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^[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 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.^[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]

If q = 6, then the Lee distance between 3140 and 2543 is 1 + 2 + 0 + 3 = 6.

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]

• 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.