Lee distance
Encyclopedia
In coding theory
, the Lee distance is a distance
between two string
s
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
.
The metric space
induced by the Lee distance is a discrete analog of the elliptic space
.
while the Hamming distance is used in case of orthogonal modulation.
Coding theory
Coding theory is the study of the properties of codes and their fitness for a specific application. Codes are used for data compression, cryptography, error-correction and more recently also for network coding...
, the Lee distance is a distance
Distance
Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance...
between two string
String (computer science)
In formal languages, which are used in mathematical logic and theoretical computer science, a string is a finite sequence of symbols that are chosen from a set or alphabet....
s
and of equal length n over the q-ary alphabetAlphabet
An alphabet is a standard set of letters—basic written symbols or graphemes—each of which represents a phoneme in a spoken language, either as it exists now or as it was in the past. There are other systems, such as logographies, in which each character represents a word, morpheme, or semantic...
{0, 1, …, q − 1} of size q ≥ 2.
It is a metric
Metric (mathematics)
In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...
, defined as
If q = 2 or q = 3 the Lee distance coincides with the Hamming distance
Hamming distance
In information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different...
.
The metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
induced by the Lee distance is a discrete analog of the elliptic space
Elliptic geometry
Elliptic geometry is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which can be interpreted as asserting that there is exactly one...
.
Example
If q = 6, then the Lee distance between 3340 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 modulationModulation
In electronics and telecommunications, modulation is the process of varying one or more properties of a high-frequency periodic waveform, called the carrier signal, with a modulating signal which typically contains information to be transmitted...
while the Hamming distance is used in case of orthogonal modulation.