# Hamming space

In statistics and coding theory, a **Hamming space** (named after American mathematician Richard Hamming) is usually the set of all [math]\displaystyle{ 2^N }[/math] binary strings of length *N*.^{[1]}^{[2]} It is used in the theory of coding signals and transmission.

More generally, a Hamming space can be defined over any alphabet (set) *Q* as the set of words of a fixed length *N* with letters from *Q*.^{[3]}^{[4]} If *Q* is a finite field, then a Hamming space over *Q* is an *N*-dimensional vector space over *Q*. In the typical, binary case, the field is thus GF(2) (also denoted by **Z**_{2}).^{[3]}

In coding theory, if *Q* has *q* elements, then any subset *C* (usually assumed of cardinality at least two) of the *N*-dimensional Hamming space over *Q* is called a **q-ary code of length N**; the elements of *C* are called **codewords**.^{[3]}^{[4]} In the case where *C* is a linear subspace of its Hamming space, it is called a linear code.^{[3]} A typical example of linear code is the Hamming code. Codes defined via a Hamming space necessarily have the same length for every codeword, so they are called block codes when it is necessary to distinguish them from variable-length codes that are defined by unique factorization on a monoid.

The Hamming distance endows a Hamming space with a metric, which is essential in defining basic notions of coding theory such as error detecting and error correcting codes.^{[3]}

Hamming spaces over non-field alphabets have also been considered, especially over finite rings (most notably over **Z**_{4}) giving rise to modules instead of vector spaces and ring-linear codes (identified with submodules) instead of linear codes. The typical metric used in this case the Lee distance. There exist a Gray isometry between [math]\displaystyle{ \mathbb{Z}_2^{2m} }[/math] (i.e. GF(2^{2m})) with the Hamming distance and [math]\displaystyle{ \mathbb{Z}_4^m }[/math] (also denoted as GR(4,m)) with the Lee distance.^{[5]}^{[6]}^{[7]}

## References

- ↑ Baylis, D. J. (1997),
*Error Correcting Codes: A Mathematical Introduction*, Chapman Hall/CRC Mathematics Series,**15**, CRC Press, p. 62, ISBN 9780412786907, https://books.google.com/books?id=ZAdDuZoJdn8C&pg=PA62 - ↑ Cohen, G.; Honkala, I.; Litsyn, S.; Lobstein, A. (1997),
*Covering Codes*, North-Holland Mathematical Library,**54**, Elsevier, p. 1, ISBN 9780080530079, https://books.google.com/books?id=7KBYOt44sugC&pg=PA1 - ↑
^{3.0}^{3.1}^{3.2}^{3.3}^{3.4}Derek J.S. Robinson (2003).*An Introduction to Abstract Algebra*. Walter de Gruyter. pp. 254–255. ISBN 978-3-11-019816-4. - ↑
^{4.0}^{4.1}Cohen et al.,*Covering Codes*, p. 15 - ↑ Marcus Greferath (2009). "An Introduction to Ring-Linear Coding Theory".
*Gröbner Bases, Coding, and Cryptography*. Springer Science & Business Media. ISBN 978-3-540-93806-4. - ↑ "Kerdock and Preparata codes - Encyclopedia of Mathematics". http://www.encyclopediaofmath.org/index.php/Kerdock_and_Preparata_codes.
- ↑ J.H. van Lint (1999).
*Introduction to Coding Theory*(3rd ed.). Springer. Chapter 8: Codes over [math]\displaystyle{ \mathbb{Z}_4 }[/math]. ISBN 978-3-540-64133-9. https://archive.org/details/introductiontoco0000lint_a3b9.

Original source: https://en.wikipedia.org/wiki/Hamming space.
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