Lee distance

In coding theory, the Lee distance is a distance between two strings and of equal length n over the q-ary alphabet {0, 1, ..., q − 1} of size q ≥ 2. It is a metric defined as If q = 2 or q = 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 with the Lee weight and with the Hamming weight. Considering the alphabet as the additive group Zq, the Lee distance between two single letters and is the length of shortest path in the Cayley graph (which is circular since the group is cyclic) between them. More generally, the Lee distance between two strings of length n is the length of the shortest path between them in the Cayley graph of . This can also be thought of as the quotient metric resulting from reducing Zn with the Manhattan distance modulo the lattice qZn. The analogous quotient metric on a quotient of Zn 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. If q = 6, then the Lee distance between 3140 and 2543 is 1 + 2 + 0 + 3 = 6. The Lee distance is named after Chester Chi Yuan Lee (李始元). 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.