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Binary Hamming codes
The Hamming(7,4) code (with r = 3)
Named after
Richard W. Hamming
Classification
Type
Linear block code
Block length
2r − 1 where r ≥ 2
Message length
2r − r − 1
Rate
1 − r/(2r − 1)
Distance
3
Alphabet size
2
Notation
[2r − 1, 2r − r − 1, 3]2-code
Properties
perfect code
v
t
e
In computer science and telecommunication, 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. By contrast, the simple parity code cannot correct errors, and can detect only an odd number of bits in error. Hamming codes are perfect codes, that is, they achieve the highest possible rate for codes with their block length and minimum distance of three.[1]
Richard W. Hamming invented Hamming codes in 1950 as a way of automatically correcting errors introduced by punched card readers. In his original paper, Hamming elaborated his general idea, but specifically focused on the Hamming(7,4) code which adds three parity bits to four bits of data.[2]
In mathematical terms, Hamming codes are a class of binary linear code. For each integer r ≥ 2 there is a code-word with block length n = 2r − 1 and message length k = 2r − r − 1. Hence the rate of Hamming codes is R = k / n = 1 − r / (2r − 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 2r − 1. The parity-check matrix of a Hamming code is constructed by listing all columns of length r that are non-zero, which means that the dual code of the Hamming code is the shortened Hadamard code, also known as a Simplex code. The parity-check matrix has the property that any two columns are pairwise linearly independent.
Due to the limited redundancy that Hamming codes add to the data, they can only detect and correct errors when the error rate is low. This is the case in computer memory (usually RAM), where bit errors are extremely rare and Hamming codes are widely used, and a RAM with this correction system is an ECC RAM (ECC memory). In this context, an extended Hamming code having one extra parity bit is often used. Extended Hamming codes achieve a Hamming distance of four, which allows the decoder to distinguish between when at most one one-bit error occurs and when any two-bit errors occur. In this sense, extended Hamming codes are single-error correcting and double-error detecting, abbreviated as SECDED.
highest possible rate for codes with their block length and minimum distance of three. Richard W. Hamming invented Hammingcodes in 1950 as a way of automatically...
capability of the code. The Hamming distance is named after Richard Hamming, who introduced the concept in his fundamental paper on Hammingcodes, Error detecting...
include the Hammingcode (which makes use of a Hamming matrix), the Hamming window, Hamming numbers, sphere-packing (or Hamming bound), Hamming graph concepts...
Hamming may refer to: Richard Hamming (1915–1998), American mathematician Hamming(7,4), in coding theory, a linear error-correcting code Overacting, or...
computer science, in the field of coding theory, the Hamming bound is a limit on the parameters of an arbitrary block code: it is also known as the sphere-packing...
classical binary [7,4,3] Hammingcode to correct for qubit flip errors (X errors) and the dual of the Hammingcode, the [7,3,4] code, to correct for phase...
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 applied...
. In fact, any binary Hammingcode of the form Ham(r, 2) is equivalent to a cyclic code, and any Hammingcode of the form Ham(r,q) with r and q-1 relatively...
In statistics and coding theory, a Hamming space (named after American mathematician Richard Hamming) is usually the set of all 2 N {\displaystyle 2^{N}}...
equivalent to Hamming weight in the binary case, in 1954. Hamming weight is used in several disciplines including information theory, coding theory, and...
block code to each other, such as its rate and its ability to detect and correct errors. Examples of block codes are Reed–Solomon codes, Hammingcodes, Hadamard...
Hadamard code while the second is called the augmented Hadamard code. The Hadamard code is unique in that each non-zero codeword has a Hamming weight of...
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...
faster than that of Hamming error correction scheme. Space satellite systems often use TMR, although satellite RAM usually uses Hamming error correction...
related to the (extended) Hammingcode H(8,4) and can, in fact, be constructed from it. The Hammingcode H(8,4) is a binary code of length 8 and rank 4;...
practice, stripes data at the bit (rather than block) level, and uses a Hammingcode for error correction. The disks are synchronized by the controller to...
Neil Sloane. The binary lexicographic codes are linear codes, and include the Hammingcodes and the binary Golay codes. A lexicode of length n and minimum...
error-detecting and error-correcting codes. He invented the concepts known as Hammingcodes, Hamming windows, Hamming numbers, and Hamming distance. In 1972, Nasir...
usually uses Hamming error correction. Some ECC memory uses triple modular redundancy hardware (rather than the more common Hammingcode), because triple...
A binary code is called an even code if the Hamming weight of each of its codewords is even. An even code should have a generator polynomial that include...
error-correcting code works by including carefully crafted redundancy with the stored (or transmitted) data. Examples include Hammingcodes, Reed–Solomon...
n}. Therefore, the minimum Hamming distance of the code equals its length n{\displaystyle n}. This gives the repetition code an error correcting capacity...
telecommunication and information theory, the code rate (or information rate) of a forward error correction code is the proportion of the data-stream that...
to a larger Hammingcode. The adjective "transverse" is most often used when it is used in combination with additional error control coding, such as a...
Cambridge University Press. p. 9. ISBN 9780521642989. Because the Hammingcode is a linear code, it can be written compactly in terms of matrices as follows...
modern development of error correction codes is credited to Richard Hamming in 1947. A description of Hamming'scode appeared in Claude Shannon's A Mathematical...
develop the 7-qubit code from the classical [7, 4] Hammingcode, led to the construction of an important class of codes called the CSS codes, named for their...
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