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[[File:Hadamard-Code.svg|thumb|250 px|Matrix of the Augmented Hadamard code [32, 6, 16] for the [[Reed–Muller code]] (1, 5) of the NASA space probe [[Mariner 9]]]]
[[File:Multigrade operator XOR.svg|thumb|250px|[[Exclusive or|XOR]] operations<br/>Here the white fields stand for 0<br/>and the red fields for 1]]
The '''Hadamard code''' is an [[error-correcting code]] named after [[Jacques Hadamard]] that is used for [[error detection and correction]] when transmitting messages over very noisy or unreliable channels. In 1971, the code was used to transmit photos of Mars back to Earth from the NASA space probe [[Mariner 9]].<ref>http://www.mcs.csueastbay.edu/~malek/TeX/Hadamard.pdf</ref> Because of its unique mathematical properties, the Hadamard code is not only used by engineers, but also intensely studied in [[coding theory]], [[mathematics]], and [[theoretical computer science]].
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Since the relative distance of the Hadamard code is 1/2, normally one can only hope to recover from at most a 1/4 fraction of error. Using [[list decoding]], however, it is possible to compute a short list of possible candidate messages as long as fewer than <math>\frac{1}{2}-\epsilon</math> of the bits in the received word have been corrupted.
In [[code division multiple access]] (CDMA) communication, the Hadamard code is referred to as Walsh Code, and is used to define individual [[telecommunications|communication]] [[channel (communications)|channels]]. It is usual in the CDMA literature to refer to codewords as “codes”. Each user will use a different codeword, or “code”, to modulate their signal. Because Walsh codewords are mathematically [[orthogonal]], a Walsh-encoded signal appears as [[random noise]] to a CDMA capable mobile [[terminal (telecommunication)|terminal]], unless that terminal uses the same codeword as the one used to encode the incoming [[signal (information theory)|signal]].<ref>{{cite web|url=http://complextoreal.com/wp-content/uploads/2013/01/CDMA.pdf|title=CDMA Tutorial: Intuitive Guide to Principles of Communications|publisher=Complex to Real|
==History==
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