Lee distance

Lee distance

= 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 qgeq 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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