edges to partitions. Permuting the vertices causes the factorizations to permute differently, in accordance with the outer automorphism. The Steiner system S(5, 8, 24) The Steiner system S(5, 8, 24), also known as the Witt design or Witt geometry, was first described by and rediscovered by . This system is connected with many of the sporadic simple groups and with the exceptional 24-dimensional lattice known as the Leech lattice. The automorphism group of S(5, 8, 24) is the Mathieu group M24, and in that context the design is denoted W24 ("W" for "Witt") Direct lexicographic generation All 8-element subsets of a 24-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. The list of octads for the elements 01, 02, 03, ..., 22, 23, 24 is then: 01 02 03 04 05 06 07 08 01 02 03 04 09 10 11 12 01 02 03 04 13 14 15 16 . . (next 753 octads omitted) . 13 14 15 16 17 18 19 20 13 14 15 16 21 22 23 24 17 18 19 20 21 22 23 24 Each single element occurs 253 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. Construction from the binary Golay code 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 positions. The result looks like this: 000000000000000000000000 000000000000000011111111 000000000000111100001111 . . (next 4090 24-bit strings omitted) . 111111111111000011110000 111111111111111100000000 111111111111111111111111 The codewords form a group under the XOR operation. Projective line construction This construction is due to Carmichael (1931). Add a new element, call it , to the 23 elements of the finite field 23 (that is, the integers mod 23). This set, , of 24 elements can be formally identified with the points of the projective line over 23. Call the following specific subset of size 8, a "block". (We can take any octad of the extended binary Golay code,