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Alphabet size: 2: Notation ... [8,4] extended Hamming code and can both detect and correct single-bit errors and detect (but not correct) double-bit errors. ...
Alphabet size: 2: Notation [7,4,3] 2-code: ... all errors with a Hamming distance of 1 can be detected and corrected, which is the point of using a Hamming code.
In 1973, Tietäväinen proved [1] that any non-trivial perfect code over a prime-power alphabet has the parameters of a Hamming code or a Golay code. A perfect code may be interpreted as one in which the balls of Hamming radius t centered on codewords exactly fill out the space (t is the covering radius = packing
A code over an alphabet Q of size |Q| = q is called q-ary R-covering code of length n if for every word there is a codeword such that the Hamming distance (,). In other words, the spheres (or balls or rook-domains) of radius R with respect to the Hamming metric around the codewords of C have to exhaust the finite metric space Q n {\displaystyle ...
The American mathematician Richard Hamming pioneered this field in the 1940s and invented the first error-correcting code in 1950: the Hamming (7,4) code. [ 5 ] FEC can be applied in situations where re-transmissions are costly or impossible, such as one-way communication links or when transmitting to multiple receivers in multicast .
The notation (,,) describes a block code over an alphabet of size , with a block length , message length , and distance . If the block code is a linear block code, then the square brackets in the notation [ n , k , d ] q {\displaystyle [n,k,d]_{q}} are used to represent that fact.
For a fixed length n, the Hamming distance is a metric on the set of the words of length n (also known as a Hamming space), as it fulfills the conditions of non-negativity, symmetry, the Hamming distance of two words is 0 if and only if the two words are identical, and it satisfies the triangle inequality as well: [2] Indeed, if we fix three words a, b and c, then whenever there is a ...
[1] [2] Hamming spaces are named after American mathematician Richard Hamming, who introduced the concept in 1950. [3] They are 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.