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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 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 ...
Golay codes, of which the Binary Golay code is of practical interest; Goppa code, ... bits long for optimal generator polynomials of degree ...
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 ...
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:
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