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Hamming space

Instatistics and coding theory, a Hamming space (named after American mathematician Richard Hamming) is usually the set of all ${\displaystyle 2^{N}}$ 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]IfQ 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 ${\displaystyle \mathbb {Z} _{2}^{2m}}$ (i.e. GF(2^2m)) with the Hamming distance and ${\displaystyle \mathbb {Z} _{4}^{m}}$ (also denoted as GR(4,m)) with the Lee distance.^[5][6][7]

^ J.H. van Lint (1999). Introduction to Coding Theory (3rd ed.). Springer. Chapter 8: Codes over ${\displaystyle \mathbb {Z} _{4}}$. ISBN 978-3-540-64133-9.

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