Lee distance

In coding theory, the Lee distance is a distance between two strings $x_1 x_2 \dots x_n$ and $y_1 y_2 \dots y_n$ of equal length n over the q-ary alphabet {0, 1, …, q − 1} of size q ≥ 2. It is a metric, defined as

$\sum_{i=1}^n \min(|x_i-y_i|,q-|x_i-y_i|).$

If q = 2 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 3140 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.