Lee distance
In coding theory, the Lee distance is a distance between two strings and of equal length n over the q-ary
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
The metric space induced by the Lee distance is a discrete analog of the elliptic space.[1]
Example
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.
The Berlekamp code is an example of code in the Lee metric.[6] 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.[2]
References
- ^ ISBN 9783662443422
- ^ ISBN 978-3-540-93806-4.
- ISBN 978-1-139-46946-3.
- (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.)
- CiteSeerX 10.1.1.398.9164. Archived (PDF) from the original on 2015-05-01. Retrieved 2020-12-16. (5/8 pages) [3]
- Thomas Strang; et al. (October 2009). "Using Gray codes as Location Identifiers". ResearchGate (Abstract).
- ISBN 978-0-521-84504-5.
- Lee, C. Y. (1958), "Some properties of nonbinary
- Berlekamp, Elwyn R. (1968), Algebraic Coding Theory, McGraw-Hill
- Voloch, Jose Felipe; Walker, Judy L. (1998). "Lee Weights of Codes from Elliptic Curves". In ISBN 978-1-4615-5121-8.