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A generator matrix for the binary Golay code is I A, where I is the 12×12 identity matrix, and A is the complement of the adjacency matrix of the icosahedron. A convenient representation [ edit ] It is convenient to use the " Miracle Octad Generator " format, with co-ordinates in an array of 4 rows, 6 columns.
The code generally known simply as the ternary Golay code is an -code, that is, it is a linear code over a ternary alphabet; the relative distance of the code is as large as it possibly can be for a ternary code, and hence, the ternary Golay code is a perfect code . The extended ternary Golay code is a [12, 6, 6] linear code obtained by adding ...
The Golay code, together with the related Witt design, features in a construction for the 196560 minimal vectors in the Leech lattice. Leech lattice (L mod 8) can be directly constructed by combination of the 3 following sets, = (+) + , (is a ones vector of size n), G - 24-bit Golay code; B - Binary integer sequence
Golay codes, of which the Binary Golay code is of practical interest; Goppa code, ... bits long for optimal generator polynomials of degree ...
In mathematics, the Miracle Octad Generator, or MOG, is a mathematical tool introduced by Rob T. Curtis for studying the Mathieu groups, binary Golay code and Leech lattice. Description [ edit ] The Miracle Octad Generator is a 4x6 array of combinations describing any point in 24-dimensional space.
The 4096 codewords of the 24-bit binary Golay code are generated, and the 759 codewords with a Hamming weight of 8 correspond to the S(5,8,24) system. The Golay code can be constructed by many methods, such as generating all 24-bit binary strings in lexicographic order and discarding those that differ from some earlier one in fewer than 8 ...
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