Hadamard code: Difference between revisions

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m Fixed typo: the error correction capability of the hadamard code is 7 bits per block rather than per word (which has only 6 bits).
m History: Clarified after re-reading some of the sources: the error-correction capability of the Hadamard code was up to 7 bits per 32-bit-word, the binary values to be transmitted were 6 bits long.
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[[James Joseph Sylvester]] developed his construction of Hadamard matrices in 1867, which actually predates Hadamard's work on Hadamard matrices. Hence the name ''Hadamard code'' is disputed and sometimes the code is called ''Walsh code'', honoring the American mathematician [[Joseph Leonard Walsh]].
 
An augmented Hadamard code was used during the 1971 [[Mariner 9]] mission to correct for picture transmission errors. The databinary wordsvalues used during this mission were 6 bits long, which represented 64 [[grayscale]] values.
 
Because of limitations of the quality of the alignment of the transmitter at the time (due to Doppler Tracking Loop issues) the maximum useful data length was about 30 bits (the encoded block). Instead of using a [[repetition code]], a [32, 6, 16] Hadamard code was used.
 
Errors of up to 7 bits per block32-bit word could be corrected using this scheme. Compared to a 5-[[repetition code]], the error correcting properties of this Hadamard code are much better, yet its rate is comparable. The efficient decoding algorithm was an important factor in the decision to use this code.
 
The circuitry used was called the "Green Machine". It employed the [[fast Fourier transform]] which can increase the decoding speed by a factor of three. Since the 1990s use of this code by space programs has more or less ceased, and the [[NASA Deep Space Network]] does not support this error correction scheme for its dishes that are greater than 26 m.