Hamming space

In statistics and coding theory, a Hamming space is usually the set of all

In the typical, binary case, the field is thus GF(2) (also denoted by Z2).

[4] 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.

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.

[4] 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 Hamming space of binary strings of length 3. The distance between vertices in the cube graph equals the Hamming distance between the strings.