12 Facts About Walsh code


Hadamard Walsh code is an error-correcting Walsh code named after Jacques Hadamard that is used for error detection and correction when transmitting messages over very noisy or unreliable channels.

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In 1971, the Walsh code was used to transmit photos of Mars back to Earth from the NASA space probe Mariner 9.

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Hadamard Walsh code is an example of a linear Walsh code of length over a binary alphabet.

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Hadamard Walsh code is a locally decodable Walsh code, which provides a way to recover parts of the original message with high probability, while only looking at a small fraction of the received word.

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Hadamard Walsh code is the name that is most commonly used for this Walsh code in the literature.

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Jacques Hadamard NASA Mars

An augmented Hadamard Walsh code was used during the 1971 Mariner 9 mission to correct for picture transmission errors.

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In particular, an equivalent way to write the inner product definition for the Hadamard Walsh code arises by using the generator matrix whose columns consist of all strings of length, that is,.

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Generator matrix of the augmented Hadamard Walsh code is obtained by restricting the matrix to the columns whose first entry is one.

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Distance of a code is the minimum Hamming distance between any two distinct codewords, i e, the minimum number of positions at which two distinct codewords differ.

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Since the Walsh–Hadamard code is a linear code, the distance is equal to the minimum Hamming weight among all of its non-zero codewords.

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All non-zero codewords of the Walsh–Hadamard code have a Hamming weight of exactly by the following argument.

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Locally decodable Walsh code is a Walsh code that allows a single bit of the original message to be recovered with high probability by only looking at a small portion of the received word.

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