[[Steiner system]]
# $\opn S(5,8,24)$

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

> [!missing]- Proof
> #missing/proof

## Construction

### From the Golay code

Let $\Omega$ denote a set of 24 points and $\mathcal{C} \leq \mathcal{E}(\Omega)$ be the $[24,12,8]_{2}$ [[extended binary Golay code]].
Then the **octads** of $\mathcal{C}$, i.e. codewords of [[Hamming distance|Hamming weight]] 8, form the octads of $\opn S(5,8,24)$.

> [!check]- Proof
> Let $S \sube \Omega$ be a 5-element subset,
> and assume there exist distinct octads $C_{1},C_{2} \in \mathcal{C}$ such that $S \sube C_{1} \cap C_{2}$.
> Then
> $$
> \begin{align*}
> \abs{C_{1} + C_{2}} &= \abs{C_{1}} + \abs{C_{2}} - 2\abs{C_{1} \cap C_{2}} \\
> &\leq 16 - 10 = 6
> \end{align*}
> $$
> which would imply that there exists codeword in $\mathcal{C}$ of weight less than 8, a contradiction. 
> 
> Now each octad accounts for ${8 \choose 5} = 56$ elements, and $759 \times 56 = 42\,502= {24\choose 5}$, 
> which exhausts all 5-element subsets.
> <span class="QED"/>


## Properties

- Let $T_{0}$ be a 4-element subset of $\Omega$.
  Then $T_{0}$ lies in exactly 5 octads $\{ T_{0} \amalg T_{i} \}_{i=1}^5$ where $\{ T_{i} \}_{i=0}^5$ forms a [[partition]] of $\Omega$ into 4-element sets called a **sextet**, and the union of any two sets in a sextet form an octad.
- The automorphisms of $\opn S(5,8,24)$ are given by [[Mathieu group M24]].

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