statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...

and

coding theory Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data transmission and data storage. Codes are studied ...

, a Hamming space (named after American mathematician

Richard Hamming Richard Wesley Hamming (February 11, 1915 – January 7, 1998) was an American mathematician whose work had many implications for computer engineering and telecommunications. His contributions include the Hamming code (which makes use of a ...

) 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 An alphabet is a standardized set of basic written graphemes (called letters) that represent the phonemes of certain spoken languages. Not all writing systems represent language in this way; in a syllabary, each character represents a syllab ...

(set) ''Q'' as the set of

words A word is a basic element of language that carries an objective or practical meaning, can be used on its own, and is uninterruptible. Despite the fact that language speakers often have an intuitive grasp of what a word is, there is no conse ...

of a fixed length ''N'' with letters from ''Q''.Cohen et al., ''Covering Codes'', p. 15 If ''Q'' is a

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...

, then a Hamming space over ''Q'' is an ''N''-dimensional

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...

over ''Q''. In the typical, binary case, the field is thus

GF(2) (also denoted \mathbb F_2, or \mathbb Z/2\mathbb Z) is the finite field of two elements (GF is the initialism of ''Galois field'', another name for finite fields). Notations and \mathbb Z_2 may be encountered although they can be confused wit ...

(also denoted by Z₂). In coding theory, if ''Q'' has ''q'' elements, then any

subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...

''C'' (usually assumed of

cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...

at least two) of the ''N''-dimensional Hamming space over ''Q'' is called a q-ary

code In communications and information processing, code is a system of rules to convert information—such as a letter, word, sound, image, or gesture—into another form, sometimes shortened or secret, for communication through a communicati ...

of length N; the elements of ''C'' are called

codeword In communication, a code word is an element of a standardized code or protocol. Each code word is assembled in accordance with the specific rules of the code and assigned a unique meaning. Code words are typically used for reasons of reliability, ...

s. In the case where ''C'' is a

linear subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, l ...

of its Hamming space, it is called a linear code. A typical example of linear code is the

Hamming code In computer science and telecommunication, Hamming codes are a family of linear error-correcting codes. Hamming codes can detect one-bit and two-bit errors, or correct one-bit errors without detection of uncorrected errors. By contrast, the sim ...

. Codes defined via a Hamming space necessarily have the same length for every codeword, so they are called

block code In coding theory, block codes are a large and important family of error-correcting codes that encode data in blocks. There is a vast number of examples for block codes, many of which have a wide range of practical applications. The abstract defini ...

s when it is necessary to distinguish them from

variable-length code In coding theory a variable-length code is a code which maps source symbols to a ''variable'' number of bits. Variable-length codes can allow sources to be compressed and decompressed with ''zero'' error ( lossless data compression) and still b ...

s that are defined by unique factorization on a monoid. The

Hamming distance In information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different. In other words, it measures the minimum number of ''substitutions'' required to chan ...

endows a Hamming space with a

metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathe ...

, which is essential in defining basic notions of coding theory such as error detecting and error correcting codes. Hamming spaces over non-field alphabets have also been considered, especially over finite rings (most notably over Z₄) giving rise to

module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modul ...

s instead of vector spaces and ring-linear codes (identified with

submodule In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mo ...

s) instead of linear codes. The typical metric used in this case the

Lee distance In coding theory, the Lee distance is a distance between two strings x_1 x_2 \dots x_n and y_1 y_2 \dots y_n of equal length ''n'' over the ''q''-ary alphabet of size . It is a metric defined as \sum_^n \min(, x_i - y_i, ,\, q - , x_i - y_i, ). If ...

. There exist a

Gray isometry The reflected binary code (RBC), also known as reflected binary (RB) or Gray code after Frank Gray, is an ordering of the binary numeral system such that two successive values differ in only one bit (binary digit). For example, the representati ...

between

\mathbb_2^

(i.e. GF(2^2m)) with the Hamming distance and

\mathbb_4^m

(also denoted as GR(4,m)) with the Lee distance.

References

Coding theory Linear algebra {{algebra-stub