Lee distance

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, defined as

${\displaystyle \sum _{i=1}^{n}\min(|x_{i}-y_{i}|,\,q-|x_{i}-y_{i}|).}$[1]

Considering the alphabet as the additive group Z_q, 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.[2]

If ${\displaystyle q=2}$ or ${\displaystyle q=3}$ 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 ${\displaystyle q>3}$ this is not the case anymore, the Lee distance can become bigger than 1.

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 C. Y. Lee. 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.[3] 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.[4]

Also, there exists a Gray isometry (bijection preserving weight) between ${\displaystyle \mathbb {Z} _{4}}$ with the Lee weight and ${\displaystyle \mathbb {Z} _{2}^{2}}$ with the Hamming weight.[4]

References

1. Deza, Elena; Deza, Michel (2014), Dictionary of Distances (3rd ed.), Elsevier, p. 52, ISBN 9783662443422
2. 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.
3. Roth, Ron (2006). Introduction to Coding Theory. Cambridge University Press. p. 314. ISBN 978-0-521-84504-5.
4. 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.

• 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 (ed.). Codes, Curves, and Signals: Common Threads in Communications. Springer Science & Business Media. ISBN 978-1-4615-5121-8.