Hamming space

In statistics and coding theory, a Hamming space is usually the set of all 2^N binary strings of length N.[1][2] 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.[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).[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]

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]

See also

References

  1. ↑ Baylis, D. J. (1997), Error Correcting Codes: A Mathematical Introduction, Chapman Hall/CRC Mathematics Series 15, CRC Press, p. 62, ISBN 9780412786907
  2. ↑ Cohen, G.; Honkala, I.; Litsyn, S.; Lobstein, A. (1997), Covering Codes, North-Holland Mathematical Library 54, Elsevier, p. 1, ISBN 9780080530079
  3. ↑ 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. ↑ 4.0 4.1 Cohen et al., Covering Codes, p. 15