Lee distance

Incoding theory, the Lee distance is a distance between two strings ${\displaystyle x_{1}x_{2}\dots x_{n}}$ and ${\displaystyle y_{1}y_{2}\dots y_{n}}$ of equal length n over the q-ary alphabet {0, 1, …, q − 1} of size q ≥ 2. It is a metric^[1] defined as $${\displaystyle \sum _{i=1}^{n}\min(|x_{i}-y_{i}|,\,q-|x_{i}-y_{i}|).}$$Ifq = 2orq = 3 the Lee distance coincides with the Hamming distance, because both distances are 0 for two single equal symbols and 1 for two single non-equal symbols. For q >3 this is not the case anymore; the Lee distance between single letters can become bigger than 1. However, there exists a Gray isometry (weight-preserving bijection) between ${\displaystyle \mathbb {Z} _{4}}$ with the Lee weight and ${\displaystyle \mathbb {Z} _{2}^{2}}$ with the Hamming weight.^[2]