Hamming space: Difference between revisions
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More generally, the Hamming space can be defined over any [[alphabet (computer science)|alphabet]] (set) ''Q'' as the set of [[word (mathematics)|words]] of a fixed length ''N'' with letters from ''Q''. If ''Q'' is a finite [[field (mathematics)|field]], then a Hamming space over ''Q'' is an ''N''-dimensional [[vector space]] over ''Q''.<ref name="Robinson2003">{{cite book|author=Derek J.S. Robinson|title=An Introduction to Abstract Algebra|year=2003|publisher=Walter de Gruyter|isbn=978-3-11-019816-4|page=254}}</ref> |
More generally, the Hamming space can be defined over any [[alphabet (computer science)|alphabet]] (set) ''Q'' as the set of [[word (mathematics)|words]] of a fixed length ''N'' with letters from ''Q''. If ''Q'' is a finite [[field (mathematics)|field]], then a Hamming space over ''Q'' is an ''N''-dimensional [[vector space]] over ''Q''.<ref name="Robinson2003">{{cite book|author=Derek J.S. Robinson|title=An Introduction to Abstract Algebra|year=2003|publisher=Walter de Gruyter|isbn=978-3-11-019816-4|page=254}}</ref> |
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[[Hamming code]]s and [[Hamming distance]] are related concepts. |
[[Hamming code]]s and [[Hamming distance]] are related concepts. The Hamming distance endows the Hamming space with a [[metric (mathematics)|metric]]. |
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==References== |
==References== |
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Revision as of 18:56, 1 May 2015
In statistics and coding theory, a Hamming space is the set of all binary strings of length N. It is used in the theory of coding signals and transmission.
More generally, the Hamming space can be defined over any alphabet (set) Q as the set of words of a fixed length N with letters from Q. If Q is a finite field, then a Hamming space over Q is an N-dimensional vector space over Q.[1]
Hamming codes and Hamming distance are related concepts. The Hamming distance endows the Hamming space with a metric.
References
- ^ Derek J.S. Robinson (2003). An Introduction to Abstract Algebra. Walter de Gruyter. p. 254. ISBN 978-3-11-019816-4.
- Baylis, D. J. (1997), Error Correcting Codes: A Mathematical Introduction, Chapman Hall/CRC Mathematics Series, vol. 15, CRC Press, p. 62, ISBN 9780412786907.
- Cohen, G.; Honkala, I.; Litsyn, S.; Lobstein, A. (1997), Covering Codes, North-Holland Mathematical Library, vol. 54, Elsevier, p. 1, ISBN 9780080530079.
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