Hadamard code
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A system used for signal error detection and correction.
The parameters of these codes are [2^n, n+1, 2^(n-1)] codes.
[edit] Construction
The code is based on Hadamard matrixes. If H is a Hadamard matrix of order 2^n the codeword are constructed by taking the rows of H and -H as codewords, where each (-1) is replaced by 0.
One can also start by creating the parity check matrix which consists of all 2^(n-1) vectors containing an odd number of 1's.
Finally the encoding process can also be done recursively.
[edit] History
A Hadamard code was used during the 1971 Mariner 9 mission to correct for picture transmission error. The data words used during this mission were 6 bits long, which represented 64 grayscale values. Because on limitation on the quality of the alignment of the transmitter the maximum data word that could be for transmission was about 30 bits. Instead of using a repetition code a [32, 6, 16] Hadamard code was used. Up to 7 errors could be corrected in each data word using this scheme. [1]
