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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.
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. At the time, Hamming worked at Bell Telephone Laboratories and was frustrated with the error-prone punched card reader, which is why he started working on error-correcting codes.
Algebraic geometry code; BCH code; BCJR algorithm; Belief propagation; Berger code; Berlekamp–Massey algorithm; Binary Golay code; Binary Goppa code; Bipolar violation; CRHF; Casting out nines; Check digit; Chien's search; Chipkill; Cksum; Coding gain; Coding theory; Constant-weight code; Convolutional code; Cross R-S code; Cryptographic hash ...
In the extended binary Golay code, all code words have Hamming weights of 0, 8, 12, 16, or 24. Code words of weight 8 are called octads and code words of weight 12 are called dodecads. Octads of the code G 24 are elements of the S(5,8,24) Steiner system. There are 759 = 3 × 11 × 23 octads and 759 complements thereof.
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 ...
Gray code; Hamming codes. Hamming(7,4): a Hamming code that encodes 4 bits of data into 7 bits by adding 3 parity bits; Hamming distance: sum number of positions which are different; Hamming weight (population count): find the number of 1 bits in a binary word; Redundancy checks. Adler-32; Cyclic redundancy check; Damm algorithm; Fletcher's ...
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.
A code which attains the Hamming bound is said to be a perfect code. Hamming codes are perfect codes. [13] [14] Returning to differential equations, Hamming studied means of numerically integrating them. A popular approach at the time was Milne's Method, attributed to Arthur Milne. [15]