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Hence the rate of Hamming codes is R = k / n = 1 − r / (2 r − 1), which is the highest possible for codes with minimum distance of three (i.e., the minimal number of bit changes needed to go from any code word to any other code word is three) and block length 2 r − 1.
The Hamming(7,4) code is closely related to the E 7 lattice and, in fact, can be used to construct it, or more precisely, its dual lattice E 7 ∗ (a similar construction for E 7 uses the dual code [7,3,4] 2).
Here is a table of all n-bit lexicode by d-bit minimal hamming distance, resulting of maximum 2 m codewords dictionnary. For example, F 4 code (n=4,d=2,m=3), extended Hamming code (n=8,d=4,m=4) and especially Golay code (n=24,d=8,m=12) shows exceptional compactness compared to neighbors.
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. [4] [5] In the case where C is a linear subspace of its Hamming space, it is called a linear code. [4]
The code rate of the octet oriented Reed Solomon block code denoted RS(204,188) is 188/204, meaning that 204 − 188 = 16 redundant octets (or bytes) are added to each block of 188 octets of useful information.
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 ...
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 radius). A quasi-perfect code is one in which the balls of Hamming radius t centered on codewords are disjoint and the balls of radius t +1 cover the space, possibly with some ...
The first error-correcting code was the Hamming(7,4) code, developed by Richard W. Hamming in 1950. This code transforms a message consisting of 4 bits into a codeword of 7 bits by adding 3 parity bits.