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Golay codes - Golay Codes Knowing only about the(7,4...

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Golay Codes Knowing only about the (7,4) Hamming code, Marcel J.E. Golay generalised Hamming's idea to perfect single-error correcting codes based on any prime number. Having done that, Golay began searching for perfect multi-error correcting codes. One of the Golay codes is of special interest since it was later used to generate a packing in the 24-dimensional space. Recall that an x -error-correcting code must have a minimum distance of at least 2 x +1. For the code to be perfect, the number of vertices of the unit n -cube inside a packing sphere of radius x must be a power of r , where r is the radix of the code. In the binary case for some integer k . This is the sum of the first x +1 entries of the n th row of the Pascal triangle. Golay found two such numbers, In the case n =90 Golay showed that no perfect double-error correcting (90,78) code could exist. For n =23 Golay found a 3- error correcting (23,12) code and gave a matrix for it, shown in Figure 6 . Figure 6: A matrix for the (23,12) Golay code By attaching I 11 to this matrix we obtain a matrix of the same form as that of the matrices for the single-error correcting codes.
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To see how the code is constructed, let [ a i , j ] be the matrix in Figure 6 . The check digits are determined from the message digits by or, since the code is binary The check digits are placed directly after the message digits, so a codeword has the form Golay supplied only a partial account of the way he constructed this matrix. As it is not at all trivial, and given that we are only interested in the result itself, it will be skipped. In mathematics and electronics engineering , a binary Golay code is a type of error-correcting code used in digital communications . The binary Golay code, along with the ternary Golay code
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