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In computer science and telecommunications, 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.
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 ...
In coding theory, Hamming(7,4) is a linear error-correcting code that encodes four bits of data into seven bits by adding three parity bits. It is a member of a larger family of Hamming codes, but the term Hamming code often refers to this specific code that Richard W. Hamming introduced in 1950.
Hamming code; Latin square based code for non-white noise (prevalent for example in broadband over powerlines) Lexicographic code; Linear Network Coding, a type of erasure correcting code across networks instead of point-to-point links; Long code; Low-density parity-check code, also known as Gallager code, as the archetype for sparse graph codes
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 only nontrivial and useful perfect codes are the distance-3 Hamming codes with parameters satisfying (2 r – 1, 2 r – 1 – r, 3), and the [23,12,7] binary and [11,6,5] ternary Golay codes. [4] [5] Another code property is the number of neighbors that a single codeword may have. [6] Again, consider pennies as an example.
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.
Hamming thereby created a family of mathematical error-correcting codes, which are called Hamming codes. This not only solved an important problem in telecommunications and computer science, it opened up a whole new field of study. [11] [12]