View source for Lee distance

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A metric on words over an alphabet $A = \{ a_1, \ldots, a_m \}$ where a single error is changing a letter one place in cyclic order.  If the alphabet is identified with $\mathbf{Z}_m = \{0, \ldots, m-1 \}$ then the Lee distance between $x, y \in \mathbf{Z}_m^n$ is
$$
d_L (x,y) = \sum_{i=1}^n \min\left(|x_i-y_i|, m-|x_i-y_y|\right) \ .
$$

When $m=2$ or $m=3$, Lee distance coincides with [[Hamming distance]].  The Lee distance on $\mathbf{Z}_4$ corresponds to Hamming distance on $\mathbf{F}_2^2$ under the [[Gray map]]
$$
0 \mapsto 00 \ ,\ \  1 \mapsto 01 \ ,\ \  2 \mapsto 11 \ ,\ \  3 \mapsto 10 \ .
$$

====References====
* Deza, Michel Marie; Deza, Elena ''Encyclopedia of distances'' (3rd ed.) Springer (2014) ISBN 978-3-662-44341-5 {{ZBL|1301.51001}}
* Roth, Ron ''Introduction to Coding Theory'', Cambridge University Press (2006)  ISBN 0-521-84504-1 {{DOI|10.1017/CBO9780511808968.011}} {{ZBL|1092.94001}}