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.[2]
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.[3] 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
latticeqZn. The analogous quotient metric on a quotient of Zn modulo an arbitrary lattice is known as a Mannheim metric or Mannheim distance.[4][5]
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 William Chi Yuan Lee (李始元). It is applied for phase
modulation while the Hamming distance is used in case of orthogonal modulation.
^Strang, Thomas; Dammann, Armin; Röckl, Matthias; Plass, Simon (October 2009).
Using Gray codes as Location Identifiers(PDF). 6. GI/ITG KuVS Fachgespräch Ortsbezogene Anwendungen und Dienste (in English and German). Oberpfaffenhofen, Germany: Institute of Communications and Navigation,
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[3]
Voloch, Jose Felipe; Walker, Judy L. (1998). "Lee Weights of Codes from Elliptic Curves". In
Vardy, Alexander (ed.). Codes, Curves, and Signals: Common Threads in Communications. Springer Science & Business Media.
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