Hamming space
In statistics and coding theory, a Hamming space is usually 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. In the typical, binary case, the field is thus GF(2).[1]
In coding theory, if Q has q elements, then any subset C of cardinality at least two of of the N-dimensional Hamming space over Q is called a q-ary code of length N; the elements of C are called codewords.[1]
The Hamming distance endows the Hamming space with a metric and is essential in defining basic notions of coding theory such a error detecting and error correcting codes.[1]
See also
References
- ^ a b c Derek J.S. Robinson (2003). An Introduction to Abstract Algebra. Walter de Gruyter. pp. 254–255. 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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