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Note that bit/s is a more widespread unit of measurement for the information rate, implying that it is synonymous with net bit rate or useful bit rate exclusive of error-correction codes. See also [ edit ]
The Hamming weight is named after the American mathematician Richard Hamming, although he did not originate the notion. [5] The Hamming weight of binary numbers was already used in 1899 by James W. L. Glaisher to give a formula for the number of odd binomial coefficients in a single row of Pascal's triangle. [6]
Each data bit is included in a unique set of 2 or more parity bits, as determined by the binary form of its bit position. Parity bit 1 covers all bit positions which have the least significant bit set: bit 1 (the parity bit itself), 3, 5, 7, 9, etc. Parity bit 2 covers all bit positions which have the second least significant bit set: bits 2 ...
The codewords in a linear block code are blocks of symbols that are encoded using more symbols than the original value to be sent. [2] A linear code of length n transmits blocks containing n symbols. For example, the [7,4,3] Hamming code is a linear binary code which represents 4-bit messages using 7-bit codewords. Two distinct codewords differ ...
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, the weight enumerator polynomial of a binary linear code specifies the number of words of each possible Hamming weight. Let be a binary linear code of length . The weight distribution is the sequence of numbers
A special case of constant weight codes are the one-of-N codes, that encode bits in a code-word of bits. The one-of-two code uses the code words 01 and 10 to encode the bits '0' and '1'. A one-of-four code can use the words 0001, 0010, 0100, 1000 in order to encode two bits 00, 01, 10, and 11.
It has minimal Hamming distance at least 7 and corrects up to three errors. Since the generator polynomial is of degree 10, this code has 5 data bits and 10 checksum bits. It is also denoted as: (15, 5) BCH code. (This particular generator polynomial has a real-world application, in the "format information" of the QR code.)