$S (5, 8, 24)$

There exists a unique $S (5, 8, 24)$ Steiner system, #m/thm/comb which is also called the Witt design.

Proof of uniqueness

#missing/proof

Construction

From the Golay code

Let $Ω$ denote a set of 24 points and $C \leq E (Ω)$ be the $[24, 12, 8] 2$ extended binary Golay code. Then the octads of $C$ , i.e. codewords of Hamming weight 8, form the octads of $S (5, 8, 24)$ .

Proof

Let $𝑆 \subseteq Ω$ be a 5-element subset, and assume there exist distinct octads $𝐶 1, 𝐶 2 \in C$ such that $𝑆 \subseteq 𝐶 1 \cap 𝐶 2$ . Then

| 𝐶 1 + 𝐶 2 | = | 𝐶 1 | + | 𝐶 2 | - 2 | 𝐶 1 \cap 𝐶 2 | \leq 16 - 10 = 6

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

Now each octad accounts for $(85) = 56$ elements, and $759 \times 56 = 42502 = (245)$ , which exhausts all 5-element subsets.

Properties

Let $𝑇 0$ be a 4-element subset of $Ω$ . Then $𝑇 0$ lies in exactly 5 octads ${𝑇 0 ⨿ 𝑇 𝑖} 5 𝑖 = 1$ where ${𝑇 𝑖} 5 𝑖 = 0$ forms a partition of $Ω$ into 4-element sets called a sextet, and the union of any two sets in a sextet form an octad.
The automorphisms of $S (5, 8, 24)$ are given by $M 24$ .

#state/develop | #lang/en | #SemBr