Lee distance

In coding theory, the Lee distance is a distance between two strings [math]\displaystyle{ x_1 x_2 \dots x_n }[/math] and [math]\displaystyle{ y_1 y_2 \dots y_n }[/math] of equal length n over the q-ary alphabet ${0, 1, \dots, q - 1$ } of size $q \geq 2$ . It is a metric^[1] defined as [math]\displaystyle{ \sum_{i=1}^n \min(|x_i - y_i|,\, q - |x_i - y_i|). }[/math] 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 [math]\displaystyle{ \mathbb{Z}_4 }[/math] with the Lee weight and [math]\displaystyle{ \mathbb{Z}_2^2 }[/math] with the Hamming weight.^[2]

Considering the alphabet as the additive group Z_q, the Lee distance between two single letters [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] 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 [math]\displaystyle{ \mathbf{Z}_q^n }[/math]. This can also be thought of as the quotient metric resulting from reducing $Z n$ with the Manhattan distance modulo the lattice $q Z n$ . The analogous quotient metric on a quotient of $Z n$ 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 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

↑ ^1.0 ^1.1 Deza, Elena; Deza, Michel (2014), Dictionary of Distances (3rd ed.), Elsevier, p. 52, ISBN 9783662443422
↑ ^2.0 ^2.1 "An Introduction to Ring-Linear Coding Theory". Gröbner Bases, Coding, and Cryptography. Springer Science & Business Media. 2009. p. 220. ISBN 978-3-540-93806-4. https://archive.org/details/grbnerbasescodin00sala.
↑ 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. https://archive.org/details/algebraiccodeson00blah_516.
↑ "Codes over Gaussian Integers". IEEE Transactions on Information Theory 40 (1): 207–216. January 1994. doi:10.1109/18.272484. IEEE Log ID 9215213.. ISSN 0018-9448. https://www.researchgate.net/publication/220036065. 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.)
↑
"Using Gray codes as Location Identifiers" (in en, de). Oberpfaffenhofen, Germany: Institute of Communications and Navigation, German Aerospace Center (DLR). October 2009. http://elib.dlr.de/60489/3/paper.pdf. Retrieved 2020-12-16. (5/8 pages) [3]
- Thomas Strang (October 2009). "Using Gray codes as Location Identifiers". https://www.researchgate.net/publication/225003251.
↑ Roth, Ron (2006). Introduction to Coding Theory. Cambridge University Press. p. 314. ISBN 978-0-521-84504-5. https://archive.org/details/introductiontoco00roth_028.

Lee, C. Y. (1958), "Some properties of nonbinary error-correcting codes", IRE Transactions on Information Theory 4 (2): 77–82, doi:10.1109/TIT.1958.1057446
Berlekamp, Elwyn R. (1968), Algebraic Coding Theory, McGraw-Hill
Voloch, Jose Felipe; Walker, Judy L. (1998). "Lee Weights of Codes from Elliptic Curves". in Vardy, Alexander. Codes, Curves, and Signals: Common Threads in Communications. Springer Science & Business Media. ISBN 978-1-4615-5121-8.

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Original source: https://en.wikipedia.org/wiki/Lee distance. Read more

[Deza-1] 1.0 ^1.1 Deza, Elena; Deza, Michel (2014), Dictionary of Distances (3rd ed.), Elsevier, p. 52, ISBN 9783662443422

[Greferath2009-2] 2.0 ^2.1 "An Introduction to Ring-Linear Coding Theory". Gröbner Bases, Coding, and Cryptography. Springer Science & Business Media. 2009. p. 220. ISBN 978-3-540-93806-4. https://archive.org/details/grbnerbasescodin00sala.

[Blahut2008-3] 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. https://archive.org/details/algebraiccodeson00blah_516.

[Huber_1994-4] "Codes over Gaussian Integers". IEEE Transactions on Information Theory 40 (1): 207–216. January 1994. doi:10.1109/18.272484. IEEE Log ID 9215213.. ISSN 0018-9448. https://www.researchgate.net/publication/220036065. 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.)

[Strang-Dammann-Roeckl-Plass_2009-5] "Using Gray codes as Location Identifiers" (in en, de). Oberpfaffenhofen, Germany: Institute of Communications and Navigation, German Aerospace Center (DLR). October 2009. http://elib.dlr.de/60489/3/paper.pdf. Retrieved 2020-12-16. (5/8 pages) [3]
Thomas Strang (October 2009). "Using Gray codes as Location Identifiers". https://www.researchgate.net/publication/225003251.

[6] Thomas Strang (October 2009). "Using Gray codes as Location Identifiers". https://www.researchgate.net/publication/225003251.

[Roth2006-6] Roth, Ron (2006). Introduction to Coding Theory. Cambridge University Press. p. 314. ISBN 978-0-521-84504-5. https://archive.org/details/introductiontoco00roth_028.

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v t e Strings
String metric	Approximate string matching Bitap algorithm Damerau–Levenshtein distance Edit distance Gestalt Pattern Matching Hamming distance Jaro–Winkler distance Lee distance Levenshtein automaton Levenshtein distance Wagner–Fischer algorithm
String-searching algorithm	Apostolico–Giancarlo algorithm Boyer–Moore string-search algorithm Boyer–Moore–Horspool algorithm Knuth–Morris–Pratt algorithm Rabin–Karp algorithm
Multiple string searching	Aho–Corasick Commentz-Walter algorithm
Regular expression	Comparison of regular expression engines Regular tree grammar Thompson's construction Nondeterministic finite automaton
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Data structures	DAFSA Suffix array Suffix automaton Suffix tree Generalized suffix tree Rope Ternary search tree Trie
Other	Parsing Pattern matching Compressed pattern matching Longest common subsequence Longest common substring Sequential pattern mining Sorting