as

$\sum_{i=1}^n min\{|x_i-y_i|,q-|x_i-y_i|\}$ .

If $q = 2$ or $q = 3$ 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 3340 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

C.Y. Lee, Some properties of nonbinary error-correcting codes, IRE Transactions on Information Theory 4 (1958) 77-82
E.R. Berlekamp, Algebraic Coding Theory, McGraw-Hill 1968
E. Deza, M.M. Deza, Dictionary of Distances, Elsevier 2006 ISBN 0-444-52087-2

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Contents

Definition and basic properties

Example

History and application

References