Steiner system

Remarks

Equivalently, a Steiner system $S(t, k, n)$ is a $t$-$(n, k, 1)$ block design. Counting $t$-sets gives the number of blocks: $\binom{n}{t} / \binom{k}{t}$. Example: on the $9$ points of the grid $(\Z/3\Z)^2$, the $12$ lines of the affine plane form an $S(2, 3, 9)$. Two distinct points $p, q$ lie on exactly one line, namely $\set{p, q, -p - q}$. Non-example: there is no $S(2, 3, 8)$. In an $S(2, 3, n)$, the blocks through a point split the other $n - 1$ points into pairs, so $n - 1$ must be even, and $8 - 1 = 7$ is odd. An $S(2, 3, n)$ is a Steiner triple system; for $n \ge 7$, one exists if and only if $n \equiv 1$ or $3 \pmod 6$. A finite projective plane of order $q$ is an $S(2, q + 1, q^2 + q + 1)$.