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The Hamming space of binary strings of length 3. The distance between vertices in the cube graph equals the Hamming distance between the strings.
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. 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).
In coding theory, if Q has q elements, then any subsetC (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. In the case where C is a linear subspace of its Hamming space, it is called a linear code. 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.
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 (i.e. GF(22m)) with the Hamming distance and (also denoted as GR(4,m)) with the Lee distance.