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If heads are interpreted as 1 and tails as 0 then moving to a codeword from the extended binary Golay code guarantees it will be possible to force a win. 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.
Used in a football pool with 11 games, the ternary Golay code corresponds to 729 bets and guarantees exactly one bet with at most 2 wrong outcomes. The set of codewords with Hamming weight 5 is a 3-(11,5,4) design. The generator matrix given by Golay (1949, Table 1.) is
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
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 ...
Golay codes, of which the Binary Golay code is of practical interest; Goppa code, ... bits long for optimal generator polynomials of degree ...
Linear code. In coding theory, a linear code is an error-correcting code for which any linear combination of codewords is also a codeword. Linear codes are traditionally partitioned into block codes and convolutional codes, although turbo codes can be seen as a hybrid of these two types. [1] Linear codes allow for more efficient encoding and ...
The Mathieu group acts on the 2048 = 1+759+1288 points of the Golay code modulo the fixed space with 3 orbits, and on the 4096 = 1+24+276+2024+1771 points of the cocode with 5 orbits, and the subgroups fixing a non-trivial point of the code or cocode give 6 of the 9 classes of maximal subgroups. The 9 classes of maximal subgroups are as follows:
The Miracle Octad Generator is a 4x6 array of combinations describing any point in 24-dimensional space. It preserves all of the symmetries and maximal subgroups of the Mathieu group M 24, namely the monad group, duad group, triad group, octad group, octern group, sextet group, trio group and duum group. It can therefore be used to study all of ...
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