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- Hamming_code abstract "In telecommunication, Hamming codes are a family of linear error-correcting codes that generalize the Hamming(7,4)-code, and were invented by Richard Hamming in 1950. Hamming codes can detect up to 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.In mathematical terms, Hamming codes are a class of binary linear codes. For each integer r ≥ 2 there is a code 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 punctured Hadamard 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 (ECC memory), where bit errors are extremely rare and Hamming codes are widely used. 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.".
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- Hamming_code wikiPageWikiLinkText "''Hamming codes''".
- Hamming_code wikiPageWikiLinkText "Hamming code".
- Hamming_code wikiPageWikiLinkText "Hamming code#General algorithm".
- Hamming_code wikiPageWikiLinkText "Hamming code#SECDED".
- Hamming_code wikiPageWikiLinkText "Hamming codes".
- Hamming_code wikiPageWikiLinkText "Hamming".
- Hamming_code wikiPageWikiLinkText "Hamming(7,4) code with an additional parity bit".
- Hamming_code wikiPageWikiLinkText "Hamming-code".
- Hamming_code wikiPageWikiLinkText "SECDED".
- Hamming_code wikiPageWikiLinkText "but not both".
- Hamming_code wikiPageWikiLinkText "extended Hamming code".
- Hamming_code wikiPageWikiLinkText "hamming code".
- Hamming_code blockLength "where".
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- Hamming_code imageCaption "The Hamming-code".
- Hamming_code name "Binary Hamming Codes".
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- Hamming_code notation "-code".
- Hamming_code properties Hamming_bound.
- Hamming_code properties Perfect_code.
- Hamming_code type Block_code.
- Hamming_code type Linear_block_code.
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