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.

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

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

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.

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

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

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,.

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

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.

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.

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

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.