# 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 $$d_L (x,y) = \sum_{i=1}^n \min\left(|x_i-y_i|, m-|x_i-y_i|\right) \ .$$
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 $$0 \mapsto 00 \ ,\ \ 1 \mapsto 01 \ ,\ \ 2 \mapsto 11 \ ,\ \ 3 \mapsto 10 \ .$$