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Lee distance

## Summary

In coding theory, the Lee distance is a distance between two strings ${\displaystyle x_{1}x_{2}\dots x_{n}}$ and ${\displaystyle y_{1}y_{2}\dots y_{n}}$ of equal length n over the q-ary alphabet {0, 1, …, q − 1} of size q ≥ 2. It is a metric[1] defined as

${\displaystyle \sum _{i=1}^{n}\min(|x_{i}-y_{i}|,\,q-|x_{i}-y_{i}|).}$
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 ${\displaystyle \mathbb {Z} _{4}}$ with the Lee weight and ${\displaystyle \mathbb {Z} _{2}^{2}}$ with the Hamming weight.[2]

Considering the alphabet as the additive group Zq, the Lee distance between two single letters ${\displaystyle x}$ and ${\displaystyle y}$ 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 ${\displaystyle \mathbf {Z} _{q}^{n}}$. 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 Mannheim metric or Mannheim distance.[4][5]

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 Dr. William C. Y. Lee (李建業). 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

1. ^ a b Deza, Elena; Deza, Michel (2014), Dictionary of Distances (3rd ed.), Elsevier, p. 52, ISBN 9783662443422
2. ^ a b Greferath, Marcus (2009). "An Introduction to Ring-Linear Coding Theory". In Sala, Massimiliano; Mora, Teo; Perret, Ludovic; Sakata, Shojiro; Traverso, Carlo (eds.). Gröbner Bases, Coding, and Cryptography. Springer Science & Business Media. p. 220. ISBN 978-3-540-93806-4.
3. ^ Blahut, Richard E. (2008). Algebraic Codes on Lines, Planes, and Curves: An Engineering Approach. Cambridge University Press. p. 108. ISBN 978-1-139-46946-3.
4. ^ Huber, Klaus (January 1994) [1993-01-17, 1992-05-21]. "Codes over Gaussian Integers". IEEE Transactions on Information Theory. 40 (1): 207–216. doi:10.1109/18.272484. eISSN 1557-9654. ISSN 0018-9448. S2CID 195866926. IEEE Log ID 9215213. Archived (PDF) from the original on 2020-12-17. Retrieved 2020-12-17. [1][2] (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. ^ 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, German Aerospace Center (DLR). 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).
6. ^ Roth, Ron (2006). Introduction to Coding Theory. Cambridge University Press. p. 314. ISBN 978-0-521-84504-5.