Lee distance

In coding theory, the Lee distance is a distance between two strings and of equal length n over the q-ary alphabet of size. It is a metric defined as
If or 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 this is not the case anymore; the Lee distance between single letters can become bigger than 1. However, there exists a Gray isometry between with the Lee weight and with the Hamming weight.
Considering the alphabet as the additive group Z_q, the Lee distance between two single letters and is the length of shortest path in the Cayley graph between them. More generally, the Lee distance between two strings of length is the length of the shortest path between them in the Cayley graph of. This can also be thought of as the metric spaces|quotient metric] resulting from reducing with the Manhattan distance modulo the lattice. The analogous quotient metric on a quotient of modulo an arbitrary lattice is known as a or Mannheim distance.
The metric space induced by the Lee distance is a discrete analog of the elliptic space.

Example

If, then the Lee distance between 3140 and 2543 is.

The Lee distance is named after William Chi Yuan Lee. It is applied for phase modulation while the Hamming distance is used in case of orthogonal modulation.
The Berlekamp code is an example of code in the Lee metric. Other significant examples are the Preparata code and Kerdock code; these codes are non-linear when considered over a field, but are linear over a ring.