Hadamard Code - History

History

Hadamard code is the name that is most commonly used for this code in the literature. Jacques Hadamard did not invent the code himself, but he defined Hadamard matrices around 1893, long before the first error-correcting code, the Hamming code, was developed in the 1940s. The Hadamard code is based on Hadamard matrices, and while there are many different Hadamard matrices that could be used here, normally only Sylvester's construction of Hadamard matrices is used to obtain the codewords of the Hadamard code. 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 not undisputed and sometimes the code is called Walsh code, honoring the American mathematician Joseph Leonard Walsh.

A Hadamard code was used during the 1971 Mariner 9 mission to correct for picture transmission errors. The data words 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 the maximum useful data length was about 30 bits. Instead of using a repetition code, a Hadamard code was used. Errors of up to 7 bits per 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 3. Since the 1990s use of this code by space programs has more or less ceased, and the Deep Space Network does not support this error correction scheme for its dishes that are greater than 26m.