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This system is connected with many of the sporadic simple groups and with the exceptional dimensional lattice known as the Leech lattice.
The automorphism group of S 5, 8, 24 is the Mathieu group M 24 , and in that context the design is denoted W 24 "W" for "Witt". All 8-element subsets of a element set are generated in lexicographic order, and any such subset which differs from some subset already found in fewer than four positions is discarded.
Each single element occurs times somewhere in some octad. Each pair occurs 77 times. Each triple occurs 21 times.
Each quadruple tetrad occurs 5 times. Each quintuple pentad occurs once. Not every hexad, heptad or octad occurs.
The codewords of the bit binary Golay code are generated, and the 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 bit binary strings in lexicographic order and discarding those that differ from some earlier one in fewer than 8 positions.
The result looks like this:. The codewords form a group under the XOR operation. The Miracle Octad Generator MOG is a tool to generate octads, such as those containing specified subsets.
It consists of a 4x6 array with certain weights assigned to the rows. In particular, an 8-subset should obey three rules in order to be an octad of S 5,8, First, each of the 6 columns should have the same parity , that is, they should all have an odd number of cells or they should all have an even number of cells.
Second, the top row should have the same parity as each of the columns. Third, the rows are respectively multiplied by the weights 0, 1, 2, and 3 over the finite field of order 4 , and column sums are calculated for the 6 columns, with multiplication and addition using the finite field arithmetic definitions.
If the column sums' parities don't match the row sum parity, or each other, or if there do not exist a, b, c such that the column sums form a valid hexacodeword, then that subset of 8 is not an octad of S 5,8, The MOG is based on creating a bijection Conwell , "The three-space PG 3,2 and its group" between the 35 ways to partition an 8-set into two different 4-sets, and the 35 lines of the Fano 3-space PG 3,2.
It is also geometrically related Cullinane, "Symmetry Invariance in a Diamond Ring", Notices of the AMS, pp A, Feb to the 35 different ways to partition a 4x4 array into 4 different groups of 4 cells each, such that if the 4x4 array represents a four-dimensional finite affine space , then the groups form a set of parallel subspaces.
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