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  2. Error correction code - Wikipedia

    en.wikipedia.org/wiki/Error_correction_code

    The American mathematician Richard Hamming pioneered this field in the 1940s and invented the first error-correcting code in 1950: the Hamming (7,4) code. [ 5 ] FEC can be applied in situations where re-transmissions are costly or impossible, such as one-way communication links or when transmitting to multiple receivers in multicast .

  3. Hamming code - Wikipedia

    en.wikipedia.org/wiki/Hamming_code

    With the addition of an overall parity bit, it becomes the [8,4] extended Hamming code and can both detect and correct single-bit errors and detect (but not correct) double-bit errors. Construction of G and H

  4. Hamming (7,4) - Wikipedia

    en.wikipedia.org/wiki/Hamming(7,4)

    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.

  5. BCH code - Wikipedia

    en.wikipedia.org/wiki/BCH_code

    This code has minimal Hamming distance 15 and corrects 7 errors. ... vector has too many errors and cannot be corrected. For example, ... of Error-Correcting Codes ...

  6. Linear code - Wikipedia

    en.wikipedia.org/wiki/Linear_code

    Hadamard code is a [,,] linear code and is capable of correcting many errors. Hadamard code could be constructed column by column : the i t h {\displaystyle i^{th}} column is the bits of the binary representation of integer i {\displaystyle i} , as shown in the following example.

  7. Block code - Wikipedia

    en.wikipedia.org/wiki/Block_code

    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. Hence this code is a block code. It turns out that it is also a linear code and that it has distance 3. In the shorthand notation above, this ...

  8. Steane code - Wikipedia

    en.wikipedia.org/wiki/Steane_code

    It is a CSS code (Calderbank-Shor-Steane), using the classical binary [7,4,3] Hamming code to correct for both qubit flip errors (X errors) and phase flip errors (Z errors). The Steane code encodes one logical qubit in 7 physical qubits and is able to correct arbitrary single qubit errors. Its check matrix in standard form is [] where H is the ...

  9. Hamming distance - Wikipedia

    en.wikipedia.org/wiki/Hamming_distance

    Since the Hamming distance between "000" and "111" is 3, and those comprise the entire set of codewords in the code, the minimum Hamming distance is 3, which satisfies 2k+1 = 3. Thus a code with minimum Hamming distance d between its codewords can detect at most d-1 errors and can correct ⌊(d-1)/2⌋ errors. [2] The latter number is also ...