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