Hamming space explained

In statistics and coding theory, a Hamming space (named after American mathematician Richard Hamming) is usually the set of all

2 ^{N} binary strings of length N. 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.^[1] 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).^[2]

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.^{[2] [1]} In the case where C is a linear subspace of its Hamming space, it is called a linear code.^[2] 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.^[2]

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

Z ^{2 m} _{2} (i.e. GF(2^2m)) with the Hamming distance and Z ^{m} _{4} (also denoted as GR(4,m)) with the Lee distance.^{[3] [4] [5]}

Notes and References

1. Cohen et al., Covering Codes, p. 15
2. Book: Derek J.S. Robinson. An Introduction to Abstract Algebra. 2003. Walter de Gruyter. 978-3-11-019816-4. 254–255.
3. Book: Massimiliano Sala . Teo Mora . Ludovic Perret . Shojiro Sakata . Carlo Traverso. Gröbner Bases, Coding, and Cryptography. 2009. Springer Science & Business Media. 978-3-540-93806-4. An Introduction to Ring-Linear Coding Theory. Marcus Greferath.
4. Web site: Kerdock and Preparata codes - Encyclopedia of Mathematics.
5. Book: J.H. van Lint. Introduction to Coding Theory. registration. 1999. Springer. 978-3-540-64133-9. 3rd. Chapter 8: Codes over Z _{4} .