S(5,8,24)

There exists a unique Steiner system, #m/thm/comb which is also called the Witt design.

Proof of uniqueness

#missing/proof

Construction

Let denote a set of 24 points and be the extended binary Golay code. Then the octads of , i.e. codewords of Hamming weight 8, form the octads of .

Proof

Let be a 5-element subset, and assume there exist distinct octads such that . Then

which would imply that there exists codeword in of weight less than 8, a contradiction.

Now each octad accounts for elements, and , which exhausts all 5-element subsets.

Let be a 4-element subset of . Then lies in exactly 5 octads where forms a partition of into 4-element sets called a sextet, and the union of any two sets in a sextet form an octad.
The automorphisms of are given by .

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