In coding theory, the Lee distance is a distance between two

string String or strings may refer to: *String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects Arts, entertainment, and media Films * ''Strings'' (1991 film), a Canadian anim ...

x_1 x_2 \dots x_n

and

y_1 y_2 \dots y_n

of equal length ''n'' over the ''q''-ary alphabet of size . It is a metric defined as

\sum_^n \min(, x_i - y_i, ,\, q - , x_i - y_i, ).

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 The reflected binary code (RBC), also known as reflected binary (RB) or Gray code after Frank Gray, is an ordering of the binary numeral system such that two successive values differ in only one bit (binary digit). For example, the representat ...

(weight-preserving bijection) between

\mathbb_4

with the Lee weight and

\mathbb_2^2

with the Hamming weight. Considering the alphabet as the additive group Z_''q'', the Lee distance between two single letters

x

and

y

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 is the length of the shortest path between them in the Cayley graph of

\mathbf_q^n

. This can also be thought of as the 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.
https://dl.acm.org/doi/10.1109/18.272484] (1+10 pages) (NB. This work was partially presented at CDS-92 Conference, Kaliningrad, Russia, on 1992-09-07 and at the IEEE Symposium on Information Theory, San Antonio, TX, USA.) (5/8 pages

* 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 .

History and application

The Lee distance is named after C. Y. Lee. It is applied for phase

modulation In electronics and telecommunications, modulation is the process of varying one or more properties of a periodic waveform, called the ''carrier signal'', with a separate signal called the ''modulation signal'' that typically contains informatio ...

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.

References

* * * {{Strings Coding theory String metrics