[[Combinatorics MOC]]
# Steiner system

An $\opn S(t,k,n)$ **Steiner system** is a set $\Omega$ of $n$ points, together with a set of $k$-element subsets $B \sube \mathcal{P}(\Omega)$, called **blocks** or **$k$-ads**, such that any set of $t$ points is contained in exactly one $k$-ad. #m/def/comb 
Excluding trivial systems, $1 < t < k < n$.

- A [[finite projective plane]] of [[Number of points in a finite projective plane|order]] $q$ is an $\opn S(2,q+1,q^2+q+1)$. 
  In particular the [[Fano plane]] is the unique [[Fano plane|$\opn S(2,3,7)$]] Steiner system.

## Special Steiner systems

- [[Fano plane|$\opn S(2,3,7)$]]
- [[S(5,8,24)]]

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