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, 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 Z₂).^[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 Z₄) 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 $\mathbb{Z}_2^{2m}$ (i.e. GF(2^2m)) with the Hamming distance and $\mathbb{Z}_4^m$ (also denoted as GR(4,m)) with the Lee distance.^[5][6][7]

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
5. ↑ Marcus Greferath (2009). "An Introduction to Ring-Linear Coding Theory". In Massimiliano Sala, Teo Mora, Ludovic Perret, Shojiro Sakata, Carlo Traverso. Gröbner Bases, Coding, and Cryptography. Springer Science & Business Media. ISBN 978-3-540-93806-4.
6. ↑ http://www.encyclopediaofmath.org/index.php/Kerdock_and_Preparata_codes
7. ↑ J.H. van Lint (1999). Introduction to Coding Theory (3rd ed.). Springer. Chapter 8: Codes over ℤ₄. ISBN 978-3-540-64133-9.