MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

A $(p, q, r)$ Steiner system is a collection of $q$-element subsets $A$ (called blocks) of an $r$-element set $S$ such that every $p$-element subset of $S$ is contained in a unique element of $A$. Good examples come from considering as blocks the set of hyperplanes in $\mathbb{A}^n$ or $\mathbb{P}^n$ over a finite field. For example, $\mathbb{A}^2$ over $\mathbb{F}_2$ \mathbb{F}_3$gives a$(2, 3, 9)$Steiner system: it contains$9$(closed)$\mathbb{F}_3$-rational) points, and let the blocks be the lines, each of which consists of$3$points. Then any$2$points are contained in a unique line. This is the unique$(2, 3, 9)$Steiner system. 1 The construction of certain Steiner systems is a good example. A$(p, q, r)$Steiner system is a collection of$q$-element subsets$A$(called blocks) of an$r$-element set$S$such that every$p$-element subset of$S$is contained in a unique element of$A$. Good examples come from considering as blocks the set of hyperplanes in$\mathbb{A}^n$or$\mathbb{P}^n$over a finite field. For example,$\mathbb{A}^2$over$\mathbb{F}_2$gives a$(2, 3, 9)$Steiner system: it contains$9$(closed) points, and let the blocks be the lines, each of which consists of$3$points. Then any$2$points are contained in a unique line. This is the unique$(2, 3, 9)\$ Steiner system.