Hadamard code

Article summary

[[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 operations Here the white fields stand for 0 and the red fields for 1 The Hadamard code is an error-correcting code named after the French mathematician 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. 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. The Hadamard code is also known under the names Walsh code, Walsh family, and Walsh–Hadamard code in recognition of the American mathematician Joseph Leonard Walsh. The Hadamard code's mathematical specification is rather involved, and is described under Constructions. It is an example of a linear code of length 2^m over a binary alphabet. Unfortunately, this term is somewhat ambiguous as some references assume a message length k = m while others assume a message length of k = m+1 . In this article, the first case is called the Hadamard code while the second is called the augmented Hadamard code.…

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