Revision as of 17:30, 17 April 2016

Gray map

A map from $\mathbf{Z}_4$ to $\mathbf{F}_2^2$, extended in the obvious way to $\mathbf{Z}_4^n$ and $\mathbf{F}_2^n$ which maps Lee distance to Hamming distance. Explicitly, $$ 0 \mapsto 00 \ ,\ \ 1 \mapsto 01 \ ,\ \ 2 \mapsto 11 \ ,\ \ 3 \mapsto 10 \ . $$

The map instantiates a Gray code in dimension 2.

How to Cite This Entry:
Richard Pinch/sandbox-5. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-5&oldid=38581

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-=Lee distance=
-A metric on words over an alphabet $A = \{ a_1, \ldots, a_m \}$ where a single error is changing a letter one place in cyclic order.  If the alphabet is identified with $\mathbf{Z}_m = \{0, \ldots, m-1 \}$ then the Lee distance between $x, y \in \mathbf{Z}_m^n$ is
-$$
-d_L (x,y) = \sum_{i=1}^n \min\left(|x_i-y_i|, m-|x_i-y_y|\right) \ .
-$$
-When $m=2$ or $m=3$, Lee distance coincides with [[Hamming distance]].  The Lee distance on $\mathbf{Z}_4$ corresponds to Hamming distance on $\mathbf{F}_2^2$ under the [[Gray map]]
-$$
-\mapsto 00 \ ,\ \  1 \mapsto 01 \ ,\ \  2 \mapsto 11 \ ,\ \  3 \mapsto 10 \ .
-$$
-====References====
-* Deza, Michel Marie; Deza, Elena ''Encyclopedia of distances'' (3rd ed.) Springer (2014) ISBN 978-3-662-44341-5 {{ZBL|1301.51001}}
-* Roth, Ron ''Introduction to Coding Theory'', Cambridge University Press (2006)  ISBN 0-521-84504-1 {{DOI|10.1017/CBO9780511808968.011}} {{ZBL|1092.94001}}
 =Gray map=
 A map from $\mathbf{Z}_4$ to $\mathbf{F}_2^2$, extended in the obvious way to $\mathbf{Z}_4^n$ and $\mathbf{F}_2^n$ which maps [[Lee distance]] to [[Hamming distance]].  Explicitly,

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Difference between revisions of "User:Richard Pinch/sandbox-5"

Revision as of 17:30, 17 April 2016

Gray map