The Hamming space of binary strings of length 3. The distance between vertices in the cube graph equals the Hamming distance between the strings.
In statistics and coding theory, a Hamming space (named after American mathematician Richard Hamming) is usually the set of all 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] 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).[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 (i.e. GF(22m)) with the Hamming distance and (also denoted as GR(4,m)) with the Lee distance.[5][6][7]
^Baylis, D. J. (1997), Error Correcting Codes: A Mathematical Introduction, Chapman Hall/CRC Mathematics Series, vol. 15, CRC Press, p. 62, ISBN 9780412786907
^Cohen, G.; Honkala, I.; Litsyn, S.; Lobstein, A. (1997), Covering Codes, North-Holland Mathematical Library, vol. 54, Elsevier, p. 1, ISBN 9780080530079
^ abcdeDerek J.S. Robinson (2003). An Introduction to Abstract Algebra. Walter de Gruyter. pp. 254–255. ISBN 978-3-11-019816-4.
^ abCohen et al., Covering Codes, p. 15
^Marcus Greferath (2009). "An Introduction to Ring-Linear Coding Theory". In Massimiliano Sala; Teo Mora; Ludovic Perret; Shojiro Sakata; Carlo Traverso (eds.). Gröbner Bases, Coding, and Cryptography. Springer Science & Business Media. ISBN 978-3-540-93806-4.
^"Kerdock and Preparata codes - Encyclopedia of Mathematics".
^J.H. van Lint (1999). Introduction to Coding Theory (3rd ed.). Springer. Chapter 8: Codes over . ISBN 978-3-540-64133-9.
subspace of its Hammingspace, it is called a linear code. A typical example of linear code is the Hamming code. Codes defined via a Hammingspace necessarily...
For a fixed length n, the Hamming distance is a metric on the set of the words of length n (also known as a Hammingspace), as it fulfills the conditions...
computer science and telecommunication, Hamming codes are a family of linear error-correcting codes. Hamming codes can detect one-bit and two-bit errors...
include the Hamming code (which makes use of a Hamming matrix), the Hamming window, Hamming numbers, sphere-packing (or Hamming bound), Hamming graph concepts...
bound from an interpretation in terms of packing balls in the Hamming metric into the space of all possible words. It gives an important limitation on the...
sum, or bit summation. The Hamming weight is named after Richard Hamming although he did not originate the notion. The Hamming weight of binary numbers...
report's release, Ham was demoted and transferred out of her management position in the Space Shuttle program. Born as Linda Hautzinger, Ham grew up outside...
of the IEEE Information Theory Society. He received the IEEE Richard W. Hamming Medal in 2006, for "contributions to the theory of error-correcting codes...
field arithmetic Finite ring Finite group Elementary abelian group Hammingspace Moore, E. H. (1896), "A doubly-infinite system of simple groups", in...
2 {\displaystyle [2^{r}-1,2^{r}-r-1,3]_{2}} Hamming code. Since d = 3 {\displaystyle d=3} , this Hamming code can correct a 1-bit error. Example : The...
mathematician Richard Hamming pioneered this field in the 1940s and invented the first error-correcting code in 1950: the Hamming (7,4) code. FEC can be...
Chimpanzees Ham and Enos also flew in the Mercury program, with Ham becoming the first great ape or Hominidae in space. The names "Sam" and "Ham" were acronyms...
block codes. The first error-correcting code was the Hamming(7,4) code, developed by Richard W. Hamming in 1950. This code transforms a message consisting...
with Wasserstein metrics on spaces of measures) or the degree of difference between two objects (for example, the Hamming distance between two strings...
carrier suitability test pilot. Ham's aviator call sign is "Hock". Ham traveled to space twice as part of the Space Shuttle program. He flew on STS-124...
Senator (Stanley Tucci) adds to the team Ham III (Andy Samberg), grandson of Ham, the first chimpanzee in space, who works as a cannonball at a circus in...
row is a sample on a two-dimensional manifold in 1024-dimensional space (a Hammingspace). The intrinsic dimensionality is two, because two variables (rotation...
the strings have the same size, the Hamming distance is an upper bound on the Levenshtein distance. The Hamming distance is the number of positions at...
error-correcting codes. He invented the concepts known as Hamming codes, Hamming windows, Hamming numbers, and Hamming distance. In 1972, Nasir Ahmed proposed the discrete...
On Saturday, February 1, 2003, Space Shuttle Columbia disintegrated as it reentered the atmosphere over Texas and Louisiana, killing all seven astronauts...
use Hamming codes, although some use triple modular redundancy (TMR). The latter is preferred because its hardware is faster than that of Hamming error...
reality. Hamming proposes that Galileo discovered the law of falling bodies not by experimenting, but by simple, though careful, thinking. Hamming imagines...
order p, and are analogous to the Heisenberg group. Elementary group Hammingspace Hans J. Zassenhaus (1999) [1958]. The Theory of Groups. Courier Corporation...
The Space Race was a 20th-century competition between two Cold War rivals, the United States and the Soviet Union, to achieve superior spaceflight capability...
The museum is the resting place of Ham, the chimpanzee who, in 1961, became the first great ape to fly into space. The museum includes exhibits about...
practice, stripes data at the bit (rather than block) level, and uses a Hamming code for error correction. The disks are synchronized by the controller...
physics, spacetime is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum....
(Element: Wind, Buddha: Amitabha, Bija mantra: YAM) Crown chakra (Element: Space, Buddha: Vairochana, Bija mantra: KHAM) Chakras clearly play a key role...
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